Analyzing exact nonlinear forced vibrations of two-phase magneto-electro-elastic nanobeams under an elliptic-type force

The present paper deals with analyzing nonlinear forced vibrational behaviors of nonlocal multi-phase piezo-magnetic beam rested on elastic substrate and subjected to an excitation of elliptic type. The applied elliptic force may be presented as a Fourier series expansion of Jacobi elliptic functions. The considered multi-phase smart material is based on a composition of piezoelectric and magnetic constituents with desirable percentages. Additionally, the equilibrium equations of nanobeam with piezo-magnetic properties are derived utilizing Hamilton's principle and von-Kármán geometric nonlinearity. Then, an exact solution based on Jacobi elliptic functions has been provided to obtain nonlinear vibrational frequencies. It is found that nonlinear vibrational behaviors of the nanobeam are dependent on the magnitudes of induced electrical voltages, magnetic field intensity, elliptic modulus, force magnitude and elastic substrate parameters.