  A novel space-discretized Finite Differences-based model reduction,
introduced in (Liu,Guo,2020) is extended to the partial differential equations
(PDE) model of a multi-layer Mead-Marcus-type sandwich beam with clamped-free
boundary conditions. The PDE model describes transverse vibrations for a
sandwich beam whose alternating outer elastic layers constrain viscoelastic
core layers, which allow transverse shear. The major goal of this project is to
design a single tip velocity sensor to control the overall dynamics on the
beam. Since the spectrum of the PDE can not be constructed analytically, the
so-called multipliers approach is adopted to prove that the PDE model is
exactly observable with sub-optimal observation time. Next, the PDE model is
reduced by the ``order-reduced'' Finite-Differences technique. This method does
not require any type of filtering though the exact observability as $h\to 0$ is
achieved by a constraint on the material constants. The main challenge here is
the strong coupling of the shear dynamics of the middle layer with overall
bending dynamics. This complicates the absorption of coupling terms in the
discrete energy estimates. This is sharply different from a single-layer
(Euler-Bernoulli) beam.
