Non-linear vibrations of a beam with a pair of piezoceramic actuators

In this study, the non-linear transverse vibrations of a system consisting of a beam with two piezoelectric actuators symmetrically bonded to its top and bottom surfaces are investigated. Due to the immovable end conditions the vibrating system is stretched. This stretching is modified by the residual in-plane stresses generated by the actuators, which are under the same in-phase voltages. A version of the Lindstedt–Poincare method is used to acquire the approximate solutions to the problem. The natural frequencies are computed from the solution to the linear problem, which is described by the equations of the first power of the small parameter. The orthogonality condition is applied to find the first correction of the linear frequency. The non-linear natural frequency of the system is derived depending on the vibration amplitude. Finally, the obtained solutions are applied to a steel beam with two piezoceramic layers. As the mass and stiffness of the piezoelectric elements are factors which affect the structure of the system, their influence on the natural linear frequency and the amplitude–frequency relationship is discussed. The main results concern the effect of the residual piezoforce on the non-linear vibration of the system.