Bending Vibrations of an Elastic Rod Controlled by Piezoelectric Forces

Bending vibrations of a thin elastic rod of rectangular cross-section are studied. A number of piezoelectric actuators (elements) is symmetrically attached without gaps to two opposite sides of the rod. Each element is glued to the neighboring ones, forming with the rod a single elastic body in the form of a rectangular parallelepiped. The body is hinged at both ends relative to the cross-sectional axis parallel to the piezoelectric layers. In opposite piezoelements, homogeneous fields of normal stresses are set antisymmetrically as functions of time. These stresses are parallel to the axis of the rod and force the elastic system to perform bending motions. Within the framework of the linear theory of elasticity for the considered system, generalized formulations of the initial-boundary value problem and the corresponding eigenvalue problem are given. These problems are defined through unknown displacements and the time integrals of mechanical stresses. An approximation of the displacement and stress fields, which is polynomial in transverse coordinates, is proposed. This approximation exactly satisfies the homogeneous boundary conditions for stresses on the lateral sides and takes into account the symmetry properties of the bending motions. For the chosen approximation, the boundary value problem for eigenvalues is exactly solved. Two branches of eigenvalues are found and used to reduce the initial-boundary value problem to a countable system of first-order ordinary differential equations with respect to complex variables. The dynamical system is decomposed into independent infinite-dimensional subsystems with a scalar control input. One of these subsystems is not controllable. For the remaining subsystems, each corresponding to a pair of piezoelectric elements, a control law for vibration damping is proposed for a specific number of the lower modes associated with the lower branch.