A quasi-two-dimensional finite element formulation for analysis of active-passive constrained layer beams

Active-passive damping is getting more popular with designers because it combines the complementary passive and active features in the control of structural vibrations. The classical three-layer structure has a viscoelastic-layer sandwiched between the host beam and a piezoelectric-layer. The more prevalent assumptions for modeling the system are the use of Euler-Bernoulli beam theory for both the host beam and piezoelectric-layer, and Timoshenko beam theory for the viscoelastic-layer. The assumption that transverse displacement is constant through the thickness limits accuracy and applicability of the model. The current formulation expresses the through-the-thickness dependency of the field variables as polynomials while their span dependency across a finite element is cubically interpolated. The versatility of the formulation is demonstrated via static and dynamic studies of examples taken from the literature. A beam treated with active-passive damping is presented and examined. The constitutive relation of the viscoelastic layer is represented using fractional derivatives and the Grünwald approximation. The extended Hamilton's principle is used to derive the system governing equations which are integrated with the Newmark time-integration system.