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

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.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.01-2017.23.
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