Developing actuator and sensor influence matrices.

The linear constitutive equations of a piezoelectric layer for the converse and direct piezoelectric effects can be written as

	(20.1)

Where and  are the stress and strain vector, E_k  is the electric field vector, D_l  is the electric displacement vector,  is the elastic stiffness matrix,  is the piezoelectric stress charge tensor and is the piezoelectric permittivity matrix. The electric field vector of the piezoelectric layer is related to the electric potential vector V by

(20.2)

The governing equation of motion neglecting material damping for this case can be written as

(20.3)

and  are the structural nodal acceleration and displacement vectors. ] and are the structural mass and stiffness matrices.  F is the global force vector;  is the structural electro-mechanical matrix and  is the actuator voltage vector. The equivalent forces   dadue to the actuator voltages can be expressed as. . The controller is designed after the placement of actuators and sensors. Therefore, the mass, damping and the stiffness matrices will not contain feedback gain elements explicitly; although change in location of actuators across the layers will change all these matrices. The analysis is done for collocated sensor and actuator. The electric voltage vector  of the   element can be expressed as

(20.4)

N_p is the number of piezoelectric layers and  is the electric voltage of the k^th piezoelectric layer in the i ^th element.