Closed form analysis of piezoelectric multilayer bending actuators using constituent equations

In this paper, we present for the first time the derivation of the constituent equations for any kind of clamped-free piezoelectric multilayer bending actuators under different excitation conditions formulated generally for any point over the entire length of the actuator. The constituent equations are presented by a 4×4 matrix M, which combines the extensive parameters mechanical moment M at the end of the bender, force F applied perpendicularly to the tip of the bender, uniform pressure load p applied over the entire length of the bender and applied electrical voltage U with the intensive parameters angular deflection α, deflection ξ, volume displacement V and electric charge Q.