Micro Cantilever Beam Theory for Transverse Dynamics Using a Continuum Mechanics Model

Abstract

The vibrational characteristics of cantilever beams with initial axial tension were studied using a nonlocal continuum Euler-Bernoulli beam model. Small size effects are essential to nanotechnology and it can not be ignored in micro or nano scale. Nonlocal elasticity theory has been proved to work well in nanomechanics and it is considered into the governing equation which can be transformed into a fourth-order ordinary differential equation together with a dispersion relation. Boundary conditions are applied so as to determine the analytical solutions of vibrational mode shape and transverse deformation through a numerical method. Relations between natural frequency and the small scale parameter are obtained, including the fundamental and the second order frequencies. It is found that both the small scale parameter and dimensionless initial axial tension play remarkable roles in dynamic behaviors of micro cantilever beams and their effects are analyzed and discussed in detail.