Viscoelastically coupled size-dependent dynamics of microbeams

This paper investigates the viscoelastically coupled size-dependent dynamics of a microbeam in the framework of the modified couple stress theory. The elastic and viscous components of the stress and deviatoric part of the symmetric couple stress tensors are obtained employing the Kelvin–Voigt viscoelastic model. The size-dependent elastic potential energy is developed based on the modified couple stress theory. The work of the viscous components of the stress and deviatoric part of the symmetric couple stress tensors are formulated in terms of the system parameters. The work of an external excitation force as well as the kinetic energy of the system is obtained as functions of the displacement field. Hamilton's principle is employed, yielding the nonlinear equations for the longitudinal and transverse motions of the viscoelastic microbeam. A weighted-residual method is then applied to the nonlinear equations of motion resulting in a high-dimensional reduced-order system with finite degrees of freedom. This high-dimensional model is solved via use of a continuation method as well as a direct time-integration technique. The viscoelastically size-dependent nonlinear frequency- and force-responses are constructed and the effects of the simultaneous presence of viscoelastic energy dissipation mechanism and the length-scale parameter are examined.