Shock and Vibration

Volume 2015, Article ID 376854, 11 pages

http://dx.doi.org/10.1155/2015/376854

## Topology Optimization for Minimizing the Resonant Response of Plates with Constrained Layer Damping Treatment

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China

Received 26 November 2014; Revised 1 March 2015; Accepted 6 March 2015

Academic Editor: Miguel Neves

Copyright © 2015 Zhanpeng Fang and Ling Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A topology optimization method is proposed to minimize the resonant response of plates with constrained layer damping (CLD) treatment under specified broadband harmonic excitations. The topology optimization problem is formulated and the square of displacement resonant response in frequency domain at the specified point is considered as the objective function. Two sensitivity analysis methods are investigated and discussed. The derivative of modal damp ratio is not considered in the conventional sensitivity analysis method. An improved sensitivity analysis method considering the derivative of modal damp ratio is developed to improve the computational accuracy of the sensitivity. The evolutionary structural optimization (ESO) method is used to search the optimal layout of CLD material on plates. Numerical examples and experimental results show that the optimal layout of CLD treatment on the plate from the proposed topology optimization using the conventional sensitivity analysis or the improved sensitivity analysis can reduce the displacement resonant response. However, the optimization method using the improved sensitivity analysis can produce a higher modal damping ratio than that using the conventional sensitivity analysis and develop a smaller displacement resonant response.

#### 1. Introduction

The CLD treatment has been regarded as an effective way to suppress structural vibrations and sound radiation. Among first studies of CLD treatments, Kerwin [1] developed a simplified theory to calculate the loss factor of a plate with CLD treatments. DiTaranto and Blasingame [2] extended Kerwin’s work by accounting for the extensional deformations in the viscoelastic layer and obtained loss factors for three and five layer beams as a function of frequency. Mead and Markus [3] derived sixth-order equation of motion for beams in terms of transverse displacements. Most of these early works dealt with full coverage passive constrained layer damping (PCLD) treatments that are evidently not practical in purpose. For instance, in the damping treatment applications to the automotive and airplane industries, the weight constraint may not allow full PCLD coverage, since the added weight induced by any vibration and noise control design modification has to be limited to a small amount. Therefore, a partial-coverage treatment is a more attractive approach to fulfill the real PCLD treatment requirements. Nokes and Nelson [4] were among the earliest investigators to provide the solution to the problem of a partially covered sandwich beam. The modal strain energy method was used to calculate modal loss factors of the treated beam for symmetric boundary conditions. Lall et al. [5, 6] performed a more thorough analytical study for partially covered planar structures. An important finding from the analysis was that for suitably chosen parameters higher values of the modal damping factor may be obtained for a partially covered beam compared to that obtained for a fully covered one under the condition of the same added weight of PCLD treatments. This encourages many researchers to investigate the optimal layout of CLD treatment in thin plates and shells.

Kodiyalam and Molnar [7] improved the modal strain energy (MSE) method through introducing a new method to account for viscoelastic material property variations with frequency. Lumsdaine [8] determined the optimal shape of a constrained damping layer on an elastic beam by means of topology optimization. The optimization objective is to maximize the system loss factor for the first mode of the beam. Zheng et al. [9] studied the optimal placement of rectangular damping patches for minimizing the structural displacement of cylindrical shells with the Genetic Algorithm. Moreira and Rodrigues [10] used MSE method to optimally locate passive constrained viscoelastic damping layers on structures. They also verified their work by comparing their results with experimental tests. Li and Liang [11] utilized the response surface method (RSM) to analyze and optimize the vibroacoustic properties of the damping structure. Ling et al. [12] studied the topology optimization for the layout of CLD in plates to suppress their vibration and sound radiation. The objective is to minimize structural modal damping ratios using the method of moving asymptotes (MMA). Lepoittevin and Kress [13] proposed a new method for enhancement of damping capabilities of segmented constrained layer damping material and developed an optimization algorithm using mathematical programming to identify a cuts arrangement that optimized the loss factor. Ansari et al. [14] studied the optimal number and location of the CLD sheets on the flat structure surface utilizing an improved gradient method and the objective was to maximize loss factor of the system. Kim et al. [15] compared the modal loss factors obtained by topology optimization to the conventional strain energy distribution (SED method) and the mode shape (MSO approach). It is found that topology optimization based on the rational approximation for material properties (RAMP) model and optimality criteria (OC) method can provide about up to 61.14 percent higher modal loss factor than SED and MSO methods. The numerical model and the topology optimization approach are also experimentally validated.

More recent attempts on the simultaneous shape and topology optimization of dynamic response have achieved some success. Rong et al. [16] developed the evolutionary structural optimization (ESO) method for the topology optimization of continuum structure under random dynamic response constraints. Zheng et al. [17] studied the optimal placement of rectangular damping patches for minimizing the structural displacement of cylindrical shells with the Genetic Algorithm. Pan and Wang [18] applied adaptive genetic algorithm to optimize truss structure with frequency domain excitations. The optimization constraints include displacement responses under force excitations and acceleration responses under foundation acceleration excitations. Alvelid [19] proposed a modified gradient of damping materials to mitigate noise and vibration. Yoon [20] proposed the topology optimization based on the SIMP method for the frequency response problem where the structure is subject to the wide excitation frequency domain. Shu et al. [21] propose level set based structural topology optimization for minimizing frequency response. The general objective function is formulated as the frequency response minimization at the specified points or surfaces with a predefined excitation frequency or a predefined frequency range under the volume constraint.

Although above optimal approaches can supply the design methods to reduce structural vibration and sound radiation in practice, the modal damping ratios for different modes are selected as the optimization objectives in these studies. In fact, the modal damping ratio is only the characteristics of vibration energy dissipation in structures rather than a controllable dynamic performance. Furthermore, the derivative of the modal damp ratio is usually neglected in conventional sensitivity analysis which will give rise to error in sensitivity analysis and produce a deviation from the optimal layout of CLD treatment in structures. This paper develops a dynamic topology optimization method of plates with CLD treatment under specified broadband harmonic excitation, which includes one or more resonance frequencies. The optimization objective is to minimize the square of displacement resonance response instead of conventional modal damping ratio. Meanwhile, an improved sensitivity analysis method with considering the derivative of modal damp ratio is developed in order to obtain the more accurate layout of CLD material on the plate and the lower displacement resonance response of CLD system.

#### 2. Statement of Optimization Formulation

##### 2.1. Dynamic Responses Using the Mode Superposition Method

The finite element equation for a structural dynamic problem under external harmonic excitations can be written in the formwhere , , and are the mass, damping, and stiffness matrices of the structure, respectively. is the vector of excitation force. and represent the frequency of the harmonic excitation and amplitude, respectively. , , and denote the displacement, velocity, and acceleration vectors, respectively.

For a plate with CLD treatment, the matrices and can be further expressed aswhere the subscript , , and denote the base plate, the viscoelastic damping layer, and the constrained layer, respectively.

For the CLD plate system, the th modal loss factor can be written by using MSE method [22]:where is the loss factor of the damping material. is the th mode strain energy of the damping material. is the total th mode strain energy of CLD system. , , and are the th mode strain energy of the base material, damping layer, and constrained layer at th element, respectively.

On the other hand, the relationship between modal loss factor and modal damping ratio is expressed as [23]

In fact, when modal loss factor is less than 0.3, the following expression can be used to calculate modal damping ratio instead of (4); the deviation is within 5%. It can be seen from Figure 1. Consider