# The influence of foundation mass on dynamic response of track-vehicle interaction

Tóm tắt The influence of foundation mass on dynamic response of track-vehicle interaction: ...k [Nv] ldξ, [Ks] = ∫ l 0 [Ns] Tks [Ns] ldξ, (17) and [Mv] = ∫ l 0 [Nv] TρA [Nv] ldξ, [Mθ ] = ∫ l 0 [Nθ ] TρI [Nθ ] ldξ, (18) [MF] = ∫ l 0 [Nv] Tm [Nv] ldξ. (19) The viscous damping property of the foundation is considered to be the dashpots system, the damping matrix of foundati...ulation show quite good agreement with numerical results in the literature. Therefore, the program which will analyze the influence of many parameters on the dynamic response of the railway track-vehicle-foundation interaction is reliable. 3.2. Numerical investigation In this section, the ef... (a) ks= 107 N, (b) ks= 109 N 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass dens...

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isplacement of the midpoint of the track Phuoc T. Nguyen, Trung D. Pham and Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analysis of the railway track-vehicle-foundation interaction. It increases time history of vertical displacement of the midpoint of the track (see in Figs. 6a and 6b) but sometimes it also decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can be seen that the mass density of dynamic foundation has an influence on the dynamic response of the railway track-vehicle-foundation interaction. To show more clearly the influence of the foundation mass parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.5 1.0 1.5 2.0 2.5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 0.6 0.8 1.0 1.2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 1.3 1.6 1.9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displacement of the railway track for various shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (a) k = 108 N/m2 Phuoc T. Nguyen, Trung D Pham and Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analys s of the railway track-vehicle-foundation nteraction. It increases time history of vertical displacement of the midpoint of the track (see in Figs. 6a and 6b) but sometimes it also decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can be seen that the mass density of dynamic foundation has an influence on the dynamic response of the railway track-vehicle-foundation interaction. To show more clearly the influence of the foundation ass parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.5 1.0 1.5 2.0 2.5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 0.6 0.8 1.0 1.2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 1.3 1.6 1.9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displac ment of the railway track fo various shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N .50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displac ment of the railway track for the various mass density f foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (b) k = 5× 108 N/m2 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness 28 Phuoc T. Nguyen, Trung D. Pham, Hoa P. Hoang Phuoc T. Nguyen, Trung D. Pham and Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analysis of the railway track-vehicle-foundation interaction. It increases time history of vertical displacement of the midpoint of the track (see in Figs. 6a and 6b) but sometimes it also decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can be seen that the mass density of dynamic foundation has an influence on the dynamic response of the railway track-vehicle-foundation interaction. To show more clearly the influence of the foundation mass parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.5 1.0 1.5 2.0 2.5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 0.6 0.8 1.0 1.2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 1.3 1.6 1.9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displacement of the railway track for various shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (a) ks = 107 N Phuoc T. Nguyen, Trung D. Pham and Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analysis of the railway track-vehicle-foundation interaction. It increases time history of vertical displacement of the midpoint of the track (see in Figs. 6a and 6b) but sometimes it also decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can be seen that the mass density of dynamic foundation has an influence on the dynamic response of the railway track-vehicle-foundation interaction. To show more clearly the influence of the foundation mass parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.5 1.0 1.5 2.0 2.5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for variou spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 0.6 0.8 1.0 1.2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 1.3 1.6 1.9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displacement of the railway track for variou shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N 0.50 0.75 1. 0 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (b) ks = 109 N Fig. 8. The DMFs of vertical displacement of the railway track for various shear layer stiffness Phuoc T. Nguyen, Trung D. Pham a Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analysis of the railway track-vehicle-foundation interaction. It increases time history of vertical displacement of the midpoint of the track (see in Figs. 6a and 6b) but sometimes it also decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can b seen that the mass de sity of dynamic foundation has an influence on t dy ami response of the railway track-vehicle-foundation interaction. To show more clearly the influence of the foundation mass parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.5 1.0 1.5 2.0 2.5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 0.6 0.8 1.0 1.2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 1.3 1.6 1.9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displacement of the railway track for various shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N 0.50 0.75 1.00 1.25 1.50 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s 0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (a) ρ f = 1700 kg/m 3 Phuoc T. Nguyen, Trung D. Pham and Hoa P. Hoang 10 As shown in Fig. 6, the mass density of dynamic foundation effects quite clear on dynamic analysis of the railway track-ve icle-foundati n interaction. It i creases time history of vertical displacement of th midpoint of the track (see in Figs. 6a and 6b) but sometimes it lso decreases the dynamic response of the track with an increase of values of the foundation mass parameter, shown in Figs. 6c and 6d. Hence, it can be seen that the mass density of dynamic foundation has an influence on the dynamic response of the railway track-vehicle-foundation interaction. T show more learly the influen e of the foundation m ss parameter on dynamic analysis of the interaction between railway track-vehicle and foundation, the effects of the mass density of foundation on dynamic magnification factor (DMF) are investigated for various values of the velocity of the moving railway vehicle, plotted in Figs. 7-9. 0.50 75 1 00 25 50 0 40 80 120 160 200 (m/s) D M F s =0 =1 =2 =5 0.5 1.0 5 2.0 .5 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 7. The DMFs of vertical displacement of the railway track for various spring stiffness: (a) k= 108N/m2, (b) k= 5x108 N/m2 0.4 .6 .8 1.0 .2 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.7 1.0 .3 .6 .9 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 8. The DMFs of vertical displacement of the rail ay track for various shear layer stiffness: (a) ks= 107 N, (b) ks= 109 N 0.50 .75 1.00 .25 . 0 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 0.4 0.7 1.0 1.3 1.6 0 40 80 120 160 200 v (m/s) D M F s =0 =1 =2 =5 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation: (a) f= 1700 kg/m3, (b) f = 1900 kg/m3 (b) ρ f = 1900 kg/m 3 Fig. 9. The DMFs of vertical displacement of the railway track for the various mass density of foundation 4. CONCLUSIONS The influences of foundation mass density on dynamic analysis of railway track un- der moving railway vehicle are investigated by means of the finite element method. The railway track modeled as a Timoshenko beam subjected to a moving vehicle modeled as a two-axle mass-spring-damper system on a dynamic foundation model including linear elastic spring, shear layer, viscous damping and mass density of foundation. The railway track, vehicle and dynamic foundation are regarded as an integrated system and the gov- erning equation of motion of the system is solved by the step-by-step integration method. The parametric analysis has been performed to investigate the effects of mass density of dynamic foundation with various values of the stiffness of foundation and velocity of the vehicle on the dynamic analysis of the system. A comparison shows that the mass The influence of foundation mass on dynamic response of track-vehicle interaction 29 density of the dynamic foundation effects significantly on the dynamic response of the system in the range of high velocity. 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