Analytical Solution for Beam Dynamics Using Modified Couple Stress Theory

The dynamic behavior of Bernoulli–Euler beams is analytically investigated using the Modified Couple Stress Theory (MCST). Governing equations, initial conditions, and boundary conditions are derived using Hamilton’s principle. Two dynamic boundary-value problems—simply supported and cantilever beams—are solved to quantify the influence of microstructural size effects on natural frequencies. Results show that natural frequencies predicted by MCST are significantly higher than those predicted by classical beam theory when the characteristic structural dimension is comparable to the internal material length-scale parameter. As the size ratio increases, MCST predictions converge to classical solutions. These findings highlight the critical role of size-dependent stiffness in micro- and nano-scale beam applications such as MEMS devices.