Analytical Solution of Non-Linear Free Vibration of Thin Rectangular Plates with Various Boundary Conditions Based On Non-Local Theory

This article deals with the small-scale effect on the nonlinear free vibration of isotropic thin nano-plate using the nonlocal elasticity plate theory. The formulations are based on the Kirchhoff's plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. To include the small scale and the geometrical nonlinearity effects, the governing differential equations are derived based on the nonlocal elasticity theory in conjunction with the von Karman geometrical model in which the effects of rotary inertia and transverse shear are neglected. With cubic non-linearities, Duffing's equation is solved by elliptic integral and natural frequencies are obtained. Also by means of Jacobi elliptic functions, some analytical solutions for deflection of plate are presented. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The effects of various parameters on the nonlinear vibrations of nanoplates are presented. Occurrence probability of internal resonance in rectangular nanoplate is investigated.