Nonlocal nonlinear chaotic and homoclinic analysis of double layered forced viscoelastic nanoplates

Abstract

The homoclinic phenomena and chaotic motions of the forced double layered nanoplates (DLNP) are investigated. By the nonlocal theory, the nonlinear equations of motion of DLNP subjected to transverse harmonic excitation are established. The buckled DLNP considered herein means that the parameter regime of the first mode are always unstable. It should be emphasized that the homoclinic Melnikov function is related to the nonlinear term which is affected by the boundary conditions. Hence two different boundary conditions, i.e. simply supported with movable and immovable edges are compared herein. The extended Melnikov method is employed to discuss the homoclinic phenomena and chaotic motions of the DLNP system. The criteria for the homoclinic motions of the four buckling cases are established. Then the results by the above global perturbation analyses are verified by the numerical integration evidences including Lyapunov exponential spectrums, Fourier spectrums and Poincaré sections. The influences of the structural parameters such as small scale effect and boundary conditions on the homoclinic behaviors are mainly discussed. From the results, the most remarkable fact can be seen that the rotary inertia term could break the symplectic symmetry of the unperturbed vibration system. The parametric regime where the chaotic motion may appear shrink with the increase of the nonlinear term r1, which means chaotic motion more likely appear for immovable edges than those with movable edges. Parametric regime where the transverse homoclinic phenomenon appears will decrease with the increase of the small scale parameter. And the homoclinic phenomena more likely appear in higher mode vibration.