Coupling of nonlinear and scale effects in the vibration response of axially moving microbeams

Abstract Microbeams are essential components for resonance, support, and energy conversion in micro/nano electromechanical systems. To address the growing demands for high performance and sensitivity in these devices, a comprehensive understanding of their vibration responses is crucial. Based on the nonlocal strain gradient theory, this paper investigates a scale-dependent moving microbeam and derives two distinct nonlinear dynamic equations by coupling the axial force with the transverse displacement. The direct multi-scale method is employed to analyze various vibration behaviors, including parametric resonance, forced vibration, and internal resonance. Special attention is paid to the influence of two scale parameters and the differences between the two nonlinear models. Numerical results reveal that the nonlocal parameter and material characteristic length parameter exhibit distinct damping weakening and strengthening effects in vibration responses. In the case of linear parametric vibration, a critical condition exists where their numerical values are equal, resulting in mutual cancellation; however, this phenomenon does not occur in the nonlinear case. For forced vibration of microbeams, scale parameters significantly influence the jump phenomenon in amplitude-frequency responses and the degree of model nonlinearity. Under 1:3 internal resonance conditions, a coupling region forms between the first- and second-order modes, with scale parameters significantly influencing internal resonance. The nonlocal parameter suppresses internal resonance, while the material characteristic length parameter promotes it.