SEMI-ANALYTICAL TRANSIENT SOLUTIONS FOR STRONG NONLINEAR SYSTEMS EXCITED BY POISSON WHITE NOISE
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摘要: 自然界与工程中都普遍存在着随机扰动, 且大多数呈现出固有的非高斯性质, 若采用高斯激励建模可能会导致巨大的误差. 泊松白噪声作为一种典型且重要的非高斯激励模型, 已引起了广泛的关注. 目前, 泊松白噪声激励下系统的动态特性分析主要集中于稳态响应的研究, 而针对瞬态响应的求解难度仍较大, 需进一步发展. 本文引入径向基神经网络, 提出了一种泊松白噪声激励下单自由度强非线性系统瞬态响应预测的高效半解析方法. 首先将广义Fokker-Plank-Kolmogorov (FPK) 方程的瞬态解表示为一组含时变待定权值系数的高斯径向基神经网络; 然后采用有限差分法离散时间导数项, 并结合随机取样技术构造含时间递推式的损失函数; 最后通过拉格朗日乘子法使得损失函数最小化获得时变最优权值系数. 作为算例, 探究了两个经典强非线性系统, 并采用蒙特卡罗模拟方法对解析结果加以验证. 结果表明: 本文方法所获得的瞬时概率密度函数与蒙特卡罗模拟数据吻合地较好, 并且算法具备较高的计算效率. 在系统响应的整个演化过程中, 本文所提方法能够非常有效地捕捉到系统响应在各个时刻下的复杂非线性特征. 此外, 本文方法所获得的高精度半解析瞬态解, 不仅可作为基准解检验其他非线性随机振动分析方法的精度, 对于结构的优化设计也存在巨大的潜在应用价值.Abstract: Random perturbations are common in nature and engineering, and most of them exhibit inherent non-Gaussian properties. Thus, it may lead to huge errors if Gaussian excitation is used for modeling. As a typical and important non-Gaussian excitation model, Poisson white noise has attracted extensive attention. At present, the dynamic characteristic analysis of the system subjected to Poisson white noise is mainly focused on the study of the stationary response, while the solution of the transient response is still difficult and needs further development. In this paper, an efficient semi-analytical method based on radial basis function neural networks (RBF-NN) are proposed for transient response prediction of single-degree-of-freedom strong nonlinear systems under Poisson white noise excitation. Firstly, the transient solution of the generalized Fokker-Plank-Kolmogorov (FPK) equation is expressed as a set of Gaussian RBF-NN with unknown time-varying weight coefficients. Then, the finite difference method is applied to discretize and approximate the time derivative term, and the loss function with time recurrence is constructed by the random sampling technique. Finally, the time-varying optimal weight coefficients can be determined by minimizing the loss function through the Lagrange multiplier method. As examples, two classical strong nonlinear systems are investigated, and the solutions are validated by the Monte Carlo simulation (MCS) method. The results show that the transient probability density functions (PDFs) obtained by the proposed scheme agree well with the MCS data, and the algorithm has high computational efficiency. In the whole evolution process of the system response, the proposed scheme can effectively capture the complex nonlinear characteristics of the system response at each moment. Furthermore, the high precision semi-analytical transient solution obtained by the proposed scheme can not only be used as a benchmark to test the accuracy of other nonlinear random vibration analysis methods, but also has great potential application value for the structural optimum design.
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图 2 不同时刻系统 (26) 的边缘概率密度函数 (实线: 径向基神经网络解, 符号: 蒙特卡罗模拟结果)
Figure 2. Marginal PDFs of system (26) at different time (lines: RBF-NN solutions, symbols: MCS results)
图 5 不同中心域网格数对算法的均方根误差与计算时间的影响
Figure 5. The effect of different number of meshes in the central domain on the root mean square (RMS) error and CPU time of the algorithm
图 6 不同时刻系统 (27) 的边缘概率密度函数 (实线: 径向基神经网络解, 符号: 蒙特卡罗模拟结果)
Figure 6. Marginal PDFs of system (27) at different time (lines: RBF-NN solutions, symbols: MCS results)
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