Solving the transient response of the randomly excited dry friction system via piecewise RBF neural networks

Over the years, practical importance and interesting dynamical features have caused a growing interest in dry friction systems.Nevertheless, an effective approach to capture the non-smooth transition behavior of such systems is still lacking. Accordingly,we propose a piecewise radial basis function neural network(RBFNN) strategy to solve the transient response of the randomly excited dry friction system. Within the established framework, the transient probability density function of the dry friction system is expressed in a piecewise form. Each segment of the solution is expressed by the sum of a series of Gaussian activation functions with time-dependent weights. These time dependent weights are solved by minimizing the loss function, which involves the residual of the Fokker-Planck-Kolmogorov equations and constraint conditions. To avoid the singularity of the initial condition being a Dirac delta function, a short-time Gaussian approximation strategy is presented to solve the initiating time-dependent weights. Based on some numerical results, the proposed scheme effectively performs. Moreover, a comparison with other existing methods reveals that the proposed scheme can completely capture the nonlinear characteristic of the dry friction system stochastic response more closely. Noteworthy, we can easily extend the proposed method to other types of non-smooth systems with piecewise response characteristics. Moreover, the semi-analytical solution provides a valuable reference for system optimization.