On the damping of flexural plate vibrations by the application of viscoelastic layers

The problem of the damping of flexural vibrations of an elastic plate due to the application of arbitrary viscoelastic layers on either side is solved within the framework of linear first order plate theory. For algebraic simplicity, only one-dimensional motion is considered, and the damping is assumed to be small.

The question of dynamic coupling between flexural and extensional motion is resolved and the physical mechanism to which such coupling is due is explained. This insight allows parametric relationships to be determined which will uncouple the problem. The coupling is seen to be a higher order effect which can be neglected within thin plate theory.

A review of various descriptions of damping is made and a discussion of three methods of introducing damping which are based upon different physical explanations is presented. From a side by side comparison of their effects on the solution of a simple problem it is seen that for small damping they are equivalent. The complex modulus representation of damping is employed where the complex plate stiffness is written as D* = D(1 + in) with D representing the ordinary flexural plate stiffness and n, the ratio of the imaginary and real parts of D*, representing the so-called loss factor.

The problem of how to apply uniformly a given quantity (weight) of viscoelastic material to a base plate in order to achieve the greatest composite loss factor is considered. It is found, contrary to what had heretofore been believed, that depending upon the magnitude of the parameters which are involved, the material should sometimes be applied in two symmetrical layers, sometimes as a single layer. Intermediate thicknesswise distribution of the viscoelastic material is seen never to result in loss factors exceeding those of the above mentioned limits. Further, the sensitivity of the composite loss factor to the actual manner of distribution of the viscoelastic material through the thickness is seen to be greatest for a single layer application and least for a symmetrical construction.

As an example of the application of the preceding results to an actual boundary value problem, the response of a three-layered, simply supported plate to a uniformly distributed, sinusoidally varying force is examined and a closed form solution is obtained. Through a simple reinterpretation of the results, the problem of a simply supported plate subjected to a sinusoidal motion of the supports is also solved. The problem of how to uniformly apply a given amount of viscoelastic material to such a plate is treated using the above mentioned solution for several different design criteria. The relative resonance amplitude (analogously the time for a given amount of amplitude decay in free vibration), the absolute midpoint deflection of the composite plate, and the maximum stress in the base plate were chosen as examples of design criteria. So-called design criteria curves, which are the locus in parametric space of cross-over points from two-layered to symmetrical three-layered viscoelastic material distribution, are given and their use to determine the most effective material distribution is indicated. From the relatively good agreement between experiment and theory for the two-layered plate, which is simply a special case of the present general solution, it is expected that similar agreement should be obtained from carefully conducted experiments on three-layered plates.