State space finite element analysis for piezoelectric precision positioning considering secondary converse piezoelectric effect

• State space element is used to solve the laminated beams with piezoelectric layers.
• Displacements and stresses were discretized by FE technique in axis direction.
• Displacements and stresses along thickness were solved using state equations.
• Influence of secondary converse piezoelectric effect was investigated.
• The parameters, thickness ratio, piezoelectric constant ratio etc., are analyzed.

A microgripper component which is made of piezo-cantilever is presented in this paper. The modified mixed variational principle including piezoelectric materials is established and then the state-vector equations of laminated piezoelectric beams are derived directly from the principle. Finally the exact expressions of mechanical and electrical variables along the thickness direction based on the state-vector equation are simply given. Based on the semi-analytical solution of the state-vector equation, a state space finite element model is established. The static analysis is carried out for a laminated beam fully covered with piezoelectric actuating layers on the top and bottom surfaces of the substrate. Both the substrate and piezoelectricity of the laminated beam are considered as one-dimensional line solids and the same linear element is used to discretize the whole beam. This semi-analytical technique can guarantee the continuity of generalized displacements and generalized stresses inter-layers and the transfer matrix method can be applied. Furthermore the exact solution in one direction ensures there is no restriction on the thickness for the smart structures. At the end of this paper the influence of secondary converse piezoelectric effect (SCPE) on the deflection of the laminated beam covered with piezoelectric driving layers for static precision displacement control is investigated. The above mentioned state space finite element method is used to predict the deflection of the beam with and without SCPE. If the influence of SCPE is eliminated, the deflections of the clamped beam will increase by ~2.7%. In addition, the driving characteristic with different PZT/steel thickness ratio considering SCPE are also studied. The maximum influence of SCPE on deflection can reach ~3% for the beam fully covered with piezoelectric active layers, which should not be neglected in practical microoperation applications.