Longitudinal vibration analysis for microbars based on strain gradient elasticity theory

TLDR

The longitudinal free vibration of micro‑scaled bars is formulated within strain gradient elasticity theory. The study investigates how additional length‑scale parameters affect vibration frequencies. The authors derive a higher‑order equation of motion via Hamilton’s principle and solve it for clamped‑clamped and clamped‑free microbars to assess length‑scale effects. The results show that size effects dominate when the diameter‑to‑length‑scale ratio is small, and the discrepancy between strain‑gradient and classical frequency predictions grows with lower slenderness ratios and higher vibration modes.

Abstract

The longitudinal free vibration problem of a micro-scaled bar is formulated using the strain gradient elasticity theory. The equation of motion together with initial conditions, classical and non-classical corresponding boundary conditions for a micro-scaled elastic bar is derived via Hamilton’s principle. The resulting higher-order equation is solved for clamped-clamped and clamped-free boundary conditions. Effects of the additional length scale parameters on the frequencies are investigated. It is observed that size effect is more significant when the ratio of the microbar diameter to the additional length scale parameter is small. It is also observed that the difference between natural frequencies predicted by current and classical models becomes more prominent for both lower values of slenderness ratio of the microbar and for higher modes.