The Transient Probability Density of a SMA Three-Potential-Well Oscillator Under Random Excitation

This study presents a stochastic dynamical model of a shape memory alloy (SMA) triple-potentialwell oscillator under Gaussian white noise excitation, which is based on Falk polynomial constitutive relations. The corresponding Fokker – Planck – Kolmogorov (FPK) equations for the system were derived, and the transient probability density was numerically solved via a Gaussian radial basis function neural network (RBFNN). This study systematically investigated the effects of cubic and quintic nonlinear stiffness, linear damping, and noise intensity on the evolution of the marginal probability density and joint probability density and verified the accuracy of the method via Monte Carlo simulation (MCS). The results indicate that the nonlinear coefficients primarily regulate the multipeak morphology and transient asymmetry of the displacement probability density by altering the symmetry of the drift field; damping determines the rate at which the probability density converges to the steady state, whereas noise intensity controls the transition efficiency between potential wells and the extent of distribution diffusion. The RBFNN method showed excellent agreement with the MCS results, with a small root-mean-square error, demonstrating the high accuracy and reliability of this method. This study revealed the multiparameter stochastic dynamical control mechanism of the SMA three-potential-well system, providing a theoretical basis and methodological support for smart structural design, vibration energy harvesting, and weak signal detection.