Dynamic behaviours of spinning pre-twisted Rayleigh micro-beams

Abstract

Developments in micro-systems’ operations rely on identifying and controlling the dynamic responses of micrometre-scale elements. Pre-twisted micro-elements feature prominently in these systems, yet very little is known about the interacting effect of motley factors on their behaviours. Presented here is a model of pre-twisted micro-beams that accounts for the coupling of scale-dependent, rotary inertia and spinning effects in vibratory and wave propagation analyses. In tackling the problem, the system’s physical domain is transformed into the mathematical realm via a scale-dependent micro-continuum theory, while its time evolution is captured by Hamiltonian mechanics. Analytic expressions are obtained for the long-wavelength limit and dispersion relations. The spectrum relations of the waves are established from an octic characteristics equation (which, being in violation of the Abel–Ruffini theorem is treated numerically). Splitting of the waves within the system is observed, and the frequencies of the split shift for altered values of the pre-twist angle and rotary inertia effect. Thinner, highly pre-twisted micro-scale beams experience a widening (narrowing) of the difference between the two phase speeds (group velocities) of the waves. Further, pre-twisting lowers the natural frequencies associated with odd-numbered modes of vibration of the element, while the small-scale effect strongly affects the higher vibration modes.

Keywords:

• Pre-twisted micro-scale beam theory
• elastic wave propagation
• spectrum curve
• spinning dynamics
• Rayleigh beam theory
• Rayleigh–Ritz method
• modified couple stress theory