Viscoelasticity has been widely studied over the past few decades as a combination of the viscous effects of energy dissipation and the elastic effects of energy storage. As an application, viscoelastic dampers are commonly used to suppress the dynamic behaviour of structures. However, in an actual physical environment, it is difficult to explore the stationary probability densities of random responses for viscoelastic systems. For this purpose, we present a numerical method to investigate the dynamic behaviour of a generalized Maxwell-type viscoelastic system under harmonic and Gaussian white noise excitations. Using approximate equivalent and stochastic averaging, we establish an averaged Itô differential equation for the amplitude of the system. For primary external resonance, we solve the reduced Fokker-Planck-Kolmogorov equation by using the successive over-relaxation technique combined with a finite difference method. Through Monte Carlo simulations, we verify the applicability, accuracy and efficiency of the proposed methodology and demonstrate that viscoelasticity has a significant impact on the dynamic behaviour of the viscoelastic systems. This work reveals the remarkable influences of viscoelasticity, excitation intensities, linear and nonlinear damping and linear and nonlinear stiffness on the probability density functions of system responses, which can guide the selection of materials, stiffnesses and structures in viscoelastic damper design.

This paper implements the parameter identification of ship roll motion based on vibration tests and the computational fluid dynamics (CFD) method. By applying sinusoidal moment excitations from a shaker, we successfully simulate the quasi-roll motion behavior of the ship model in the air. We conduct two vibration tests, including a constant-amplitude angular displacement test and a constant-amplitude angular velocity test, to identify the restoring moment and the damping moment coefficients of the quasi-roll motion. In addition, we perform the approximate equivalence of the fluid around the ship through the CFD method and the fluid-solid coupling technique. Thus, we obtain the hydrodynamic coefficients, such as the added mass moment of inertia and the equivalent damping, to accomplish the parameter identification of the ship's real roll motion. Finally, the accuracy of the identified parameters is verified by means of numerical simulations based on the Runge-Kutta method.