research article free vibration and hardening behavior of

Research ArticleFree Vibration and Hardening Behavior ofTruss Core Sandwich Beam

J E Chen1 W Zhang2 M Sun3 and M H Yao2

1Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical SystemTianjin University of Technology Tianjin 300384 China2College of Mechanical Engineering Beijing University of Technology Beijing 100124 China3School of Science Tianjin Chengjian University Tianjin 300384 China

Correspondence should be addressed to M Sun sunmin0537163com

Received 1 October 2015 Revised 19 February 2016 Accepted 6 March 2016

Academic Editor Emiliano Mucchi

Copyright copy 2016 J E Chen et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The dynamic characteristics of simply supported pyramidal truss core sandwich beam are investigated The nonlinear governingequation of motion for the beam is obtained by using a Zig-Zag theory The averaged equations of the beam with primarysubharmonic and superharmonic resonances are derived by using the method of multiple scales and then the correspondingfrequency response equations are obtained The influences of strut radius and core height on the linear natural frequencies andhardening behaviors of the beam are studied It is illustrated that the first-order natural frequency decreases continuously and thesecond-order and third-order natural frequencies initially increase and then decrease with the increase of strut radius and the firstthree natural frequencies all increase with the rise of the core height Furthermore the results indicate that the hardening behaviorsof the beam become weaker with the increase of the rise of strut radius and core heightThemechanisms of variations in hardeningbehavior of the sandwich beam with the three types of resonances are detailed and discussed

1 Introduction

Truss core sandwich structures are drawing more and moreattention from researchers due to their excellent mechanicalproperties andmany functional applications Cellularmateri-als have always played a significant role in many engineeringfields As a new member of the family of cellular materialstruss-like material has broad application prospects Evanset al [1] compared the integrated performances of the differ-ent types of cellular materials the advantages of mechanicalcharacteristics andmultifunction of the lattice materials werehighlighted Deshpande and Fleck [2] investigated the col-lapse responses of the sandwich beams with truss coresTheyrevealed that truss core sandwich beams were significantlylighter than the sandwich beams with metallic foam coreagainst the design constraint of collapse load Wallach andGibson [3] analyzed the elastic moduli uniaxial compressivestrengths and shear strength of three-dimensional trussstructures It was concluded that the truss material hadimproved properties over the closed-cell aluminum foam

Next the manufacturing techniques mechanical behaviorsand functionalities of the sandwich structures with differenttruss cores were paid close attention to by researchers fromdifferent research fields [4ndash9]

Unfortunately the investigations on vibration responsesof the truss core sandwich structures are extremely limitedLou et al [10] studied the free vibration of the simplysupported truss core sandwich beams the natural frequenciesof the sandwich beams were obtained and the theoreticalresults agree well with the numerical results Xu and Qiu[11] investigated the free vibration of the truss core sandwichbeams with interval parameters The effects of geometric andmaterial parameters on the natural frequencies were ana-lyzed Lou et al [12] analyzed the effects of local damage onvibration characteristics of the truss core sandwich structuresby numerical and experimental methods Li and Lyu [13]investigated the active vibration control of lattice sandwichbeams with pyramidal lattice core using the piezoelectricactuatorsensor pairs Song and Li [14] studied the nonlinearaeroelastic characteristics and the active flutter control of

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7348518 13 pageshttpdxdoiorg10115520167348518

2 Shock and Vibration

sandwich beams The face sheets and piezoelectric materialwere modeled by the Euler-Bernoulli beam theory and thecore was modeled by the Reddy third-order shear deforma-tion theory

Moreover the vibration behaviors of the truss coresandwich structures can be deduced with the help of a largenumber of dynamical studies on the sandwich structureswith honeycomb core foam core and other cores FrostigandThomsen [15] used high-order sandwich panel theory toanalyze the free vibration of sandwich panels with a flexiblecore Biglari and Jafari [16] studied the free vibrations ofdoubly curved sandwich panels based on a refined three-layered theory Jam et al [17] investigated the free vibrationof sandwich panels by using an improved high-order theoryIn several papers like the aforementioned three referencesthe three-dimensional elasticity theory was used for the corewhich has a better accuracy to predict the response of thethick sandwich structures

For thin and some moderately thick sandwich struc-tures the equivalent single layer theory and the layerwisetheory were widely used E Nilsson and A C Nilsson [18]investigated the dynamic properties of sandwich structureswith honeycomb and foam cores Li and Zhu [19] studiedthe free vibration of honeycomb sandwich plates by usingthe improved Reddy third-order shear deformation platetheory J H Zhang andW Zhang [20] investigated the globalbifurcations and multipulse chaotic dynamics of a simplysupported honeycomb sandwich rectangular plate by usingthe extended Melnikov method Yang et al [21] investigatedthe transverse vibrations and stability of an axially movingsandwich beam Sahoo and Singh [22] analyzed the bucklingand free vibration analysis of the laminated composite andsandwich plates by using a new trigonometric Zig-Zag theoryKhare et al [23] presented the free vibration analysis ofcomposite and sandwich laminates by using a high-ordershear deformation theory Sakiyama et al [24] studied the freevibration of a three-layer sandwich beam with an elastic orviscoelastic core and arbitrary boundary conditions Banerjee[25] analyzed the free vibration of three-layered symmetricsandwich beams by using the dynamic stiffness methodCetkovic and Vuksanovic [26] studied the bending freevibrations and buckling of the laminated composite andsandwich plates by using the generalized layerwise theoryof Reddy Ferreira [27] analyzed the mechanical behaviorsof the laminated composite and sandwich plates by using alayerwise shear deformation theory and the multiquadricsdiscretization method

Although the studies on the vibrations of the sandwichstructures are relatively detailed the unique dynamic charac-teristics especially the nonlinear behaviors of the sandwichstructures with truss core are still required to be exploredThe main aim of this paper is to particularly study the linearand nonlinear dynamic behaviors of the truss core sandwichbeam In this paper the nonlinear governing equation ofmotion for the simply supported truss core sandwich beamwith immovable edges is obtained by using the von Karmantype equation Hamiltonrsquos principle and a Zig-Zag theoryThe influences of strut radius and core height on the linear

r

a

120579

Figure 1 Sketches of a unit cell of the sandwich beam withpyramidal truss core

natural frequencies and hardening behaviors of the sandwichbeam are investigated The nonlinear frequency responsesof the sandwich beam with primary subharmonic andsuperharmonic resonances are obtained and the numericalresults are discussed

2 Formulation

The simply supported sandwich beam with pyramidal trusscore subjected to a transverse uniform load is considered inthe paper A unit cell of the pyramidal truss core is shown inFigure 1The length radius and inclination angle of the trussare represented by 119886 119903 and 120579 respectively Thus the relativedensity 120588 and the equivalent shear modulus 119866 are expressedas follows [2]

120588 =

120588119888

120588

=

2120587

cos2120579 sin 120579(

119903

119886

)

2

(1a)

119866 = 120587 sin 120579 ( 119903119886

)

2

119864 (1b)

where120588119888is the equivalent density of the truss core and120588 and119864

are the density and elasticity modulus of the mother materialof the truss

The truss core sandwich beam can be modeled as alaminated beam which is composed of two solid layers andan equivalent continuum core layer Generally for a sandwichstructure the following assumptions [28] can be given

(1) The thickness of the truss core sandwich beamremains constant during deformation

(2) Only bending deformation is considered for both thinface sheets and only shear deformation is consideredfor the core

(3) The face sheets and truss core are combined closelyThus the deflections between each layer are continu-ous

These assumptions are widely used to study the sandwichstructures which have thin face sheets and thick core Based

Shock and Vibration 3

z

hhc x

w0

u0

120601x

120597w0

120597x

Figure 2 Undeformed and deformed geometries of the truss coresandwich beam

on undeformed and deformed geometries of the truss coresandwich beam as shown in Figure 2 the displacement fieldcan be expressed in terms of the midplane displacements 119906

0

and1199080and rotations 120601

119909 the face sheets of the sandwich beam

adopt the Kirchhoff hypothesis and the truss core adopts thefirst-order shear deformation theory Hence

minus

ℎ

2

le 119911 le minus

ℎ119888

2

119906119905= 1199060minus 119911

1205971199080

120597119909

+

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119905= 1199080

(2a)

minus

ℎ119888

2

le 119911 le

ℎ119888

2

119906119888= 1199060minus 119911120601119909

119908119888= 1199080

(2b)

ℎ119888

2

le 119911 le

ℎ

2

119906119887= 1199060minus 119911

1205971199080

120597119909

minus

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119887= 1199080

(2c)

where the subscripts 119905 119888 and 119887 represent the top face sheetthe truss core and the bottom face sheet respectively ℎrepresents the total thickness of the truss core sandwich beamand ℎ119888represents the height of the truss core

Using von Karmanrsquos theory for the geometric nonlinear-ity the strain-displacement relations have the form

1205761199091199091=

120597119906119905

120597119909

+

1

2

(

120597119908119905

120597119909

)

2

120574119909119911=

120597119906119888

120597119911

+

120597119908119888

120597119909

1205761199091199093=

120597119906119887

120597119909

+

1

2

(

120597119908119887

120597119909

)

2

(3)

where 1205761199091199091

and 1205761199091199093

represent the strains of the face sheets and120574119909119911

represents the strain of the truss coreSubstituting (2a) (2b) and (2c) into (3) we have

1205761199091199091

1205761199091199093

120574119909119911

=

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

+ 119911

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

=

1205971199060

120597119909

+

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

1205971199060

120597119909

minus

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

minus120601119909+

1205971199080

120597119909

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

=

minus

1205972

1199080

1205971199092

minus

1205972

1199080

1205971199092

0

(4)

The mathematical statement of Hamiltonrsquos principle isgiven as

int

1199052

1199051

(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)

The virtual kinetic energy 120575119870 virtual strain energy 120575119880and the virtual work 120575119882 are respectively given by

120575119880 = 1205751198801+ 1205751198802+ 1205751198803

120575119870 = 1205751198701+ 1205751198702+ 1205751198703

120575119882 = minusint

119860

1198651205751199080minus 12058301205751199080119889119909 119889119910

(6a)

120575119880 = int

119897

0

int

minusℎ1198882

minusℎ2

12059011990911990911205751205761199091199091119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

12059011990911990931205751205761199091199093119889119909 119889119911

= int

119897

0

int

minusℎ1198882

minusℎ2

1205901199091199091(120575120576(0)

1199091199091+ 119911120575120576(1)

1199091199091) 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

1205901199091199093(120575120576(0)

1199091199093

+ 119911120575120576(1)

1199091199093) 119889119909 119889119911 = int

119897

0

1198731199091199091120575120576(0)

1199091199091

+1198721199091199091120575120576(1)

1199091199091119889119909 + int

119897

0

int

ℎ1198882

minusℎ1198882

119876119909120575120574119909119911119889119909 + int

119897

0

1198731199091199093

sdot 120575120576(0)

1199091199093+1198721199091199093120575120576(1)

1199091199093119889119909 119889119911

(6b)


120575119870 = int

119897

0

int

minusℎ1198882

minusℎ2

120588 [(0minus 119911

1205970

120597119909

+

ℎ119888

2

120601119909minus

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

+

ℎ119888

2

120575120601119909minus

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120588119888[(0minus 119911

120601119909) (1205750minus 119911120575

120601119909)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ2

ℎ1198882

120588 [(0minus 119911

1205970

120597119909

minus

ℎ119888

2

120601119909+

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

minus

ℎ119888

2

120575120601119909+

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

(6c)

where the superposed dot on a variable indicates its firstderivative with respect to time

0= 1205971199060120597119905 and so on

Substituting (6a) (6b) and (6c) into (5) the nonlineargoverning equations of motion for the truss core sandwichbeam are obtained

(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812

120601119909 (7a)

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(7b)

(1198731199091199091

1205971199080

120597119909

+ 1198731199091199093

1205971199080

120597119909

)

119909

+

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909119909

+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(7c)

where

11986801

11986811

11986821

= int

minusℎ1198882

minusℎ2

1

119911

1199112

120588119889119911

11986802

11986812

11986822

= int

ℎ1198882

minusℎ1198882

1

119911

1199112

120588119888119889119911

(8a)

1198731199091199091= int

minusℎ1198882

minusℎ2

1205901199091199091119889119911

1198731199091199093= int

ℎ2

ℎ1198882

1205901199091199093119889119911

119876119909= int

ℎ1198882

minusℎ1198882

120590119909119911119889119911

1198721199091199091= int

minusℎ1198882

minusℎ2

1199111205901199091199091119889119911

1198721199091199093= int

ℎ2

ℎ1198882

1199111205901199091199093119889119911

(8b)

We consider the face sheets and the truss core aremade ofthe same isotropy material The stress-strain relations of thetruss core sandwich beam can be represented by

1205901199091199091= 1198641205761199091199091

1205901199091199093= 1198641205761199091199093

120590119909119911= 119866120574119909119911

(9)

The stress-displacement relations can be expressed as

1198731199091199091= 1198601120576(0)

1199091199091+ 1198611120576(1)

1199091199091

1198731199091199093= 1198603120576(0)

1199091199093+ 1198613120576(1)

1199091199093

1198721199091199091= 1198611120576(0)

1199091199091+ 1198631120576(1)

1199091199091

1198721199091199093= 1198613120576(0)

1199091199093+ 1198633120576(1)

1199091199093

119876119909= 119862120574(0)

119909119911

(10a)

where

(1198601 1198611 1198631) = int

minusℎ1198882

minusℎ2

119864 (1 119911 1199112

) 119889119911

(1198603 1198613 1198633) = int

ℎ2

ℎ1198882

119864 (1 119911 1199112

) 119889119911

119862 = int

ℎ1198882

minusℎ1198882

119866119889119911

(10b)

1198601and 119860

3are called extensional stiffness 119861

1and 119861

3are

called bending-extensional stiffness 1198631and 119863

3are called

bending stiffness 119862 is called shear stiffness and 119868119894119895are the

mass moments of inertiaConsidering the symmetry of the two face sheets of the

truss core sandwich beam about the 119909-axis we have

1198601= 1198603

1198611= minus1198613

1198631= 1198633

(11)


Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements

21198601

1205972

1199060

1205971199092+ 21198601

1205971199080

120597119909

1205972

1199080

1205971199092= (211986801+ 11986802) 0minus 11986812

120601119909 (12a)

ℎ2

119888

2

1198601

1205972

120601119909

1205971199092minus

ℎ2

119888

2

1198601

1205973

1199080

1205971199093minus ℎ1198881198611

1205973

1199080

1205971199093minus 119862120601119909

+ 119862

1205971199080

120597119909

= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(12b)

(

ℎ2

119888

2

1198601+ 1198611ℎ119888)

1205973

120601119909

1205971199093

+ (minus

ℎ2

119888

2

1198601minus 21198611ℎ119888minus 21198631)

1205974

1199080

1205971199094minus 119862

120597120601119909

120597119909

+ 119862

1205972

1199080

1205971199092+ 31198601(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092+ 21198601

1205972

1199060

1205971199092

1205971199080

120597119909

+ 21198601

1205971199060

120597119909

1205972

1199080

1205971199092+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(12c)

3 Validation

In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as

1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)

The displacements 120601119909and 119908

0which satisfy (13) can be

represented as

120601119909=

119873

sum

119899=1

120601119899(119905) cos119899120587119909

119897

(14a)

1199080=

119873

sum

119899=1

119908119899(119905) sin119899120587119909

119897

(14b)

We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of

Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters

Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669

Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters

Mode Lou Present1 13774 137812 52309 523953 109138 109411

(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method

The dimensions and material properties in Table 1 are

119864119891= 689GPa

ℎ119891= 04572mm

120588119891= 2680 kgm3

119866119888= 008268GPa

ℎ119888= 127mm

120588119888= 328 kgm3

119897 = 09144m

(15)


119864119891= 210GPa

ℎ119891= 1mm

120588119891= 7930 kgm3

119866119888= 103667GPa

ℎ119888= 15mm

120588119888= 3131739 kgm3

119897 = 06364m

(16)

4 Nonlinear Frequency Response Equation

In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply


supported boundary conditions with immoveable edges ofthe beam can be expressed as

1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)

The transformations of variables and parameters areintroduced to obtain the dimensionless equation

1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)

Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906

0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)

In addition the excitation has the form

119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)

Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906

0 120601119909 and 119908

0into the resulting equations the

ordinary differential equation is derived

119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908

119873) represents the coupled relation

between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix

The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam

120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)

where 120576 is a small dimensionless parameterBecause the coupled term 119892

119899(1199081 1199082 119908

119873) has no

contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation

119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)

The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form

119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905

Then we have the differential operators

119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905

+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)

1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863

01198631+ sdot sdot sdot (24b)

where1198630= 120597120597119879

0and119863

1= 120597120597119879

1

We have the following relation

Ω119899= 120596119899+ 120576120590119899 (25)

120590119899

is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency

Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations


1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)

The solution of (26a) in the complex form can beexpressed as

1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)

where 119860119899is the complex conjugate of 119860

119899

Substituting (27) into (26b) yields

1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)

where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms

Function 119860119899may be denoted in the complex form

119860119899=

1

2

119886119899119890119894120593119899 (29)

Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)

Steady-state motions occur when 119899and

119899equal zero

which correspond to the singular points of averaged (32a) and(32b) Thus 119886

119899and 120574119899are constants Eliminating 120574

119899by using

the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows

(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)

Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired

120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)

For subharmonic resonance the detuning parameter 120590119899

is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)

Similarly the averaged equations in the case are obtained

119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1

The frequency response equation of the truss core sand-wich beam with subharmonic resonance is

(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)

For superharmonic resonance the detuning parameter120590119899

is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)

The averaged equations in the case are obtained

119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)

The frequency response equation of the truss core sand-wich beam with superharmonic resonance is

(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)

The plot of 119886119899as a function of 120590

119899for given 119891 is called

a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix


120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010

Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency

5 Numerical Results

In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =

2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ

119888= 001m 120579 = 1205874 and 119897 = 07m

Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m

The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known

120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092

Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency

from 120572119899in the Appendix that the nonlinear term is related to

the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different

At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4

Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen

Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height


a1

25

2

15

1

05

1205901

minus10 minus5 0 5 10

(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)

Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006

will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses

Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses

Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same


a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)

Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ

119888ℎ = 074 dash line ℎ

119888ℎ = 082 dot-dash line ℎ

119888ℎ = 09

as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4

6 Conclusions

The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with

the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of


a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)

Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ


119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)

Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ


119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior

Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller

the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed


It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics

Appendix

The relations between the in-plane displacements and trans-verse displacement are

119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)

The relations between the rotation angles and transversedisplacement are

120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)

The coefficients in (20) are given as follows

120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)

References

[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001


[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001

[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001

[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003

[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004

[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006

[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007

[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009

[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011

[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012

[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013

[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014

[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014

[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014

[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004

[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010

[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010

[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002

[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009

[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012

[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013

[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014

[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004

[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996

[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003

[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009

[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005

[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969

[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



sandwich beams The face sheets and piezoelectric materialwere modeled by the Euler-Bernoulli beam theory and thecore was modeled by the Reddy third-order shear deforma-tion theory

Moreover the vibration behaviors of the truss coresandwich structures can be deduced with the help of a largenumber of dynamical studies on the sandwich structureswith honeycomb core foam core and other cores FrostigandThomsen [15] used high-order sandwich panel theory toanalyze the free vibration of sandwich panels with a flexiblecore Biglari and Jafari [16] studied the free vibrations ofdoubly curved sandwich panels based on a refined three-layered theory Jam et al [17] investigated the free vibrationof sandwich panels by using an improved high-order theoryIn several papers like the aforementioned three referencesthe three-dimensional elasticity theory was used for the corewhich has a better accuracy to predict the response of thethick sandwich structures

For thin and some moderately thick sandwich struc-tures the equivalent single layer theory and the layerwisetheory were widely used E Nilsson and A C Nilsson [18]investigated the dynamic properties of sandwich structureswith honeycomb and foam cores Li and Zhu [19] studiedthe free vibration of honeycomb sandwich plates by usingthe improved Reddy third-order shear deformation platetheory J H Zhang andW Zhang [20] investigated the globalbifurcations and multipulse chaotic dynamics of a simplysupported honeycomb sandwich rectangular plate by usingthe extended Melnikov method Yang et al [21] investigatedthe transverse vibrations and stability of an axially movingsandwich beam Sahoo and Singh [22] analyzed the bucklingand free vibration analysis of the laminated composite andsandwich plates by using a new trigonometric Zig-Zag theoryKhare et al [23] presented the free vibration analysis ofcomposite and sandwich laminates by using a high-ordershear deformation theory Sakiyama et al [24] studied the freevibration of a three-layer sandwich beam with an elastic orviscoelastic core and arbitrary boundary conditions Banerjee[25] analyzed the free vibration of three-layered symmetricsandwich beams by using the dynamic stiffness methodCetkovic and Vuksanovic [26] studied the bending freevibrations and buckling of the laminated composite andsandwich plates by using the generalized layerwise theoryof Reddy Ferreira [27] analyzed the mechanical behaviorsof the laminated composite and sandwich plates by using alayerwise shear deformation theory and the multiquadricsdiscretization method

Although the studies on the vibrations of the sandwichstructures are relatively detailed the unique dynamic charac-teristics especially the nonlinear behaviors of the sandwichstructures with truss core are still required to be exploredThe main aim of this paper is to particularly study the linearand nonlinear dynamic behaviors of the truss core sandwichbeam In this paper the nonlinear governing equation ofmotion for the simply supported truss core sandwich beamwith immovable edges is obtained by using the von Karmantype equation Hamiltonrsquos principle and a Zig-Zag theoryThe influences of strut radius and core height on the linear

r

a

120579

Figure 1 Sketches of a unit cell of the sandwich beam withpyramidal truss core

natural frequencies and hardening behaviors of the sandwichbeam are investigated The nonlinear frequency responsesof the sandwich beam with primary subharmonic andsuperharmonic resonances are obtained and the numericalresults are discussed

2 Formulation

The simply supported sandwich beam with pyramidal trusscore subjected to a transverse uniform load is considered inthe paper A unit cell of the pyramidal truss core is shown inFigure 1The length radius and inclination angle of the trussare represented by 119886 119903 and 120579 respectively Thus the relativedensity 120588 and the equivalent shear modulus 119866 are expressedas follows [2]

120588 =

120588119888

120588

=

2120587

cos2120579 sin 120579(

119903

119886

)

2

(1a)

119866 = 120587 sin 120579 ( 119903119886

)

2

119864 (1b)

where120588119888is the equivalent density of the truss core and120588 and119864

are the density and elasticity modulus of the mother materialof the truss

The truss core sandwich beam can be modeled as alaminated beam which is composed of two solid layers andan equivalent continuum core layer Generally for a sandwichstructure the following assumptions [28] can be given

(1) The thickness of the truss core sandwich beamremains constant during deformation

(2) Only bending deformation is considered for both thinface sheets and only shear deformation is consideredfor the core

(3) The face sheets and truss core are combined closelyThus the deflections between each layer are continu-ous

These assumptions are widely used to study the sandwichstructures which have thin face sheets and thick core Based


z

hhc x

w0

u0

120601x

120597w0

120597x



0




minus

ℎ

2

le 119911 le minus

ℎ119888

2

119906119905= 1199060minus 119911

1205971199080

120597119909

+

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119905= 1199080

(2a)

minus

ℎ119888

2

le 119911 le

ℎ119888

2

119906119888= 1199060minus 119911120601119909

119908119888= 1199080

(2b)

ℎ119888

2

le 119911 le

ℎ

2

119906119887= 1199060minus 119911

1205971199080

120597119909

minus

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119887= 1199080

(2c)



1205761199091199091=

120597119906119905

120597119909

+

1

2

(

120597119908119905

120597119909

)

2

120574119909119911=

120597119906119888

120597119911

+

120597119908119888

120597119909

1205761199091199093=

120597119906119887

120597119909

+

1

2

(

120597119908119887

120597119909

)

2

(3)

where 1205761199091199091

and 1205761199091199093



1205761199091199091

1205761199091199093

120574119909119911

=

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

+ 119911

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

=

1205971199060

120597119909

+

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

1205971199060

120597119909

minus

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

minus120601119909+

1205971199080

120597119909

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

=

minus

1205972

1199080

1205971199092

minus

1205972

1199080

1205971199092

0

(4)


int

1199052

1199051

(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)


120575119880 = 1205751198801+ 1205751198802+ 1205751198803

120575119870 = 1205751198701+ 1205751198702+ 1205751198703

120575119882 = minusint

119860

1198651205751199080minus 12058301205751199080119889119909 119889119910

(6a)

120575119880 = int

119897

0

int

minusℎ1198882

minusℎ2

12059011990911990911205751205761199091199091119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

12059011990911990931205751205761199091199093119889119909 119889119911

= int

119897

0

int

minusℎ1198882

minusℎ2

1205901199091199091(120575120576(0)

1199091199091+ 119911120575120576(1)

1199091199091) 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

1205901199091199093(120575120576(0)

1199091199093

+ 119911120575120576(1)

1199091199093) 119889119909 119889119911 = int

119897

0

1198731199091199091120575120576(0)

1199091199091

+1198721199091199091120575120576(1)

1199091199091119889119909 + int

119897

0

int

ℎ1198882

minusℎ1198882

119876119909120575120574119909119911119889119909 + int

119897

0

1198731199091199093

sdot 120575120576(0)

1199091199093+1198721199091199093120575120576(1)

1199091199093119889119909 119889119911

(6b)


120575119870 = int

119897

0

int

minusℎ1198882

minusℎ2

120588 [(0minus 119911

1205970

120597119909

+

ℎ119888

2

120601119909minus

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

+

ℎ119888

2

120575120601119909minus

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120588119888[(0minus 119911

120601119909) (1205750minus 119911120575

120601119909)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ2

ℎ1198882

120588 [(0minus 119911

1205970

120597119909

minus

ℎ119888

2

120601119909+

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

minus

ℎ119888

2

120575120601119909+

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

(6c)


0= 1205971199060120597119905 and so on


(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812

120601119909 (7a)

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(7b)

(1198731199091199091

1205971199080

120597119909

+ 1198731199091199093

1205971199080

120597119909

)

119909

+

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909119909

+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(7c)

where

11986801

11986811

11986821

= int

minusℎ1198882

minusℎ2

1

119911

1199112

120588119889119911

11986802

11986812

11986822

= int

ℎ1198882

minusℎ1198882

1

119911

1199112

120588119888119889119911

(8a)

1198731199091199091= int

minusℎ1198882

minusℎ2

1205901199091199091119889119911

1198731199091199093= int

ℎ2

ℎ1198882

1205901199091199093119889119911

119876119909= int

ℎ1198882

minusℎ1198882

120590119909119911119889119911

1198721199091199091= int

minusℎ1198882

minusℎ2

1199111205901199091199091119889119911

1198721199091199093= int

ℎ2

ℎ1198882

1199111205901199091199093119889119911

(8b)


1205901199091199091= 1198641205761199091199091

1205901199091199093= 1198641205761199091199093

120590119909119911= 119866120574119909119911

(9)


1198731199091199091= 1198601120576(0)

1199091199091+ 1198611120576(1)

1199091199091

1198731199091199093= 1198603120576(0)

1199091199093+ 1198613120576(1)

1199091199093

1198721199091199091= 1198611120576(0)

1199091199091+ 1198631120576(1)

1199091199091

1198721199091199093= 1198613120576(0)

1199091199093+ 1198633120576(1)

1199091199093

119876119909= 119862120574(0)

119909119911

(10a)

where

(1198601 1198611 1198631) = int

minusℎ1198882

minusℎ2

119864 (1 119911 1199112

) 119889119911

(1198603 1198613 1198633) = int

ℎ2

ℎ1198882

119864 (1 119911 1199112

) 119889119911

119862 = int

ℎ1198882

minusℎ1198882

119866119889119911

(10b)

1198601and 119860


1and 119861

3are


3are called




1198601= 1198603

1198611= minus1198613

1198631= 1198633

(11)



21198601

1205972

1199060

1205971199092+ 21198601

1205971199080

120597119909

1205972

1199080

1205971199092= (211986801+ 11986802) 0minus 11986812

120601119909 (12a)

ℎ2

119888

2

1198601

1205972

120601119909

1205971199092minus

ℎ2

119888

2

1198601

1205973

1199080

1205971199093minus ℎ1198881198611

1205973

1199080

1205971199093minus 119862120601119909

+ 119862

1205971199080

120597119909

= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(12b)

(

ℎ2

119888

2

1198601+ 1198611ℎ119888)

1205973

120601119909

1205971199093

+ (minus

ℎ2

119888

2

1198601minus 21198611ℎ119888minus 21198631)

1205974

1199080

1205971199094minus 119862

120597120601119909

120597119909

+ 119862

1205972

1199080

1205971199092+ 31198601(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092+ 21198601

1205972

1199060

1205971199092

1205971199080

120597119909

+ 21198601

1205971199060

120597119909

1205972

1199080

1205971199092+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(12c)

3 Validation


1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)



represented as

120601119909=

119873

sum

119899=1

120601119899(119905) cos119899120587119909

119897

(14a)

1199080=

119873

sum

119899=1

119908119899(119905) sin119899120587119909

119897

(14b)





Mode Lou Present1 13774 137812 52309 523953 109138 109411



119864119891= 689GPa

ℎ119891= 04572mm

120588119891= 2680 kgm3

119866119888= 008268GPa

ℎ119888= 127mm

120588119888= 328 kgm3

119897 = 09144m

(15)


119864119891= 210GPa

ℎ119891= 1mm

120588119891= 7930 kgm3

119866119888= 103667GPa

ℎ119888= 15mm

120588119888= 3131739 kgm3

119897 = 06364m

(16)





1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)


1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)


0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)


119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)


0 120601119909 and 119908



119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908




120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)


119899(1199081 1199082 119908

119873) has no


119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)


119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905


119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905


1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863


where1198630= 120597120597119879

0and119863

1= 120597120597119879

1


Ω119899= 120596119899+ 120576120590119899 (25)

120590119899




1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

hhc x

w0

u0

120601x

120597w0

120597x



0




minus

ℎ

2

le 119911 le minus

ℎ119888

2

119906119905= 1199060minus 119911

1205971199080

120597119909

+

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119905= 1199080

(2a)

minus

ℎ119888

2

le 119911 le

ℎ119888

2

119906119888= 1199060minus 119911120601119909

119908119888= 1199080

(2b)

ℎ119888

2

le 119911 le

ℎ

2

119906119887= 1199060minus 119911

1205971199080

120597119909

minus

ℎ119888

2

(120601119909minus

1205971199080

120597119909

)

119908119887= 1199080

(2c)



1205761199091199091=

120597119906119905

120597119909

+

1

2

(

120597119908119905

120597119909

)

2

120574119909119911=

120597119906119888

120597119911

+

120597119908119888

120597119909

1205761199091199093=

120597119906119887

120597119909

+

1

2

(

120597119908119887

120597119909

)

2

(3)

where 1205761199091199091

and 1205761199091199093



1205761199091199091

1205761199091199093

120574119909119911

=

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

+ 119911

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

120576(0)

1199091199091

120576(0)

1199091199093

120574(0)

119909119911

=

1205971199060

120597119909

+

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

1205971199060

120597119909

minus

ℎ119888

2

(

120597120601119909

120597119909

minus

1205972

1199080

1205971199092) +

1

2

(

1205971199080

120597119909

)

2

minus120601119909+

1205971199080

120597119909

120576(1)

1199091199091

120576(1)

1199091199093

120574(1)

119909119911

=

minus

1205972

1199080

1205971199092

minus

1205972

1199080

1205971199092

0

(4)


int

1199052

1199051

(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)


120575119880 = 1205751198801+ 1205751198802+ 1205751198803

120575119870 = 1205751198701+ 1205751198702+ 1205751198703

120575119882 = minusint

119860

1198651205751199080minus 12058301205751199080119889119909 119889119910

(6a)

120575119880 = int

119897

0

int

minusℎ1198882

minusℎ2

12059011990911990911205751205761199091199091119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

12059011990911990931205751205761199091199093119889119909 119889119911

= int

119897

0

int

minusℎ1198882

minusℎ2

1205901199091199091(120575120576(0)

1199091199091+ 119911120575120576(1)

1199091199091) 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120590119909119911120575120574119909119911119889119909 119889119911 + int

119897

0

int

ℎ2

ℎ1198882

1205901199091199093(120575120576(0)

1199091199093

+ 119911120575120576(1)

1199091199093) 119889119909 119889119911 = int

119897

0

1198731199091199091120575120576(0)

1199091199091

+1198721199091199091120575120576(1)

1199091199091119889119909 + int

119897

0

int

ℎ1198882

minusℎ1198882

119876119909120575120574119909119911119889119909 + int

119897

0

1198731199091199093

sdot 120575120576(0)

1199091199093+1198721199091199093120575120576(1)

1199091199093119889119909 119889119911

(6b)


120575119870 = int

119897

0

int

minusℎ1198882

minusℎ2

120588 [(0minus 119911

1205970

120597119909

+

ℎ119888

2

120601119909minus

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

+

ℎ119888

2

120575120601119909minus

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120588119888[(0minus 119911

120601119909) (1205750minus 119911120575

120601119909)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ2

ℎ1198882

120588 [(0minus 119911

1205970

120597119909

minus

ℎ119888

2

120601119909+

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

minus

ℎ119888

2

120575120601119909+

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

(6c)


0= 1205971199060120597119905 and so on


(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812

120601119909 (7a)

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(7b)

(1198731199091199091

1205971199080

120597119909

+ 1198731199091199093

1205971199080

120597119909

)

119909

+

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909119909

+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(7c)

where

11986801

11986811

11986821

= int

minusℎ1198882

minusℎ2

1

119911

1199112

120588119889119911

11986802

11986812

11986822

= int

ℎ1198882

minusℎ1198882

1

119911

1199112

120588119888119889119911

(8a)

1198731199091199091= int

minusℎ1198882

minusℎ2

1205901199091199091119889119911

1198731199091199093= int

ℎ2

ℎ1198882

1205901199091199093119889119911

119876119909= int

ℎ1198882

minusℎ1198882

120590119909119911119889119911

1198721199091199091= int

minusℎ1198882

minusℎ2

1199111205901199091199091119889119911

1198721199091199093= int

ℎ2

ℎ1198882

1199111205901199091199093119889119911

(8b)


1205901199091199091= 1198641205761199091199091

1205901199091199093= 1198641205761199091199093

120590119909119911= 119866120574119909119911

(9)


1198731199091199091= 1198601120576(0)

1199091199091+ 1198611120576(1)

1199091199091

1198731199091199093= 1198603120576(0)

1199091199093+ 1198613120576(1)

1199091199093

1198721199091199091= 1198611120576(0)

1199091199091+ 1198631120576(1)

1199091199091

1198721199091199093= 1198613120576(0)

1199091199093+ 1198633120576(1)

1199091199093

119876119909= 119862120574(0)

119909119911

(10a)

where

(1198601 1198611 1198631) = int

minusℎ1198882

minusℎ2

119864 (1 119911 1199112

) 119889119911

(1198603 1198613 1198633) = int

ℎ2

ℎ1198882

119864 (1 119911 1199112

) 119889119911

119862 = int

ℎ1198882

minusℎ1198882

119866119889119911

(10b)

1198601and 119860


1and 119861

3are


3are called




1198601= 1198603

1198611= minus1198613

1198631= 1198633

(11)



21198601

1205972

1199060

1205971199092+ 21198601

1205971199080

120597119909

1205972

1199080

1205971199092= (211986801+ 11986802) 0minus 11986812

120601119909 (12a)

ℎ2

119888

2

1198601

1205972

120601119909

1205971199092minus

ℎ2

119888

2

1198601

1205973

1199080

1205971199093minus ℎ1198881198611

1205973

1199080

1205971199093minus 119862120601119909

+ 119862

1205971199080

120597119909

= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(12b)

(

ℎ2

119888

2

1198601+ 1198611ℎ119888)

1205973

120601119909

1205971199093

+ (minus

ℎ2

119888

2

1198601minus 21198611ℎ119888minus 21198631)

1205974

1199080

1205971199094minus 119862

120597120601119909

120597119909

+ 119862

1205972

1199080

1205971199092+ 31198601(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092+ 21198601

1205972

1199060

1205971199092

1205971199080

120597119909

+ 21198601

1205971199060

120597119909

1205972

1199080

1205971199092+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(12c)

3 Validation


1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)



represented as

120601119909=

119873

sum

119899=1

120601119899(119905) cos119899120587119909

119897

(14a)

1199080=

119873

sum

119899=1

119908119899(119905) sin119899120587119909

119897

(14b)





Mode Lou Present1 13774 137812 52309 523953 109138 109411



119864119891= 689GPa

ℎ119891= 04572mm

120588119891= 2680 kgm3

119866119888= 008268GPa

ℎ119888= 127mm

120588119888= 328 kgm3

119897 = 09144m

(15)


119864119891= 210GPa

ℎ119891= 1mm

120588119891= 7930 kgm3

119866119888= 103667GPa

ℎ119888= 15mm

120588119888= 3131739 kgm3

119897 = 06364m

(16)





1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)


1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)


0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)


119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)


0 120601119909 and 119908



119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908




120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)


119899(1199081 1199082 119908

119873) has no


119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)


119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905


119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905


1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863


where1198630= 120597120597119879

0and119863

1= 120597120597119879

1


Ω119899= 120596119899+ 120576120590119899 (25)

120590119899




1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120575119870 = int

119897

0

int

minusℎ1198882

minusℎ2

120588 [(0minus 119911

1205970

120597119909

+

ℎ119888

2

120601119909minus

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

+

ℎ119888

2

120575120601119909minus

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ1198882

minusℎ1198882

120588119888[(0minus 119911

120601119909) (1205750minus 119911120575

120601119909)

+ 01205750] 119889119909 119889119911

+ int

119897

0

int

ℎ2

ℎ1198882

120588 [(0minus 119911

1205970

120597119909

minus

ℎ119888

2

120601119909+

ℎ119888

2

1205970

120597119909

)

sdot (1205750minus 119911

1205971205750

120597119909

minus

ℎ119888

2

120575120601119909+

ℎ119888

2

1205971205750

120597119909

)

+ 01205750] 119889119909 119889119911

(6c)


0= 1205971199060120597119905 and so on


(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812

120601119909 (7a)

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(7b)

(1198731199091199091

1205971199080

120597119909

+ 1198731199091199093

1205971199080

120597119909

)

119909

+

ℎ119888

2

(1198731199091199091minus 1198731199091199093)119909119909

+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(7c)

where

11986801

11986811

11986821

= int

minusℎ1198882

minusℎ2

1

119911

1199112

120588119889119911

11986802

11986812

11986822

= int

ℎ1198882

minusℎ1198882

1

119911

1199112

120588119888119889119911

(8a)

1198731199091199091= int

minusℎ1198882

minusℎ2

1205901199091199091119889119911

1198731199091199093= int

ℎ2

ℎ1198882

1205901199091199093119889119911

119876119909= int

ℎ1198882

minusℎ1198882

120590119909119911119889119911

1198721199091199091= int

minusℎ1198882

minusℎ2

1199111205901199091199091119889119911

1198721199091199093= int

ℎ2

ℎ1198882

1199111205901199091199093119889119911

(8b)


1205901199091199091= 1198641205761199091199091

1205901199091199093= 1198641205761199091199093

120590119909119911= 119866120574119909119911

(9)


1198731199091199091= 1198601120576(0)

1199091199091+ 1198611120576(1)

1199091199091

1198731199091199093= 1198603120576(0)

1199091199093+ 1198613120576(1)

1199091199093

1198721199091199091= 1198611120576(0)

1199091199091+ 1198631120576(1)

1199091199091

1198721199091199093= 1198613120576(0)

1199091199093+ 1198633120576(1)

1199091199093

119876119909= 119862120574(0)

119909119911

(10a)

where

(1198601 1198611 1198631) = int

minusℎ1198882

minusℎ2

119864 (1 119911 1199112

) 119889119911

(1198603 1198613 1198633) = int

ℎ2

ℎ1198882

119864 (1 119911 1199112

) 119889119911

119862 = int

ℎ1198882

minusℎ1198882

119866119889119911

(10b)

1198601and 119860


1and 119861

3are


3are called




1198601= 1198603

1198611= minus1198613

1198631= 1198633

(11)



21198601

1205972

1199060

1205971199092+ 21198601

1205971199080

120597119909

1205972

1199080

1205971199092= (211986801+ 11986802) 0minus 11986812

120601119909 (12a)

ℎ2

119888

2

1198601

1205972

120601119909

1205971199092minus

ℎ2

119888

2

1198601

1205973

1199080

1205971199093minus ℎ1198881198611

1205973

1199080

1205971199093minus 119862120601119909

+ 119862

1205971199080

120597119909

= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(12b)

(

ℎ2

119888

2

1198601+ 1198611ℎ119888)

1205973

120601119909

1205971199093

+ (minus

ℎ2

119888

2

1198601minus 21198611ℎ119888minus 21198631)

1205974

1199080

1205971199094minus 119862

120597120601119909

120597119909

+ 119862

1205972

1199080

1205971199092+ 31198601(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092+ 21198601

1205972

1199060

1205971199092

1205971199080

120597119909

+ 21198601

1205971199060

120597119909

1205972

1199080

1205971199092+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(12c)

3 Validation


1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)



represented as

120601119909=

119873

sum

119899=1

120601119899(119905) cos119899120587119909

119897

(14a)

1199080=

119873

sum

119899=1

119908119899(119905) sin119899120587119909

119897

(14b)





Mode Lou Present1 13774 137812 52309 523953 109138 109411



119864119891= 689GPa

ℎ119891= 04572mm

120588119891= 2680 kgm3

119866119888= 008268GPa

ℎ119888= 127mm

120588119888= 328 kgm3

119897 = 09144m

(15)


119864119891= 210GPa

ℎ119891= 1mm

120588119891= 7930 kgm3

119866119888= 103667GPa

ℎ119888= 15mm

120588119888= 3131739 kgm3

119897 = 06364m

(16)





1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)


1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)


0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)


119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)


0 120601119909 and 119908



119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908




120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)


119899(1199081 1199082 119908

119873) has no


119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)


119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905


119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905


1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863


where1198630= 120597120597119879

0and119863

1= 120597120597119879

1


Ω119899= 120596119899+ 120576120590119899 (25)

120590119899




1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











21198601

1205972

1199060

1205971199092+ 21198601

1205971199080

120597119909

1205972

1199080

1205971199092= (211986801+ 11986802) 0minus 11986812

120601119909 (12a)

ℎ2

119888

2

1198601

1205972

120601119909

1205971199092minus

ℎ2

119888

2

1198601

1205973

1199080

1205971199093minus ℎ1198881198611

1205973

1199080

1205971199093minus 119862120601119909

+ 119862

1205971199080

120597119909

= (

ℎ2

119888

2

11986801+ 11986822)120601119909

minus (

ℎ2

119888

2

11986801+ ℎ11988811986811)

1205970

120597119909

minus 119868120

(12b)

(

ℎ2

119888

2

1198601+ 1198611ℎ119888)

1205973

120601119909

1205971199093

+ (minus

ℎ2

119888

2

1198601minus 21198611ℎ119888minus 21198631)

1205974

1199080

1205971199094minus 119862

120597120601119909

120597119909

+ 119862

1205972

1199080

1205971199092+ 31198601(

1205971199080

120597119909

)

2

1205972

1199080

1205971199092+ 21198601

1205972

1199060

1205971199092

1205971199080

120597119909

+ 21198601

1205971199060

120597119909

1205972

1199080

1205971199092+ 119865

minus 1205830= (

ℎ2

119888

2

11986801+ ℎ11988811986811)

120597120601119909

120597119909

minus (

ℎ2

119888

2

11986801+ 2ℎ11988811986811+ 211986821)

1205972

0

1205971199092

+ (211986801+ 11986802) 0

(12c)

3 Validation


1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)



represented as

120601119909=

119873

sum

119899=1

120601119899(119905) cos119899120587119909

119897

(14a)

1199080=

119873

sum

119899=1

119908119899(119905) sin119899120587119909

119897

(14b)





Mode Lou Present1 13774 137812 52309 523953 109138 109411



119864119891= 689GPa

ℎ119891= 04572mm

120588119891= 2680 kgm3

119866119888= 008268GPa

ℎ119888= 127mm

120588119888= 328 kgm3

119897 = 09144m

(15)


119864119891= 210GPa

ℎ119891= 1mm

120588119891= 7930 kgm3

119866119888= 103667GPa

ℎ119888= 15mm

120588119888= 3131739 kgm3

119897 = 06364m

(16)





1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)


1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)


0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)


119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)


0 120601119909 and 119908



119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908




120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)


119899(1199081 1199082 119908

119873) has no


119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)


119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905


119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905


1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863


where1198630= 120597120597119879

0and119863

1= 120597120597119879

1


Ω119899= 120596119899+ 120576120590119899 (25)

120590119899




1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1199060= 1199080=

120597120601119909

120597119909

= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)


1199060=

1199060

119897

1199080=

1199080

ℎ

120601119909= 120601119909

119909 =

119909

119897

119865 =

1198974

119864ℎ4119865

120583 =

119897

ℎ

(

1

120588119864

)

12

120583

119905 =

ℎ

1198972(

119864

120588

)

12

119905

Ω =

1198972

ℎ

(

120588

119864

)

12

Ω

ℎ119888=

ℎ119888

ℎ

119860119894=

119897

119864ℎ2119860119894

119861119894=

119897

119864ℎ3119861119894

119862 =

119897

119864ℎ2119862

119863119894=

119897

119864ℎ4119863119894

119868119905119894=

1

119897(119905+1)120588

119868119905119894 119905 = 0 1 2

(18)


0 120601119909 and 119908


represented as

1199060=

119873

sum

119899=1

119906 (119905) sin (2119899120587119909) (19a)

120601119909=

119873

sum

119899=1

120601 (119905) cos (119899120587119909) (19b)

1199080=

119873

sum

119899=1

119908 (119905) sin (119899120587119909) (19c)


119865 =

119873

sum

119899=1

119891 (119905) sin (119899120587119909) (19d)


0 120601119909 and 119908



119899+ 120583119899119899+ 1205962

119899119908119899+ 1205721198991199083

119899+ 119892119899(1199081 1199082 119908

119873)

= 120573119899119891119899cos (Ω

119899119905) 119899 = 1 2 119873

(20)

where 119892119899(1199081 1199082 119908




120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

120573119899997888rarr 120576120573

119899

(21)


119899(1199081 1199082 119908

119873) has no


119899+ 120576120583119899119899+ 1205962

119899119908119899+ 1205761205721198991199083

119899= 120576120573119899119891119899cos (Ω

119899119905) (22)


119908119899(119905 120576) = 119909

1198990(1198790 1198791) + 120576119909

1198991(1198790 1198791) + sdot sdot sdot (23)

where 1198790= 119905 and 119879

1= 120576119905


119889

119889119905

=

120597

1205971198790

1205971198790

120597119905

+

120597

1205971198791

1205971198791

120597119905


1198892

1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)

2

= 1198632

0+ 2120576119863


where1198630= 120597120597119879

0and119863

1= 120597120597119879

1


Ω119899= 120596119899+ 120576120590119899 (25)

120590119899




1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1205760 order

1198632

01199091198990+ 1205962

1198991199091198990= 0 (26a)

1205761 order

1198632

01199091198991+ 1205962

1198991199091198991= minus2119863

011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093

1198990

+ 120573119899119891119899cos (120596

1198991198790+ 1205901198991198791)

(26b)


1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)


119899


1198632

01199091198991+ 1205962

1198991199091198991= (minus2119894120596

1198991198631119860119899minus 120583119899119894120596119899119860119899

minus 31205721198991198602

119899119860119899+

120573119899119891119899

2

1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST

(28)



119860119899=

1

2

119886119899119890119894120593119899 (29)


119899= minus

120583119899119886119899

2

+

120573119899119891119899sin (120590

1198991198791minus 120593119899)

2120596119899

(30a)

119886119899119899=

31205721198991198863

119899

8120596119899

minus

120573119899119891119899cos (120590

1198991198791minus 120593119899)

2120596119899

(30b)

Letting

120574119899= 1205901198991198791minus 120593119899 (31)

we have

119899= minus

120583119899119886119899

2

+

120573119899119891119899sin 120574119899

2120596119899

(32a)

119886119899119899= 119886119899120590119899minus

31205721198991198863

119899

8120596119899

+

120573119899119891119899cos 120574119899

2120596119899

(32b)


119899equal zero



119899by using


(

120583119899119886119899

2

)

2

+ (120590119899119886119899minus

31205721198991198863

119899

8120596119899

)

2

= (

120573119899119891119899

2120596119899

)

2

(33)


120583119899997888rarr 120576120583

119899

120572119899997888rarr 120576120572

119899

(34)




119899= minus

120583119899119886119899

2

minus

31205721198991205731198991198862

119899Λ sin 120574

119899

4120596119899

(36a)

119886119899119899= 119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

minus

9120572119899120573119899Λ1198862

119899cos 120574119899

4120596119899

(36b)

where Λ = (12)119891119899(1205962

119899minus Ω2

119899)minus1


(minus

120583119899119886119899

2

)

2

+

1

9

[119886119899(120590119899minus

91205721198991205732

119899Λ2

120596119899

) minus

91205721198991198863

119899

8120596119899

]

2

=

91205722

1198991205732

1198991198864

119899Λ2

161205962

119899

(37)




119899= minus

120583119899119886119899

2

minus

1205721198991205733

119899Λ3 sin 120574

119899

120596119899

(39a)

119886119899119899= 119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

minus

1205721198991205733

119899Λ3 cos 120574

119899

120596119899

(39b)


(minus

120583119899119886119899

2

)

2

+ [119886119899(120590119899minus

31205721198991205732

119899Λ2

120596119899

) minus

31205721198991198863

119899

8120596119899

]

2

=

1205722

1198991205736

119899Λ6

1205962

119899

(40)





120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120596

32

28

24

20

16

12

8

4

120588c120588

000 002 004 006 008 010


5 Numerical Results



119888= 001m 120579 = 1205874 and 119897 = 07m



120596

32

28

24

20

16

12

8

4

hch

072 076 080 084 088 092








a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a1

25

2

15

1

05

1205901


(a)a2

10

12

06

04

08

02

1205902


(b)

a3

07

08

05

04

06

03

02

01

1205903


(c)






a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a1

14

12

10

08

06

04

02

1205901


(a)a2

06

05

04

03

02

01

1205902


(b)

a3

04

03

02

01

1205903


(c)




119888ℎ = 09


6 Conclusions




a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a1

15

1

05

0

1205901


(a)a1

1205901


08

07

06

05

04

03

02

01

0

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)

1205901

minus4 minus2 0 2 4

a1

16

14

12

10

08

06

04

02

(a)

1205901

minus5 0 5 10 15

a1

14

12

10

08

06

04

02

(b)



119888ℎ = 082 dot-dash

line ℎ119888ℎ = 09)






Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Appendix


119906119899= 1198781198991199082

119899

119878119899= minus

1198991198861120587

8

(A1)


120601119899= 119870119899119908119899

119870119899=

1198991198873120587 minus 1198993

11988721205873

119899211988711205872minus 1

(A2)


120574119899=

1198792119899

1198791119899

1205962

119899= minus

1198793119899

1198791119899

120572119899= minus

1198794119899

1198791119899

120573119899=

1198795119899

1198791119899

1198791119899=

1198889

2

1198792119899=

1198888

2

1198793119899=

1198994

1205874

1198882

2

minus

1198992

1205872

1198884

2

minus

1198991205871198701198991198883

2

+

1198993

1205873

1198701198991198881

2

1198794119899= minus

1198994

1205874

8

+

1198993

1205873

1198781198991198886

2

minus 1198993

1205873

1198781198991198885

1198795119899=

1198887

2

(A3)

where

1198861=

ℎ2

1198972

1198871= minus

ℎ2

ℎ

2

119888

21198972

1198601

119862

1198872=

ℎ3

ℎ

2

119888

21198973

1198601

119862

+

ℎ3

ℎ119888

1198973

1198611

119862

1198873= minus

ℎ

119897

1198881=

119897

3ℎ

(

ℎ119888

2

+ ℎ

1198611

1198601

)

1198882= minus

1

3

(

ℎ

2

119888

2

+ 2ℎ

1198611

1198601

+ 2

1198631

1198601

)

1198883= minus

1198973

3ℎ3

119862

1198601

1198884=

1198972

3ℎ2

119862

1198601

1198885=

21198972

3ℎ2

1198886=

21198972

3ℎ2

1198887=

119897

3ℎ1198601

1198888=

1198972

3ℎ21198601

1198889=

1198972

3ℎ21198601

(211986801+ 11986802)

(A4)

Competing Interests


Acknowledgments


References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article free vibration and hardening behavior of

Documents