The response of the train–bridge system has an obvious random behavior. A high traffic density and a long maintenance period of a track will result in a substantial increase in the number of trains running on a bridge, and there is small likelihood that the maximum responses of the train and bridge happen in the total maintenance period of the track. Firstly, the coupling model of train–bridge systems is reviewed. Then, an ensemble method is presented, which can estimate the small probabilities of a dynamic system with stochastic excitations. The main idea of the ensemble method is to use the NARX (nonlinear autoregressive with exogenous input) model to replace the physical model and apply subset simulation with splitting to obtain the extreme distribution. Finally, the efficiency of the suggested method is compared with the direct Monte Carlo simulation method, and the probability exceedance of train responses under the vertical track irregularity is discussed. The results show that when the small probability of train responses under vertical track irregularity is estimated, the ensemble method can reduce both the calculation time of a single sample and the required number of samples.

火车-桥梁系统的响应具有明显的随机行为。高的交通密度和较长的轨道维护周期将导致桥上行驶的火车数量大量增加，并且火车和桥梁的最大响应发生在铁路总维护周期的可能性很小。跟踪。首先，回顾了火车-桥梁系统的耦合模型。然后，提出了一种集成方法，该方法可以估计具有随机激励的动态系统的小概率。集成方法的主要思想是使用NARX（带有外生输入的非线性自回归）模型代替物理模型，并应用带有分裂的子集模拟来获得极限分布。最后，将所提方法的效率与直接蒙特卡罗模拟方法进行了比较，并讨论了垂直轨道不平顺下列车响应的概率超标问题。结果表明，当估计垂直轨道不平顺下列车响应的可能性很小时，集成方法既可以减少单个样本的计算时间，又可以减少所需样本数。