This paper presents a new optimization method for coupled vehicle–bridge systems subjected to uneven road surface excitation. The vehicle system is simplified as a multiple rigid-body model and the single-span bridge is modeled as a simply supported Bernoulli–Euler beam. The pseudo-excitation method transforms the random surface roughness into the superposition of a series of deterministic pseudo-harmonic excitations, which enables convenient and accurate computation of first and second order sensitivity information. The precise integration method is used to compute the vertical random vibrations for both the vehicle and the bridge. The sensitivities are used to find the optimal solution, with vehicle ride comfort taken as the objective function. Optimization efficiency and computational accuracy are demonstrated numerically.
Recently, dynamic analysis of coupled vehicle–bridge systems has received much attention because it relates to many engineering fields, e.g. interaction forces, dynamic effects and bridge design [1]. For such systems, it is well known that road irregularity is the most important factor causing random vibration. Analysis of such random vibration has long been regarded as very difficult. Optimum design is even more difficult but it is a matter of great concern and significance.
Fryba [2] pointed out that irregularities sometimes had a very appreciable random component for moving loads and for structures associated with transport, Silva [3] regarded the pavement roughness as a probabilistic model in order to investigate the dynamic behavior of a reinforced concrete highway bridge deck crossed by a heavy train and Xia [4] took the track irregularity as a major excitation when analysing the vibrations of train–bridge coupled systems. Vibration analysis of such coupled systems could be undertaken by conventional random vibration theory approaches [5], but unfortunately these are generally too complicated and inefficient. However very few research papers have been published which are relevant to efficient solution, which is the topic of this paper.
In general, difficulties mainly arise in the two areas identified in, respectively, the next two paragraphs.
Conventional methods for random vibration analysis take one or more sample curves as time-history input functions of road irregularity when estimating the random response of vehicle–bridge systems by means of time-history analysis and statistical processing [6], [7] and [8]. This approach is obviously quite complicated and costly and so becomes particularly unacceptable for optimization problems because many re-analyses are required. Much work has been done to avoid costly computation, but unfortunately this results in limited accuracy and reliability because very few time-history curve samples are used in the analysis and optimization.
Any conventional numerical integration method, e.g. the Newmark method, requires the time-step size to be very small. This is because the vehicle is assumed to “jump” suddenly from one point to the next point at intervals, during which the magnitude of the contact force remains unchanged. However, in reality all wheels move continuously and every contact force may vary within each time step. Therefore significant errors will arise unless the time step is so small that it results in excessive computation costs.
To overcome the difficulties identified in the previous two paragraphs, the pseudo-excitation method (PEM) [9] and [10] has been applied to considerably simplify the solution of the dynamic equations by transforming the random surface roughness into the superposition of a series of deterministic pseudo-harmonic surface unevennesses. The precise integration method (PIM) [11] and [12] has also been extended to simulate the continuous variation within each time step of the magnitudes and positions of the contact forces. Hence the uniformly modulated, multi-point, different-phase, non-stationary random excitations of the road acting on the wheels are transformed into a deterministic pseudo-excitation vector by using extended PEM with the phase-lags between the wheels taken into account, so that the solution can be obtained efficiently by means of PIM.
In general, the dynamic optimization is efficient based on sensitivity information, but sensitivity analysis of random dynamic problems is more difficult. It is known that the derivatives of eigenvalues and eigenvectors have been studied by many scholars [13], [14] and [15]. The MISA method [16] was proposed for optimizing structures subjected to dynamic stress and displacement constraints. The least-squares iteration method has been used to compute the first-order eigenvalue sensitivity and sometimes the second-order sensitivity [17]. Sensitivity analysis of structures with transient dynamic load under stress constraints was investigated by Durbin [18]. The optimization design of a wing structure excited by random gust loads was investigated by Rao [19] with the sensitivity formulas of mean-square responses being derived. A matrix perturbation method [20] for sensitivity analysis of structural dynamic responses was developed. Some methods for dynamic optimization designs of structures have been discussed and compared [21]. Unfortunately, relatively few papers relevant to the algorithm and application of second order flexibility, particularly with respect to random vibration, can be found in the literature. In fact, the first-order sensitivity analysis only indicates the local optimal direction, whereas the second-order sensitivity analysis is sometimes quite useful for obtaining the global optimal solution. Although the Hessian matrix is a well-known means for calculating general second-order flexibilities, the enormous effort required by it seriously restricts its practical applications. These sensitivity analysis method are both based on conventional inefficient random vibration methods, and are naturally ineffective as well.
Our paper presents a new method for first and second order sensitivity analyses of structural random responses, which is based on PEM–PIM and on previous optimality research work [22]. The method replaces the right-hand side random excitation of the random equations of motion by pseudo-excitations, in order to enable the convenient and accurate derivation of various first and second order sensitivity formulae. Numerical examples demonstrate the correctness and high efficiency of this new method, which is then used in the optimum design of a bus, with ride comfort as the objective function.
