Analysis of Asymmetric Piezoelectric Composite Beam
  This paper deals with the vibration analysis of an asymmetric composite beam composed of glass a piezoelectric material. The Bernoulli's beam theory is adopted for mechanical deformations, and the electric potential field of the piezoelectric material is assumed such that the divergence-free requirement of the electrical displacements is satisfied. The accuracy of the analytic model is assessed by comparing the resonance frequencies obtained by the analytic model with those obtained by the finite element method. The model developed can be used as a tool for designing piezoelectric actuators such as micro-pumps. 
J.-S. Chen, S.-H. Chen, K.-C. Wu