On the free and forced vibration of a sandwich beam resting on a viscoelastic foundation

The introduction of porosities within materials enables structural components to achieve enhanced performance in practice, particularly under conditions requiring sound insulation, weight reduction, or thermal protection. In this study, a weighted averaging approach combined with the equivalent linearization method is utilized to obtain closed-form expressions for the nonlinear static bending, nonlinear free vibration, and nonlinear stochastic forced vibration responses of multilayer beams with both macro- and micro-scale porosities. The distinctive and compelling aspect of this work lies in the analysis of nonlinear forced vibration under random excitation for beams resting on a viscoelastic foundation. The governing equations are formulated based on an improved high-order shear deformation theory and the principle of virtual displacements, resulting in the equilibrium equations for the structure. Based on the detailed numerical results, several noteworthy phenomena have been identified. In particular, material porosity reduces both the mass and stiffness of the beam, leading to an increase in the nonlinear static deflection, while the nonlinear natural frequency decreases. However, the nonlinear dynamic response is found to depend on the specific type of sandwich configuration. Moreover, as the vibration amplitude increases, the natural frequency exhibits a nonlinear increasing trend. These findings provide valuable insights for the engineering design of sandwich beams.