On the Internal Resonances of Size-Dependent Clamped–Hinged Microbeams: Continuum Modeling and Numerical Simulations

Abstract

The nonlinear forced vibrations of size-dependent clamped–hinged microbeams to fundamental excitations of respectively the first mode and second mode in the presence of three-to-one internal resonance are investigated both analytically and numerically for the first time. Equation of motion incorporating large deflection, along with the symmetric part of couple stress is derived by virtue of Hamilton’s principle. Utilizing Galerkin discretization with the first two modes, the discretized ordinary differential equations (ODEs) are then handled analytically with multi-dimensional Lindstedt–Poincaré (MDLP) method. The frequency–response relationships in the fundamental resonance for the first mode and the second mode are presented as well as compared with the classical solution. Results reveal that the size-dependent internal resonances are significantly different from the classical situation whenever it is of the first mode or the second mode. Furthermore, the influences of material length scale parameter, excitation force and damping on the performance of nonlinear system are discussed for fundamental excitation of the first order. The frequency–response relationships are illustrated for the first two modes in each case. Moreover, numerical modelings are conducted to compare to the analytical solutions. The numerical results fully support the analytical predictions. Also, simulations indicate the appearance of chaos under relatively large excitation force whether it is the vibration of the first mode or the second mode.