vibration of rail vehicles and modelling of system by using matlab

3.1. Six Degrees of Freedom Rail Vehicle ModelBased on the model given in figure 3, m g is the car body mass, m b is the bogie mass and m a is the axlemass, J g is the car body inertia, J b is the bogie inertia and J a is the axle inertia.kes 1 , kes 2 , c 1 and c 2 arethe secondary suspension stiffness and damping coefficients and k 1 , k 2 , c 3 and c 4 are the primarysuspension stiffness and damping coefficients. kh 1 , kh 2 , are the Hertzian spring stiffness.The rail vehicle model has six degrees of freedom; Z g , Z b , Z a , ø g , ø b , ø a . Z g is the vertical movement ofcar body, Z b is the vertical movement of bogie and Z a is the vertical movement of axle. θ g , θ b , θ a are theyaw movement of car body, bogie and axle respectively. Zy 1 and Zy 2 are the irregularities on the trackand regarded as a sinus function [6].L 12L 2Z az gkes 1m g , J gc 1 c 2m b , J bθ bθ gz bkes 2kh 2c 3 k 2k1 c 4m a , J a θ a, k akh 1z y1z y2Figure 3. Quarter rail vehicle model[6].The equations of motion of the rail vehicle are obtained by the use of the Lagrange Equation. Theequation of motion of the rail vehicle system is:[ M ]& z&+ [ C] z&+[ K] z = Fz(8)[M], [C] and [K] are mass, damping and stiffness matrices and [Fz] is the force resulting trackirregularity motion.4. MATLAB/SIMULINKSimulink able setting up dynamic system models by using block diagrams. So it is possible to examinethe systems as in a laboratory. The equations of motion obtained from the former part are given in theMATLAB form in equations (9)-(14):(-1/mg)*(kes1*((u[1]-u[3])-L1*u[7])+kes2*((u[1]-u[3])+L1*u[7])+c1*((u[2]-u[4])-L1*u[8])+c2*((u[2]-u[4])+L1*u[8])) (9)(-1/mb)*(-kes1*((u[1]-u[3])-L1*u[7])-kes2*((u[1]-u[3])+L1*u[7])+k1*((u[3]-u[5])-L1*u[9])+k2*((u[3]-u[5])+L1*u[9])-c1*((u[2]-u[4])-L1*u[8])-c2*((u[2]-u[4])+L1*u[8])+c3*((u[4]-u[6])-L1*u[10])+c4*((u[4]-u[6])+L1*u[10])) (10)(-1/ma)*(-k1*((u[3]-u[5])-L1*u[9])-k2*((u[3]-u[5])+L1*u[9])+kh1*((u[5]-u[13])-L2*u[11])+kh2*((u[5]-u[14])+L2*u[11])-c3*((u[4]-u[6])-L1*u[10])-c4*((u[4]-u[6])+L1*u[10])) (11)(-1/Jg)*(-kes1*L1*((u[1]-u[3])-L1*u[7])+kes2*L1*((u[1]-u[3])+L1*u[7]) -c1*L1*((u[2]-u[4])-L1*u[8])+c2*L1*((u[2]-u[4])+L1*u[8])) (12)827

(-1/Jb)*(-k1*L1*((u[3]-u[5])-L1*u[9])+k2*L1*((u[3]-u[5])+L1*u[9]) -c3*L1*((u[4]-u[6])-L1*u[10])+c4*L1*((u[4]-u[6])+L1*u[10])) (13)(-1/Ja)*(-kh1*L2*((u[5]-u[13])-L2*u[11])+kh2*L2*((u[5]-u[14])+L2*u[11])) (14)These equations (9)-(14) are written in the boxes named as vertical motion of car body, bogie andaxle(wheelset) and yaw of car body, bogie and axle (wheelset) respectively. And then, with the help ofm.file it is possible to realize the simulations for asked parameters.4.1. Create a Model in SimulinkFigure 4. Simulink block diagram of quarter rail vehicle model.5. RESULTSThe application of the MATLAB as environment gives us a chance to build a user friendly interfaceand using the possibilities of the MATLAB/Simulink made the package open for inserting new optionsby the user as well as decreasing the time of preparing and carrying out the simulation process. In thelight of this paper it is possible to obtain motion equations for all degrees of freedom systems and torealize the simulation in MATLAB/Simulink.6. REFERENCES[1] Chudzikiewicz A.: Simulation of Rail Vehicle Dynamics in MATLAB Environment, Vehicle SystemDynamics, 33, pp. 107-119, 2000,[2] Esveld, C.: Modern Railway Track, MRT Productions, 2001,[3] Knothe, K.: Gleisdynamik, Technische Universitat,2001,[4] Jenkins, H., Stephenson, J., Clayton, G.A., Morland, G.W., Lylon, D.,: The effect of track and vehicleparameters on Wheel/rail vertical dynamic forces, Railway engineering Journal, 2-16, 1974,[5] Yalcin, N.S., Guclu, R., Metin, M., Yazici, H.: Analyses of railway induced vibrations for different tracktypes, Inter-Noise 2007, İstanbul, Turkey.[6] Bayraktar, M., Guclu, R., Metin, M., Yazici, H.: Vibration Analysis of Light Rail Vehicle Due to AxleModelling, UMTS 2009, KKTC.828