Two-step data-driven identification of probability densities for random vibrating systems with implicit Hamiltonian functions

Nonlinear random vibration is a common phenomenon, and predicting its probability density is an essential component of vibration engineering. This paper proposes a data-driven method for identifying explicit expressions of response probability densities in random vibrating systems with implicit Hamiltonian functions. The process concludes with two steps, identifying the Hamiltonian function and calculating the probability density of the stationary response. The former uses the differential equations of the motion of the quasi-Hamiltonian systems to identify Hamiltonian functions from the simulated data, while the latter estimates the logarithm of the probability density from the identified Hamiltonian functions and acquires an explicit expression. Their unknown coefficients can be attributed to the solution of a set of undetermined equations. The proposed method is applied to systems in which the Hamiltonian functions cannot be simply derived, such as those with complicated stiffness. Two examples are presented to demonstrate the applicability and effectiveness of our method, i.e., the nonlinear vibration energy harvester (VEH) and the Duffing oscillator with LuGre friction. The proposed technique outperforms Monte Carlo simulations (MCSs) in efficiency. The results show that our method is insensitive to parameters and can be used for identifying transient probability density. Its application scope is wider than the stochastic averaging method.