Shape control of laminated cantilevered beams with piezoelectric actuators

Treated herein is the shape control of a composite laminated beam using piezoelectric patch actuators. The first order shear deformation beam theory is used for the analysis to ensure accurate bending solutions. Optimality conditions and analytical expressions are derived for determining the optimal voltages to be applied to the piezoelectric actuators so as to bend the beam to the desired shape. The formulation incorporates upper and lower bounds on the voltages as these voltages are, in practice, constrained due to cost and material constraints. To verify the correctness of the formulation and the optimality conditions, the optimal voltage for the well-known Lee and Moon's piezoelectric cantilevered beam is determined and compared with existing results in the literature. Several shape control problems of laminated beams are solved which include symmetric and asymmetric beams with two sided and one sided piezoelectric patch actuator configurations. As a special case of the more general patch actuator formulation, the problem of determining optimal voltages for a symmetric beam with piezoelectric layered actuators is also solved. The obtained optimal voltages for the various beam problems considered herein may serve also as benchmark results for checking the validity, convergence and accuracy of numerical results obtained by other numerical methods such as the Finite Element Method