Optimization of segmented constrained layer damping with mathematical programming using strain energy analysis and modal data
Introduction
For many years vibration damping has been an important criterion in the design phase of many engineering applications. Large dissipation capabilities over a wide frequency range are desired. A common solution is to use a constrained viscoelastic material bonded to the vibrating structure. The damping material is applied in a sandwiched configuration. The inner surface is attached to the host structure and the outer surface is constrained by a stiff cover. With such configuration, the damping layer is mainly deformed in shear. The energy is dissipated into heat because of the relaxation process occurring in the long molecule chains [1], [2]. Further improvement can be achieved by cutting the whole damping treatment. As a result, the number and the volume of high-shear regions is increased leading to a higher damping rate. Additionally, structural topology optimization can be a powerful tool to further improve the design of damping treatment.
In 1959, Kerwin [3] published his observation that a stiff constraining layer, placed on top of the viscoelastic damping layer, can significantly increase the structural damping rate. Ungar and Kerwin [4] re-examinated the concept of loss factor applied for viscoelastic systems. Their main conclusion is that the stored energy can be estimated only if the energy storage and dissipation mechanisms are known. DiTaranto [5] determined the loss factor of a freely vibrating laminated beam having any possible boundary conditions using an analytical model. Mead and Markus [6] derived a mathematical expression for the transverse displacement of a three-layered sandwich beam with a viscoelastic core. They assumed different boundary conditions at one end of the beam such as no transverse displacement, no rotation, no bending moment, or no shear force. Rao [7] also presented a formula for the frequency and loss factor of a sandwich beam under the following boundary conditions: clamped–free, clamped–simply supported, clamped–clamped, simply supported–simply supported, and free–free. Plunkett and Lee [8] invented the concept of segmenting the constraining layer. Their study included experiments and derivation of a formula for optimum distance of their equidistant cuts arrangement. Kress [9] solved a shear-lag model for simulating segmented constrained layer damping treatment, similar to the investigation by Plunkett and Lee, with a transfer-matrix method. An illustration of the shear stress distribution over the whole length of the beam was given. A good agreement between simulations and measurements was observed. He also derived a simple formula for optimum spacing of his equidistant cuts arrangement that is different from the formula of Plunkett and Lee. Nevertheless, the effect of cuts were only investigated at the first bending mode of a cantilever beam. Torvik and Strickland [10] investigated a structure consisting of a base plate with a multiple-layer damping treatment with unanchored constrained layers, attached to one side of it. The constraining layers were segmented in two-dimensions. In the field of laminated composites, Mantena et al. [11] also investigated the optimal side length of constrained layer damping material. They considered various geometric arrangements of a load-carrying structure in terms of the clamping situation with special regard to the damping material. The main limitation of their work is that they focused their investigation on a single mode and considered just one segment. Alam and Asmani [12] sought optimal damping treatment design by considering as parameter damping material’s thickness. Huang et al. [13] did a similar study. Kung and Singh [14] developed an energy-based approach of multiple constrained layer damping patches. They only looked at the effect of constrained layer damping patches at several modes separately. In the field of active vibration control, Lesieutre and Lee [15] performed a finite element analysis on segmented active constrained layer damping. Liu and Wang [16] investigated the distribution of passive and active constrained layer damping patches. In both papers, no length optimization of the damping treatment was performed. In the field of structural optimization, genetic algorithms were used by Trompette and Fatemi [17], and by Al-Ajmi and Bourisli [18] to optimize the segments’ length. They were only able to identify a distribution of segments for a single mode and considered only one optimization technique. No comparison with other optimization methods on the efficiency of the selected algorithm to find the best solution was realised. As a general remark concerning all the papers above mentioned, the main limitation is that the different studies did not take into account a large frequency range. Additionaly, many of them just assumed either a fixed length for each segment or a fixed number of segments.
The objective of the present work is to develop an optimization algorithm based on mathematical programming that enables to find a single cuts arrangement for optimum damping of all modes within a selected frequency range. Additionally, a new mode-shaped based technique is proposed for the initial conditions. It aims at facilitating the finding of the best design.
In Section 2, the finite element modeling is addressed. The material data and geometrical parameters are also presented. The method to estimate the modal loss factor is described in Section 3. A finite element analysis is performed on a cantilever beam with a segmented constrained layer damping in Section 4. It enables to observe high-shear deformation regions and to have a clear understanding of the phenomena under interest. The dissipated energy is also quantified. Section 5 discusses the efficiency of segmented constrained layer damping material with equally spaced cuts. In Section 6, the optimization method is described. The convexity properties of the objective function are analyzed to investigate whether the solutions are unique. Results are discussed for a single mode optimization and over a large frequency range in Section 7. In Section 8, a guideline for the enhancement of constrained layer damping material via segmentation is proposed as a conclusion.
Section snippets
Overview
The structure of interest is a cantilever elastic beam on which is bonded a constrained damping layer. The method used for the finite element modeling is illustrated in Fig. 1.
The beam and the constraining layer are modeled with two-dimensional structural solid elements called plane42 in ANSYS 11.0. The viscoelastic core is modeled with two-dimensional structural solid elements called plane182. All of these utilize two translations at each of the four nodes. An important feature of the plane182
Modal loss factor calculation
An energy formulation is used to estimate the modal loss factor of the kth mode of interest. The modal loss factor is obtained by using the modal strain energy method [19]where is the material loss factor of the layer i at mode k, and is the modal strain energy of the layer i at mode k. is the total modal strain energy at mode kHowever, the method gives good estimates only if the following conditions are fulfilled:
- •
The structure must
Mechanism of a segmented constrained layer damping material
This section explains how the damping mechanism of a segmented constrained layer damping treatment works.
Equally segmented constrained layer damping
In Section 4, the effect of the position of a single cut on the modal loss factor was investigated at a given mode. From this result, the authors deduce as an hypothetical rule-of-thumb namely that a cut has to be located at the maximum of the bending moment. The rule-of-thumb can be applied if a modal analysis obtains information on mode shape and/or the shear strain distribution in the viscoelastic material is known. However, this Section investigates the influence of the number of equally
Optimization with mathematical programming
In Section 4.2, the cut’s position has been identified as the main parameter for the design of a segmented constrained layer damping material. The identification of the position requires an accurate knowledge of either the flexural displacement or the shear strain distribution. This section presents the algorithm that optimizes the cuts arrangement.
Regarding the problem of interest, the optimization task is nonlinear which requires an iterative solution process for finding the minimum of the
Optimization of a single mode
Section 4.2 discusses the hypothetical rule-of-thumb that a cut has to be positioned at a maximum of bending moment to have a significant increase of energy dissipation. Therefore, a knowledge on the mode shape is a valuable information for the selection of the initial cut arrangement of the optimization process. Fig. 12 presents the deflection lines for the first 4 modes. For the optimization at a given mode and one single cut, the proposed method is following: the cut is initially placed at
Conclusions
This paper has presented a new approach for finding optimum damping treatment by sectioning the constraining and constrained layers. The effect of cutting the damping treatment, which leads to segmentation, has been explained. The position of the cut is the main parameter that influences the modal loss factor. At the core of the new approach is a rule-of-thumb for the position of a single cut: it has to be placed at the positions where the bending of the respective modal deflection line is the
Acknowledgements
The authors would like to thank D. Keller for the valuable input on structural optimization. The authors would also like to thank P. Ermanni for supporting this research.
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