Research ArticleEfficient Method for Calculating the CompositeStiffness of Parabolic Leaf Springs with Variable Stiffness forVehicle Rear Suspension

Wen-ku Shi Cheng Liu Zhi-yong Chen Wei He and Qing-hua Zu

State Key Laboratory of Automotive Simulation and Control Jilin University Changchun 130025 China

Correspondence should be addressed to Zhi-yong Chen chen zyjlueducn

Received 7 October 2015 Revised 28 January 2016 Accepted 2 February 2016

Academic Editor Reza Jazar

Copyright copy 2016 Wen-ku Shi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The composite stiffness of parabolic leaf springs with variable stiffness is difficult to calculate using traditional integral equationsNumerical integration or FEAmaybe used butwill require computer-aided software and long calculation times An efficientmethodfor calculating the composite stiffness of parabolic leaf springs with variable stiffness is developed and evaluated to reduce thecomplexity of calculation and shorten the calculation time A simplifiedmodel for double-leaf springs with variable stiffness is builtand a composite stiffness calculation method for the model is derived using displacement superposition and material deformationcontinuity The proposed method can be applied on triple-leaf and multileaf springs The accuracy of the calculation method isverified by the rig test and FEA analysis Finally several parameters that should be considered during the design process of springsare discussed The rig test and FEA analytical results indicate that the calculated results are acceptable The proposed method canprovide guidance for the design and production of parabolic leaf springs with variable stiffness The composite stiffness of the leafspring can be calculated quickly and accurately when the basic parameters of the leaf spring are known

1 Introduction

The development of lightweight technology and energy con-servation has resulted in the extensive application of leafsprings with variable stiffness on cars Leaf springs withvariable stiffness are one of the focal points in automobileleaf springs because of their advantages over traditional leafsprings [1ndash5] Parabolic leaf springs are springs that haveleaves with constant widths but have varying cross-sectionalthicknesses along the longitudinal direction following aparabolic law This type of leaf spring has many advantagessuch as lightness of weight mono-leaf springs have beenapplied in car parts However the reliability of multileafsprings cannot be fully guaranteed because of manufacturinglimitations therefore this type of spring has not been widelyapplied on cars Double-leaf parabolic springs with variablestiffness are the simplest form of multileaf springs withmono-leaf springs being applied in car parts However thereliability of multileaf springs cannot be fully guaranteedbecause of manufacturing limitations therefore this type

of spring has not been widely applied on cars Double-leafparabolic springs with variable stiffness are the simplest formof multileaf springs with variable stiffness and are also thefinal generation of this type of multileaf spring At presenta second main spring is added under the first main springto protect the latter for safety considerations thus forming atriple-leaf spring with variable stiffness

Stiffness is an important design parameter for leaf springswith variable stiffnessThis parameter can be calculated usingthree methods namely formula method FEA method andrig testThe formula and FEAmethods are preferred over therig test because of the highmanpower and time requirementsof the latter The formula method is commonly used incalculating leaf spring stiffness In [1] the main and auxiliarysprings are modeled as multileaf cantilever beams and anefficient method for calculating the nonlinear stiffness of aprogressive multileaf spring is developed and evaluated In[2] the stiffness of the leaf spring is calculated by an in-housesoftware based on mathematical calculations using the thick-ness profile of the leaves In [3] the deformations of the main

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5169018 12 pageshttpdxdoiorg10115520165169018

2 Mathematical Problems in Engineering

F1

F2

l0f

l1f

l2f

l3f

lf

h3

h1

h4

h6

h2

h5 2s

l1rl2r

l3r l0r

lr

Figure 1 Two levels of parabolic leaf spring with variable stiffness

and auxiliary springs are considered simultaneously to calcu-late the stiffness of the leaf springwith variable stiffness by themethod of common curvature In [4] a force model of onetype of leaf spring with variable stiffness is established and acurvature-force hybrid method for calculating the propertiesof such a leaf spring is developed In [5 6] an equation forcalculating the stiffness of a leaf spring with large deflection isderived The FEA method is also frequently used to calculatethe leaf spring stiffness In [7ndash9] leaf spring stiffness is calcu-lated using the FEAmethod and the result is verified by a rigtest In [10] the finite element method was used to calculatethe stiffness of the parabolic leaf spring and the effects ofmaterials dimensions load and other factors on the stiffnessare discussed In [11] an explicit nonlinear finite elementgeometric analysis of parabolic leaf springs under variousloads is performed The vertical stiffness wind-up stiffnessand roll stiffness of the spring are also calculated The FEAmethod is more commonly used than the formula method inthe calculation of leaf spring stiffness However the former ismore complex than the latter and requires specialized finiteelement software as a secondary step Engineers who use thismethod should also have some work experience By contrastdesigners can easilymaster the formulamethod in calculatingthe stiffness of the leaf spring because it is simple and doesnot require special software and computer aidThus derivinga stiffness calculation formula for leaf springs is significantHowever the type and size of leaf springs vary Thus nouniform formula can be applied to all types of leaf springsVarious equations can only fit a particular type of leaf springCurrent studies on calculating the stiffness of parabolic leafsprings with variable stiffness mostly considered the FEAmethod Hence a simple composite stiffness calculationformula should be derived for this method

The main spring bears the load alone when the load issmall in parabolic leaf springs with variable stiffness in theform of main and auxiliary springs Thereafter the main andauxiliary springs come into contact because of the increasedload Finally the main and auxiliary springs bear the loadtogether when the main and auxiliary springs are under fullcontactTherefore the stiffness of the spring changes in threestagesDuring the first stage the stiffness of the spring is equalto the stiffness of the main spring before the main springand secondary springs come into contact During the secondstage stiffness increases when the main and auxiliary springscome into contact During the third stage the stiffness of thespring has a composite stiffness generated by both the mainand auxiliary springs when the springs are under full contact

Changes in the stiffness of the leaf spring during stage twoare nonlinear and the stiffness is difficult to calculate Threecalculation equations are currently used [12ndash14] to calculatethe stiffness of mono-leaf parabolic springs However onlyintegral equations are available for calculating the compositestiffness of parabolic leaf springwith variable stiffness [12 13]However the direct calculation of the composite stiffness of ataper-leaf spring using integral equations is quite challengingand these equations cannot be applied during the design of aleaf spring In this paper a simple and practical equation forcalculating the composite stiffness of parabolic leaf springswith variable stiffness is derived The equation can be usednot only for calculating the composite stiffness but also fordesigning such a type of leaf spring

In this paper firstly the leaf spring assembly modelwas simplified A simple model of double-leaf parabolicspringwith variable stiffness (referred to as double-leaf springmodel) was considered The difficulties in the calculation ofthe composite stiffness of double-leaf spring were analyzedand the equation for calculating the composite stiffness wasderived using the method of material mechanics Thereafterthe superposition principle was used to derive the equationfor triple-leaf springs This equation can be extended tocalculate the stiffness of multileaf springs The correctnessof the equation was verified by a rig test and finite elementsimulation Finally the parameters that should be consideredwhen using the equations for leaf spring design were dis-cussed

2 Model of Two-Level ParabolicLeaf Spring with Variable Stiffness andIts Simplification

21 Leaf Spring Assembly Model The leaf spring studied inthis paper is a parabolic leaf spring with variable stiffness andunequal arm length A triple-leaf spring model is analyzed(Figure 1) The triple-leaf spring consists of two main springsand an auxiliary springThe stiffness of the spring is designedas a two-level variable stiffness after the requirements for acomfortable ride under different loads were considered Thefirst-level stiffness is the stiffness of the main spring and thesecond-level stiffness is a composite stiffness determined bythe stiffness of both the main and the auxiliary springs Therear shackle is designed as an underneath shackle that makesthe length of the auxiliary spring significantly less than thatof the main spring because of the limited installation space

Mathematical Problems in Engineering 3

for the spring security (leaf spring reverse bend) and vehicleperformance requirements Thus the composite stiffness ofthe multileaf spring is not a simple stiffness superposition ofmultiple parabolic spring leaves with equal lengths

22 Model Simplification The composition of the structureof the parabolic leaf spring should be analyzed first to studyits stiffness The two main springs have the same widths andthicknesses The primary function of the second main springis to protect the first main spring increasing the reliabilityof the leaf spring The spring leaves contact by end and anantifriction material is placed at the contact areaThus inter-leaf friction is not considered in the simplified model Theleaf spring forcemodel can be simplified to a cantilevermodelaccording to the theory of material mechanics (Figure 2)Theblue dashed line in Figure 2 shows an imaginary parabolicportion of the leaf spring to demonstrate the cross-sectionalshape variation law of parabolic leaf springs The triple-leafspring model in this paper is a combination of a mono-leaf (Figure 3) and a double-leaf spring model (Figure 4)The stiffness of the mono-leaf spring model can be easilyobtained thus this study focuses on calculating the stiffnessof the model shown in Figure 4Themodel of a two-level leafspring with variable stiffness will eventually develop into adouble-leaf spring model (Figure 4) The highest utilizationefficiency of the leaf spring material and lowest assemblyweight are considered

Several assumptions are developed according to thetheory of material mechanics as follows

(1) The leaf spring is generally considered a cantileverbeam A fixed constraint is applied at one of the endsA concentration force perpendicular to the surface ofthe main spring is applied at the other end

(2) The unique type of force between the leaves is avertical force

(3) The boundaries between the main and auxiliarysprings should have the same curvatures and deflec-tions

(4) The thickness of the leaves is negligible and the mainand auxiliary springs are in full contact

3 Derivation of Equation for Calculating theComposite Stiffness for Two-Level ParabolicLeaf Springs with Variable Thickness

A model for the parabolic double-leaf spring with variablestiffness (referred to as double-leaf spring model) is firstexamined according to the theory in Section 2 Thereafteraccording to the theory of superimposition a triple-leafspring model was studied Finally the results are applied toa multileaf spring

31 Challenges in Composite Stiffness Calculation Varioustypes of equations are used to calculate the composite stiffnessof parabolic leaf springswith variable stiffnessThedesign partof the automobile engineering manual [7] and Spring Manual

F

Figure 2 Cantilever beam model showing the two levels of theparabolic leaf spring with variable stiffness

F

Figure 3 Cantilever beam model of a parabolic mono-leaf spring

F

Figure 4 Cantilever beam model of a parabolic double-leaf springwith variable stiffness

[8] provide the following integral equation for calculatingcomposite stiffness

1

119870=

1

2120585int

119897119898

0

1199092

119864 (119868119886 (119909) + 119868119898 (119909))119889119909 (1)

where 119870 stands for the spring composite stiffness 119864 standsfor the material elastic modulus 120585 is the distortion correctioncoefficient 119868119886(119909) is the sectional moment of inertia of theauxiliary spring 119868119898(119909) is the sectional moment of inertia ofthe main spring and 119897119898 stands for half-length of the mainspring

The integral equation is difficult to use in the directcalculation of the composite stiffness of the spring becauseof the variable cross sections of the main and auxiliarysprings Numerical integration is used to solve the problemwhich cannot be performed without a computer This studyis conducted to determine whether a simple and accurateformula can be derived to calculate the composite stiffnessof a parabolic leaf spring with variable stiffness The theoryof material mechanics is used to derive the equation forcalculating the composite stiffness

32 Derivation of Equation for Calculating the Deflection ofa Mono-Leaf Spring The deflection of the single parabolicleaf spring (mono-leaf spring) is the basis for calculating thedeflection of a double-leaf spring First the mono-leaf springequation is theoretically derived to obtain the equation forthe deflection of any point along the leaf The deformationof the leaf is considered when a force is applied at the end ormiddle part of the leaf However the leaf forced at the end is aspecial case compared with that when the leaf is forced at the

4 Mathematical Problems in Engineering

O

DCB

A

s

l

xF

l1

l0

l2

h1

h1x

h2

Figure 5 Mono-leaf spring (half-spring) with a force at its midsection

midsection First the mono-leaf spring with an applied forceat its midsection is considered

321 Theoretical Derivation of Mono-Leaf Spring withan Applied Force at Its Midsection Differential equationsare established segmentally and integration constants areobtained from boundary conditions on the basis of material-bend deformation theory A force model of a mono-leafspring (half-spring) with an applied force at its midsectionis built and shown in Figure 5 Point 119863 is selected as thecoordinate origin The 119909-axis is located in the horizontalright whereas the 119910-axis is vertically downward 119891(119909) is thedeflection of the point with a distance of 119909 from point119863 ℎ1119909is the thickness function of the parabolic part on the leaf andℎ1119909 = ℎ2radic(119909 + 1198970)1198972

(1) Differential equations are built segmentally usingmaterial-bend deformation theory

Step 1 For CD section with bending moment 119872(1199091) =

1198651199091 (0 le 1199091 le 1198971 minus 1198970) and sectional moment of inertia 1198681 thedeflection curve differential equation is expressed as follows

119864119868111989110158401015840

1(1199091) = 1198651199091 (2)

The equation is integrated once and the double integral of 1199091is as follows

11986411986811198911015840

1(1199091) =

1

211986511990912+ 1198621

11986411986811198911 (1199091) =1

611986511990913+ 11986211199091 + 1198631

(3)

Step 2 For the BC section with bending moment 119872(1199092) =

1198651199092 (1198971 minus 1198970 le 1199092 le 1198972 minus 1198970) and sectional moment of inertia1198682119909(119909) = 119887ℎ

3

111990912 = (119887ℎ

3

212)((1199092 + 1198970)1198972)

32= 1198682((1199092 +

1198970)1198972)32 then the deflection curve differential equation is

expressed as follows

119864119868211989110158401015840

2(1199092) = 119872(1199092)

= 11989732

2119865 [(1199092 + 1198970)

minus12minus 1198970 (1199092 + 1198970)

minus32]

(4)

Integrating will result in the following equation

11986411986821198911015840

2(1199092) = 2119865119897

32

2[(1199092 + 1198970)

12+ 1198970 (1199092 + 1198970)

minus12]

+ 1198622

11986411986821198912 (1199092) = 411986511989732

2[1

3(1199092 + 1198970)

32+ 1198970 (1199092 + 1198970)

12]

+ 11986221199092 + 1198632

(5)

Step 3 For the 119860119861 section with bending moment of 119872(119909) =

1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) and sectional moment of inertia of1198682 the deflection curve differential equation is as follows

119864119868211989110158401015840

3(1199093) = 1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) (6)

Integrating will yield the following equations

11986411986821198911015840

3(1199093) =

1

21198651199092

3+ 1198623

11986411986821198913 (1199093) =1

61198651199093

3+ 11986231199093 + 1198633

(7)

(2) Deflection and rotation angle are determined basedon the boundary conditions and continuity of deflection tosolve the constants in the differential equations

Step 1 For the boundary condition in which the spring isfixed at point 119862 (1199093 = 119897 minus 1198970)

1198911015840

3(119897 minus 1198970) = 1198913 (119897 minus 1198970) = 0 (8)

Constants 1198623 and1198633 are calculated by (7)-(8) as follows

1198623 = minus119865

2(119897 minus 1198970)

2

1198633 =119865

3(119897 minus 1198970)

3

(9)

Step 2 The continuity of deformation at point 119861 (1199092 = 1199093 =

1198972 minus 1198970) is considered

1198911015840

2(1198972 minus 1198970) = 119891

1015840

3(1198972 minus 1198970)

1198912 (1198972 minus 1198970) = 1198913 (1198972 minus 1198970)

(10)

Mathematical Problems in Engineering 5

O

DCBA

l

x

F

s

l1

l0

l2

h1

h1x

h2

Figure 6 Mono-leaf spring (half-spring) with applied force at its end

Constants 1198622 and 1198632 are calculated by (5) (7) and (10)as follows

1198622 = 119865(minus3

21198972

2minus 311989701198972 +

1

21198972

0) + 1198623

1198632 = 119865(1

31198973

2minus 311989701198972

2minus 31198972

01198972 +

1

31198973

0) + 1198633

(11)

Step 3 The continuity of deformation at point 119861 (1199091 = 1199092 =

1198971 minus 1198970) is considered Thus

1198911015840

1(1198971 minus 1198970) = 119891

1015840

2(1198971 minus 1198970)

1198911 (1198971 minus 1198970) = 1198912 (1198971 minus 1198970)

(12)

Constants 1198621 and 1198631 are calculated by (3) (5) and (12) asfollows

1198621 = 119865 [minus3

21205723(1 minus 120572) 119897

2

2minus 31205722(120572 minus 1) 11989701198972

+1

2(1205723minus 1) 1198972

0] + 1198623120572

3

1198631 = 119865 [1

31205723(1 minus 120572

3) 1198973

2minus 31205723(1 minus 120572) 1198970119897

2

2

minus 31205722(120572 minus 1) 119897

2

01198972 +

1

3(1205723minus 1) 1198973

0] + 1198633120572

3

(13)

(3) The deflection curve function of the 119862119863 section (0 le

1199091 le 1198971 minus 1198970) is as follows

1198641198681119891 (1199091) =1

61198651199093

1+ 11986211989811199091 + 1198631198981 (14)

where 1198621198981 = 1198621119865 1198631198981 = 1198631119865The deflection and the rotation angle at point 119863 are

obtained by setting 1199091 to zero

1198911198632 = 1198911 (1199092 = 0) =1198631198981119865

1198641198681

1198911015840

1198632=

1198621198981119865

1198641198681

(15)

The deflection at point 119874 is as follows

1198911198742 =119865 (minus11986211989811198970 + 1198631198981)

1198641198681

(16)

322Theoretical Derivation ofMono-Leaf Spring with AppliedForce at Its End A force model of a mono-leaf spring (half-spring) with an applied force at its end is shown in Figure 6This model is a special case of the model shown in Figure 5

Thus (14) can be used and 1198970 in (14) is set to zero to obtainthe deflection curve function of 119862119874 section (0 le 119909 le 1198971) asfollows

1198641198681119891 (119909) =1

61198651199093+ 119865[minus

1198972

21205723minus

3

2(1 minus 120572) 120572

31198972

2]119909

+ 119865 [1

311989731205723+

1

31205723(1 minus 120572

3) 1198973

2]

(17)

defining11986311989810 = (12057233)[1198973+(1minus120572

3)1198973

2] and11986211989810 = 120572

3[minus11989722minus

(32)(1 minus 120572)1198972

2]

Thus the deflection at the end (119909 = 0) is as follows

1198911198741 =11986511986311989810

1198641198681

(18)

The deflection at point119863 (119909 = 1198970) is expressed as follows

1198911198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (19)

Thus the deflection equations for a leaf with appliedforces at the end andmidsection are derivedThese equationswill be used for the calculation of the deflection of the double-leaf spring

33 Derivation of Equation for Composite Stiffness for Double-Leaf Spring A force model of a double-leaf spring (half-spring) with applied force at its end is shown in Figure 7 119865and 1198651 represent the forces applied at the end of the mainand auxiliary springs respectively 1198651015840

1is the reaction force of

1198651 and the deflection at point 119874 is a superposition of thedeflection caused by 119865

1015840

1and 1198651 The force 1198651015840

1is initially deter-

mined using the boundary condition at point 119863 Thereafterthe deflection at point119874 is calculated using the superpositionprinciple of displacement Finally the composite stiffness ofthe double-leaf spring is obtained

(1) Calculation of the Force at the Contact Point The auxiliaryspring has contact with the main spring only at its end when

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

F1

F2

l0f

l1f

l2f

l3f

lf

h3

h1

h4

h6

h2

h5 2s

l1rl2r

l3r l0r

lr

Figure 1 Two levels of parabolic leaf spring with variable stiffness

and auxiliary springs are considered simultaneously to calcu-late the stiffness of the leaf springwith variable stiffness by themethod of common curvature In [4] a force model of onetype of leaf spring with variable stiffness is established and acurvature-force hybrid method for calculating the propertiesof such a leaf spring is developed In [5 6] an equation forcalculating the stiffness of a leaf spring with large deflection isderived The FEA method is also frequently used to calculatethe leaf spring stiffness In [7ndash9] leaf spring stiffness is calcu-lated using the FEAmethod and the result is verified by a rigtest In [10] the finite element method was used to calculatethe stiffness of the parabolic leaf spring and the effects ofmaterials dimensions load and other factors on the stiffnessare discussed In [11] an explicit nonlinear finite elementgeometric analysis of parabolic leaf springs under variousloads is performed The vertical stiffness wind-up stiffnessand roll stiffness of the spring are also calculated The FEAmethod is more commonly used than the formula method inthe calculation of leaf spring stiffness However the former ismore complex than the latter and requires specialized finiteelement software as a secondary step Engineers who use thismethod should also have some work experience By contrastdesigners can easilymaster the formulamethod in calculatingthe stiffness of the leaf spring because it is simple and doesnot require special software and computer aidThus derivinga stiffness calculation formula for leaf springs is significantHowever the type and size of leaf springs vary Thus nouniform formula can be applied to all types of leaf springsVarious equations can only fit a particular type of leaf springCurrent studies on calculating the stiffness of parabolic leafsprings with variable stiffness mostly considered the FEAmethod Hence a simple composite stiffness calculationformula should be derived for this method

The main spring bears the load alone when the load issmall in parabolic leaf springs with variable stiffness in theform of main and auxiliary springs Thereafter the main andauxiliary springs come into contact because of the increasedload Finally the main and auxiliary springs bear the loadtogether when the main and auxiliary springs are under fullcontactTherefore the stiffness of the spring changes in threestagesDuring the first stage the stiffness of the spring is equalto the stiffness of the main spring before the main springand secondary springs come into contact During the secondstage stiffness increases when the main and auxiliary springscome into contact During the third stage the stiffness of thespring has a composite stiffness generated by both the mainand auxiliary springs when the springs are under full contact

Changes in the stiffness of the leaf spring during stage twoare nonlinear and the stiffness is difficult to calculate Threecalculation equations are currently used [12ndash14] to calculatethe stiffness of mono-leaf parabolic springs However onlyintegral equations are available for calculating the compositestiffness of parabolic leaf springwith variable stiffness [12 13]However the direct calculation of the composite stiffness of ataper-leaf spring using integral equations is quite challengingand these equations cannot be applied during the design of aleaf spring In this paper a simple and practical equation forcalculating the composite stiffness of parabolic leaf springswith variable stiffness is derived The equation can be usednot only for calculating the composite stiffness but also fordesigning such a type of leaf spring

In this paper firstly the leaf spring assembly modelwas simplified A simple model of double-leaf parabolicspringwith variable stiffness (referred to as double-leaf springmodel) was considered The difficulties in the calculation ofthe composite stiffness of double-leaf spring were analyzedand the equation for calculating the composite stiffness wasderived using the method of material mechanics Thereafterthe superposition principle was used to derive the equationfor triple-leaf springs This equation can be extended tocalculate the stiffness of multileaf springs The correctnessof the equation was verified by a rig test and finite elementsimulation Finally the parameters that should be consideredwhen using the equations for leaf spring design were dis-cussed

2 Model of Two-Level ParabolicLeaf Spring with Variable Stiffness andIts Simplification

21 Leaf Spring Assembly Model The leaf spring studied inthis paper is a parabolic leaf spring with variable stiffness andunequal arm length A triple-leaf spring model is analyzed(Figure 1) The triple-leaf spring consists of two main springsand an auxiliary springThe stiffness of the spring is designedas a two-level variable stiffness after the requirements for acomfortable ride under different loads were considered Thefirst-level stiffness is the stiffness of the main spring and thesecond-level stiffness is a composite stiffness determined bythe stiffness of both the main and the auxiliary springs Therear shackle is designed as an underneath shackle that makesthe length of the auxiliary spring significantly less than thatof the main spring because of the limited installation space

Mathematical Problems in Engineering 3

for the spring security (leaf spring reverse bend) and vehicleperformance requirements Thus the composite stiffness ofthe multileaf spring is not a simple stiffness superposition ofmultiple parabolic spring leaves with equal lengths

22 Model Simplification The composition of the structureof the parabolic leaf spring should be analyzed first to studyits stiffness The two main springs have the same widths andthicknesses The primary function of the second main springis to protect the first main spring increasing the reliabilityof the leaf spring The spring leaves contact by end and anantifriction material is placed at the contact areaThus inter-leaf friction is not considered in the simplified model Theleaf spring forcemodel can be simplified to a cantilevermodelaccording to the theory of material mechanics (Figure 2)Theblue dashed line in Figure 2 shows an imaginary parabolicportion of the leaf spring to demonstrate the cross-sectionalshape variation law of parabolic leaf springs The triple-leafspring model in this paper is a combination of a mono-leaf (Figure 3) and a double-leaf spring model (Figure 4)The stiffness of the mono-leaf spring model can be easilyobtained thus this study focuses on calculating the stiffnessof the model shown in Figure 4Themodel of a two-level leafspring with variable stiffness will eventually develop into adouble-leaf spring model (Figure 4) The highest utilizationefficiency of the leaf spring material and lowest assemblyweight are considered

Several assumptions are developed according to thetheory of material mechanics as follows

(1) The leaf spring is generally considered a cantileverbeam A fixed constraint is applied at one of the endsA concentration force perpendicular to the surface ofthe main spring is applied at the other end

(2) The unique type of force between the leaves is avertical force

(3) The boundaries between the main and auxiliarysprings should have the same curvatures and deflec-tions

(4) The thickness of the leaves is negligible and the mainand auxiliary springs are in full contact

3 Derivation of Equation for Calculating theComposite Stiffness for Two-Level ParabolicLeaf Springs with Variable Thickness

A model for the parabolic double-leaf spring with variablestiffness (referred to as double-leaf spring model) is firstexamined according to the theory in Section 2 Thereafteraccording to the theory of superimposition a triple-leafspring model was studied Finally the results are applied toa multileaf spring

31 Challenges in Composite Stiffness Calculation Varioustypes of equations are used to calculate the composite stiffnessof parabolic leaf springswith variable stiffnessThedesign partof the automobile engineering manual [7] and Spring Manual

F

Figure 2 Cantilever beam model showing the two levels of theparabolic leaf spring with variable stiffness

F

Figure 3 Cantilever beam model of a parabolic mono-leaf spring

F

Figure 4 Cantilever beam model of a parabolic double-leaf springwith variable stiffness

[8] provide the following integral equation for calculatingcomposite stiffness

1

119870=

1

2120585int

119897119898

0

1199092

119864 (119868119886 (119909) + 119868119898 (119909))119889119909 (1)

where 119870 stands for the spring composite stiffness 119864 standsfor the material elastic modulus 120585 is the distortion correctioncoefficient 119868119886(119909) is the sectional moment of inertia of theauxiliary spring 119868119898(119909) is the sectional moment of inertia ofthe main spring and 119897119898 stands for half-length of the mainspring

The integral equation is difficult to use in the directcalculation of the composite stiffness of the spring becauseof the variable cross sections of the main and auxiliarysprings Numerical integration is used to solve the problemwhich cannot be performed without a computer This studyis conducted to determine whether a simple and accurateformula can be derived to calculate the composite stiffnessof a parabolic leaf spring with variable stiffness The theoryof material mechanics is used to derive the equation forcalculating the composite stiffness

32 Derivation of Equation for Calculating the Deflection ofa Mono-Leaf Spring The deflection of the single parabolicleaf spring (mono-leaf spring) is the basis for calculating thedeflection of a double-leaf spring First the mono-leaf springequation is theoretically derived to obtain the equation forthe deflection of any point along the leaf The deformationof the leaf is considered when a force is applied at the end ormiddle part of the leaf However the leaf forced at the end is aspecial case compared with that when the leaf is forced at the

4 Mathematical Problems in Engineering

O

DCB

A

s

l

xF

l1

l0

l2

h1

h1x

h2

Figure 5 Mono-leaf spring (half-spring) with a force at its midsection

midsection First the mono-leaf spring with an applied forceat its midsection is considered

321 Theoretical Derivation of Mono-Leaf Spring withan Applied Force at Its Midsection Differential equationsare established segmentally and integration constants areobtained from boundary conditions on the basis of material-bend deformation theory A force model of a mono-leafspring (half-spring) with an applied force at its midsectionis built and shown in Figure 5 Point 119863 is selected as thecoordinate origin The 119909-axis is located in the horizontalright whereas the 119910-axis is vertically downward 119891(119909) is thedeflection of the point with a distance of 119909 from point119863 ℎ1119909is the thickness function of the parabolic part on the leaf andℎ1119909 = ℎ2radic(119909 + 1198970)1198972

(1) Differential equations are built segmentally usingmaterial-bend deformation theory

Step 1 For CD section with bending moment 119872(1199091) =

1198651199091 (0 le 1199091 le 1198971 minus 1198970) and sectional moment of inertia 1198681 thedeflection curve differential equation is expressed as follows

119864119868111989110158401015840

1(1199091) = 1198651199091 (2)

The equation is integrated once and the double integral of 1199091is as follows

11986411986811198911015840

1(1199091) =

1

211986511990912+ 1198621

11986411986811198911 (1199091) =1

611986511990913+ 11986211199091 + 1198631

(3)

Step 2 For the BC section with bending moment 119872(1199092) =

1198651199092 (1198971 minus 1198970 le 1199092 le 1198972 minus 1198970) and sectional moment of inertia1198682119909(119909) = 119887ℎ

3

111990912 = (119887ℎ

3

212)((1199092 + 1198970)1198972)

32= 1198682((1199092 +

1198970)1198972)32 then the deflection curve differential equation is

expressed as follows

119864119868211989110158401015840

2(1199092) = 119872(1199092)

= 11989732

2119865 [(1199092 + 1198970)

minus12minus 1198970 (1199092 + 1198970)

minus32]

(4)

Integrating will result in the following equation

11986411986821198911015840

2(1199092) = 2119865119897

32

2[(1199092 + 1198970)

12+ 1198970 (1199092 + 1198970)

minus12]

+ 1198622

11986411986821198912 (1199092) = 411986511989732

2[1

3(1199092 + 1198970)

32+ 1198970 (1199092 + 1198970)

12]

+ 11986221199092 + 1198632

(5)

Step 3 For the 119860119861 section with bending moment of 119872(119909) =

1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) and sectional moment of inertia of1198682 the deflection curve differential equation is as follows

119864119868211989110158401015840

3(1199093) = 1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) (6)

Integrating will yield the following equations

11986411986821198911015840

3(1199093) =

1

21198651199092

3+ 1198623

11986411986821198913 (1199093) =1

61198651199093

3+ 11986231199093 + 1198633

(7)

(2) Deflection and rotation angle are determined basedon the boundary conditions and continuity of deflection tosolve the constants in the differential equations

Step 1 For the boundary condition in which the spring isfixed at point 119862 (1199093 = 119897 minus 1198970)

1198911015840

3(119897 minus 1198970) = 1198913 (119897 minus 1198970) = 0 (8)

Constants 1198623 and1198633 are calculated by (7)-(8) as follows

1198623 = minus119865

2(119897 minus 1198970)

2

1198633 =119865

3(119897 minus 1198970)

3

(9)

Step 2 The continuity of deformation at point 119861 (1199092 = 1199093 =

1198972 minus 1198970) is considered

1198911015840

2(1198972 minus 1198970) = 119891

1015840

3(1198972 minus 1198970)

1198912 (1198972 minus 1198970) = 1198913 (1198972 minus 1198970)

(10)

Mathematical Problems in Engineering 5

O

DCBA

l

x

F

s

l1

l0

l2

h1

h1x

h2

Figure 6 Mono-leaf spring (half-spring) with applied force at its end

Constants 1198622 and 1198632 are calculated by (5) (7) and (10)as follows

1198622 = 119865(minus3

21198972

2minus 311989701198972 +

1

21198972

0) + 1198623

1198632 = 119865(1

31198973

2minus 311989701198972

2minus 31198972

01198972 +

1

31198973

0) + 1198633

(11)

Step 3 The continuity of deformation at point 119861 (1199091 = 1199092 =

1198971 minus 1198970) is considered Thus

1198911015840

1(1198971 minus 1198970) = 119891

1015840

2(1198971 minus 1198970)

1198911 (1198971 minus 1198970) = 1198912 (1198971 minus 1198970)

(12)

Constants 1198621 and 1198631 are calculated by (3) (5) and (12) asfollows

1198621 = 119865 [minus3

21205723(1 minus 120572) 119897

2

2minus 31205722(120572 minus 1) 11989701198972

+1

2(1205723minus 1) 1198972

0] + 1198623120572

3

1198631 = 119865 [1

31205723(1 minus 120572

3) 1198973

2minus 31205723(1 minus 120572) 1198970119897

2

2

minus 31205722(120572 minus 1) 119897

2

01198972 +

1

3(1205723minus 1) 1198973

0] + 1198633120572

3

(13)

(3) The deflection curve function of the 119862119863 section (0 le

1199091 le 1198971 minus 1198970) is as follows

1198641198681119891 (1199091) =1

61198651199093

1+ 11986211989811199091 + 1198631198981 (14)

where 1198621198981 = 1198621119865 1198631198981 = 1198631119865The deflection and the rotation angle at point 119863 are

obtained by setting 1199091 to zero

1198911198632 = 1198911 (1199092 = 0) =1198631198981119865

1198641198681

1198911015840

1198632=

1198621198981119865

1198641198681

(15)

The deflection at point 119874 is as follows

1198911198742 =119865 (minus11986211989811198970 + 1198631198981)

1198641198681

(16)

322Theoretical Derivation ofMono-Leaf Spring with AppliedForce at Its End A force model of a mono-leaf spring (half-spring) with an applied force at its end is shown in Figure 6This model is a special case of the model shown in Figure 5

Thus (14) can be used and 1198970 in (14) is set to zero to obtainthe deflection curve function of 119862119874 section (0 le 119909 le 1198971) asfollows

1198641198681119891 (119909) =1

61198651199093+ 119865[minus

1198972

21205723minus

3

2(1 minus 120572) 120572

31198972

2]119909

+ 119865 [1

311989731205723+

1

31205723(1 minus 120572

3) 1198973

2]

(17)

defining11986311989810 = (12057233)[1198973+(1minus120572

3)1198973

2] and11986211989810 = 120572

3[minus11989722minus

(32)(1 minus 120572)1198972

2]

Thus the deflection at the end (119909 = 0) is as follows

1198911198741 =11986511986311989810

1198641198681

(18)

The deflection at point119863 (119909 = 1198970) is expressed as follows

1198911198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (19)

Thus the deflection equations for a leaf with appliedforces at the end andmidsection are derivedThese equationswill be used for the calculation of the deflection of the double-leaf spring

33 Derivation of Equation for Composite Stiffness for Double-Leaf Spring A force model of a double-leaf spring (half-spring) with applied force at its end is shown in Figure 7 119865and 1198651 represent the forces applied at the end of the mainand auxiliary springs respectively 1198651015840

1is the reaction force of

1198651 and the deflection at point 119874 is a superposition of thedeflection caused by 119865

1015840

1and 1198651 The force 1198651015840

1is initially deter-

mined using the boundary condition at point 119863 Thereafterthe deflection at point119874 is calculated using the superpositionprinciple of displacement Finally the composite stiffness ofthe double-leaf spring is obtained

(1) Calculation of the Force at the Contact Point The auxiliaryspring has contact with the main spring only at its end when

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

for the spring security (leaf spring reverse bend) and vehicleperformance requirements Thus the composite stiffness ofthe multileaf spring is not a simple stiffness superposition ofmultiple parabolic spring leaves with equal lengths

22 Model Simplification The composition of the structureof the parabolic leaf spring should be analyzed first to studyits stiffness The two main springs have the same widths andthicknesses The primary function of the second main springis to protect the first main spring increasing the reliabilityof the leaf spring The spring leaves contact by end and anantifriction material is placed at the contact areaThus inter-leaf friction is not considered in the simplified model Theleaf spring forcemodel can be simplified to a cantilevermodelaccording to the theory of material mechanics (Figure 2)Theblue dashed line in Figure 2 shows an imaginary parabolicportion of the leaf spring to demonstrate the cross-sectionalshape variation law of parabolic leaf springs The triple-leafspring model in this paper is a combination of a mono-leaf (Figure 3) and a double-leaf spring model (Figure 4)The stiffness of the mono-leaf spring model can be easilyobtained thus this study focuses on calculating the stiffnessof the model shown in Figure 4Themodel of a two-level leafspring with variable stiffness will eventually develop into adouble-leaf spring model (Figure 4) The highest utilizationefficiency of the leaf spring material and lowest assemblyweight are considered

Several assumptions are developed according to thetheory of material mechanics as follows

(1) The leaf spring is generally considered a cantileverbeam A fixed constraint is applied at one of the endsA concentration force perpendicular to the surface ofthe main spring is applied at the other end

(2) The unique type of force between the leaves is avertical force

(3) The boundaries between the main and auxiliarysprings should have the same curvatures and deflec-tions

(4) The thickness of the leaves is negligible and the mainand auxiliary springs are in full contact

3 Derivation of Equation for Calculating theComposite Stiffness for Two-Level ParabolicLeaf Springs with Variable Thickness

A model for the parabolic double-leaf spring with variablestiffness (referred to as double-leaf spring model) is firstexamined according to the theory in Section 2 Thereafteraccording to the theory of superimposition a triple-leafspring model was studied Finally the results are applied toa multileaf spring

31 Challenges in Composite Stiffness Calculation Varioustypes of equations are used to calculate the composite stiffnessof parabolic leaf springswith variable stiffnessThedesign partof the automobile engineering manual [7] and Spring Manual

F

Figure 2 Cantilever beam model showing the two levels of theparabolic leaf spring with variable stiffness

F

Figure 3 Cantilever beam model of a parabolic mono-leaf spring

F

Figure 4 Cantilever beam model of a parabolic double-leaf springwith variable stiffness

[8] provide the following integral equation for calculatingcomposite stiffness

1

119870=

1

2120585int

119897119898

0

1199092

119864 (119868119886 (119909) + 119868119898 (119909))119889119909 (1)

where 119870 stands for the spring composite stiffness 119864 standsfor the material elastic modulus 120585 is the distortion correctioncoefficient 119868119886(119909) is the sectional moment of inertia of theauxiliary spring 119868119898(119909) is the sectional moment of inertia ofthe main spring and 119897119898 stands for half-length of the mainspring

The integral equation is difficult to use in the directcalculation of the composite stiffness of the spring becauseof the variable cross sections of the main and auxiliarysprings Numerical integration is used to solve the problemwhich cannot be performed without a computer This studyis conducted to determine whether a simple and accurateformula can be derived to calculate the composite stiffnessof a parabolic leaf spring with variable stiffness The theoryof material mechanics is used to derive the equation forcalculating the composite stiffness

32 Derivation of Equation for Calculating the Deflection ofa Mono-Leaf Spring The deflection of the single parabolicleaf spring (mono-leaf spring) is the basis for calculating thedeflection of a double-leaf spring First the mono-leaf springequation is theoretically derived to obtain the equation forthe deflection of any point along the leaf The deformationof the leaf is considered when a force is applied at the end ormiddle part of the leaf However the leaf forced at the end is aspecial case compared with that when the leaf is forced at the

4 Mathematical Problems in Engineering

O

DCB

A

s

l

xF

l1

l0

l2

h1

h1x

h2

Figure 5 Mono-leaf spring (half-spring) with a force at its midsection

midsection First the mono-leaf spring with an applied forceat its midsection is considered

321 Theoretical Derivation of Mono-Leaf Spring withan Applied Force at Its Midsection Differential equationsare established segmentally and integration constants areobtained from boundary conditions on the basis of material-bend deformation theory A force model of a mono-leafspring (half-spring) with an applied force at its midsectionis built and shown in Figure 5 Point 119863 is selected as thecoordinate origin The 119909-axis is located in the horizontalright whereas the 119910-axis is vertically downward 119891(119909) is thedeflection of the point with a distance of 119909 from point119863 ℎ1119909is the thickness function of the parabolic part on the leaf andℎ1119909 = ℎ2radic(119909 + 1198970)1198972

(1) Differential equations are built segmentally usingmaterial-bend deformation theory

Step 1 For CD section with bending moment 119872(1199091) =

1198651199091 (0 le 1199091 le 1198971 minus 1198970) and sectional moment of inertia 1198681 thedeflection curve differential equation is expressed as follows

119864119868111989110158401015840

1(1199091) = 1198651199091 (2)

The equation is integrated once and the double integral of 1199091is as follows

11986411986811198911015840

1(1199091) =

1

211986511990912+ 1198621

11986411986811198911 (1199091) =1

611986511990913+ 11986211199091 + 1198631

(3)

Step 2 For the BC section with bending moment 119872(1199092) =

1198651199092 (1198971 minus 1198970 le 1199092 le 1198972 minus 1198970) and sectional moment of inertia1198682119909(119909) = 119887ℎ

3

111990912 = (119887ℎ

3

212)((1199092 + 1198970)1198972)

32= 1198682((1199092 +

1198970)1198972)32 then the deflection curve differential equation is

expressed as follows

119864119868211989110158401015840

2(1199092) = 119872(1199092)

= 11989732

2119865 [(1199092 + 1198970)

minus12minus 1198970 (1199092 + 1198970)

minus32]

(4)

Integrating will result in the following equation

11986411986821198911015840

2(1199092) = 2119865119897

32

2[(1199092 + 1198970)

12+ 1198970 (1199092 + 1198970)

minus12]

+ 1198622

11986411986821198912 (1199092) = 411986511989732

2[1

3(1199092 + 1198970)

32+ 1198970 (1199092 + 1198970)

12]

+ 11986221199092 + 1198632

(5)

Step 3 For the 119860119861 section with bending moment of 119872(119909) =

1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) and sectional moment of inertia of1198682 the deflection curve differential equation is as follows

119864119868211989110158401015840

3(1199093) = 1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) (6)

Integrating will yield the following equations

11986411986821198911015840

3(1199093) =

1

21198651199092

3+ 1198623

11986411986821198913 (1199093) =1

61198651199093

3+ 11986231199093 + 1198633

(7)

(2) Deflection and rotation angle are determined basedon the boundary conditions and continuity of deflection tosolve the constants in the differential equations

Step 1 For the boundary condition in which the spring isfixed at point 119862 (1199093 = 119897 minus 1198970)

1198911015840

3(119897 minus 1198970) = 1198913 (119897 minus 1198970) = 0 (8)

Constants 1198623 and1198633 are calculated by (7)-(8) as follows

1198623 = minus119865

2(119897 minus 1198970)

2

1198633 =119865

3(119897 minus 1198970)

3

(9)

Step 2 The continuity of deformation at point 119861 (1199092 = 1199093 =

1198972 minus 1198970) is considered

1198911015840

2(1198972 minus 1198970) = 119891

1015840

3(1198972 minus 1198970)

1198912 (1198972 minus 1198970) = 1198913 (1198972 minus 1198970)

(10)

Mathematical Problems in Engineering 5

O

DCBA

l

x

F

s

l1

l0

l2

h1

h1x

h2

Figure 6 Mono-leaf spring (half-spring) with applied force at its end

Constants 1198622 and 1198632 are calculated by (5) (7) and (10)as follows

1198622 = 119865(minus3

21198972

2minus 311989701198972 +

1

21198972

0) + 1198623

1198632 = 119865(1

31198973

2minus 311989701198972

2minus 31198972

01198972 +

1

31198973

0) + 1198633

(11)

Step 3 The continuity of deformation at point 119861 (1199091 = 1199092 =

1198971 minus 1198970) is considered Thus

1198911015840

1(1198971 minus 1198970) = 119891

1015840

2(1198971 minus 1198970)

1198911 (1198971 minus 1198970) = 1198912 (1198971 minus 1198970)

(12)

Constants 1198621 and 1198631 are calculated by (3) (5) and (12) asfollows

1198621 = 119865 [minus3

21205723(1 minus 120572) 119897

2

2minus 31205722(120572 minus 1) 11989701198972

+1

2(1205723minus 1) 1198972

0] + 1198623120572

3

1198631 = 119865 [1

31205723(1 minus 120572

3) 1198973

2minus 31205723(1 minus 120572) 1198970119897

2

2

minus 31205722(120572 minus 1) 119897

2

01198972 +

1

3(1205723minus 1) 1198973

0] + 1198633120572

3

(13)

(3) The deflection curve function of the 119862119863 section (0 le

1199091 le 1198971 minus 1198970) is as follows

1198641198681119891 (1199091) =1

61198651199093

1+ 11986211989811199091 + 1198631198981 (14)

where 1198621198981 = 1198621119865 1198631198981 = 1198631119865The deflection and the rotation angle at point 119863 are

obtained by setting 1199091 to zero

1198911198632 = 1198911 (1199092 = 0) =1198631198981119865

1198641198681

1198911015840

1198632=

1198621198981119865

1198641198681

(15)

The deflection at point 119874 is as follows

1198911198742 =119865 (minus11986211989811198970 + 1198631198981)

1198641198681

(16)

322Theoretical Derivation ofMono-Leaf Spring with AppliedForce at Its End A force model of a mono-leaf spring (half-spring) with an applied force at its end is shown in Figure 6This model is a special case of the model shown in Figure 5

Thus (14) can be used and 1198970 in (14) is set to zero to obtainthe deflection curve function of 119862119874 section (0 le 119909 le 1198971) asfollows

1198641198681119891 (119909) =1

61198651199093+ 119865[minus

1198972

21205723minus

3

2(1 minus 120572) 120572

31198972

2]119909

+ 119865 [1

311989731205723+

1

31205723(1 minus 120572

3) 1198973

2]

(17)

defining11986311989810 = (12057233)[1198973+(1minus120572

3)1198973

2] and11986211989810 = 120572

3[minus11989722minus

(32)(1 minus 120572)1198972

2]

Thus the deflection at the end (119909 = 0) is as follows

1198911198741 =11986511986311989810

1198641198681

(18)

The deflection at point119863 (119909 = 1198970) is expressed as follows

1198911198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (19)

Thus the deflection equations for a leaf with appliedforces at the end andmidsection are derivedThese equationswill be used for the calculation of the deflection of the double-leaf spring

33 Derivation of Equation for Composite Stiffness for Double-Leaf Spring A force model of a double-leaf spring (half-spring) with applied force at its end is shown in Figure 7 119865and 1198651 represent the forces applied at the end of the mainand auxiliary springs respectively 1198651015840

1is the reaction force of

1198651 and the deflection at point 119874 is a superposition of thedeflection caused by 119865

1015840

1and 1198651 The force 1198651015840

1is initially deter-

mined using the boundary condition at point 119863 Thereafterthe deflection at point119874 is calculated using the superpositionprinciple of displacement Finally the composite stiffness ofthe double-leaf spring is obtained

(1) Calculation of the Force at the Contact Point The auxiliaryspring has contact with the main spring only at its end when

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

O

DCB

A

s

l

xF

l1

l0

l2

h1

h1x

h2

Figure 5 Mono-leaf spring (half-spring) with a force at its midsection

midsection First the mono-leaf spring with an applied forceat its midsection is considered

321 Theoretical Derivation of Mono-Leaf Spring withan Applied Force at Its Midsection Differential equationsare established segmentally and integration constants areobtained from boundary conditions on the basis of material-bend deformation theory A force model of a mono-leafspring (half-spring) with an applied force at its midsectionis built and shown in Figure 5 Point 119863 is selected as thecoordinate origin The 119909-axis is located in the horizontalright whereas the 119910-axis is vertically downward 119891(119909) is thedeflection of the point with a distance of 119909 from point119863 ℎ1119909is the thickness function of the parabolic part on the leaf andℎ1119909 = ℎ2radic(119909 + 1198970)1198972

(1) Differential equations are built segmentally usingmaterial-bend deformation theory

Step 1 For CD section with bending moment 119872(1199091) =

1198651199091 (0 le 1199091 le 1198971 minus 1198970) and sectional moment of inertia 1198681 thedeflection curve differential equation is expressed as follows

119864119868111989110158401015840

1(1199091) = 1198651199091 (2)

The equation is integrated once and the double integral of 1199091is as follows

11986411986811198911015840

1(1199091) =

1

211986511990912+ 1198621

11986411986811198911 (1199091) =1

611986511990913+ 11986211199091 + 1198631

(3)

Step 2 For the BC section with bending moment 119872(1199092) =

1198651199092 (1198971 minus 1198970 le 1199092 le 1198972 minus 1198970) and sectional moment of inertia1198682119909(119909) = 119887ℎ

3

111990912 = (119887ℎ

3

212)((1199092 + 1198970)1198972)

32= 1198682((1199092 +

1198970)1198972)32 then the deflection curve differential equation is

expressed as follows

119864119868211989110158401015840

2(1199092) = 119872(1199092)

= 11989732

2119865 [(1199092 + 1198970)

minus12minus 1198970 (1199092 + 1198970)

minus32]

(4)

Integrating will result in the following equation

11986411986821198911015840

2(1199092) = 2119865119897

32

2[(1199092 + 1198970)

12+ 1198970 (1199092 + 1198970)

minus12]

+ 1198622

11986411986821198912 (1199092) = 411986511989732

2[1

3(1199092 + 1198970)

32+ 1198970 (1199092 + 1198970)

12]

+ 11986221199092 + 1198632

(5)

Step 3 For the 119860119861 section with bending moment of 119872(119909) =

1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) and sectional moment of inertia of1198682 the deflection curve differential equation is as follows

119864119868211989110158401015840

3(1199093) = 1198651199093 (1198972 minus 1198970 le 1199093 le 119897 minus 1198970) (6)

Integrating will yield the following equations

11986411986821198911015840

3(1199093) =

1

21198651199092

3+ 1198623

11986411986821198913 (1199093) =1

61198651199093

3+ 11986231199093 + 1198633

(7)

(2) Deflection and rotation angle are determined basedon the boundary conditions and continuity of deflection tosolve the constants in the differential equations

Step 1 For the boundary condition in which the spring isfixed at point 119862 (1199093 = 119897 minus 1198970)

1198911015840

3(119897 minus 1198970) = 1198913 (119897 minus 1198970) = 0 (8)

Constants 1198623 and1198633 are calculated by (7)-(8) as follows

1198623 = minus119865

2(119897 minus 1198970)

2

1198633 =119865

3(119897 minus 1198970)

3

(9)

Step 2 The continuity of deformation at point 119861 (1199092 = 1199093 =

1198972 minus 1198970) is considered

1198911015840

2(1198972 minus 1198970) = 119891

1015840

3(1198972 minus 1198970)

1198912 (1198972 minus 1198970) = 1198913 (1198972 minus 1198970)

(10)

Mathematical Problems in Engineering 5

O

DCBA

l

x

F

s

l1

l0

l2

h1

h1x

h2

Figure 6 Mono-leaf spring (half-spring) with applied force at its end

Constants 1198622 and 1198632 are calculated by (5) (7) and (10)as follows

1198622 = 119865(minus3

21198972

2minus 311989701198972 +

1

21198972

0) + 1198623

1198632 = 119865(1

31198973

2minus 311989701198972

2minus 31198972

01198972 +

1

31198973

0) + 1198633

(11)

Step 3 The continuity of deformation at point 119861 (1199091 = 1199092 =

1198971 minus 1198970) is considered Thus

1198911015840

1(1198971 minus 1198970) = 119891

1015840

2(1198971 minus 1198970)

1198911 (1198971 minus 1198970) = 1198912 (1198971 minus 1198970)

(12)

Constants 1198621 and 1198631 are calculated by (3) (5) and (12) asfollows

1198621 = 119865 [minus3

21205723(1 minus 120572) 119897

2

2minus 31205722(120572 minus 1) 11989701198972

+1

2(1205723minus 1) 1198972

0] + 1198623120572

3

1198631 = 119865 [1

31205723(1 minus 120572

3) 1198973

2minus 31205723(1 minus 120572) 1198970119897

2

2

minus 31205722(120572 minus 1) 119897

2

01198972 +

1

3(1205723minus 1) 1198973

0] + 1198633120572

3

(13)

(3) The deflection curve function of the 119862119863 section (0 le

1199091 le 1198971 minus 1198970) is as follows

1198641198681119891 (1199091) =1

61198651199093

1+ 11986211989811199091 + 1198631198981 (14)

where 1198621198981 = 1198621119865 1198631198981 = 1198631119865The deflection and the rotation angle at point 119863 are

obtained by setting 1199091 to zero

1198911198632 = 1198911 (1199092 = 0) =1198631198981119865

1198641198681

1198911015840

1198632=

1198621198981119865

1198641198681

(15)

The deflection at point 119874 is as follows

1198911198742 =119865 (minus11986211989811198970 + 1198631198981)

1198641198681

(16)

322Theoretical Derivation ofMono-Leaf Spring with AppliedForce at Its End A force model of a mono-leaf spring (half-spring) with an applied force at its end is shown in Figure 6This model is a special case of the model shown in Figure 5

Thus (14) can be used and 1198970 in (14) is set to zero to obtainthe deflection curve function of 119862119874 section (0 le 119909 le 1198971) asfollows

1198641198681119891 (119909) =1

61198651199093+ 119865[minus

1198972

21205723minus

3

2(1 minus 120572) 120572

31198972

2]119909

+ 119865 [1

311989731205723+

1

31205723(1 minus 120572

3) 1198973

2]

(17)

defining11986311989810 = (12057233)[1198973+(1minus120572

3)1198973

2] and11986211989810 = 120572

3[minus11989722minus

(32)(1 minus 120572)1198972

2]

Thus the deflection at the end (119909 = 0) is as follows

1198911198741 =11986511986311989810

1198641198681

(18)

The deflection at point119863 (119909 = 1198970) is expressed as follows

1198911198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (19)

Thus the deflection equations for a leaf with appliedforces at the end andmidsection are derivedThese equationswill be used for the calculation of the deflection of the double-leaf spring

33 Derivation of Equation for Composite Stiffness for Double-Leaf Spring A force model of a double-leaf spring (half-spring) with applied force at its end is shown in Figure 7 119865and 1198651 represent the forces applied at the end of the mainand auxiliary springs respectively 1198651015840

1is the reaction force of

1198651 and the deflection at point 119874 is a superposition of thedeflection caused by 119865

1015840

1and 1198651 The force 1198651015840

1is initially deter-

mined using the boundary condition at point 119863 Thereafterthe deflection at point119874 is calculated using the superpositionprinciple of displacement Finally the composite stiffness ofthe double-leaf spring is obtained

(1) Calculation of the Force at the Contact Point The auxiliaryspring has contact with the main spring only at its end when

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

O

DCBA

l

x

F

s

l1

l0

l2

h1

h1x

h2

Figure 6 Mono-leaf spring (half-spring) with applied force at its end

Constants 1198622 and 1198632 are calculated by (5) (7) and (10)as follows

1198622 = 119865(minus3

21198972

2minus 311989701198972 +

1

21198972

0) + 1198623

1198632 = 119865(1

31198973

2minus 311989701198972

2minus 31198972

01198972 +

1

31198973

0) + 1198633

(11)

Step 3 The continuity of deformation at point 119861 (1199091 = 1199092 =

1198971 minus 1198970) is considered Thus

1198911015840

1(1198971 minus 1198970) = 119891

1015840

2(1198971 minus 1198970)

1198911 (1198971 minus 1198970) = 1198912 (1198971 minus 1198970)

(12)

Constants 1198621 and 1198631 are calculated by (3) (5) and (12) asfollows

1198621 = 119865 [minus3

21205723(1 minus 120572) 119897

2

2minus 31205722(120572 minus 1) 11989701198972

+1

2(1205723minus 1) 1198972

0] + 1198623120572

3

1198631 = 119865 [1

31205723(1 minus 120572

3) 1198973

2minus 31205723(1 minus 120572) 1198970119897

2

2

minus 31205722(120572 minus 1) 119897

2

01198972 +

1

3(1205723minus 1) 1198973

0] + 1198633120572

3

(13)

(3) The deflection curve function of the 119862119863 section (0 le

1199091 le 1198971 minus 1198970) is as follows

1198641198681119891 (1199091) =1

61198651199093

1+ 11986211989811199091 + 1198631198981 (14)

where 1198621198981 = 1198621119865 1198631198981 = 1198631119865The deflection and the rotation angle at point 119863 are

obtained by setting 1199091 to zero

1198911198632 = 1198911 (1199092 = 0) =1198631198981119865

1198641198681

1198911015840

1198632=

1198621198981119865

1198641198681

(15)

The deflection at point 119874 is as follows

1198911198742 =119865 (minus11986211989811198970 + 1198631198981)

1198641198681

(16)

322Theoretical Derivation ofMono-Leaf Spring with AppliedForce at Its End A force model of a mono-leaf spring (half-spring) with an applied force at its end is shown in Figure 6This model is a special case of the model shown in Figure 5

Thus (14) can be used and 1198970 in (14) is set to zero to obtainthe deflection curve function of 119862119874 section (0 le 119909 le 1198971) asfollows

1198641198681119891 (119909) =1

61198651199093+ 119865[minus

1198972

21205723minus

3

2(1 minus 120572) 120572

31198972

2]119909

+ 119865 [1

311989731205723+

1

31205723(1 minus 120572

3) 1198973

2]

(17)

defining11986311989810 = (12057233)[1198973+(1minus120572

3)1198973

2] and11986211989810 = 120572

3[minus11989722minus

(32)(1 minus 120572)1198972

2]

Thus the deflection at the end (119909 = 0) is as follows

1198911198741 =11986511986311989810

1198641198681

(18)

The deflection at point119863 (119909 = 1198970) is expressed as follows

1198911198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (19)

Thus the deflection equations for a leaf with appliedforces at the end andmidsection are derivedThese equationswill be used for the calculation of the deflection of the double-leaf spring

33 Derivation of Equation for Composite Stiffness for Double-Leaf Spring A force model of a double-leaf spring (half-spring) with applied force at its end is shown in Figure 7 119865and 1198651 represent the forces applied at the end of the mainand auxiliary springs respectively 1198651015840

1is the reaction force of

1198651 and the deflection at point 119874 is a superposition of thedeflection caused by 119865

1015840

1and 1198651 The force 1198651015840

1is initially deter-

mined using the boundary condition at point 119863 Thereafterthe deflection at point119874 is calculated using the superpositionprinciple of displacement Finally the composite stiffness ofthe double-leaf spring is obtained

(1) Calculation of the Force at the Contact Point The auxiliaryspring has contact with the main spring only at its end when

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

DC

BA

l

F

x

s

l1

l0

l2

l3

h4

h1x

h2x

h5

h1h

2

F1

Figure 7 Force model of a double-leaf spring (half-spring)

a force is applied at the end of a double-leaf spring The forcebetween the two leaves is transmitted through the contactpoint The deflection and rotation angle between two leavesat the contact point do not remarkably varyThus the contactpoint has two boundary conditions namely equal deflectionsand equal rotation angles For each boundary condition theforce between the two leaves can be calculated

Step 1 (the two leaves have the same deflections at the contactpoint) By using (18) and (19) the deflection at point119863 of themain spring caused by force 119865 is as follows

1198911198981198631 =1198973

06 + 11986311989810 + 119862119898101198970

1198641198681

119865 (20)

The deflection at point 119863 of the main spring caused by force1198651015840

1is as follows

1198911198981198632 =11986511198631198981

1198641198681

(21)

The total deflection at point119863 of themain spring is as follows119891119898119863 = 1198911198981198631 minus 1198911198981198632

The deflection at point 119863 of the auxiliary spring causedby 1198651 is as follows

119891119886119863 =119865111986311988610

1198641198684

(22)

where11986311988610 = (12057331198653)[(119897 minus 1198970)

3+ (1 minus 120573

3)1198973

3]

For 119891119898119863 = 119891119886119863 the force at the contact point because ofthe same deflection is as follows

1198651119891 =

(1198973

06 + 11986311989810 + 119862119898101198970)

120574311986311988610 + 1198631198981

119865 (23)

Step 2 (the two leaves have the same rotation angles at thecontact point) The force between two leaves can be easilycalculated when they have the same rotation angles at thecontact point

1198651120579 =1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

119865 (24)

where 11986211988610 = 1205733[minus(119897 minus 1198970)

22 minus (32)(1 minus 120573)119897

2

3]

The deflections or rotation angles of the two springsat their contact area are not exactly the same because ofthe rubber between the main and auxiliary springs at theircontact area The force at the contact point is neither 1198651119891 nor1198651120579 The force is a combined effect of the two forces Thusforce 1198651 at the contact point is assumed to be as follows

1198651 = 1205961198651119891 + (1 minus 120596) 1198651120579 (25)

where 120596 is a weight coefficient with a value that ranges fromzero to one

(2) The Deflection at the End Point 119874 Is Calculated

Step 1 The deflection caused by force 119865 is expressed asfollows

1198911198741 =11986511986311989810

1198641198681

(26)

Step 2 The deflection caused by force 1198651 is expressed asfollows

1198911198742 =1198651 (1198631198981 minus 11986211989811198970)

1198641198681

(27)

Step 3 The total deflection at point 119874 is defined as follows

119891119874 = 1198911198741 minus 1198911198742 =119865

1198641198681

11986311989810 minus (1198631198981 minus 11986211989811198970)

sdot [1205961198973

06 + 11986311989810 + 119862119898101198970

120574311986311988610 + 1198631198981

+ (1 minus 120596)1198972

02 + 11986211989810

120574311986211988610 + 11986211989810

] =

119865120581equ1198973

1198641198681

(28)

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

F

l

x

s

l1

l4

l2

l3

h4

h1x

h2x

h3x

h5 h7

h8

h1

h2

Figure 8 Force model of a triple-leaf spring (half-spring)

where

120581equ (120582 120583 120572 120573 120574) = 11988911989810 minus (1198891198981 minus 1198881198981120583)

sdot [120596(16) 120583

3+ 11988911989810 + 11988811989810120583

120574311988911988610 + 1198891198981

+ (1 minus 120596)12058322 + 11988811989810

120574311988811988610 + 1198881198981

]

11988911989810 = 12057231 + (1 minus 120572

3) 1205823

3

11988811989810 = minus1205723 1 + 3 (1 minus 120572) 120582

2

2

1198891198981 =1

3(1 minus 120583)

31205723+

1

31205723(1 minus 120572

3) 1205823minus 31205723(1 minus 120572)

sdot 1205831205822minus 31205722(120572 minus 1) 120583

2120582 +

1

3(1205723minus 1) 120583

3= 11988911989810

minus 31205723(1 minus 120572) 120583120582

2minus 31205722(120572 minus 1) 120583

2120582 minus

1

31205833+ 12057231205832

minus 1205723120583

1198881198981 = minus(1 minus 120583)

2

21205723minus

3

2(1 minus 120572) 120572

31205822minus 3 (120572 minus 1) 120572

2120583120582

+1

2(1205723minus 1) 120583

2= 11988811989810 minus 3 (120572 minus 1) 120572

2120583120582 minus

1

21205832

+ 1205723120583

(29)

Finally an equation for calculating the composite stiffnessof a double parabolic leaf spring is obtained as follows

119870two =119865

119891119874

=1198641198681

120581equ1198973 (30)

Twomethods are used to calculate the composite stiffnessof a parabolic double-leaf spring to further refine the valuerange of 120596 in the equations First (30) is used to calculatedirectly whereas the second approach is to use (1) and per-form numerical integration to obtain the composite stiffnessThe results from the two methods are compared when 120596 isset to different values When 120596 is 05 to 07 the result errorbetween two methods is less than 5

34 Derivation of an Equation for Calculating the CompositeStiffness of Triple-Leaf and Multileaf Springs Compared withthe double-leaf spring triple- and multileaf springs have twoor more main leaves with similar lengths A simplified modelis built as discussed in Section 22 Thus the equation forcalculating the composite stiffness of triple- and multileafsprings can be easily derived A force model of a triple-leaf spring (half-spring) with an applied force at its end isbuilt and shown in Figure 8 ℎ3119909 stands for the thicknessof the auxiliary spring at the point with a distance of 119909 tothe parabola vertex (end point of the main spring) ℎ3119909 =

ℎ7radic1199091198972 1198974 stands for the length of the section with equalthickness at the end of the second main spring 1198974 =

1198972(ℎ7ℎ8)2

First twomain springs with different parameters are con-sideredThe stiffness of the addedmain spring is expressed asfollows

119870119904 =1198641198687

1198892119898101198973 (31)

where 119889211989810(120578 1205722) = 1205723

2((1 + 120582

3(1 minus 120572

3

2))3) 1205722 = ℎ7ℎ8 and

1198687 = (112)119887ℎ3

7

The composite stiffness of a triple-leaf spring is defined asfollows

119870three = 119870two + 119870119904 =1198641198681

120581equ1198973+

1198641198687

1198892119898101198973 (32)

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

The equation formultileaf spring can be derived similarlyThe parameters of the multi-main springs are generally thesame Thus the equation for the composite stiffness of atriple-leaf spring can be simplified as follows

119870three =1198641198681

120581equ1198973+

1198641198681

119889119898101198973 (33)

By contrast the equation for the multileaf spring isexpressed as follows

119870ℎ = 119870two + (119899 minus 1)119870119905 = 120585 [1198641198681

120581equ1198973+

(119899 minus 1) 1198641198681

119889119898101198973

] (34)

where 119899 stands for the number of the main springs and 120585 is acorrection factor ranging from 092 to 099

35 Calculation of Composite Stiffness of Taper-Leaf Springwith Front and Rear Halves of Unequal Lengths All previ-ously derived equations for composite stiffness calculationwere based on half-spring models The composite stiffnessof an entire leaf spring is calculated by determining thecomposite stiffness at the front and rear halves (usually theyare not of equal lengths at the front and rear halves of thetaper-leaf spring are are not equal just as the leaf-springstudied in this paper) which should be calculated first byusing the equations shown above

119870119891 and 119870119903 represent the composite and the stiffness ofthe front and rear half springs of a taper-leaf spring Thecomposite stiffness of the entire spring can be calculatedusing (33)

119870 = (119870119891 + 119870119903)120575 (1 + 120577)

2

(1 + 120575) (1 + 1205751205772) (35)

where 120575 = 119870119891119870119903 120577 = 119897119891119897119903The composite stiffness of the front half spring is

expressed as follows

119870119891 =1198641198681

120581equ1198973

119891

+1198641198681

119889119898101198973

119891

(36)

The composite stiffness of the rear half spring is definedas follows

119870119903 =1198641198681

120581equ1198973119903

+1198641198681

119889119898101198973119903

(37)

This calculation method is not limited to the triple-leafspring studied in this paper The method can also be used formultileaf springs

36 Experimental and FEA Assessments The correctness ofthe theoretical formula is verified by testing the mechanicalproperties of a fabricated triple-leaf spring (Figure 9)The twomain leaves in the triple-leaf spring have similar geometricparameters and the front and rear halves of each leaf havethe same root thicknesses and end thicknessesThe geometricparameters of the triple-leaf spring are listed in Table 1

Hydraulic actuator

Leaf spring

Track

PusherSliding car

Figure 9 Experimental apparatus for measuring the stiffness of thetriple-leaf spring

Table 1 Geometric parameters of a triple-leaf spring

Front RearLength ofhalf-spring(mm)

119897119891 = 692 119897119903 = 718

Springwidth 119887

(mm)60

Main spring Auxiliary springFront Rear Front Rear

Length ofparabolicportion (mm)

1198972119891 = 633 1198972119903 = 654 1198973119891 = 492 1198973119903 = 501

End thickness(mm) ℎ1 = 102 ℎ4 = 108

Root thickness(mm) ℎ2 = 142 ℎ5 = 234

The leaf spring is tested on a static stiffness test rig Thetest rig consists of a hydraulic actuator a pusher used to loadthe spring rail base tested leaf spring sliding car and soon Two spring eyes are fixed to the sliding cars and theseeyes can only slide along the track when the spring is loadedThe leaf spring is loaded vertically by the actuator pusherThe load is gradually increased to 245 kN from 0N and thenreloaded to 0N The loading process which is as long as thereloading process is 120 s long The load and displacementduring testing are recorded and their relationship is shownin Figure 10

The experimental results show that hysteresis loss appearsduring the process of loading and reloading because of thefriction between the main and auxiliary springs Thus ahysteresis loop is found in the displacement-force curve Thechange trend of leaf spring stiffness with increasing load issimilar to our predictions in the previous section When themain and auxiliary springs are under full contact the stiffnessachieves a maximum and constant value

The proposed formula is derived on the basis of thecondition that the main and auxiliary springs are under fullcontact Thus the formula is only suitable for calculatingthe stiffness of the leaf spring when the main and auxiliarysprings are under full contact The simulation experimentaland calculated equation (30) (120596 = 06) results are shown inTable 2 The error between the calculated and experimental

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Displacement (mm)

0

7

14

21

28

Forc

e (kN

)

0 45 90 135 180 225

Figure 10 Stiffness of the triple-leaf spring

Table 2 Comparative results

Stiffness (Nmm) Relative errorwith test result

Test result 1838Simulation result 1756 45Calculation result 1798 22

results is small (within 5) The calculated value is smallerthan the experimental value because the friction betweenthe main and auxiliary springs is neglected during equationderivation processThus the equation for composite stiffnesscalculation derived in this paper can fully meet the needsfor engineering application Moreover the simulation resultwhich is close to the calculated result verifies the correctnessof the calculation equation

4 Attention of the Derived Equation in theSpring Design Process

The correctness of the derived equation is confirmed by theresults Thus the equation not only can be used to calculatethe stiffness of existing leaf springs but also can be used inthe design of a leaf spring During the designing process thegeometric parameters of the spring leaves can be designedon the basis of the desired stiffness of the spring Howeverin the actual machining process of the leaf spring someactual dimensions of the spring leaf vary from the calculateddimensions when reliability stress concentration and otherfactors are considered The effect of these differences on thecomposite stiffness value should be determined

41 RootThickness The value of the root thickness of the leafspring used in the formula is not equal to the onemeasured ona real leaf springThis value is defined to be the vertical offsetbetween the vertex of the parabolic leaf spring and the pointin which the parabola reaches the U-bolt (ℎ in Figure 11)However in the production of a leaf spring the root thicknessof a leaf spring is designed to be equal to the vertical offset

Table 3 Geometric parameters of the mono-leaf spring model

Mono-leaf springSpring width 119887 (mm) 80Length of half spring 119897 (mm) 800Length of parabolic portion 1198972 (mm) 720End thickness ℎ1 (mm) 12Root thickness ℎ2 (mm) 25

Table 4 Geometric parameters of mono-leaf spring model

Length of half-spring 119897 (mm) 700Spring width 119887 (mm) 80

Main spring Auxiliary springLength of parabolic portion (mm) 1198972 = 620 1198973 = 500

End thickness (mm) ℎ1 = 10 ℎ4 = 20

Root thickness (mm) ℎ2 = 8 ℎ5 = 25

between the vertex of the spring leaf parabola and the point inwhich the parabola reaches the center bolt (ℎ+Δℎ in Figure 11)considering the reliability of the U bolt and reducing thestress of the spring at the U-bolt

42 Transition Region between the Isopachous and ParabolicPortions of the Leaf Spring During the actual processing of aleaf spring the designer tends to increase at an arc transitionregion at the junction to reduce the stress concentration atthe junction between isopachous and parabolic portions ofthe leaf spring (blue area in Figure 12) The finite elementsimulation analytical method is used because the shape of thetransition region is difficult to use to describe the function

The effect of this transition region on stiffness calculationis examined A group of mono-leaf spring models and agroup of double-leaf spring modes are used for FEA Theirgeometric parameters are shown in Tables 3 and 4

Both groups of models contain a model without a transi-tion region and a model with arc transition region

The transition region should not be too large or too smallIt matches the size of the spring leaf (as shown in Table 5)Two groups of simulation results from ABAQUS are shownin Table 5 The existence of the transition region does notremarkably affect stiffness calculation so the error of less than2 can be neglected

43 End Thickness of Auxiliary Leaf The end thickness ofan auxiliary leaf is the thickness of the uniform thickness atthe end of the auxiliary leaf (ℎ4 in Figure 1) The auxiliaryspring not the main spring generally bears only the verticalload Thus in the actual structure its end thickness can besmall enough to be close to zero If a clip exists at the end ofthe auxiliary spring to transmit the lateral load or a rubberblock to cushion stiffness mutation the end thickness of theauxiliary spring is minimized as long as it satisfies certainneeds of the lateral load and bearing reliability of the rubberblockThis characteristic is also in line with the requirementsof lightness of weight

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 5 Simulation results of models with or without transition region

GroupThe radius of thetransition region

(mm)Simulation results (mm) Stiffness

(Nmm)Relativeerror

Mono-leafspring

Without transitionregion

+1426e minus 02

minus4596e + 01

minus9193e + 01

minus1379e + 02

minus1839e + 02

minus2298e + 02

minus2758e + 02

minus3218e + 02

minus3678e + 02

minus4137e + 02

minus4597e + 02

UU2

5438

6010

UU2

+1554e minus 02

minus4555e + 01

minus9112e + 01

minus1367e + 02

minus1823e + 02

minus2278e + 02

minus2734e + 02

minus3190e + 02

minus3645e + 02

minus4101e + 02

minus4557e + 02

5486 088

Double-leafspring

Without transitionregion

UU2+2779e minus 03

minus9774e + 00

minus1955e + 01

minus2933e + 01

minus3910e + 01

minus4888e + 01

minus5866e + 01

minus6843e + 01

minus7821e + 01

minus8798e + 01

minus9776e + 01

13298

Main spring 2600Auxiliary spring 500

UU2+2779e minus 03

minus9750e + 00

minus1950e + 01

minus2926e + 01

minus3901e + 01

minus4876e + 01

minus5852e + 01

minus6827e + 01

minus7802e + 01

minus8777e + 01

minus9753e + 01

13329 023

44 Other Factors Other factors such as U-bolt preloadleaf spring arc height and surface treatment also affect thestiffness value The manner in which these factors affect thestiffness is similar to traditional leaf springs with uniformthickness

5 Conclusion

The conclusion is as follows(1) An equation for calculating the composite stiffness for

multileaf springs when the main and auxiliary spring

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

l

h

l1

l2

h+Δh

U-bolts

Arc transitionRed dotted line isthe extent of leafsprings parabolicarea

Figure 11 Root thickness of spring leaf

l

l1l2

Arc transition area

Red dotted line isthe extent of leafsprings parabolic

Figure 12 Transition region

are under full contact is derived The correctness ofthe calculation method is verified by the rig test andsimulation

(2) Parameters that should be considered for designingparabolic leaf springs are discussed to provide guid-ance for the design and manufacture of such leafsprings

Nomenclature

119897 The length of the first main spring (subscripts119891 and 119903 indicate the front half or rear half ofthe spring) (see 119897119891 and 119897119903 in Figure 1)

1198970 The distance between the ends of the mainand auxiliary springs (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198970119891 and 1198970119903 in Figure 1)

120583 Ratio of 1198970 to 119897

1198972 The length of the parabolic portion of thefirst main spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198972119891 and 1198972119903 in Figure 1)

120582 Ratio of 1198972 to 119897

1198973 Length of the parabolic portion of theauxiliary spring (subscripts 119891 and 119903 indicatethe front half or rear half of the spring) (see1198973119891 and 1198973119903 in Figure 1)

1198971 Length of the isopachous portion of thefirst main spring (subscripts 119891 and 119903

indicate the front half or rear half of thespring) (see 1198971119891 and 1198971119903 in Figure 1)

ℎ1 ℎ2 ℎ3 Front end thickness root thickness andrear end thickness of the first mainspring

120572 Ratio of ℎ1 to ℎ2

ℎ4 ℎ5 ℎ6 Front end thickness root thickness andrear end thickness of the auxiliary spring

120573 Ratio of ℎ4 to ℎ3

120574 Ratio of ℎ1 to ℎ4

ℎ7 ℎ8 Root thickness and rear end thickness ofthe second main spring

ℎ1119909 ℎ2119909 ℎ3119909 Thickness functions of the cross sectionin the parabolic portion of the first mainspring second main spring and auxil-iary spring

119865 Force acting at the end of the leaf spring(subscripts119891 and 119903 indicate the front halfor rear half of the spring) (see 119865119891 and 119865119903

in Figure 1)119904 The half-length of the isopachous por-

tion at the spring root119887 Spring width120585 Distortion correction coefficient1198681 Sectional moment of inertia at the end

of the first main spring consider 1198681 =

119887ℎ3

112

1198682 Sectional moment of inertia at the rootof the first main spring consider 1198682 =

119887ℎ3

212

1198683 Sectional moment of inertia at the endof the auxiliary spring consider 1198683 =

119887ℎ3

312

1198684 Sectional moment of inertia at the rootof the auxiliary spring consider 1198684 =

119887ℎ3

412

Conflict of Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The authors would like to thank the School of AutomotiveEngineering Changchun Jilin China and the NationalNatural Science Foundation of China for supporting theproject (Grant no 51205158)

References

[1] S Kim W Moon and Y Yoo ldquoAn efficient method for calcu-lating the nonlinear stiffness of progressive multi-leaf springsrdquo

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

International Journal of Vehicle Design vol 29 no 4 pp 403ndash422 2002

[2] M Bakir M Siktas and S Atamer ldquoComprehensive durabilityassessment of leaf springs with CAE methodsrdquo SAE TechnicalPapers 2014-01-2297 2014

[3] R Liu R Zheng and B Tang ldquoTheoretical calculations andexperimental study of gradually variable rigidity leaf springsrdquoAutomobile Technology vol 11 pp 12ndash15 1993

[4] G Hu P Xia and J Yang ldquoCurvature-force hybrid methodfor calculating properties of leaf springs with variable stiffnessrdquoJournal of Nanjing University of Aeronautics amp Astronautics vol40 no 1 pp 46ndash50 2008

[5] T Horibe and N Asano ldquoLarge deflection analysis of beams ontwo-parameter elastic foundation using the boundary integralequation methodrdquo JSME International Journal Series A SolidMechanics and Material Engineering vol 44 no 2 pp 231ndash2362001

[6] D K Roy and K N Saha ldquoNonlinear analysis of leaf springs offunctionally gradedmaterialsrdquo Procedia Engineering vol 51 pp538ndash543 2013

[7] G Savaidis L Riebeck and K Feitzelmayer ldquoFatigue lifeimprovement of parabolic leaf springsrdquo Materials Testing vol41 no 6 pp 234ndash240 1999

[8] M M Shokrieh and D Rezaei ldquoAnalysis and optimization of acomposite leaf springrdquo Composite Structures vol 60 no 3 pp317ndash325 2003

[9] Y S Kong M Z Omar L B Chua and S Abdullah ldquoStressbehavior of a novel parabolic spring for light duty vehiclerdquoInternational Review ofMechanical Engineering vol 6 no 3 pp617ndash620 2012

[10] M Soner N Guven A Kanbolat T Erdogus and M KOlguncelik ldquoParabolic leaf spring design optimization consid-ering FEA amp Rig test correlationrdquo SAE Technical Paper 2011-01-2167 2011

[11] Y S Kong M Z Omar L B Chua and S Abdullah ldquoExplicitnonlinear finite element geometric analysis of parabolic leafsprings under various loadsrdquo The Scientific World Journal vol2013 Article ID 261926 11 pages 2013

[12] W Liu Automotive Design Tsinghua University Press BeijingChina 2001

[13] Editorial BoardThe Design Part of the Automobile EngineeringManual Peoplersquos Communications Press Beijing China 2001

[14] Y Zhang H Liu and D Wang Spring Manual MachineryIndustry Press Beijing China 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of