Analytical solution for nonlinear forced response of a viscoelastic piezoelectric cantilever beam resting on a nonlinear elastic foundation to an external harmonic excitation

The nonlinear free and forced vibrations of viscoelastic cantilevers with a piecewise piezoelectric actuator layer bonded on the top surface and resting on a nonlinear elastic foundation are analyzed. The cantilever is an Euler–Bernoulli beam and its viscoelastic property complies with Kelvin–Voigt model. The equation of motion is obtained by using the Hamilton principle and then by employing the Galerkin procedure, the governing equation of motion is reduced to a second order nonlinear ordinary differential equation in time. The nonlinearities of the system appear in stiffness, inertia and damping terms and are arisen from nonlinear stiffness of the elastic foundation and piezoelectricity, viscoelasticity, and geometry of the structure. In forced vibrations, a harmonic transverse mechanical load is applied to the structure and the primary resonance of the system is investigated. The Multiple Time Scales perturbation method is used to perform the nonlinear dynamical analysis. Analytical expressions for nonlinear natural frequency, amplitude of free vibration and frequency response of the system are presented. The effects of various parameters including linear, nonlinear, and shearing stiffness of the elastic foundation, damping coefficient, length of the piezoelectric layer, and the piezoelectric coefficient on the free and forced nonlinear responses of the system are discussed in detail.