  Models for piezoelectric beams using Euler-Bernoulli small displacement
theory predict the dynamics of slender beams at the low frequency accurately
but are insufficient for beams vibrating at high frequencies or beams with low
length-to-width aspect ratios. A more thorough model that includes the effects
of rotational inertia and shear strain, Mindlin-Timoshenko small displacement
theory, is needed to predict the dynamics more accurately for these cases.
Moreover, existing models ignore the magnetic effects since the magnetic
effects are relatively small. However, it was shown recently \cite{O-M1} that
these effects can substantially change the controllability and stabilizability
properties of even a single piezoelectric beam. In this paper, we use a
variational approach to derive models that include magnetic effects for an
elastic beam with two piezoelectric patches actuated by different voltage
sources. Both Euler-Bernoulli and Mindlin-Timoshenko small displacement
theories are considered. Due to the magnetic effects, the equations are quite
different from the standard equations.
