Vibration characteristics analysis of CLD/plate based on the multi-objective optimization

1. Introduction

Vibration control of flexible structure is a common subject of engineering community. Active Constrained Layer Damping (ACLD) treatment has been used widely for damping the vibration of beams [1, 2], plates [3] and shells [4]. Such a system generally comprises a layer of viscoelastic material (VEM) sandwiched between a host structure and an active cover sheet, such as a piezoelectric layer.

The overall performance of the ACLD system is governed by performance of the passive (open-loop) and the active controls (closed-loop). Thus, the optimization for the ACLD system consists of two parts, specifically, optimum design of the open-loop ACLD system, i.e. CLD system, and the closed-loop system. Optimization for CLD system is essential to enhance the performance of ACLD system. Furthermore, with optimally designed CLD treatments, the performance is guaranteed to be robust even if the active component of the ACLD ceases to operate or fail [5]. In the case of optimum design of CLD system, extensive efforts have been exerted to optimally design CLD treatment subject to space or weight limitations [6-8]. Parametric studies regarding variations in the natural frequencies and the mode damping ratios or loss factors have been implemented. Kung and Singh [9] developed an energy-based approach of multiple constrained layer damping patches. They only investigated the effect of constrained layer damping patches on suppressing vibration of several modes separately. Baz [5] optimized the placement of ACLD patches using the mode strain energy (MSE) method. In this study, the total weight of the damping treatments was taken as the objective function while satisfying constraints imposed on the mode damping ratios. Zheng and Cai [10] employed four different nonlinear optimization methods/algorithms (sub-problem approximation method, the first-order method, sequential quadratic programming (SQP) and genetic algorithm (GA)) to find the optimal locations and lengths of the CLD patches, to minimize displacement amplitude at the middle beam. Al-Ajmi and Bourisli [11] optimized CLD segments’ length using the genetic algorithm (GA). The optimization procedure was only able to identify a distribution of segments for a single mode. Zheng Ling [12] proposed an optimality criterion to find the optimal placement of CLD patches on the cylindrical shell. Li Yinong and Xie Ronglu [13] considered the CLD structure optimization as a topology optimization problem, and the evolutionary structural optimization (ESO) method was employed to find the optimal configuration of the CLD patches. Both papers took the mode loss factors as objective functions. Lepoittevin Grégoire and Kress, Gerald [14] proposed a new arrangement method for enhancement of damping capabilities of segmented CLD treatments, and a mathematical programming method was developed in order to get the optimal arrangement that optimized the mode loss factor. As mentioned above, locations of the CLD patches, thicknesses of the CLD and VEM were optimized, but there is a limitation that all the parameters were rarely taken as the optimal design variables at the same time. Additionally, many of them just took a single objective function to find the optimal configurations of CLD treatments.

In the present work, the objective is to develop a multi-objective optimization procedure for CLD structure with the ability to optimize locations of the CLD patches and thicknesses of the VEM and CLM simultaneously. Furthermore, effect of the parameters on vibration suppression performance can be analyzed based on Pareto optimal solutions set.

This paper is organized in six sections. In section 1, the brief introduction is given. In section 2, the finite element model of the CLD/plate is developed. In section 3, a multi-objective optimization model is established and two optimization strategies are proposed. Locations of CLD patches expressed by positive integers, and thicknesses of VEM and CLM which are presented by positive consequent-real numbers, constitute the hybrid variables. The mode loss factors are served as objective functions, the added-mass and frequency shift are taken as constraint conditions. To obtain the optimal configuration of CLD treatments, the improved NSGA-II (INSGA-II) is proposed and explained in section 4. In section 5, the multi-objective optimization processes for the CLD system are carried out, and the effectiveness of the INSGA-II is verified. The vibration characteristics of CLD system are analyzed for different multi-objective optimization configurations. Furthermore, the influences of algorithm parameters on optimization procedure are also discussed. In section 6, a brief summary of the conclusions is given.

2. Finite element model

Fig. 1 shows a schematic drawing of a cantilever plate treated partially with the constrained layer damping (CLD) treatment. ZN-1 polymer is used as the viscoelastic material (VEM) layer which is boned to the base plate (aluminum). P-5H piezoelectric ceramics patch is selected as the constrained layer (CL) to enhance and control vibration energy dissipation. Their geometric and physical properties are shown in Table 1. It is assumed that shear strains in the constrained layer and base plate are negligible. The transverse displacements of any points on any cross section in the CLD/plate are considered to be the same. Besides, all layers are considered to be boned together perfectly.

Fig. 1The cantilever plate treated with CLD patches

2.1. Degrees of freedom and shape functions

The CLD/plate is divided into $N$ elements and a two-dimensional integrated element with four nodes shown in Fig. 2 is considered to model the CLD/plate system. Each node has seven degrees of freedom, including the longitudinal displacements $u_c$ and $v_c$ of the constrained layer, the longitudinal displacements $u_b$, $v_b$ of the base plate, the lateral displacement $w$ of the whole structure, and the slopes $w{,}_x$, $w{,}_y$ of lateral displacement. The displacement of an arbitrary point in an element can be interpolated using the following polynomials:

1

$u_c={\alpha }_1+{\alpha }_{2}x+{\alpha }_{3}y+{\alpha }_{4}xy,v_c={\alpha }_5+{\alpha }_{6}x+{\alpha }_{7}y+{\alpha }_{8}xy,$
$u_b={\alpha }_9+{\alpha }_{10}x+{\alpha }_{11}y+{\alpha }_{12}xy,v_b={\alpha }_{13}+{\alpha }_{14}x+{\alpha }_{15}y+{\alpha }_{16}xy,$
$\begin{array}{c}w={\alpha }_{17}+{\alpha }_{18}x+{\alpha }_{19}y+{\alpha }_{20}{x}^2+{\alpha }_{21}xy+{\alpha }_{22}{y}^2+{\alpha }_{23}{x}^3\end{array}$
$+{\alpha }_{24}{x}^{2}y+{\alpha }_{25}x{y}^2+{\alpha }_{26}{y}^3+{\alpha }_{27}{x}^{3}y+{\alpha }_{28}x{y}^3,$
$w_{,x}=\frac{\partial w}{\partial y},w_{,y}=-\frac{\partial w}{\partial x},$

where the constants $\alpha =\{{\alpha }_1\dots {\alpha }_{28}\}$ are determined in terms of twenty-eight components of the node displacements vector ${\mathbf{q}}^e$ of the $i$th element. The element displacement vector ${\mathbf{q}}^e$ is given by:

2

${\mathbf{q}}^e={\left\{\begin{array}{llll}{\mathbf{q}}_1& {\mathbf{q}}_2& {\mathbf{q}}_3& {\mathbf{q}}_4\end{array}\right\}}^T,$

where $q_i=\left\{u_{ci}\mathrm{}\mathrm{}\mathrm{}{v}_{ci}\mathrm{}\mathrm{}\mathrm{}{u}_{bi}\mathrm{}\mathrm{}{v}_{bi}\mathrm{}\mathrm{}\mathrm{}{w}_i\mathrm{}\mathrm{}\mathrm{}{w}_{i,x}\mathrm{}\mathrm{}\mathrm{}{w}_{i,y}\right\}$, for $i=\mathrm{}$1, 2, 3, 4.

Substituting $\alpha$ expressed by the nodes displacements and the nodes coordinates into Eq. (1), the displacement at any location ($x$, $y$) inside the $i$th element can be described by:

3

$u_c={\sum }_{i=1}^4{s}_i{u}_{ci},v_c={\sum }_{i=1}^4{s}_i{v}_{ci},u_p={\sum }_{i=1}^4{s}_i{u}_{bi},v_p={\sum }_{i=1}^4{s}_i{v}_{bi},$
$w={\sum }_{i=1}^4\left(s_{wi}{w}_i+s_{wi,x}{w}_{i,x}+s_{wi,y}{w}_{i,y}\right),$

where:

$s_i=\frac{1}{4}\left(1+\frac{x}{x_i}\right)\left(1+\frac{y}{y_i}\right),s_{wi}=\frac{1}{8}\left(1+\frac{x}{x_i}\right)\left(1+\frac{y}{y_i}\right)\left[2+\frac{x}{x_i}\left(1-\frac{x}{x_i}\right)+\frac{y}{y_i}\left(1-\frac{y}{y_i}\right)\right],$
$s_{wi,x}=-\frac{y_i}{8}\left(1+\frac{x}{x_i}\right){\left(1+\frac{y}{y_i}\right)}^2\left(1-\frac{x}{x_i}\right),s_{wi,y}=\frac{x_i}{8}{\left(1+\frac{x}{x_i}\right)}^2\left(1+\frac{y}{y_i}\right)\left(1-\frac{x}{x_i}\right),$

are termed as the shape function, and $x_i$, $y_i$ are the coordinates of the nodes in the $i$th element coordinates system. Eq. (3) can be rewritten in the matrix form as:

4

${\left\{u_c{v}_c{u}_b{v}_{b}w{w}_{,x}{w}_{,y}\right\}}^T={\left\{{\mathbf{S}}_{uc}{\mathbf{S}}_{vc}{\mathbf{S}}_{ub}{\mathbf{S}}_{vb}{\mathbf{S}}_w{\mathbf{S}}_{w,x}{\mathbf{S}}_{w,y}\right\}}^T{\mathbf{q}}^e,$

where ${\mathbf{S}}_{uc}$, ${\mathbf{S}}_{vc}$, ${\mathbf{S}}_{ub}$, ${\mathbf{S}}_{vb}$, ${\mathbf{S}}_w$, ${\mathbf{S}}_{w,x}$, ${\mathbf{S}}_{w,y}$ are called the shape function for $u_c$, $v_c$, $u_b$, $v_b$, $w$, $w_{,x}$, $w_{,y}$.

The deformation relationship of the CLD patches and the base plate is shown in Fig. 3. Then, the displacements in $x$-direction and $y$-direction of the viscoelastic layer (VL) can be formulated as:

5

$u_v=\frac{1}{2}\left(u_c+u_b+\frac{h_c-h_b}{2}\frac{\partial w}{\partial x}\right),v_v=\frac{1}{2}\left(v_c+v_b+\frac{h_c-h_b}{2}\frac{\partial w}{\partial y}\right).$

Fig. 2The CLD element

Fig. 3The deformation relationship

And the shear deformation of the VL can be derived as:

6

${\gamma }_x=\frac{u_c-u_b}{h_v}+\frac{d}{h_v}\frac{\partial w}{\partial x},{\gamma }_y=\frac{v_c-v_b}{h_v}+\frac{d}{h_v}\frac{\partial w}{\partial y},$

where $d=(h_b+h_c)/2+h_v$, and $h_b$, $h_c$, $h_v$ denote the thickness of the base plate, the constrained layer and the viscoelastic layer respectively.

Hence, the shape functions for the longitudinal displacements $u_v\text{,}$$v_v$ and the shear deformations ${\gamma }_x$, ${\gamma }_y$ of the VEM core can be derived as [4]:

7

${\mathbf{S}}_{uv}=\frac{1}{2}\left[\left({\mathbf{S}}_{uc}+{\mathbf{S}}_{ub}\right)-\frac{h_c-h_b}{2}{\mathbf{S}}_{w,y}\right],$
${\mathbf{S}}_{vv}=\frac{1}{2}\left[\left({\mathbf{S}}_{uc}+{\mathbf{S}}_{ub}\right)+\frac{h_c-h_b}{2}{\mathbf{S}}_{w,x}\right],$
${\mathbf{S}}_{\gamma x}=\frac{1}{h_v}\left[\left({\mathbf{S}}_{uc}-{\mathbf{S}}_{ub}\right)-\left(\frac{h_c+h_b}{2}+h_v\right){\mathbf{S}}_{w,y}\right],$
${\mathbf{S}}_{\gamma y}=\frac{1}{h_v}\left[\left({\mathbf{S}}_{uc}-{\mathbf{S}}_{ub}\right)+\left(\frac{h_c+h_b}{2}+h_v\right){\mathbf{S}}_{wx}\right].$

So the displacement vector of the VEM can be expressed as:

8

$\left\{u_v{v}_v\right\}=\left\{\begin{array}{ll}{\mathbf{S}}_{uv}& {\mathbf{S}}_{vv}\end{array}\right\}{\mathbf{q}}^e,$

9

$\left\{{\gamma }_x{\gamma }_y\right\}=\left\{{\mathbf{S}}_{\gamma x}{\mathbf{S}}_{\gamma y}\right\}{\mathbf{q}}^e.$

3. Problem formulations

Most real-world search and optimization problems naturally involve multiple objectives. For the optimization problem of the CLD structures, it requires that several trade-off CLD configurations, using which the selected resonance peaks can be decreased simultaneously, can be obtained. Thus, it can be considered as a multi-objective optimization problem. The standard form of the multi-objective optimization problem for CLD/plate can be described as:

where $\mathbf{d}$ is the vector of the design variables.

4. Multi-objective optimization algorithm-improved NSGA-II

For multi-objective optimization for CLD structure, the relationship between the design variables and the objectives is hard to describe by the use of explicit mathematical equations. Meanwhile, the design variables space composes of two different types of design variables: discrete and continuous. Besides, due to the difficulty of characterizing visco-elastic properties of VEM accurately, that makes the optimization problem more complex. As well known, the traditional optimization strategies, such as linear programming algorithm, are powerful to solve single objective optimization problem described by explicit mathematical equations, but difficult to solve the complex multi-objective problem. However, GA is very useful and powerful to deal with such multi-objective optimization problem. In this paper, the non-dominated sorting genetic algorithm (NSGA-II), which has an excellent robustness, is introduced and improved to solve the hybrid variables multi-objective optimization problem.

The basic flow chart of the NSGA-II algorithm which is proposed by Deb [16], considering the constrained condition, is employed to carry out the optimization shown in Fig. 2. Some basic ideals are explained as follows:

(1) Chromosome representation. In this investigation, to overcome the ‘Hanming Cliff’ in the binary codes, all the values of the design variables are encoded by decimal code that is different from the original NSGA-II algorithm. For simplicity, the chromosome constructions of hybrid variables, objective function value, constraint violation value, the rank value, the crowding distance, are shown in Fig. 3.

(2) Constraint handling. In the multi-optimization problem, the constraint violation is described as:

In the constraint handling approaches, a tournament selection is employed where two solutions are compared at a time, and the following criteria are always enforced [17]:

a) Any feasible solution is preferred to any infeasible solution.

b) Among two feasible solutions, the one having smaller constraint violation is preferred.

c) Among two infeasible solutions, the one having smaller constraint violation is preferred.

(3) Non-dominated sorting. The constrain-domination condition [18] for any two solutions is described as follows:

If a solution $d_i$ is said to ‘constrain-dominate’ a solution $d_j$, then any of the following conditions are true:

a) Solutions $d_i$ is feasible and solution $d_j$ is not.

b) Solution $d_i$ and $d_j$ are both infeasible, but solution $d_i$ has a smaller constraint violation.

c) Solution $d_i$ and $d_j$ are feasible and solution $d_j$ dominates solution $d_j$ in the usual sense.

In the non-dominated sorting procedure, for each solution, two entities will be calculated firstly: 1) domination count $n_d$, the number of solutions which dominate the solution $d$, and 2) $S_d$, a set of solutions that the solution $d$ dominates. Then the following procedure [16] is used to carry out the non-dominated sorting.

Step 1: Find all the solutions with the domination count $n_d$ being zero, and set these solutions in the first non-dominated front. The rank value for these solutions is set to be one.

Step 2: For each solution $d$ with $n_d=\text{0}$, each member $q$ of its set $S_d$ is visited and its domination count is reduced by one.

Step 3: In doing so, if for any member $q$, the domination count becomes zero, it will be put in a separate set $Q_d$. These members belong to the second non-domination front. The rank value for these solutions is set to be two.

Step 4: The above procedure is continued with each member of $Q_d$ and the third front is identified. And the process continues until all fronts are identified.

(4) Crowding distance. The crowding distance of the $i$th solution (marked with solid circles) in its front which is shown in Fig. 4, is the average side-length of the cuboids (shown by a dashed box). It denotes the diversity distribution of the solution in its front.

Fig. 4Flow chart of the optimization algorithm

(5) The genetic operator

In the standard NSGA-II, simulated binary crossover operator and polynomial mutation operator are served as genetic operators. But they can be used to solve real coded optimization problem. Herein, the new genetic operators are introduced and improved for the special optimization problem of the CLD/plate.

a) The new crossover operator

The Laplace crossover operator which is proposed in [19], is introduced in the NSGA-II to deal with the hybrid/mixed design variables optimization problems. According to the Laplace crossover operator, the two offspring produced by the two parents, $x_i^1$, $x_i^2$ are computed as:

where:

$u_i$, $r_i$ are uniform random numbers between 0 and 1, $a\in R$ is called location parameter, and $b>\text{0}$ is termed as scale parameter. For integer design variables, $b=b_{int}$, is usually assigned to be an integer, otherwise, $b=b_{real}$ is set to be a positive real value below 1. For smaller value of $b$, offspring are expected to be produced near the parents, and for lager $b$, offspring are likely to be produced far from parents.

Because two or more CLD patches can not be bonded on the same location on the base plate, i.e., $l_i\ne l_j$ for $i\ne j$, the procedure of the original crossover operator will be modified. Based on the smallest distance principle, the crossover operator will be carried out for every pair variables denoting the locations of CLD patches in the parent. The procedure is expressed as follows:

Step 1: The crossover operator is carried out for the first pair variables $l_1^{p1}$ and $l_1^{p2}$ in the parent and the first two offspring are got as follows: