Spectral element analysis of a non-classical model of a spinning micro beam embedded in an elastic medium

A mathematical model, the formulation of which is premised on the higher-order modified couple stress theory, is presented for the free vibration analysis of a circular doubly-symmetric spinning micro beam embedded in an elastic medium. Based on the higher-order constitutive model, the extended Hamiltonʹs principle is applied to the size-dependent energy expressions to arrive at the coupled elastodynamics governing equations of the micro beam in two orthogonal planes. The spectral element method (SEM) is then used for detailed frequency analyzes of the micro beam in the presence of the material length scale, Poissonʹs ratio and an elastic medium of the Winkler-type. The accuracy of the SEM predictions is numerically ascertained by contrasting with exact solutions of the classical model. Numerical results from the frequency analyzes reveal the doubling of the forward critical speed of the spinning micro beam under a range of the size effect. Parametric studies through an applied statistical tool of Pareto charts demonstrate the existence of interaction effects under nonlinear variations of the material length scale, the Poissonʹs ratio, the elastic foundation stiffness and the spinning rate.