PARAMETER IDENTIFICATION OF A FORK-LIFT TRUCK DYNAMIC MODEL

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1 11 th National Congress on Theoretical and Applied Mechanics, -5 Sept. 9, Borovets, Bulgaria PARAMETER IDENTIFICATION OF A FORK-IFT TRUCK DYNAMIC MODE E. S. CHANKOV Strength of Materials Dep., Technical University of Sofia, Sofia, 1, Bulgaria chankov@tu-sofia.bg G. B. STOYCHEV Strength of Materials Dep., Technical University of Sofia, Sofia, 1, Bulgaria gstojch@tu-sofia.bg V. V. PASHEVA Analysis and Numerical Methods Dep., Technical University of Sofia, Sofia, 1, Bulgaria vvp@tu-sofia.bg G. I. VENKOV Analysis and Numerical Methods Dep., Technical University of Sofia, Sofia, 1, Bulgaria giv@tu-sofia.bg ABSTRACT. The paper studies the dynamics of a fork-lift truck lifting installation represented as an elastic beam attached to a spring rigid bar of two degrees of freedom. An approach which proposes an effective simultaneous solution of the partial differential equation describing the beam and the ordinary differential equations concerning the rigid bar is developed. Eperimental data from a real forklift truck is used to perform parameter identification. KEY WORDS: Fork-lift truck, dynamic model, optimization 1. Introduction Fork-lift trucks play significant role in logistics. Optimization of their construction is important for store effectiveness. Several decades ago simplified models, which consist of rigid bodies, have been introduced [1, ]. Modeling of fork-lifts is constantly developing and multibody D [3, 4] and 3D [5, 6] models have been investigated. One cannot estimate the stresses in the lifting installation until representing it as an elastic body. For a D model it can be regarded as an elastic beam.

2 E. Chankov, G. Stoychev, V. Pasheva, G. Venkov There are numerous papers dedicated to the dynamics of systems containing rigid and fleible bodies [7, 8, 9]. Solution of such systems can be accomplished by applying different approaches such as the one which utilizes Green s functions [8, 9] or by using the Finite Element (FE) Method [1, 11]. Using appropriate values for the parameters of a fork-lift truck dynamic model, such as spring and damper characteristics and mass properties, is also of significant importance. Eperimentally determined values of these parameters vary in wide ranges [3, 5]. The purpose of the current paper is to investigate a fork-lift truck dynamic model in which the lifting installation is represented as an elastic beam and to define an optimization problem aiming parameter identification based on eperimental data.. Dynamic model This paper is focused on the dynamic model shown in Fig. 1. It consists of a rigid body with two degrees of freedom (DOF) and an elastic beam attached to it. l 6 m c 3 C m ψ w k 3 z φ α m C h 3 B k l 3 c k 1 c 1 l l 1 Fig. 1. The rigid body has its center of gravity at point C and two DOFs vertical z- translation and φ-rotation about C. There is a joint connection between the lower end of the beam and the rigid body - point B. The beam has an offset tip heavy mass at the free end and is able to rotate (shown by ψ) about B. There is an additional elastic connection between the oscillator and the beam. The horizontal deflection of the fleible beam is denoted by w. The model in Fig.1 presents an adequate scheme of study of a fork-lift truck dynamics when the elasticity of the lifting structure is taken into account. The study could be divided into two parts. The first one assumes that the influence of the beam elasticity over the dynamic response of the concentrated parameters system is insignificant. The beam and the tip mass are considered as a rigid body with a center of gravity at point C m (Fig. 1). Thus, the model is now regarded as a three DOF system consisting of two rigid bodies. The governing equations of the system are derived through the Newton s Second aw. For small vibrations, the dynamic response is described by the following linear system of ODEs h 4

3 Parameter Identification of a Fork-lift Truck Model, (.1) ( m m ) z lm 1 ϕ c z c ϕ k z k ϕ = (.) ( ) (.3) M M K K lm z l m I c z c ch k z k kh, 1 M 1 M C ϕ ϕ 3 ψ ϕ 3 ψ = m BC sin β z lm BC sin βϕ I ψ h c cos αψ h k cos αψ =, M m 1 M m B mm = m1 m ck = c1 c kk = k1 k, c lc 1 1 lc , k = l1k1 lk, h,5( l3 l1) h4 sin α h3h4 cos α where,, k = lk l k =, c = l c l c, =, m is the oscillator s mass and I C its mass moment of inertia with respect to С; m 1 and m are the beam s mass and the tip mass, respectively; I B is the mass moment of inertia of the beam and m with respect to point B; β is the angle between the beam s ais and the line through points В and C m ; angle α and dimensions l 1, l, l 3, l 6, h 3, h 4, are shown in Fig. 1; k 1, k, k 3, c 1, c, c 3, are spring and damping coefficients of the spring-damper elements (Fig. 1). Dots denote differentiation with respect to time. A unit impulse defined by the following initial conditions is applied at t = : (.4) z () =, z () =, ϕ () =, ϕ () =, ψ () = ψ, ψ () =. The second part of the problem refers to the dynamic response of the elastic beam due to the motion of the rigid body system (Fig. ). l 6 F v F h M b F h w w k 3 c 3 B α h 4 F s B h 4 (a) Fig. (b) The response is given by 4 wt (, ) wt (, ) (.5) EI ρa = F () ( 4 s t δ h4 ), t where ρ and E are the density and modulus of elasticity of the material; A and I area and moment of inertia of the beam s cross section; is a current coordinate; δ is the Dirac delta function; F s is the horizontal component of the spring-damper force given by the equation ( 4, ) (.6) () h h wh t Fs t = k3. w( h4,) t c3 t

4 E. Chankov, G. Stoychev, V. Pasheva, G. Venkov h where cos h k3 = k3 α and c cos 3 = c3 α. Based on the solution of the system (.1),(.),(.3) one can obtain the horizontal F h and the vertical F v inertial force which act on the mass m (Fig. a) (.7) F () h t = mψ, Fv () t = m( z l 1ϕ l 6ψ ). It is assumed that the mass m 1 of the beam is several times smaller than the mass m. This allows the distributed inertial load acting on the beam to be neglected. Reduction of the heavy tip mass to the ais of the beam results in a concentrated moment M b at the free end (Fig. b) M () t = Fl = m l z l ϕ l ψ. (.8) ( ) b v The beam boundary conditions (BCs) for the pin supported end are w(, t) (.9) w(, t ) =, =. and for the free end are 3 wt (, ) w( t, ) (.1) EI = m () 3 F h t, t wt (, ) wt (, ) = ( ) t M t. b where I is the mass moment of inertia of the heavy tip mass m. The initial conditions (ICs) are w (,) =, w (,) =. Simultaneous solution of Eqs. (.1), (.), (.3) and (.5) can be obtained by FE discretization. For that purpose the beam is divided into N elements with two DOF per node deflection wt (, ) and slope θ (, t). Eq. (.5) can be written as a system of N second order ODEs of the type M u C u K u = f (.11) EI ( I ml6 ) (1) [ ]{ } [ ]{ } [ ]{ } { } where [M], [C] and [K] are the global mass, damping and stiffness matrices respectively; {u} is the vector of the nodal parameters w and θ; {f} is the vector of the nodal loads. The horizontal component of the force in the spring-damper element F s, the concentrated heavy mass m, the force F h and the moment M b can be incorporated in the mass [M], stiffness [K] and damping [C] matrices by modifying them, which is shown in details in [11]. This in terms leads to a homogenous system (.13) M * { u } C * { u } K * { u} = { } * * * where [ M ], [ C ] and [K ] are the mass, damping and stiffness matrices respectively after modification. Epressing the second-order system (.1), (.), (.3), (.13) in first-order form will lead to the following matri-vector equation: v = D v, (.14) { } [ ]{ }

5 Parameter Identification of a Fork-lift Truck Model T where {} v = { u... u z ϕ ψ u... u z } 1 n 1 ϕ ψ, u i (i = 1...n) are the elements of the vector {u}, n = N1 is the number of ordinary differential equations used to represent the partial equation (.5) after accountiong the boundary conditions (.9). The matri [D] is consisting of 4 submatrices [11]: zero, identity, modified stiffness and dampig matricess, all with dimension (n3) (n3). By the use of the matri eponential, the solution of Eq. (.14) becomes D. t t (.15) vt () e. vt ( ) t t T, where { ( )} [ ]( ) { } = { }, [, ] vt is the initial conditions vector; T denotes the end time, t =. 3. Parameter identification The model studied in the paper can be used for parameter identification. The paper deals with determination of the elasticity and damping characteristics of the tires and the tilting cylinder. Information for the values of these parameters is available [3, 5] but they vary in wide ranges: Front tires k 1 = 1, ,.1 6 Nm -1, c 1 = Nsm -1 ; Rear tires k =, ,.1 6 Nm -1, c = Nsm -1 ; Tilting cylinder k 3 = Nm -1, c 3 = Nsm -1. A real fork-lift truck model EB produced by Balkancar Record Bulgaria is used for the investigation. Its lifting capacity is 1 kg and the maimum lift height is 3,3 m. The parameters of the truck which are needed for the solution of the problem have the following values: m= 3 kg, m 1 =5 kg, m =6 kg, I C =11 kg m, I B =95 kg m, l 1 =1,15 m, l =, m, l 3 =,9 m, l 6 =,5 m, h 3 =,4 m, h 4 =,5 m, =3 m, α=18,β =7,5,BC m =,5 m, E=.1 11 Pa, A= m, I= m 4, ρ=785 kg m -3, N=3, ψ =,8 m s -1. Eperimental data about the stresses near by the point of attachment of the tilting cylinder ( = h 4 ) is available (Fig. 3). Using Eq (.15) one can easily find the stress at this point. Neglecting the shearing and normal internal forces, the stress can be determined by the relation M ( ) ( ) (3.16) ( ) y h4, t wh4, t σ h4, t =. zd = Ezd I y where I y = I, M y is the bending moment and z d is the depth of the cross-section. The derivative in (.16) can be found by differentiation of the element s form functions. A procedure, which minimizes the mean square difference between the numerically calculated σ and the eperimentally determined stress σ is applied T e (3.17) S = q σ h4, t σ h4, t dt, ( ) ( ) where q is a scale factor and the optimization parameters are k 1, k, k 3, c 1, c, c 3. Fig.3 shows the result after optimization. The achieved values of the parameters are k 1 = 3,8.1 6 Nm -1 ; k = 4,.1 6 Nm -1 ; k 3 = 8,8.1 6 Nm -1 ; c 1 = 3,.1 3 Nsm -1 ; c =,8.1 3 Nsm -1 ; c 3 = Nsm -1. n e

6 E. Chankov, G. Stoychev, V. Pasheva, G. Venkov σ, MPa Mathlab Eperiment Time, s Fig. 3 R E F E R E N C E S [1] BACHVAROV, S., N. MAINOV, K. ARNAUDOV, D. CHOTOVA, Non Resonance Dynamics of an Electric Fork-lift Truck at Transportation, Appl. Math., Vol. (1984), No., (In Bulgarian). [] PISAREV, A., G. GEORGIEV, V OVCHAROV, K. KOSTOV. Dynamic oading in the Supporting Construction of Fork-lift Trucks. Proc. VI National on Theoretical and Appl. Mech., Vol. 1 (1989), (In Bulgarian). [3] BEHA, E., Dynamische Beanspruchung und Bewegungsverhalten von Gabelstaplern, Dissertation, Universität Stuttgart, 1989 (In German). [4] PETROV, P., Dynamic Modeling of Fork-lift Trucks at Transportation Ativities, Dissertation, Technical University of Sofia, 1997 (In Bulgarian). [5] TODOROV, M., Vibration Study and Parameter Optimization of Fork-lift Trucks, Dissertation, Technical University of Sofia, 1996 (In Bulgarian). [6] ANGEOV, I., 3D Matri Modeling in Kinematics, Dynamics and Mechanical and Multi Body Systems Vibration, Dissertation, Technical University of Sofia, (In Bulgarian). [7] MITCHE, T., J. BRUCH MITCHE, T., BRUCH, J., Free Vibrations of a Fleible Arm Attached to a Complaint Finite Hub, ASME Journal of Vibration, Acoustics, Stress, Reliability in Design, Vol. 11 (1988), [8] NICHOSON, J. W.,. A. BERGMAN, Free Vibrations of Combined Dynamical Systems. Journal of Eng. Mech., Vol. 11 (1986), 1-13 [9] KUKA, S., Application of Green Functions in Frequency Analysis of Timoshenko Beams With Oscillators. Journal of Sound and Vibration, Vol. 5 (1997), [1] IN, Y. H, M.W. TRETHEWEY, Finite Element Analysis of Elastic Beams Subjected to Moving Dynamic oads. Journal of Sound and Vibration, Vol. 136 (199), [11] CHANKOV, E. S., G. I. VENKOV, G. B. STOYCHEV, An Elastic Beam Mounted to a Spring-Mass Dynamic System. Proc.33th International Conference of Applications of Mathematics in Engineering and Economics, 7,

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