We consider a multilayer Rao-Nakra sandwich beam with shear damping included in alternate layers. In the case
that the wave speeds of the layers are distinct, we show that by "tuning" the damping in each layer appropriately,
it is possible to construct a scalar boundary control that exactly controls the entire layered system in a control
time related to the sum of the control times of each separate layer. In the case where some waves speeds coincide,
one can reduce the number of controls needed, but not to a single scalar control.
We consider a three layer Rao-Nakra sandwich beam with damping proportional to shear included in the core layer. We prove that eigenvectors of the beam form a Riesz basis for the natural energy space. In the damped case, we are able to give precise conditions under which solutions decay at a uniform exponential rate. We also consider the problem of boundary control using bending moment and lateral force control at one end. We prove that the space of exact controllability has finite co-dimension and provide sufficient conditions (related to small damping) for exact controllability to a zero energy state.
We describe some possible models for a multilayer sandwich beam consisting of alternating stiff and compliant beam layers. The stiff layers are modeled under Euler-Bernoulli assumptions while the compliant layers essentially only carry the shear. We include viscous damping in the compliant layers and consider the optimization problem of choosing the damping parameteres for each layer so that the maximal asymptotic damping angle in the system eigenvalues is obtained. The solution is obtained analytically as a closed-form function of the various material parameters.
A dynamic model for a multilayered laminated plate is developed. The laminated plate consists of 2n plate layers and 2n - 1 adhesive layers. The layers (both plate and adhesive layers) are assumed to be homogeneous, transversely isotropic and perfectly bonded to one another. In the initial modeling, the Reissner-Mindlin theory of shear deformable plates is applied to each layer, resulting in a high-order plate theory in which the shear motions of the layers are completely independent. Simpler, lower-order models can then be obtained from this initial model from asymptotic limits based upon the assumptions that (1) the adhesive layers are very thin, (2) the elastic modulii of the adhesive layers are small compared to those of the plate layers, (3) the shear stiffnesses of the plate layers are very large, (4) the rotational moments of inertia of the individual plate layers are very small.
Three models for two-layered beams in which slip can occur at the interface are described. In the first, beam layers are modelled under the assumptions of Timoshenko beam theory. Along the interface a `glue layer' of negligible thickness bonds the surfaces so that a small amount of slip is possible. Dissipation is assumed to be proportional to the rate of slip. The second is obtained from the first by letting the shear stiffness of each beam tend to infinity. The third is obtained from the second by assuming that the moment of inertia parameter is negligible. In this case an analog of the Euler-Bernoulli beam is obtained which exhibits frequency- proportional damping characteristics.
Conference Committee Involvement (3)
Modeling, Signal Processing, and Control III
11 March 2009 | San Diego, California, United States
Modeling, Signal Processing, and Control II
10 March 2008 | San Diego, California, United States
Modeling, Signal Processing, and Control
19 March 2007 | San Diego, California, United States
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