Using Energy Methods to Derive Beam Finite Elements Incorporating Piezoelectric Materials

We apply the principle of virtual work to derive the equations of motion for arbitrarily shaped linear piezoelectric materials which are either embedded in or perfectly bonded to the surface of an elastic substructure. Using an Euler-Bernoulli strain model, the equations of motion are subsequently specialized to obtain finite elements for an elastic beam incorporating piezoelectric materials. This formulation leads to a general consistent electrical stiffness matrix that characterizes the electromechanical coupling of the piezoelectric material; thus a finite element for any particular beam cross section geometry can be obtained by computing the areas of cross sections of the substructure and the piezoelectric material and the first and second moments of their cross sections. Actuator finite elements are discussed for cases where the piezoelectric material is connected to a voltage source and a current source. We also discuss sensor finite elements when the piezoelectric is connected to two types of amplifying circuits to measure the induced electric potential across the piezoelectric material. Several examples are presented to compare the results in this paper to the results of other researchers and to ascertain the numerical properties of the finite element for a specific example.