Semi-analytical solution to the steady-state periodic dynamic response of an infinite beam carrying a moving vehicle

Vehicle-induced vibration is among the most important aspects for riding comfort and structural durability. It is well known, however, that the span of structure is much larger than the size of vehicle, resulting in a huge computational cost for dynamic analysis, and meanwhile, the multi-degree-of-freedom (MDOF) vehicle and the nonlinear interaction between beam and vehicle are difficult to be involved in an explicit solution. In the present work, a joint harmonic balance and Fourier transform (HBFT) method is proposed for the steady-state periodic dynamic response of an infinite beam carrying a vehicle taking into account the surface roughness and the nonlinear interaction between beam and vehicle, and the semi-analytical steady-state periodic (SASP) solution is then obtained. First, the vehicle and beam are separated as two subsystems, where the infinite beam is supported by a viscoelastic foundation and the vehicle is described by a MDOF discrete model. The response of the beam is analytically solved from the Fourier transform approach and the response of the vehicle is explicitly formulated considering the Fourier expansion of the nonlinear interaction forces. Next, the Fourier coefficients for such interaction forces are numerically solved from the harmonic balance method based on the compatibility between the subsystems. By substituting the solved interaction forces back into the closed-form analytical solutions, the responses of both the subsystems are evaluated explicitly. This allows for accurate and efficient analysis for the steady-state periodic response of vehicle–beam system considering the infinite beam, the discrete vehicle and the nonlinear interaction between them. After validating the SASP solution by comparing with the assumed modal method, the effect from the vehicle speed on the response of the vehicle–beam system is discussed, indicating the existence of a few resonant speeds, having potential value for the optimal design of vehicles and road pavements. Apart from the present example, the SASP solution can be applied in broader cases and the proposed HBFT method is generally feasible to a variety of dynamic coupling problems consisting of a continuum and a discrete model.