Research on the application of the parameter freezing precise exponential integrator in vehicle-road coupling vibration

doi:10.1007/s10483-025-3220-7

Abstract

Abstract:

The vehicle-road coupling dynamics problem is a prominent issue in transportation, drawing significant attention in recent years. These dynamic equations are characterized by high-dimensionality, coupling, and time-varying dynamics, making the exact solutions challenging to obtain. As a result, numerical integration methods are typically employed. However, conventional methods often suffer from low computational efficiency. To address this, this paper explores the application of the parameter freezing precise exponential integrator to vehicle-road coupling models. The model accounts for road roughness irregularities, incorporating all terms unrelated to the linear part into the algorithm's inhomogeneous vector. The general construction process of the algorithm is detailed. The validity of numerical results is verified through approximate analytical solutions (AASs), and the advantages of this method over traditional numerical integration methods are demonstrated. Multiple parameter freezing precise exponential integrator schemes are constructed based on the Runge-Kutta framework, with the fourth-order four-stage scheme identified as the optimal one. The study indicates that this method can quickly and accurately capture the dynamic system's vibration response, offering a new, efficient approach for numerical studies of high-dimensional vehicle-road coupling systems.

Key words: vehicle-road coupled dynamics, dynamic response, parameter freezing precise exponential integrator, Newmark-β integration

2010 MSC Number:

O242

Yu ZHANG, Chao ZHANG, Shaohua LI, Shaopu YANG. Research on the application of the parameter freezing precise exponential integrator in vehicle-road coupling vibration. Applied Mathematics and Mechanics (English Edition), 2025, 46(2): 373-390.

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Figures/Tables 17

Fig. 1

Table 1

Vehicle, road and subgrade parameters"

Parameter	Unit	Numerical value	Parameter	Unit	Numerical value
$m 1$	kg	190	$L$	m	100
$m 2$	kg	10 109	$b$	m	6
$k 1$	$N ⋅ m − 1$	36 000 000	$h$	m	0.1
$k 2$	$N ⋅ m − 1$	9 000 000	$E$	GPa	$1.6 × 109$
$c 1$	$N ⋅ s ⋅ m − 1$	700	$ρ$	$kg ⋅ m − 3$	$2.5 × 103$
$c 2$	$N ⋅ s ⋅ m − 1$	80 000	$k r$	$N ⋅ m − 2$	$4.8 × 107$
			$c r$	$N ⋅ s ⋅ m − 2$	$3 × 105$

Table 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 2

Comparison of calculation results with different multiple time steps of Newmark-β method"

FPEI4-4 time step obtained $t$ /s	Multiple	Calculation time $t$ /s		Error level of the result
FPEI4-4 time step obtained $t$ /s	Multiple	FPEI4-4	Newmark- $β$	Error level of the result
0.001	10	338	242	Small
0.01	100	16	242	Small
0.025	250	7	242	Large

Table 2

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Table 3

The comparison of the calculation results of different modal truncation orders"

Modal truncation order $N M$	Calculation time $t$ /s		Error level of the result
Modal truncation order $N M$	FPEI4-4	Newmark- $β$	Error level of the result
50	2	26	Small
80	4	64	Small
100	7	104	Small
150	16	242	Small

Table 3

Fig. 11

Fig. 12

Fig. 13

Fig. 14

References 35

[1]	Editorial Department of China Journal of Highway and Transport. Summary of academic research on pavement engineering in China (in Chinese). China Journal of Highway and Transport, 37(3), 1–81 (2024)
[2]	MAHAJAN, G. R., RADHIKA, B., and BILIGIRI, K. P. A critical review of vehicle-pavement interaction mechanism in evaluating flexible pavement performance characteristics. Road Materials and Pavement Design, 23(4), 735–769 (2022)
[3]	SNEHASAGAR, G., KRISHNANUNNI, C. G., and RAO, B. N. Dynamics of vehicle-pavement system based on a viscoelastic Euler-Bernoulli beam model. International Journal of Pavement Engineering, 21(13), 1669–1682 (2020)
[4]	KHAVASSEFAT, P., JELAGIN, D., and BIRGISSON, B. Dynamic response of flexible pavements at vehicle-road interaction. Road Materials and Pavement Design, 16(2), 256–276 (2015)
[5]	FENG, G. Z., LI, S. H., and YANG, S. P. Coupling vibration analysis of hub motor driving electric vehicle and sandwich plate road. Applied Mathematical Modelling, 137, 115668 (2025)
[6]	SHAO, X. X., NAGHDY, F., DU, H., and QIN, Y. Coupling effect between road excitation and an in-wheel switched reluctance motor on vehicle ride comfort and active suspension control. Journal of Sound and Vibration, 433, 683–702 (2019)
[7]	LU, Z., HU, Z., YAO, H., LIU, J., and ZHAN, Y. An analytical method for evaluating highway embankment responses with consideration of dynamic wheel-pavement interactions. Soil Dynamics and Earthquake Engineering, 83, 135–147 (2016)
[8]	LI, Z., LI, X., LIU, B., and WANG, J. Influence of vehicle body vibration induced by road excitation on the performance of a vehicle-mounted piezoelectric-electromagnetic hybrid energy harvester. Smart Materials and Structures, 30(5), 055019 (2019)
[9]	LU, Y. J., YANG, S. P., and LI, S. H. Research on dynamics of a class of heavy vehicle-tire-road coupling system. Science China Technological Sciences, 54, 2054–2063 (2019)
[10]	ELNASHAR, G., BHAT, R. B., and SEDAGHATI, R. Modeling and dynamic analysis of a vehicle-flexible pavement coupled system subjected to road surface excitation. Journal of Mechanical Science and Technology, 33, 3115–3125 (2019)
[11]	MA, X., QUAN, W., DONG, Z., DONG, Y., and SI, C. Dynamic response analysis of vehicle and asphalt pavement coupled system with the excitation of road surface unevenness. Applied Mathematical Modelling, 104, 421–438 (2022)
[12]	KRISHNANUNNI, C. G. and RAO, B. N. Decoupled technique for dynamic response of vehicle pavement systems. Engineering Structures, 191, 264–279 (2019)
[13]	WANG, Y., CAO, J. D., HUANG, W., MA, T., and ZHOU, Z. Dynamic response analysis of vehicle-pavement-coupled system based on RIOHTrack full-size accelerated loading test. Construction and Building Materials, 402, 132744 (2023)
[14]	LI, S. H., YANG, S. P., and CHEN, L. Q. A nonlinear vehicle-road coupled model for dynamics research. Journal of Computational and Nonlinear Dynamics, 8(2), 021001 (2013)
[15]	GUO, Z. Q., WU, W., and YUAN, S. H. Longitudinal-vertical dynamics of wheeled vehicle under off-road conditions. Vehicle System Dynamics, 60(2), 470–490 (2022)
[16]	CHEN, H. Y., DING, H., and CHEN, L. Q. Study on Galerkin truncation convergence term in the vehicle-pavement system (in Chinese). Journal of Mechanical Engineering, 57(12), 31–39 (2021)
[17]	CHEN, H. Y., DING, H., LI, S. H., and CHEN, L. Q. The scheme to determine the convergence term of the Galerkin method for dynamic analysis of sandwich plates on nonlinear foundations. Acta Mechanica Solida Sinica, 34, 1–11 (2021)
[18]	ZHANG, J. N., YANG, S. P., LI, S. H., LU, Y. J., and DING, H. Influence of vehicle-road coupled vibration on tire adhesion based on nonlinear foundation. Applied Mathematics and Mechanics (English Edition), 42(5), 607–624 (2021) https://doi.org/10.1007/s10483-021-2724-6
[19]	ZHONG, W. X. and YANG, Z. S. On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem. Applied Mathematics and Mechanics (English Edition), 12(1), 49–54 (1991)
[20]	ZHONG, W. X. Precise time-history integration method for structural dynamic equation (in Chinese). Journal of Dalian University of Technology, 42(5), 607–624 (1994)
[21]	YAO, W. A., GAO, Q., TAN, S. J., and WU, F. Development and extended application of precise integration method (in Chinese). Chinese Journal of Computational Mechanics, 41(1), 2–25 (2024)
[22]	HUANG, Y. X., CUI, Y., and YANG, C. G. Adaptive time-stepping increment-dimensional precise integration method for solving structural dynamic equations (in Chinese). Journal of Vibration and Shock, 42(14), 198–203 (2023)
[23]	DENG, Z. C. and LI, Q. J. Precise exponential integrator and its application in dynamics of spacecraft formation flying (in Chinese). Acta Scientiarum Naturalium Universitatis Pekinensis, 52(4), 669–675 (2021)
[24]	ZHANG, Y., LI, S. H., and REN, J. Y. Parameter freezing precise exponential integrator and its application in nonlinear vehicle-bridge coupled vibration (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 56(1), 259–273 (2024)
[25]	CHEN, H. Y, DING, H., LI, S. H., and CHEN, L. Q. Convergent term of the Galerkin truncation for dynamic response of sandwich beams on nonlinear foundations. Journal of Sound and Vibration, 483, 115514 (2020)
[26]	DING, H., CHEN, L. Q., and YANG, S. P. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load. Journal of Sound and Vibration, 331, 2426–2442 (2012)
[27]	YOUNESIAN, D., HOSSEINKHANI, A., ASKARI, H., and ESMAILZADEH, E. Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications. Nonlinear Dynamics, 97(1), 853–895 (2019)
[28]	ZHANG, J. N., YANG, S. P., LI, S. H., DING, H., LU, Y. J., and SI, C. D. Study on crack propagation path of asphalt pavement under vehicle-road coupled vibration. Applied Mathematical Modelling, 101, 481–502 (2022)
[29]	XU, Y., ZHANG, B. Y., WANG, Y., ZHANG, X. L., and XU, X. Performance analysis and optimization for ISD suspension considering the vertical vibrational interaction of vehicle and bridge (in Chinese). Chinese Journal of Automotive Engineering, 6, 1–9 (2024)
[30]	FENG, G. Z., LI, S. H., and LU, Y. J. Analysis of electromechanical coupling vibration characteristics of electric vehicle-road system considering elastically supported boundary conditions (in Chinese). China Journal of Highway and Transport, 33(8), 81–91 (2020)
[31]	LAWSON, J. D. Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM Journal on Numerical Analysis, 4(3), 372–380 (1967)
[32]	ZHANG, H., YAN, J. Y., QIAN, X., and SONG, S. H. Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation. Applied Numerical Mathematics, 161, 372–390 (2021)
[33]	HUANG, J., JU, L., and XU, Y. Efficient exponential integrator finite element method for semilinear parabolic equations. SIAM Journal on Scientific Computing, 45(4), 1545–1570 (2023)
[34]	YAU, J. D., YANG, Y. B., and KUO, S. R. Impact response of high speed rail bridges and riding comfort of rail cars. Engineering Structures, 21(9), 836–844 (1999)
[35]	DING, H., YANG, Y., CHEN, L. Q., and YANG, S. P. Vibration of vehicle-pavement coupled system based on a Timoshenko beam on a nonlinear foundation. Journal of Sound and Vibration, 333, 6623–6636 (2014)