A Comment on Boundary Conditions in the Modeling of Beams with Constrained Layer Damping Treatments (2000)

Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received Jan. 2000; revised Sept. 2000. Associate Editor: J. Q. Sun. The most commonly used mathematical formulation for a beam with a constrained layer damping treatment was developed by Mead and Markus 1. In their formulation, the base beam and the constraining layer are set to undergo identical transverse deflections and the longitudinal displacements of the base beam and the constraining layer are set to be related via the thickness and Young’s modulus of each layer. As a result, the Mead-Markus formulation only applies to a class of systems with boundary conditions described in Markus et al. 2, Rao 3, and Lifshitz and Leibowitz 4. Trompette et al. 5 were first to investigate the practical implication of this issue by investigating a specific constrained layer damped cantilevered beam. The purpose of this study was to investigate a broader range of boundary and damping treatment configurations. The error of the Mead-Markus formulation was investigated as a function of the thickness of viscoelastic layer and was shown to be large for certain common boundary conditions. A modified Mead-Markus formulation that allows the longitudinal motion of the base beam and constraining layer to be independent from each other is suggested as a remedy. Figure 1 shows the displacement variables fields associated with the Mead-Markus model for an Euler-Bernoulli beam. The transverse displacement is represented by wx,t, and longitudinal displacement of the j-th layer is represented by ujx,t, where x is the position along the beam and t is time. The subscripts 1, 2 and 3 refer to the base beam, the viscoelastic layer, and the constraining layer, respectively. All the layers have length L and width b. The j-th layer has thickness hj, and storage moduli Ej and Gj. In the Mead-Markus model, assuming equilibrium of axial forces results in (1)E1h1b∂u1∂x=−E3h3b∂u3∂xwhere E1h1b and E3h3b are axial rigidities of the base beam and the constraining layer, respectively. Integration of Eq. (1) with respect to x leads to(2)E1h1u1=−E3h3u3+Cwhere C is an arbitrary function of time. In the original derivation by Mead and Markus 1, it is assumed that C=0, resulting in (3)u1=E3h3E1h1u3Note that u1 and u3 are no longer independent. The presence of Eq. (3) has a profound implication in that the dependence of u1 and u3 limits the boundary conditions allowed in the Mead-Markus formulation. Figure 2 illustrates a simple lab setup consisting of a cantilevered base beam and a free-free constraining layer. The proper boundary conditions at the left end should be (4)∂u10,t∂x=0andu30,t=0which implies that u10,t≠0 for the free-free constraining layer and violates Eq. (3). To incorporate these boundary conditions properly, one can modify the Mead-Markus formulation by assuming that u1 and u3 are independent as was done by Shen 6 (and experimentally validated by Huang et al. 7). Figure 3(a) shows the comparison between predictions given by the Mead-Markus formulation (where we are forced to set u10,t=u30,t=0 at the left end) and the modified Mead-Markus formulation, and experimental results (both with boundary conditions at the left end given by Eq. (4)). Good agreement between the modified theory and experiments can be seen and a significant error in the first natural frequency and corresponding amplitude predicted by the Mead-Markus model can be observed. For the same boundary conditions, Fig. 3(b) shows the error produced by not being able to use correct boundary conditions in Mead-Markus model as compared to the modified theory. As the thickness of the viscoelastic layer was increased, it was found that the error in amplitude of the first mode increased significantly. The errors in the higher modes were found to be insignificant. Many boundary conditions are compatible with the Mead-Markus formulation. Table 1 lists a range of boundary conditions, most of which can be modeled using Mead-Markus. No significant difference was detected between the predicted frequency response using Mead-Markus and the modified theory when modeling boundary conditions compatible with the Mead-Markus formulation. As soon as the boundary conditions are not compatible, however, significant error in the prediction can occur when using a Mead-Markus model. Figures 4 and 5 show results for when the base beam was subjected to clamped-clamped and pinned-pinned boundary conditions and the constraining layer was free at both ends. Neither boundary condition is compatible with Mead-Markus and could therefore not be modeled using this theory without modification. The clamped-clamped case was modified such that the constraining layer was also constrained at the ends. For the second case, the pinned ends of the base beam were released such that they were free to move axially. Figures 4 and 5 show the error produced by Mead-Markus model with these modified boundary conditions as compared to using the actual boundary conditions with the modified formulation. For both types of boundary conditions, the difference between the two formulations was shown to increase significantly with decreasing thickness of the viscoelastic layer. The incompatibility of Mead-Markus with these two boundary conditions did not result in significant error in the higher modes. No appreciable difference between the two formulations was seen for mode two and higher. Careful attention to the boundary conditions must be exercised when predicting the behavior of the first mode using the Mead-Markus formulation. For certain boundary conditions, more accurate results can be obtained by using a formulation that allows independent longitudinal motion of the beam and the constraining layer. This material is based upon work supported by Army Research Office under agreement of No. DAAG 55-98-1-0387.