Torsional vibration induced by gyroscopic effect in the modified couple stress based micro-rotors

Abstract

In this research, the small-scale effects in the torsional vibration of the micro-rotors with eccentric micro-disks are investigated based on the modified couple stress theory. The torsional deformation of the micro-shaft described by function φ(x,t) is considered to be independent of the flexural deformation described by functions v(x,t) and w(x,t). Using Hamilton's principle, the system of coupled nonlinear governing partial differential equations of motion and the associated boundary conditions are derived. The system of equations includes one corresponding to the torsional deformation and two others corresponding to the flexural deformation. By employing the Galerkin method, the system of governing equations is transformed into three coupled ordinary differential equations. The flexural-torsional nonlinear coupling due to the gyroscopic effect can excite the torsional mode. So, by utilizing the multiple scales method the non-resonant excitation and the torsional super-harmonic resonance are analytically investigated and in the special cases, the analytical results are compared with the numerical ones. Some parametric studies are presented to investigate the effects of various parameters, including the material length scale parameter, on the torsional natural frequency and the torsional super-harmonic resonance amplitude of the micro-rotors.