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DYNAMIC LOADING OF A FORKLIFT

TRUCK LIFTING INSTALLATION

Emanuil CHANKOV

Georgi STOYCHEV

Abstract: Investigation of dynamic stresses in the lifting

installation of a fork-lift truck is in consideration in this

paper. A two dimensional model of the truck is proposed

in which the lifting installation is represented as an

elastic beam. The constitutive equations for the system

are formulated.Numerical calculations are obtained and

analyzed. A university license version of finite element

code COSMOS/M is used to perform the numerical

experiments. On the other hand fork-lift truck lifting

installation is investigated under laboratory conditions.

The numerical and experimental data are compared.

There is a good agreement between computed results and

the experimental data from the tensometric tests at

different loading conditions.

Key words: Forklift truck, dynamic model, stress, FEA

1. INTRODUCTIONThe lifting installation of fork-lift trucks is a complicatedstructure subjected to various static and dynamic loads.The optimal design of the structural is of significanteconomic and technical efficiency mean. The problem

concerning the exact determination of the stresses anddeformations in lifting installations arises from thebeginning of fork-lift truck production.In some earlier papers [10] the static deformations of themast at the case when the load is lifted to maximumheight are studied. It is represented as a construction ofbeams with a constant cross-section. Its lower end isattached to a pin support and the tilting hydraulic cylinderis represented as a rigid support. The determination of thedeformations is done by methods of classical Mechanics.Various loading cases for a fork-lift truck are studied in[12]. The mast is represented as a beam construction andthe dynamic loads acting on it are received using a

dynamic coefficient.Similar methods for calculation of the deformations andstresses in the lifting installation of fork-lift trucks aredemonstrated in [8].

The constitutive equation of the static deflection line isused in [2] in order to determine the deformations of afork-lift truck mast. The must is modeled as a beam withvariable cross-section. The forces are applied in the masscentre of the load which is lifted to maximum height.A method for the design of the lifting installation of ahoisting machine in which the static forces are multiplied

by a dynamic coefficient is proposed in [7].The stress distribution in a fork is analyzed in [11]. A 2DFE model is proposed for that purpose.A quasi-static investigation of the stresses in the liftinginstallation of a fork lift truck is done in [5]. A detaileddynamic model is built. The velocities and accelerationswhich are acting on the lifting installation at a particularmoment of time are calculated and then added to a 3D FEmodel of the mast.Determination of dynamic stresses in the liftinginstallation requires an appropriate dynamic model of thetruck. In the present publication a combined model withdistributed and concentrated parameters is used. The mast

is represented as a flexible beam and the other parts of thetruck are rigid bodies.The analysis of the dynamic response of elastic beams hasbeen a subject of study for a long time. There is a vastnumber of papers concerning the natural frequencies andmode shapes of beams with different boundary conditions[4], [6], [9], [13], [15], [20].Problems concerning combined dynamic systemsconsisting of flexible and rigid bodies are also wellstudied. An approximate method is used in [16] for thedetermination of the free vibrations of a flexible robotarm fixed to a spring supported mass.The response of a cantilever beam carrying spring-masssystems is examined in [3] with the use of Greensfunctions.There are different methods which are used for studyingthe dynamics of combined systems. The Finite ElementMethod is one of the most wide-spread and general ones[19]. In the paper [14] it is used for dynamic analysis ofelastic beams with arbitrary moving spring-mass-dampersystems. The same method is applied in [1], [17] and [18]where similar problems are being analyzed.In the present paper the stress distribution in the liftinginstallation of a fork-lift truck under dynamic loading isanalyzed. A 2D model of the truck is proposed. The finite

element analysis is obtained by finite element codeCOSMOS/M. Beam elements are used for the finitemodel of the mast. The verification of the proposed finiteelement model was realized by the tensometric tests underdifferent loading conditions.

2. DYNAMIC MODELThe model of the truck is shown in Fig. 1. This is a 2Dmodel with 5 degrees of freedom:y andz horizontal andvertical translation of the trucks chassis, rotation ofthe chassis around its mass centre, rotation of themast as a rigid body around the point of attachment to the

truck, w the horizontal deflection of the points from theelastic mast.The constitutive dynamic equations of the trucks chassisare

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1 2 3

1 2 3

1 1 2 2 3 3

1 1 2 2 3 3 1 1

. . .

. . . . .

h h h h

v v v v

h h h

v v v h v

my F F F R

mz F F F R

I F h F h F h

F L F L F L R h R L

= + +

= + +

= +

+ + +

&&

&&

&&

(1)

Fig. 1. A 2D dynamic model of the truck

where m and I are the mass and the mass moment ofinertia of the chassis, the dimensions L1, L2,L3, h1, h2, h3

are shown in Fig. 1, hR and vR are the horizontal andvertical component of the reaction at the point of

attachment of the mast to the chassis respectively, 1hF ,

1vF , 2

hF , 2vF , 3

hF , 3vF are the horizontal and vertical

components of the forces in the elastic elements, the frontand rear tyres and the tilting cylinder respectively. Theforces may be expressed as

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

1 1 1 1 1

1 1 1 1 1

2 2 2 2 2

2 2 2 2 2

3 4 4 3

44 3

3 4 3 4 3

. .

. .

. .

. .

. ( , ) .cos

( , ). .cos

. . .sin . . .sin

h h h

v v v

h h h

v v v

h

v

F y h k y h c

F z l k z L c

F y h k y h c

F z l k z L c

F h w h t k

w h th c

t

F L k L c

= + + +

= + +

= + + +

=

= + +

+ +

= +

&&

&&

&&

&&

&

&

(2)

where 1hk , 1

vk , 2hk , 2

vk , 1hc , 1

vc , 2hc , 2

vc are the horizontal

and vertical component of the elasticity and damping

coefficients of the front and rear tyres, 3k and 3c are the

elasticity and damping coefficient of the tilting cylinder, is the angle between the cylinder and the horizontal axis,

L4 and h4 are shown on Fig. 1.The equations for the lifting installation represented as arigid body have the following form

( ) ( )

( ) ( )

( ) ( )

( )( )

1 2 1 1 2 3

1 2 1 1 2 3

1 2 1 2

1 1 1 2

3 4 3 4

cos . sin .

.cos sin .

. .

h h

v v

B

h v

m m y h m m F R

m m z L m m F R

m m BC y m m BC z

h L m m BC I

F h F l

+ + + = +

+ + + = +

+ + +

+ + + =

= +

&&&&

&&&&

&& &&

&& &&

(3)

where , m1 and m2 are the masses of the load and the mastrespectively,IB is the mass moment of inertia of the liftinginstallation (together with the load) with respect to thepoint of attachment B, point Cis the mass centre of thelifting installation (together with the load), is the anglebetween the lineBCand the axis of the mast.The partial differential equation concerning the transversevibrations of the mast represented as a flexible beam witha constant cross-section has the expression

( )( ) ( ) ( )

4 2

3 44 2

*1 6 5 1 6 1 6

( , ) ( , )( , ) . ( )

. . .

h

m m m

w x t w x t EI A q x t F x h

x tF x h h F x h F x h

+ = + +

+ +

(4)

where, and Eare the density and modulus of elasticityof the material, A and I the area and the moment ofinertia of the beams cross-section respectively, x is thecurrent coordinate along the axis of the beam, q(x,t) is theperpendicular to the axis of the mast component of theinertial distributed load which is due to the motion of themast as a rigid body (Fig. 2), is delta function for theconcentrated forces along the beam, Fm,1 is a force couplewhich is used for representation of the concentrated

bending moment Mb at the tip end of the beam, receivedby transferring the mass of the load to the axis of the

beam (Fig. 2), *,1mF is the inertial force due to the motion

of the mass of the load as a part of the system withconcentrated parameters, the dimensions h5 and h6 areshown on Fig. 2.

Fig.2. Dynamic loads acting on the elastic mast

The expressions of the distributed load q(x,t) and the

forces Fm1 and*1mF are the following

Mb

B

w

w

m1, I1

B

h6

*1mF

3hF

q(x,t)

Fm1

Fm1h5

h4x

m1, I1

L6

h6

m, I

w

zc3

h4k3

x

y

L1L2

h3h1h2

L4

1vk

2hk

2vk 1

vc 2hc

1hc

2vc

m2, I2

B

C

L3

1hk

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( )

( )( )

( )( )

24 1

26 11 4 6 6 1

5 5

* 21 1 6 4 6 1

( , ) . .

..

.

b

m

m

q x t A x L y h

M L mF L L h z L

h h

F m h L L y h

= + +

= = + + +

= + + +

&& & &&&&

&& & &&&&

&& & &&&&

(5)

whereL6 is the distance between the load and the axis of

the beam (Fig. 1).

3. FE SOLUTIONThe finite element method is applied to solve the systemof equations (1), (3) and (4). A standard 2D beam elementwith two degrees of freedom per node is used. The mast,represented as a beam is divided into Nelements. In thisway the partial differential equation (4) is substituted by asystem of2N+2 ordinary differential equations which inmatrix form has the following expression

[ ]{ } [ ]{ } [ ]{ } { }M u C u K u F + + =&& & (6)

where [M], [C] and [K] are the global mass, damping andstiffness matrices respectively, {u} is the vector of thedegrees of freedom in the nodes, {F} is the vector of theexternal loads. It should be noted that the mass matrix ismodified by adding the mass of the load and its momentof inertia as follows

[ ]

11 1( 1) 1

( 1)1 ( 1)( 1) 2 ( 1)

1 ( 1) 2

...

... ... ... ...

...

...

n n

n n n n n

n n n nn

m m m

Mm m m m

m m m I

= + +

(n=2N+2).In a similar way the stiffness and damping matrices aremodified by adding the elasticity and dampingcoefficients respectively

[ ]

11 1 1

1 3

1

... ...

... ... ... ... ...

... ...

... ... ... ... ...

... ...

s n

h

s ss sn

n ns nn

k k k

K k k k k

k k k

= +

,

[ ]

11 1 1

1 3

1

... ...

... ... ... ... ...

... ...

... ... ... ... ...

... ...

s n

h

s ss sn

n ns nn

c c c

C c c c c

c c c

= +

,

where by the subscript s is denoted the point ofattachment of the tilting cylinder to the mast.The solution of the problem is done by combining thesystems (1), (3) and (6).For the numerical solution of the problem a finite element

code COSMOS/M is used. The values of the elasticityand damping coefficients of the tyres and the tiltingcylinder are taken from [2]. The excitation is modelledwith initial velocity applied to the load.

4. EXPERIMENTAL INVESTIGATIONThe verification of the proposed dynamic model and theFE implementation was realized on a fork-lift truck modelEB 687.33.10 produced by Balkancar Record - Bulgaria.The lifting capacity is 1000 kg and the maximum liftheight is 3,3 m..

Fig. 3.

The zones where stains are measured can be seen in Fig.3, where: 1 truck chassis, 2 mast, 3 load, 4 liftingcylinder, 5 tilting cylinder, 6 strain gauge.The flow chart of the measurements is shown in Fig. 4,where SG is stain gauge, A amplifier, ADC analog-digital converter, C computer. An electronic measuringsystem SPIDER 8 manufactured by HBM-Germany forelectric measurement of mechanical variables was used.

Fig. 4.

The excitation of the system is done by rapid dropdownand stop of the load when it is lifted at maximum height.The results from the numerical calculations andexperimental data are given in Fig. 5. A good agreementin frequency and amplitude values may be observed.

0,0 0,6 1,2 1,8 2,4 3,0 3,6 4,2 4,8 5,4 6,0-25

-20

-15

-10

-5

0

5

10

15

2025

Stress,

MPa

Time, s

Numerical solution

Experiment

Fig. 5.

1

2

3

4

5 6

SG A ADC

C

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5. CONCLUSIONThe proposed dynamic model is proper for investigationof the stresses in the lifting installation of a fork-lift truckand to analyze the spectra response of the structure. The2D dynamic model of the truck in which the liftinginstallation is represented as an elastic beam permits

structural assessment with acceptable accuracy for theengineering practice. Based on the constitutive equationsof the model one can predict the main tendencies in thestructural behavior. The verification of the proposeddynamic model realized by the tensometric tests of fork-lift truck under different loading conditions permits toavoid the mistakes during the modeling and design. Theapproach shown may be successfully applied in theengineering practice for design and assessment of fork-lifttruck structures.The advantage of the proposed model is the possibility toanalyze and optimize the dynamic behavior of the fork-liftstructure. Further investigations should be directed to the

verification of the model for other cases of loading. It isof great interest to analyze the influence of differentdamping characteristics and elastic parameters on theresponse spectraof the truck structure.

REFERENCES

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[2] BEHA, E., Dynamische Beanspruchung und

Bewegungsverhalten von Gabelstaplern, Dissertation,Universitt Stuttgart, 1989[3] BERGMAN, L., NICHOLSON, J., Forced vibrations

of a damped combined linear system, ASME Journalof Vibration, Acoustics, Stress, and Reliability inDesign Vol. 107, 1985, pp 275-281

[4] CHANKOV, E., Transverse Vibrations of aCantilever Beam With a Concentrated Mass and

Moment at the Free End, Proceedings of The ThirdInternational Conference Challenges in HigherEducation and Research in the 21st Century,Sozopol, 2005, pp. 102-106

[5] CHANKOV, E., STOYCHEV, G., GENOV, J.,

Structural analysis of a fork lift truck liftinginstallation at dynamic loading, Mechanics ofMachines, Vol. 45, 2003, pp. 512-56

[6] CHANKOV, E., STOYCHEV, G., VENKOV, G.,GENOV, J.,A Continuous Dynamic Model of a Fork-lift Truck Lifting Installation, Proceedings of the 10thNational Congress of Theoretical and AppliedMechanics, Varna, 2005, pp. 11-16

[7] FIGUEIREDO, M., OLIVEIRA, F., GONCALVES,J., CASTRO, P., FERNANDES, A., Fracture Analysis of Forks of a Heavy Duty Lift Truck,Engineering Failure Analysis, No. 8, 2001, pp 411-421

[8] GEORGIEV, G., Design and Calculations for Fork-lift Trucks, Technics, Sofia, 1980

[9] GURGOZE, M., BATAN, H., A note on thevibrations of restrained cantilever beam carrying a

heavy tip body, Journal of Sound and Vibrations,Vol. 106, No. 3, 1986, pp 533-536

[10] KOLAROV, I., Deformation of The Mast of a Fork-lift Truck, Mashinostroene, No. 5, 1971, pp 211-213

[11] KOLAROV, I., SLAVCHEV, C., Loading andCalculation Cases for Design of Lifting Installations

of Fork-lift Trucks, Mashinostroene, No. 6, 1974, pp21-31

[12] KHN, I., Untersuchung der Vertikalschwingungenvon Regalbediengerten, Dissertation, UniversittKarlsruhe, 2001

[13] LAURA, P., POMBO, J., SUSEMIHL, E., A note onthe vibrations of a clamped free beam with a massat the free end, Journal of Sound and Vibrations, Vol.37, No. 2, 1974, pp 161-168

[14] LIN, Y., TRETHEWEY, M., Finite element analysisof elastic beams subjected to moving dynamic loads,Journal of Sound and Vibrations, Vol. 136, No. 2,1990, pp 323-342

[15] MAURIZI, M., BAMBILL DE ROSSIT, D., Free

vibrations of a clamped clamped beam with anintermediate elastic support, Journal of Sound andVibrations, Vol. 119, No. 1, 1987, pp 173-176

[16] MITCHEL, T., BRUCH, J., Free vibrations of a flexible arm attached to a complaint finite hub,ASME Journal of Vibration, Acoustics, Stress, andReliability in Design Vol. 110, 1988, pp 118-120

[17] PASHEVA, V., VENKOV, G., CHANKOV, E.,GENOV, J., Mathematical modeling of an elasticbeam fixed to a simple oscillator, Proceedings of The32nd International Conference Applications ofMathematics in Engineering and Economics, ,Sozopol, 2007, pp. 175-181

[18] PASHEVA, V., VENKOV, G., STOYCHEV, G.,CHANKOV, E., Dynamic modeling of systems withconcentrated and distributed parameters, Mechanicsof Machines, Vol. 65, 2006, pp. 52-55

[19] READY, J., An Introduction to the Finite ElementMethod, Mc Graw-Hill, 1984

[20] VERNIERE DE IRASSAR, P., FICCADENTI, M.,LAURA, P., Dynamic analysis of abeam with anintermediate elastic support, Journal of Sound andVibrations, Vol. 96, No. 3, 1984, pp 381-389

CORRESPONDENCE

Emanuil CHANKOV, M.Sc. Eng.Technical University of SofiaDepartment Strength of materialsKliment Ohridski 81000 Sofia, Bulgariachankov@tu-sofia.bg

Georgy STOYCHEV, Assoc. Prof. Ph.D.Technical University of Sofia

Department Strength of materialsKliment Ohridski 81000 Sofia, Bulgariagstojch@tu-sofia.bg