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.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.01-2017.23.
REFERENCES
[1] E. Winckler. Die lehre von elastizitat und festigkeit. Dominicus, Prague, (1867).
[2] D. Younesian, Z. Saadatnia, and H. Askari. Analytical solutions for free oscillations of beams
on nonlinear elastic foundations using the variational iteration method. Journal of Theoretical
and Applied Mechanics, 50, (2), (2012), pp. 639–652.
[3] R. U. A. Uzzal, R. B. Bhat, and W. Ahmed. Dynamic response of a beam subjected to mov-
ing load and moving mass supported by Pasternak foundation. Shock and Vibration, 19, (2),
(2012), pp. 205–220. https://doi.org/10.1155/2012/919512.
[4] K. Q. Do and T. C. Nguyen. Dynamic response of plate on visco-elastic foundation consid-
ering the mass of moving object. In International Symposium on Dynamics and Control, Hanoi,
(2012), pp. 215–227.
[5] J. S. Kim and M. K. Kim. The dynamic response of an Euler-Bernoulli beam on an elastic
foundation by finite element analysis using the exact stiffness matrix. Journal of Physics: Con-
ference Series, 382, (1), (2012). https://doi.org/10.1088/1742-6596/382/1/012008.
[6] D. K. Nguyen, T. H. Trinh, and G. B. Sthenly. Post-buckling response of elastic-plastic beam
resting on an elastic foundation to eccentric axial load. The IES Journal Part A: Civil & Struc-
tural Engineering, 5, (1), (2012), pp. 43–49. https://doi.org/10.1080/19373260.2012.652769.
[7] L. Wang, J. Ma, J. Peng, and L. Li. Large amplitude vibration and parametric instability of
inextensional beams on the elastic foundation. International Journal of Mechanical Sciences, 67,
(2013), pp. 1–9. https://doi.org/10.1016/j.ijmecsci.2012.12.002.
[8] S. B. Cos¸kun, B. O¨ztu¨rk, and U. Mutman. Adomian decomposition method for vibration of
nonuniform Euler beams on elastic foundation. In Proceedings of the 9th International Confer-
ence on Structural Dynamics, EURODYN 2014, Porto, Portugal, (2014), pp. 1935–1940.
[9] P. C. Jorge, F. M. F. Simo˜es, and A. P. Da Costa. Dynamics of beams on non-uniform non-
linear foundations subjected to moving loads. Computers & Structures, 148, (2015), pp. 26–34.
https://doi.org/10.1016/j.compstruc.2014.11.002.
[10] D. Froio, R. G. Moioli, and E. Rizzi. Numerical dynamic analysis of beams on nonlinear
elastic foundations under harmonic moving load. Eccomas Proceedia, 2149, (2016), pp. 4784–
4809. https://doi.org/10.7712/100016.2149.7515.
[11] I.-B. Teodoru. Beams on elastic foundation the simplified continuum approach. Buletinul In-
stitutului Politehnic din lasi. Sectia Constructii, Arhitectura, 55, (4), (2009), pp. 37–45.
[12] M. M. Filonenko-Borodich. Some approximate theories of elastic foundation. Uchenyie Za-
piski Moskovkogo Gosudarstuennogo Universiteta Mekhanika, Moscow, 46, (1940), pp. 3–18. (in
Russian).
[13] M. Hete´nyi. Beams on elastic foundation: theory with applications in the fields of civil and mechani-
cal engineering. University of Michigan Press, Ann Arbor, (1946).
[14] M. Hete´nyi. Beams on elastic foundation. University of Michigan Press, (1950).
[15] P. L. Pasternak. On a new method of analysis of an elastic foundation by means of two constants.
Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvui Arkhitekture, (1954). (in Russian).
30 Phuoc T. Nguyen, Trung D. Pham, Hoa P. Hoang
[16] E. Reissner. A note on deflections of plates on a viscoelastic foundation. Journal of Applied
Mechanics, 25, (1958), pp. 144–145.
[17] E. Reissner. Note on the formulation of the problem of the plate on an elastic foundation.
Acta Mechanica, 4, (1), (1967), pp. 88–91. https://doi.org/10.1007/bf01291090.
[18] A. D. Kerr. Elastic and viscoelastic foundation models. Journal of Applied Mechanics, 31, (3),
(1964), pp. 491–498. https://doi.org/10.1115/1.3629667.
[19] A. D. Kerr. A study of a new foundation model. Acta Mechanica, 1, (2), (1965), pp. 135–147.
https://doi.org/10.1007/bf01174308.
[20] V. Z. Vlasov and U. N. Leont’ev. Beams, plates and shells on elastic foundation. Israel Program
for Scientific Translation, (1966).
[21] R. Jones and J. Xenophontos. The Vlasov foundation model. International Journal of Mechanical
Sciences, 19, (6), (1977), pp. 317–323. https://doi.org/10.1016/0020-7403(77)90084-4.
[22] K. Ozgan. Dynamic analysis of thick plates including deep beams on elastic foun-
dations using modified Vlasov model. Shock and Vibration, 20, (1), (2013), pp. 29–41.
https://doi.org/10.1155/2013/856101.
[23] D. T. Pham, P. H. Hoang, and T. P. Nguyen. Dynamic response of beam on a new foundation
model subjected to a moving oscillator by finite element method. In 16th Asia Pacific Vibration
Conference, Hanoi, Vietnam, (2015), pp. 244–250.
[24] T. P. Nguyen, D. T. Pham, and P. H. Hoang. A new foundation model for dynamic analysis of
beams on nonlinear foundation subjected to a moving mass. Procedia Engineering, 142, (2016),
pp. 166–173. https://doi.org/10.1016/j.proeng.2016.02.028.
[25] P. T. Nguyen, T. D. Pham, and H. P. Hoang. A dynamic foundation model for the analysis
of plates on foundation to a moving oscillator. Structural Engineering and Mechanics, 59, (6),
(2016), pp. 1019–1035. https://doi.org/10.12989/sem.2016.59.6.1019.
[26] D. T. Pham, P. H. Hoang, and T. P. Nguyen. Dynamic response of beam on a new non-uniform
dynamic foundation subjected to a moving vehicle using finite element method. IJERT, 6, (3),
(2017), pp. 279–285. https://doi.org/10.17577/ijertv6is030244.
[27] T. D. Pham, P. H. Hoang, and T. P. Nguyen. Experiments on influence of foundation mass
on dynamic characteristic of structures. Structural Engineering and Mechanics, 65, (5), (2018),
pp. 505–511.
[28] P. Lou. A vehicle-track-bridge interaction element considering vehicle’s pitch-
ing effect. Finite Elements in Analysis and Design, 41, (4), (2005), pp. 397–427.
https://doi.org/10.1016/j.finel.2004.07.004.
[29] S. H. Ju. Finite element investigation of traffic induced vibrations. Journal of Sound and Vibra-
tion, 321, (3-5), (2009), pp. 837–853. https://doi.org/10.1016/j.jsv.2008.10.031.
[30] T. Yokoyama. Vibration analysis of Timoshenko beam-columns on two-parameter
elastic foundations. Computers & Structures, 61, (6), (1996), pp. 995–1007.
https://doi.org/10.1016/0045-7949(96)00107-1.
[31] H. Matsunaga. Vibration and buckling of deep beam-columns on two-parameter
elastic foundations. Journal of Sound and Vibration, 228, (2), (1999), pp. 359–376.
https://doi.org/10.1006/jsvi.1999.2415.
[32] H. Ding, K. L. Shi, L. Q. Chen, and S. P. Yang. Dynamic response of an infinite Timoshenko
beam on a nonlinear viscoelastic foundation to a moving load. Nonlinear Dynamics, 73, (1-2),
(2013), pp. 285–298. https://doi.org/10.1007/s11071-013-0784-0.

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