INVESTIGATION OF DYNAMIC STRESSES IN A FORKLIFT TRUCK LIFTING INSTALLATION Georgi STOYCHEV Emanuil CHANKOV

Abstract: The dynamic stresses which appear in the lifting installation of a fork-lift truck at loading-unloading operations are investigated in this paper. The strains at a zone nearby the attachment of the tilting cylinder are measured. A record of the accelerations which act on the truck is also done. The elasticity and damping characteristics of the tilting cylinder are determined. The experimental data is acquired in laboratory conditions and is used for verification of a two dimensional numerical model of the truck which is analyzed by a university license version of finite element code COSMOS/M. There is good agreement between the computed results and the experimental data. Key words: Forklift truck, experimental study, stress, FEA

1. INTRODUCTION The lifting installation of fork-lift trucks is a complicated structure subjected to various static and dynamic loads. The optimal design of this structure is of significant economic and technical efficiency importance. The problem concerning the exact determination of the stresses and deformations in lifting installations arises from the beginning of fork-lift truck production. In some earlier papers [11] the static deformations of the mast at the case when the load is lifted to maximum height are studied. It is represented as a construction of beams with a constant cross-section. Its lower end is attached to a pin support and the tilting hydraulic cylinder is represented as a rigid support. The determination of the deformations 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 and the dynamic loads acting on it are received using a dynamic coefficient. Similar methods for calculation of the deformations and stresses in the lifting installation of fork-lift trucks are demonstrated in [10].

In some cases the deflection function of a fork-lift truck mast is used in order to determine the deformations [2]. The mast is modeled as a beam with variable crosssection and the forces are applied at the mass centre of the load lifted to maximum height. In order to investigate the elastic vibrations the authors [5] solve analytically a partial differential equation which describes the behavior of a fork-lift truck mast represented as a uniform elastic beam. A method for the design of the lifting installation of a stacker crane in which the static forces are multiplied by a dynamic coefficient is proposed in [13]. The stress distribution in a fork is analyzed in [9] where a 2D FE model is proposed. The stress distribution in the lifting installation of a fork lift truck can be also accomplished by quasi-static calculations [4]. The authors build a detailed dynamic model to calculate the velocities and accelerations which are acting on the lifting installation at a particular moment of time and apply them in a 3D FE model of the mast. Calculation of dynamic stresses in the lifting installation requires an appropriate dynamic model of the truck. In some works [2], [4], [10], [18], [21] complicated 2D and 3D fork-lift truck models with concentrated parameters are investigated while in others [6], [7], [16], [17] the dynamic models consist of flexible and rigid bodies. Problems concerning combined dynamic systems consisting of flexible and rigid bodies are well studied and different methods are used to solve such systems. An approximate method is used in [15] for the determination of the free vibrations of a flexible robot arm fixed to a spring supported mass. The response of a cantilever beam carrying spring-mass systems is examined in [3] with the use of Green’s functions. The Finite Element Method is one of the most widespread and general methods [8], [19] which is applied for investigation of combined dynamic systems. In the paper [14] it is used for dynamic analysis of elastic beams with arbitrary moving spring-mass-damper systems. The same method is applied in [1], [16], [17] where similar problems are being analyzed. The mass, spring and damping characteristics of a forklift truck dynamic model can differ depending on the type of the truck. In the literature [2], [18], [21] one can find the range in which this characteristics can vary. Experimental study of tire elasticity and damping is conducted in [21]. The characteristics of hydraulic elements are determined in [2]. The tilting cylinder has significant influence over the stresses in the lifting installation. In the present publication the authors propose a method, developed in [20], for determination of the spring and damping characteristics of this element. The dynamic stresses due to load-unload operations are experimentally measured and used for verification of a two dimensional numerical model of the truck solved by finite element code.

2. DYNAMIC MODEL The model of the truck is shown in Fig. 1. This is a 2D model with 5 degrees of freedom: y and z – horizontal and 75

vertical translation of the truck’s chassis, φ – rotation of the chassis around its mass centre, ψ – rotation of the mast as a rigid body around the point of attachment to the truck, w – the horizontal deflection of the points from the elastic mast. The constitutive dynamic equations of the truck’s chassis are my&& = − F − F + F + R h 1

h 2

h 3

h

mz&& = F1v + F2v + F3v − R v

(1)

I ϕ&& = − F1h .h1 − F2h .h2 − F3h .h3 + + F1v .L1 − F2v .L2 + F3v .L3 + R h .h1 − R v .L1

m1, I1 C w ψ

m2, I2

φ k

y

c2h k2v

k3

h3

h2 m, I

h 2

k1h

h1

c1h

L3

v 2

c

L1

L2

h6

c3

α

β

h4

B

x

v k1v c1 L4

where m and I are the mass and the mass moment of inertia of the chassis, the dimensions L1, L2, L3, h1, h2, h3 are shown in Fig. 1, R h and R v are the horizontal and vertical component of the reaction at the point of attachment of the mast to the chassis respectively, F1h , h 2

v 2

h 3

v 3

F , F , F , F , F are the horizontal and vertical components of the forces in the elastic elements, the front and rear tyres and the tilting cylinder respectively. The forces can be expressed as F1h = ( y + h1 .ϕ ) k1h + ( y& + h1 .ϕ& ) c1h F1v = − ( z + l1 .ϕ ) k1v − ( z& + L1 .ϕ& ) c1v (2)

F3h = ( h4 .ψ + w(h4 , t ) ) k3 .cos α + ∂w(h4 , t ) ⎞ ⎛ + ⎜ h4 .ψ& + ⎟ c3 .cos α ∂t ⎝ ⎠ F3v = L4 .ψ .k3 .sin α + L4 .ψ& .c3 .sin α

where k1h , k1v , k 2h , k2v , c1h , c1v , c2h , c2v are the horizontal and vertical component of the elasticity and damping 76

n ∂ 4 w( x, t ) ∂ 2 w( x, t ) + ρA = q ( x, t ) + ∑ Fi .δ ( x − xi ) 4 2 ∂x ∂t i =1

(4)

where, ρ and E are the density and modulus of elasticity of the material, A and I – the area and the moment of inertia of the beam’s cross-section respectively, x is the current coordinate along the axis of the beam, q(x,t) and Fi are the inertial distributed and concentrated loads which are due to the motion of the truck as a rigid body system, δ is the Dirac delta function. Using the finite element method, equation (4) can be represented as an ordinary differential equations system which in matrix form has the following expression

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

(5)

where [M], [C] and [K] are the global mass, damping and stiffness matrices respectively, {u} is the vector of the degrees of freedom in the nodes, {F} is the vector of the external loads. The approach for a simultaneous solution of the systems (1), (3) and (5) is discussed in [6].

3. EXPERIMENTAL INVESTIGATION

F2h = ( y + h2 .ϕ ) k2h + ( y& + h2 .ϕ& ) c2h F2v = − ( z − l2 .ϕ ) k2v − ( z& − L2 .ϕ& ) c2v

(3)

where , m1 and m2 are the masses of the load and the mast respectively, IB is the mass moment of inertia of the lifting installation (together with the load) with respect to the point of attachment – B, point C is the mass centre of the lifting installation (together with the load), β is the angle between the line BC and the axis of the mast. The systems of equations (1) and (3) are written assuming that the dynamics of the rigid body model is not influenced by the elastic behavior of the lifting installation and that the angles φ and ψ are small. The partial differential equation concerning the transverse vibrations of the mast represented as a flexible uniform beam has the expression EI

Fig.1. 2D dynamic model of the truck

v 1

( m1 + m2 ) &&y + h1 ( m1 + m2 ) ϕ&& = − F3h + R h ( m1 + m2 ) &&z + L1 ( m1 + m2 ) ϕ&& = − F3v + R v ( m1 + m2 ) BC cos β .&&y − ( m1 + m2 ) BC sin β .&&z + + ( h1 .cos β − L1 sin β )( m1 + m2 ) BC.ϕ&& + I Bψ&& = = − F3h .h4 + F3v .l4

L6

z

coefficients of the front and rear tyres, k3 and c3 are the elasticity and damping coefficient of the tilting cylinder, α 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 a rigid body have the following form

The experiment was realized in laboratory conditions on a fork-lift truck model EB 687.33.10 produced by Balkancar Record - Bulgaria. The lifting capacity is 1000 kg and the maximum lift height is 3,3 m. The load in use is m1 = 600 kg. The measuring devices used in the experiment are HBM SPIDER 8 and NI USB-6008/6009 OEM device, connected to a computer were used for measurement of acceleration, velocity, displacement and strain. The zones at which the sensors were mounted can be seen in Fig. 2,

where 1 is the truck chassis, 2 – the mast, 3 – the load, 4 – the lifting cylinder, 5 – the tilting cylinder, 6 – the strain gages, 7 – the displacement transducer, 8 – dual-axis accelerometer, 9 and 10 - the velocity sensors.

c3 =

8

3

6

7

.ln

y4 (t1 ) , y4 (t2 )

(7)

The moment of inertia IB and the coordinates of C can be found by modeling the lifting installation and the load in a 3D CAD software (in this case SolidWorks was used (Fig. 5)).

9 5

2I B

τ .h42

(6)

where y4 is the horizontal displacement of the point where the tilting cylinder is attached to the mast, τ = t2-t1, g is the Earth’s acceleration, l5 is the position of the mass centre C (Fig. 4).

2

4

1

2 I B ⎡ 2 ⎛ y4 (t1 ) ⎞ ⎤ ( m1 + m2 ) g .l5 , k3 = 2 2 ⎢ 4.π + ⎜ ln ⎟ ⎥+ h4 .τ ⎢ h42 ⎝ y4 (t2 ) ⎠ ⎥⎦ ⎣

For the truck and load used in the experiment the following values of the constants are determined:

10

m1 = 600 kg, m2 = 340 kg, IB = 7500 kg.m2, h4 = 0.35 m, l5 = 0.45 m. Fig. 2. Positioning of the sensors The flow chart of the measurements is shown in Fig. 3, where A is amplifier, ADC – analog to digital converter, C – computer.

Sensor

A

ADC C

Fig. 3. Flow chart of the measurements

3.1. Determination properties

of

the

tilting

cylinder

The properties of the tilting cylinder are obtained by displacement transducer situated as shown in Fig. 4.

l5 C ψ h6

Chassis

Displace ment

(m1+ m2)g

y4

h5

h4

Chassis

Fig. 5. 3D model of the lifting installation with the load B Fig. 4. Experimental scheme for determination of the tilting cylinder properties The following formulas, which derivation has been discussed in [20], can be used to determine the spring and damping coefficients of the cylinder respectively

The experiment was conducted by applying to the construction an impulse while the load was lifted at maximum. A record of the deflection y4 is shown in Fig. 5. Average values of the spring and damping coefficients determined are:

k3 = 6.106 N/m, c3 = 150.103 Ns/m. 77

1,5

0.6

Acceleration, m/s2

, mm

2,0

τ =0.63 s

0.28

y

4

1,0 0,8 0,6 0,4 0,2 0,0 -0,2 -0,4 -0,6 -0,8 -1,0 0,0

t1 t2

1,0 0,5 0,0 -0,5 -1,0 2.85 s

-1,5 -2,0

0,5

1,0

1,5

-2,5

2,0

0

Time, s

2

4

6

8

10

12

Time, s

Fig. 5. Record of the displacement at the point of attachment of the tilting cylinder

Fig. 7. Record of the fork-lift truck velocity

3.2. Determination of the stresses and verification of the model Three dangerous cases at loading-unloading operations are studied. The first one is when the truck accelerates forwards with the load lifted to maximum height and then stops rapidly. The free vibrations of the system are observed. For the adequate numerical model the right initial conditions have to be given. The time between the start and the end of the forward motion may be obtained by observation of measured velocity represented in Fig. 6.

30 Numerical solution Experiment

20 Stress, MPa

Comparison between these results and the values given in the literature [2] shows that the spring constant k3 is in the appropriate range, while the damping coefficient c3 has much bigger value. This can be explained by the fact that the obtained coefficient accounts for the damping of the system as a whole, i. e. this is the damping coefficient of the system reduced to the tilting cylinder.

10 0 -10 -20 -30 0

1

2

3

4

5

6

7

Time, s

Fig. 8. Comparison of the stresses from the first load case 30 Numerical solution Experiment

20

0,8 Stress, MPa

10

Velocity, m/s

0,6 0,4 0,2

0 -10 -20 -30 -40

0,0

2.85 s

-50 0

-0,2 0

1

2

3

4

5

Time, s

Fig. 6. Record of the fork-lift truck velocity The horizontal acceleration of the load (Fig. 7) for this period is applied in the numerical calculations. The stresses at the zone nearby the attachment of the tilting cylinder to the mast are measured by strain gages. Comparison between the experimentally obtained and the calculated stresses is shown in Fig. 8. One can observe good agreement. 78

1

2

3

4

5

6

7

Time, s

Fig. 9. Comparison of the stresses due to backwards motion and rapid stop of the fork-lift truck Similarly the stresses are measured and calculated for the other two load cases:  backwards motion and stop with load lifted to maximum height (Fig. 9);  drop down of the load followed by rapid stop when the load is lifted to maximum height (Fig. 10).

Good agreement can be observed when the numerical and experimental results from the third load case are compared. There is no so good agreement when comparing the results acquired for the backwards motion load case (Fig. 9). It means that the model is not very appropriate for this case due to big stiffness of the truck for this motion. 30 Numerical solution Experiment

Stress, MPa

20 10 0 -10 -20 -30 0,0

0,5

1,0

1,5

2,0

2,5

Time, s

Fig. 10. Comparison of the stresses due to rapid drop down and stop of the load

4. CONCLUSION The proposed dynamic model is proper for investigation of the stresses in the lifting installation of a fork-lift truck when analyzing the spectra response of the structure in the case of loading-unloading operations. The 2D dynamic model of the truck allows stress assessment with acceptable accuracy for the engineering practice. The comparison of the experimentally obtained and calculated results shows that this simplified model is applicable in the engineering practice for design and assessment of fork-lift truck structures. The advantage of the proposed model is the usage of a reduced damping coefficient which on one hand simplifies the model and on the other hand allows easy optimization of the dynamic behavior of the fork-lift structure. Further investigations should be directed to the verification of the model for other cases of loading.

REFERENCES [1] ASHRAFIUON, H., Optimal design of vibration absorber system supported by elastic base, ASME Journal of Vibration and Acoustics, Vol. 114, 1992, pp 280-283 [2] BEHA, E., Dynamische Beanspruchung und Bewegungsverhalten von Gabelstaplern, Dissertation, Universität Stuttgart, 1989

[3] BERGMAN, L., NICHOLSON, J., Forced vibrations of a damped combined linear system, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design Vol. 107, 1985, pp 275-281 [4] CHANKOV, E., STOYCHEV, G., GENOV, J., Structural analysis of a fork lift truck lifting installation at dynamic loading, Mechanics of Machines, Vol. 45, 2003, pp. 512-56 [5] CHANKOV, E., STOYCHEV, G., VENKOV, G., GENOV, J., A Continuous Dynamic Model of a Forklift Truck Lifting Installation, Proceedings of the 10th National Congress of Theoretical and Applied Mechanics, Varna, 2005, pp. 11-16 [6] CHANKOV, E., VENKOV G., STOYCHEV G., An Elastic Beam Mounted to a Spring-Mass Dynamic System, Proceedings of The American Institute of Physics, 2007, pp.145-152 [7] CHANKOV, E., VENKOV G., STOYCHEV G., Fork-lift truck dynamic model with concentrated and distributed parameters, Mechanics of Machines, Vol. 70, 2007, pp. 94-97 [8] COOK, R., MALKUS, D., PLESHA, M., Concepts and application of finite element analysis, John Wiley & Sons, 1989 [9] 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 411421 [10] GEORGIEV, G., Design and Calculations for Forklift Trucks, Technics, Sofia, 1980 [11] KOLAROV, I., Deformation of The Mast of a Forklift Truck, Mashinostroene, No. 5, 1971, pp 211-213 [12] KOLAROV, I., SLAVCHEV, C., Loading and Calculation Cases for Design of Lifting Installations of Fork-lift Trucks, Mashinostroene, No. 6, 1974, pp 21-31 [13] KÜHN, I., Untersuchung der Vertikalschwingungen von Regalbediengeräten, Dissertation, Universität Karlsruhe, 2001 [14] LIN, Y., TRETHEWEY, M., Finite element analysis of elastic beams subjected to moving dynamic loads, Journal of Sound and Vibrations, Vol. 136, No. 2, 1990, pp 323-342 [15] MITCHEL, T., BRUCH, J., Free vibrations of a flexible arm attached to a complaint finite hub, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design Vol. 110, 1988, pp 118-120 [16] PASHEVA, V., VENKOV, G., CHANKOV, E., GENOV, J., Mathematical modeling of an elastic beam fixed to a simple oscillator, Proceedings of The 32nd International Conference Applications of Mathematics in Engineering and Economics, , Sozopol, 2007, pp. 175-181 [17] PASHEVA, V., VENKOV, G., STOYCHEV, G., CHANKOV, E., Dynamic modeling of systems with concentrated and distributed parameters, Mechanics of Machines, Vol. 65, 2006, pp. 52-55 [18] PETROV, P., Dynamic modeling of fork-lift trucks at transportation activities, Dissertation, Technical University of Sofia, 1997 [19] READY, J., An Introduction to the Finite Element Method, Mc Graw-Hill, 1984 79

[20] STOYCHEV, G., CHANKOV, E., Experimental study of a fork-lift truck at dynamic loading, Mechanics of Machines, Vol. 82, 2009, pp. 96-100 [21] TODOROV, M., Vibration study and parameter optimization of fork-lift trucks, Dissertation, Technical University of Sofia, 1996

CORRESPONDENCE Georgy STOYCHEV, Assoc. Prof. Ph.D. Technical University of Sofia Department Strength of materials Kliment Ohridski 8 1000 Sofia, Bulgaria [email protected] Emanuil CHANKOV, M.Sc. Eng. Technical University of Sofia Department Strength of materials Kliment Ohridski 8 1000 Sofia, Bulgaria [email protected]

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