Analytical solution for forced vibration of piezoelectrically actuated Timoshenko beam

Forced vibrations of a Timoshenko beam covered with a piezoelectric actuator on its top surface were investigated in this article. As the proposed beam model complied with Timoshenko beam theory, the effects of both rotary inertia and shear deformation were considered. Hamilton principle in conjunction with the Galerkin procedure were applied to derive the governing equation of motion resulting in a second-order ordinary differential equation in time. A sinusoidal electric voltage was applied to the piezoelectric actuator, and a spatially distributed harmonic mechanical force was exerted to the beam. The response of the system to the force stimulation gave an analytical relation between natural frequency and amplitude of the vibration. Using the obtained analytical relation, the effects of different factors and material properties including the modulus of elasticity of the piezoelectric layer and the piezoelectric coefficient on the vibrational response of the beam were examined. The results indicated that the piezoelectric layer as an actuator provided an effective tool for active control of vibration. Increasing the piezoelectric coefficient as well as the electric voltage applied on the piezoelectric actuator increased the amplitude of vibration, while the amplitude decreased by increasing the modulus of elasticity of the piezoelectric actuator. The results were also verified by finite element analysis.