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Research ArticleTopology Optimization for Minimizing the Resonant Responseof Plates with Constrained Layer Damping Treatment

Zhanpeng Fang and Ling Zheng

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

Correspondence should be addressed to Ling Zheng; [email protected]

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

Academic Editor: Miguel Neves

Copyright © 2015 Z. Fang and L. Zheng. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

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 ofdisplacement resonant response in frequency domain at the specified point is considered as the objective function. Two sensitivityanalysismethods are investigated and discussed.Thederivative ofmodal damp ratio is not considered in the conventional sensitivityanalysis method. An improved sensitivity analysis method considering the derivative of modal damp ratio is developed to improvethe computational accuracy of the sensitivity.The evolutionary structural optimization (ESO)method is used to search the optimallayout of CLDmaterial on plates. Numerical examples and experimental results show that the optimal layout of CLD treatment onthe plate from the proposed topology optimization using the conventional sensitivity analysis or the improved sensitivity analysiscan reduce the displacement resonant response. However, the optimization method using the improved sensitivity analysis canproduce a higher modal damping ratio than that using the conventional sensitivity analysis and develop a smaller displacementresonant response.

1. Introduction

The CLD treatment has been regarded as an effective way tosuppress structural vibrations and sound radiation. Amongfirst studies of CLD treatments, Kerwin [1] developed asimplified theory to calculate the loss factor of a plate withCLD treatments. DiTaranto and Blasingame [2] extendedKerwin’s work by accounting for the extensional deforma-tions in the viscoelastic layer and obtained loss factors forthree and five layer beams as a function of frequency. Meadand Markus [3] derived sixth-order equation of motion forbeams in terms of transverse displacements. Most of theseearly works dealt with full coverage passive constrainedlayer damping (PCLD) treatments that are evidently notpractical in purpose. For instance, in the damping treatmentapplications to the automotive and airplane industries, theweight constraint may not allow full PCLD coverage, sincethe added weight induced by any vibration and noise controldesign modification has to be limited to a small amount.

Therefore, a partial-coverage treatment is a more attractiveapproach to fulfill the real PCLD treatment requirements.Nokes and Nelson [4] were among the earliest investigatorsto provide the solution to the problem of a partially coveredsandwich beam. The modal strain energy method was usedto calculate modal loss factors of the treated beam forsymmetric boundary conditions. Lall et al. [5, 6] performeda more thorough analytical study for partially covered planarstructures. An important finding from the analysis was thatfor suitably chosen parameters higher values of the modaldamping factor may be obtained for a partially covered beamcompared to that obtained for a fully covered one under thecondition of the same added weight of PCLD treatments.This encourages many researchers to investigate the optimallayout of CLD treatment in thin plates and shells.

Kodiyalam and Molnar [7] improved the modal strainenergy (MSE) method through introducing a new methodto account for viscoelastic material property variations withfrequency. Lumsdaine [8] determined the optimal shape of

Hindawi Publishing CorporationShock and VibrationVolume 2015, Article ID 376854, 11 pageshttp://dx.doi.org/10.1155/2015/376854

2 Shock and Vibration

a constrained damping layer on an elastic beam by meansof topology optimization. The optimization objective is tomaximize the system loss factor for the first mode of thebeam. Zheng et al. [9] studied the optimal placement of rect-angular damping patches for minimizing the structural dis-placement of cylindrical shells with the Genetic Algorithm.Moreira and Rodrigues [10] used MSE method to optimallylocate passive constrained viscoelastic damping layers onstructures. They also verified their work by comparing theirresults with experimental tests. Li and Liang [11] utilized theresponse surface method (RSM) to analyze and optimize thevibroacoustic properties of the damping structure. Ling et al.[12] studied the topology optimization for the layout ofCLD in plates to suppress their vibration and sound radia-tion. The objective is to minimize structural modal damp-ing ratios using the method of moving asymptotes (MMA).Lepoittevin and Kress [13] proposed a new method forenhancement of damping capabilities of segmented con-strained layer damping material and developed an optimiza-tion algorithm using mathematical programming to identifya cuts arrangement that optimized the loss factor. Ansari etal. [14] studied the optimal number and location of the CLDsheets on the flat structure surface utilizing an improved gra-dient method and the objective was to maximize loss factorof the system. Kim et al. [15] compared the modal loss factorsobtained by topology optimization to the conventional strainenergy distribution (SEDmethod) and themode shape (MSOapproach). It is found that topology optimization based onthe rational approximation for material properties (RAMP)model and optimality criteria (OC) method can provideabout up to 61.14 percent higher modal loss factor than SEDand MSO methods. The numerical model and the topologyoptimization approach are also experimentally validated.

More recent attempts on the simultaneous shape andtopology optimization of dynamic response have achievedsome success. Rong et al. [16] developed the evolutionarystructural optimization (ESO) method for the topologyoptimization of continuum structure under randomdynamicresponse constraints. Zheng et al. [17] studied the optimalplacement of rectangular damping patches for minimizingthe structural displacement of cylindrical shells with theGenetic Algorithm. Pan and Wang [18] applied adaptivegenetic algorithm to optimize truss structure with frequencydomain excitations.The optimization constraints include dis-placement responses under force excitations and accelerationresponses under foundation acceleration excitations. Alvelid[19] proposed a modified gradient of damping materialsto mitigate noise and vibration. Yoon [20] proposed thetopology optimization based on the SIMP method for thefrequency response problem where the structure is subjectto the wide excitation frequency domain. Shu et al. [21]propose level set based structural topology optimization forminimizing frequency response. The general objective func-tion is formulated as the frequency response minimization atthe specified points or surfaces with a predefined excitationfrequency or a predefined frequency range under the volumeconstraint.

Although above optimal approaches can supply thedesign methods to reduce structural vibration and sound

radiation in practice, the modal damping ratios for differentmodes are selected as the optimization objectives in thesestudies. In fact, the modal damping ratio is only the charac-teristics of vibration energy dissipation in structures ratherthan a controllable dynamic performance. Furthermore, thederivative of the modal damp ratio is usually neglectedin conventional sensitivity analysis which will give rise toerror in sensitivity analysis and produce a deviation fromthe optimal layout of CLD treatment in structures. Thispaper develops a dynamic topology optimization methodof plates with CLD treatment under specified broadbandharmonic excitation, which includes one or more resonancefrequencies. The optimization objective is to minimize thesquare of displacement resonance response instead of con-ventional modal damping ratio. Meanwhile, an improvedsensitivity analysis method with considering the derivative ofmodal damp ratio is developed in order to obtain the moreaccurate layout of CLD material on the plate and the lowerdisplacement resonance response of CLD system.

2. Statement of Optimization Formulation

2.1. Dynamic Responses Using theMode SuperpositionMethod.Thefinite element equation for a structural dynamic problemunder external harmonic excitations can be written in theform

M

𝑋 + C

𝑋 + K𝑋 = 𝑓 (𝑡) , (1)

where M, C, and K are the mass, damping, and stiffnessmatrices of the structure, respectively. 𝑓(𝑡) = 𝐹𝑒𝑖𝜔𝑓𝑡 is thevector of excitation force. 𝜔

𝑓and 𝐹 represent the frequency

of the harmonic excitation and amplitude, respectively. 𝑋,

𝑋, and

𝑋 denote the displacement, velocity, and accelerationvectors, respectively.

For a plate with CLD treatment, thematricesM andK canbe further expressed as

M = M𝑏+MV +M𝑐,

K = K𝑏+ KV + K𝑐,

(2)

where the subscript 𝑏, V, and 𝑐 denote the base plate,the viscoelastic damping layer, and the constrained layer,respectively.

For the CLD plate system, the 𝑖thmodal loss factor can bewritten by using MSE method [22]:

𝜂

𝑖=

𝜂V𝐸𝑖

𝑠V

𝐸

𝑖

𝑠

= 𝜂V

∑

𝑛

𝑗=1𝐸

𝑖𝑗

𝑠V

∑

𝑛

𝑗=1(𝐸

𝑖𝑗

𝑠𝑝 + 𝐸𝑖𝑗

𝑠V + 𝐸𝑖𝑗

𝑠𝑐)

, (3)

where 𝜂V is the loss factor of the damping material. 𝐸𝑖𝑠V is

the 𝑖th mode strain energy of the damping material. 𝐸𝑖𝑠is the

total 𝑖th mode strain energy of CLD system. 𝐸𝑖𝑗𝑠𝑝, 𝐸𝑖𝑗𝑠V, and 𝐸𝑖𝑗

𝑠𝑐

are the 𝑖th mode strain energy of the base material, dampinglayer, and constrained layer at 𝑗th element, respectively.

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

𝜂 = 2𝜁

√

1 − 𝜁

2.

(4)

Shock and Vibration 3

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

Modal damp ratio

Mod

al lo

ss fa

ctor

𝜂 = 2𝜁

𝜂 = 2𝜁(1 − 𝜁2)1/2

Figure 1: Relationship between loss factor 𝜂 and damping ratio 𝜉.

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

𝜂 = 2𝜁. (5)

Based on above analysis, it is assumed that the dampingin CLD plate system can be expressed by using a proportionaldamping form. Equation (1) can be uncoupled as follows:

𝑦

𝑖+ 2𝜔

𝑖𝜁

𝑖𝑦

𝑖+ 𝜔

2

𝑖𝑦

𝑖= 𝑞

𝑖 (𝑖 = 1, 2, . . . , 𝑁) ,(6)

where 𝑦𝑖is the modal coordinates defined by

𝑋 = [𝜑

1, 𝜑

2, . . . , 𝜑

𝑛]

{

{

{

{

{

{

{

{

{

{

{

{

{

𝑦

1

𝑦

2

.

.

.

𝑦

𝑛

}

}

}

}

}

}

}

}

}

}

}

}

}

= Φ𝑦. (7)

The displacement response in time domain is furtherexpressed by using the mode superposition method:

𝑋(𝜔

𝑓) =

𝑚

∑

𝑖=1

𝜑

𝑖𝜑

𝑇

𝑖𝐹

√

(𝜔

2

𝑖− 𝜔

2

𝑓)

2

+ (2𝜁

𝑖𝜔

𝑓𝜔

𝑖)

2

. (8)

In the above equations, 𝜔𝑖and 𝜑

𝑖are the 𝑖th circular

frequency and mode shape normalized with respect to themass matrix. 𝜁

𝑖denotes the mode damping ratio of 𝑖th mode.

𝑞

𝑖is the 𝑖th element ofΦ𝑇𝑓(𝑡) and is called the 𝑖th generalized

excitation.𝑚 is the number of truncated modes.When CLD plate system is subjected to harmonic exci-

tations, the exciting frequency range is assigned to [𝜔𝑙, 𝜔

𝑡],

which includes one or several resonant frequencies. Onthe other hand, it is assumed that the system has a smalldamping and no repeated frequency in the range. Therefore,the displacement response at frequency 𝜔

𝑖is similar to single

modal response, which can be expressed as follows:

𝑋(𝜔

𝑖) =

𝜑

𝑖𝜑

𝑇

𝑖𝐹

2𝜁

𝑖𝜔

2

𝑖

. (9)

2.2. Formulation of the Optimization Problem. In previousstudies, the dynamic compliance is often considered as theobjective function of dynamic optimization [24, 25]. It con-centrates the frequency response over the whole structure.However, the local frequency response is quite importantfor some practical problems [21]. The frequency responseat specified points or surfaces should be taken into theobjective function of dynamic structural optimization tosuppress the vibration on these points or surfaces. In thispaper, minimizing the square of resonance amplitude at thespecified point is selected as the optimization objective toachieve the aim. On the other hand, the consumption ofCLD material is limited strictly to reach the light weightof the structure. The topology optimization problem is thusformulated as

min: 𝑠 =

𝑚

∑

𝑖=1

𝑤

𝑖𝑋

2(𝜔

𝑖)

s.t: 𝑉

0−

𝑛

∑

𝑒=1

𝑝

𝑒𝑉

𝑒= 0

𝑝

𝑒= {0, 1} , 𝑒 = 1, 2, . . . , 𝑛,

(10)

where 𝑝𝑒is the design variables, and it can be assigned 1 or

0. 𝑝𝑒= 1 denotes that the 𝑒th element is bonded with CLD

material, while 𝑝𝑒= 0 denotes that 𝑒th element is not bonded

with CLD material. 𝑛 is the number of elements and𝑚 is thenumber of points. 𝑤

𝑖is the 𝑖th weighting coefficient. 𝑠 is the

objective function. 𝑉𝑒is the volume of an individual element

and 𝑉0is the volume constraint for the consumption of CLD

material.

3. The Sensitivity Analysis

The sensitivity analysis of objective function with respectto the design variables is required to solve the previousoptimization problem. From (9), it can be seen that the dis-placement response at a specified point depends on resonantfrequency and the mode damping ratio and mode shapenormalized with respect to the mass matrix. Furthermore,the sensitivity of objective function with respect to the designvariables can be derived from (10):

𝜕𝑠

𝜕𝑝

𝑒

= 2

𝑎

∑

𝑖=1

𝑤

𝑖𝑋(𝜔

𝑖)((

𝜕𝜑

𝑖

𝜕𝑝

𝑒

𝜑

𝑇

𝑖+ 𝜑

𝑖

𝜕𝜑

𝑇

𝑖

𝜕𝑝

𝑒

)

𝐹

2𝜁

𝑖𝜔

2

𝑖

− 2𝜁

𝑖

𝜕𝜔

2

𝑖

𝜕𝑝

𝑒

𝜑

𝑖𝜑

𝑇

𝑖𝐹

(2𝜁

𝑖𝜔

2

𝑖)

2

−2𝜔

2

𝑖

𝜕𝜁

𝑖

𝜕𝑝

𝑒

𝜑

𝑖𝜑

𝑇

𝑖𝐹

(2𝜁

𝑖𝜔

2

𝑖)

2) .

(11)

In (11), the first term represents the partial derivativeof eigenvector to the design variable. The second termrepresents the partial derivative of eigenvalue to the designvariable. The third term denotes the partial derivative ofmodal damping ratio to the design variable. In the conven-tional sensitivity analysis, the modal damping ratio is often


assumed as a constant [16]. Therefore, the third term is notconsidered in

𝜕𝑠

𝜕𝑝

𝑒

= 2

𝑎

∑

𝑖=1

𝑤

𝑖𝑋(𝜔

𝑖)((

𝜕𝜑

𝑖

𝜕𝑝

𝑒

𝜑

𝑇

𝑖+ 𝜑

𝑖

𝜕𝜑

𝑇

𝑖

𝜕𝑝

𝑒

)

𝐹

2𝜁

𝑖𝜔

2

𝑖

−2𝜁

𝑖

𝜕𝜔

2

𝑖

𝜕𝑝

𝑒

𝜑

𝑖𝜑

𝑇

𝑖𝐹

(2𝜁

𝑖𝜔

2

𝑖)

2) .

(12)

Equation (12) is the conventional sensitivity analysis. Infact, the modal damping ratio in CLD system always variesin optimization process due to the layout modification ofCLD treatment on structural surface. In view of that, theimproved sensitivity analysis method is developed and themodal damping ratio is not a constant in iteration process anymore. Equation (11) is used to calculate the sensitivity for eachelement accurately.

So far, some efficient methods have been used to calculatepartial derivative of eigenvector, such as the finite differencemethod, the modal method, the modal truncation method,Nelson’s method, and an improved modal method. In thispaper, the modal truncation method is used to calculatepartial derivative of eigenvector. According to the modaltruncation method, the derivative of eigenvector is written as

𝜕𝜑

𝑖

𝜕𝑝

𝑒

=

𝑚

∑

𝑘=1

𝛼

𝑖𝑘𝜑

𝑘, (13)

where

𝛼

𝑖𝑖= −

1

2

𝜑

𝑇

𝑖

𝜕M𝜕𝑝

𝑒

𝜑

𝑖= −

1

2

𝜑

𝑇

𝑖(M𝑒V +M

𝑒

𝑐) 𝜑

𝑖,

𝛼

𝑖𝑘=

𝜑

𝑇

𝑘(𝜕K/𝜕𝑝

𝑒− 𝜔

2

𝑖(𝜕M/𝜕𝑝

𝑒)) 𝜑

𝑖

𝜔

2

𝑖− 𝜔

2

𝑘

=

𝜑

𝑇

𝑘((K𝑒V + K

𝑒

𝑐) − 𝜔

2

𝑖(M𝑒V +M

𝑒

𝑐)) 𝜑

𝑖

𝜔

2

𝑖− 𝜔

2

𝑘

, 𝑖 = 𝑘.

(14)

The eigenvalue sensitivity is

𝜕𝜔

2

𝑖

𝜕𝑝

𝑒

= 𝜑

𝑇

𝑖(

𝜕K𝜕𝑝

𝑒

− 𝜔

2

𝑖

𝜕M𝜕𝑝

𝑒

)𝜑

𝑖

= 𝜑

𝑇

𝑖((K𝑒V + K

𝑒

𝑐) − 𝜔

2

𝑖(M𝑒V +M

𝑒

𝑐)) 𝜑

𝑖.

(15)

The derivative of the 𝑖thmodal loss factor can be obtainedfrom (3):

𝜕𝜂

𝑖

𝜕𝑝

𝑒

= 𝜂V𝐸

𝑖𝑒

𝑠V𝐸𝑖

𝑠− (𝐸

𝑖𝑒

𝑠V + 𝐸𝑖𝑒

𝑠𝑐) 𝐸

𝑖

𝑠V

(𝐸

𝑖

𝑠)

2. (16)

Thus, the derivative of the 𝑖th modal damping ratio isexpressed by applying (5):

𝜕𝜁

𝑖

𝜕𝑝

𝑒

=

𝜂V

2

𝐸

𝑖𝑒

𝑠V𝐸𝑖

𝑠− (𝐸

𝑖𝑒

𝑠V + 𝐸𝑖𝑒

𝑠𝑐) 𝐸

𝑖

𝑠V

(𝐸

𝑖

𝑠)

2. (17)

In order to eliminate checkerboard patterns in layoutoptimization, the mesh-independency filter [26] is used tomodify the element sensitivities.

Yes

State

Finite element model of theCLD plate is established

The modal analysisis carried out

Sensitivity analysis andmesh-independency filter

Is the constraintcondition satisfied?

Stop

No

The initial value ofdesign variables is set

sensitivities are deletedRR0 elements with maximum

RR0=RR0+ER

Figure 2: Flowchart of the topology optimization.

4. Optimization Strategy

The evolutionary structural optimization (ESO) is used tosolve the proposed optimization problem. The ESO methodhas been widely used in structural optimization since Xie andSteven [27] presented this method. The advantages of ESOmethod lie in its simplicity in achieving shape and topologyoptimization for both static and dynamic problems. Theflowchart for implementation of the topology optimizationprocedures is shown in Figure 2.The evolutionary procedurefor topology optimization of CLD structures to minimize thesquare of displacement response at resonant frequency for thespecified point is summarized as follows.

A The finite element model of the plate with CLDtreatment is established.

B The initial value of design variables is set. The con-straint consumption of CLD material 𝑉∗, the initialremoved elements of CLD material 𝑅𝑅

0, and evolu-

tionary ratio ER are determined, respectively.

C Themodal analysis of the plate with CLD treatment iscompleted.

D The sensitivities of objective function to design vari-able are calculated and the mesh-independency filtermethod is used to modify the sensitivity of theelement.

E Delete 𝑅𝑅0elements with minimum sensitivity val-

ues.


Table 1: Physical and geometrical parameters of CLD plate.

Length Width Thickness Young’s modulus Density Poisson’s ratio(m) (m) (mm) (MPa) (kg/m3)

Base layer 0.2 0.1 0.8 70 2800 0.3Damping layer 0.2 0.1 0.05 12 1200 0.495Constrained layer 0.2 0.1 0.13 70 2700 0.3

Table 2: A comparison between CLD system with full coverage and CLD system with optimal layout of CLD material on plate.

Full coverage The conventional sensitivity analysis The improved sensitivity analysisFrequency (Hz) 19.80 20.99 19.02Modal damp ratio 1.53 × 10

−33.18 × 10

−37.16 × 10

−3

Objective function (m2) 3.40 × 10

−39.05 × 10

−42.18 × 10

−4

f(t)

Figure 3: The cantilever CLD plate.

F Check the constraint condition; if the constraintcondition is satisfied, then stop calculation; or update𝑅𝑅

0by the formula of 𝑅𝑅

0= 𝑅𝑅

0+ ER.

G Repeat procedures C–G until the constraint condi-tion is satisfied.

5. Numerical Examples

5.1. The Cantilever Plate/CLD System. Figure 3 is a cantileverplate/CLD system. The main physical and geometricalparameters of the base plate, the damping layer, and con-strained layer are listed in Table 1.The loss factor of viscoelas-tic material is 0.5. A unit harmonic force is applied at themidpoint of the free edge. The range of excitation frequencyis [10Hz–40Hz], which includes the first-order resonantfrequency (19.80Hz). The design objective is considered asthe square of the first-order resonant amplitude at the loadingpoint.

The volume consumption of CLD material is limited to50% of full coverage in optimization process. The objectivefunction is to minimize the square of displacement responseat first-order resonant frequency for the specified loadingpoint. In order to obtain the symmetric results, only thesensitivity for the upper half of the cantilever CLD plate iscalculated and the sensitivity for the whole cantilever CLDplate is obtained through the symmetry constraint. The ESOmethod is used to solve the proposed optimization problem.According to ESOmethod, when iteration number increases,the rejection ratio also increases.

Figures 4 and 5 demonstrate the topology evolvingprocesses by using conventional sensitivity analysis and theimproved sensitivity analysis, respectively. It is seen that someholes appear in the optimal layout of CLD material whenthe improved sensitivity analysis is used and the boundaryline of CLD material is not smooth anymore. Figures 6, 7,and 8 show iteration histories of the first-mode frequency,modal damping ratio, and the objective function, respectively.It is noted that the modal damping ratio in CLD system byusing improved sensitivity analysis increases more rapidlythan by using conventional sensitivity analysis. Meanwhile,the objective function decreases rapidly to a lower value thanthat obtained from the conventional sensitivity analysis. Edgeeffect should be considered as the main reason to explainthe phenomenon. The shear deformation occurring in thedamping layer is responsible for the dissipation of energy andcutting both the damping layer and constrained layer willincrease the shear deformation at relative position [13]. Thisimplies that the improved sensitivity analysis ismore accurateon the layout optimization of CLDmaterial on the cantileverplate, which also validates the effectiveness of the proposedimproved sensitivity analysis.

Table 2 compares information before and after opti-mization. Data demonstrates that the first-mode frequencydecreases by 0.3%, the modal damping ratio with optimallayout is 4.67 times that with full coverage, and the objectivefunction is only 0.06% of CLD system with full coverage.It is noticed that, in optimization process, only 50% CLDmaterial is applied on the surface of the cantilever plate. Thismeans that dynamic response in firstmode can be suppressedsignificantly with an optimal layout of CLD material andmuch less extra weight. It is usually required in engineeringapplications due to structural light weight trend.

5.2. The Clamped Plate/CLD System. Figure 9 shows aclamped plate/CLD system with all clamped plates on foursides.The base plate is made of steel, with the length of 0.6m,width of 0.4m, and thickness of 0.0015m. Young’s modulus,density, and Poisson’s ratio are 212GPa, 7860 kg/m3, and 0.3,respectively.The same CLDmaterial as the previous exampleis applied on the plate. A unit harmonic force is applied atthe center of the plate. The range of the exciting frequency


(a) (b)

(c)

Figure 4: Topology evolving process using conventional sensitivity analysis. (a) Iteration 7 (rejection ratio 17.5%). (b) Iteration 14 (rejectionratio 35%). (c) Iteration 20 (rejection ratio 50%).

(a) (b)

(c)

Figure 5: Topology evolving process using improved sensitivity analysis. (a) Iteration 7 (rejection ratio 17.5%). (b) Iteration 14 (rejection ratio35%). (c) Iteration 20 (rejection ratio 50%).

is selected from 50Hz to 150Hz, which includes the first-and second-mode frequency. The objective function is thesum of the amplitude squares for the first-order and second-order displacement response at the loading point. Settingthe weighted factors for both the first and second mode as0.5, the objective function can be calculated by 𝑤

1𝑋

2(𝜔

1) +

𝑤

2𝑋

2(𝜔

2), 𝑤1= 𝑤

2= 0.5. In this example, only the

sensitivity for the upper left quadrant of the clamped CLD

plate is calculated and the sensitivity for the whole clampedCLD plate is obtained through the symmetry constraint. Thevolume consumption of CLDmaterial is limited to 50%of fullcoverage in optimization process. According to ESOmethod,when iteration number increases, the rejection ratio alsoincreases.

The topology evolving processes by conventional sensitiv-ity analysis and improved sensitivity analysis are diagrammed


Table 3: A comparison between CLD system with full coverage and the layout of CLD material.

Full coverage The conventional sensitivity analysis The improved sensitivity analysisWeighted frequency (Hz) 83.44 82.11 81.50Weighted modal damp ratio 2.4 × 10

−32.57 × 10

−33.91 × 10

−3

Objective function (m2) 3.84 × 10

−73.35 × 10

−71.79 × 10

−7

0 5 10 15 2019

19.5

20

20.5

21

21.5

Iteration number

Freq

uenc

y (H

z)

Conventional sensitivity analysisImproved sensitivity analysis

Figure 6: Iteration histories of the first-order frequency.

0 5 10 15 200

2

4

6

8

Iteration number

Mod

al d

amp

ratio


×10−3

Figure 7: Iteration histories of modal damp ratio.

0 5 10 15 2019

19.5

20

20.5

21

21.5

Iteration number

Freq

uenc

y (H

z)


Figure 8: Iteration histories of objective function value.

f(t)

Figure 9: The clamped plate/CLD system.

in Figures 10 and 11, respectively. Figure 12 demonstrates theiteration histories of the weighted frequency (𝑤

1𝜔

1+ 𝑤

2𝜔

2).

Figure 13 shows the iteration histories of the weighted modaldamping ratio (𝑤

1𝜁

1+ 𝑤

2𝜁

2). Figure 14 shows the iteration

histories of the objective function. It is obvious that theweighted frequency by using the improved sensitivity analysisdecreases much more than that using the conventionalsensitivity analysis. It should be corresponding with actualsituation. Additionally, the weighted modal damping ratioof CLD system increases rapidly and reaches a higher valuethan that using the conventional sensitivity analysis. In thesame way, the objective function decreases to a lower valuethan that using the conventional sensitivity analysis. Edgeeffect is also the main reason to explain the phenomenon.Some holes appear in the optimal layout of CLD materialusing the improved sensitivity analysis and they increase theshear deformation at those positions. Therefore, the layoutof CLD material obtained by improved sensitivity analysis ismore accurate in details than that obtained by conventionalsensitivity analysis. The influence of the improved sensitivityanalysis on the layout of CLDmaterial and vibration suppres-sion in plates is significant.

Table 3 shows that the improved sensitivity analysisresults in obvious increase in the modal damping ratio ofthe clamped plate with CLD treatment and a rapid dropin the displacement response for the first and the secondmode. It is seen that the weighted modal damping ratio usingimproved sensitivity analysis increases 52% more than thatusing conventional sensitivity analysis. Of course, whetherthe conventional or improved sensitivity analysis is used intopology optimization process, it is obvious that the optimallayout of CLDmaterial on the plate may result in the effectivevibration suppression at a cost of 50% CLD material applied.


(a) (b)

(c)

Figure 10: Topology evolving process using conventional sensitivity analysis. (a) Iteration 14 (rejection ratio 17.5%). (b) Iteration 28 (rejectionratio 35%). (c) Iteration 40 (rejection ratio 50%).

(a) (b)

(c)

Figure 11: Topology evolving process using improved sensitivity analysis. (a) Iteration 14 (rejection ratio 17.5%). (b) Iteration 28 (rejectionratio 35%). (c) Iteration 40 (rejection ratio 50%).


0 10 20 30 4081.5

82

82.5

83

83.5

Iteration number

Wei

ght f

requ

ency

(Hz)


Figure 12: Iteration histories of the frequency.

0 5 10 15 20 25 30 352

2.5

3

3.5

4

Iteration number

Wei

ghte

d m

odal

dam

p ra

tio

×10−3


Figure 13: Iteration histories of the modal damp ratio.

0 5 10 15 20 25 30 351

2

3

4

5

Iteration number

×10−7


Obj

ectiv

e fun

ctio

n (m

2 )

Figure 14: Iteration histories of the modal damp ratio.

6. Experimental Verification

The three samples are prepared and the vibration responsesare measured in order to verify the effectiveness of theoptimal layout of CLD material for the first example shownin Figure 3. These measured samples are shown in Figure 15.

Figure 15(a) is the cantilever plate with full coverage. Fig-ure 15(b) is the optimal layout by using conventional sen-sitivity analysis. Figure 15(c) is the optimal layout by usingimproved sensitivity analysis.Three holes are drilled to clampthe sample.

In experimental process, a hammer “PCB086C03” is usedto supply an excitation for these cantilever plates with CLDtreatment. The displacement response is measured by usinga laser displacement sensor “KEYENCE LK- H080” and alldates are recorded and analyzed by LMS test system. Theexcitation point and the measure point are at the same place,which locate at the middle of the free end of cantilever plate.The experimental setup is shown in Figure 16.

The frequency response comparison between numericaland experimental results of the cantilever plates is shown inFigure 17.The results obtained indicate that the experimentalresults are in agreement with numerical calculations. Onthe other hand, it is noted that the optimal layout of CLDmaterial on the cantilever plate can significantly reducethe displacement response at the peak of the first naturalfrequency. Meanwhile, the consumption of CLD material isonly 50% of full coverage CLDmaterial. In addition, it is seenthat the optimal layout of CLD treatment on the cantileverplate using the improved sensitivity analysis can indeed resultin more effective vibration suppression than that using theconventional sensitivity analysis.

7. Conclusion

In this paper, an improved sensitivity analysis method isproposed and used to obtain the optimal layout of CLDmaterial on plates.The optimizationmodel is established andthe square of the displacement response at the first resonantfrequency or other resonant frequencies is considered as theobjective function to suppress structural vibration in a smallextra mass of CLD material. The ESO method is adopted tosearch the optimal layout of CLD material on the plates. Thenumerical examples are given and the experiments are carriedout to verify the effectiveness of the improved sensitivityanalysis. The experimental and theoretical results show that


(a) (b)

(c)

Figure 15: The experimental samples. (a) Full coverage. (b) The optimal layout by using conventional sensitivity analysis. (c) The optimallayout by using improved sensitivity analysis.

LMS SCADAS III dataacquisition front-end

Fixture

Laser displacement

sensor

Plate/CLD systemComputer with LMS Test. Lab

Force signal

Displacement signal

Hammer

(a) (b)

Figure 16: The experimental setup. (a) Schematic drawing. (b) The picture of the plate clamped in its fixture.

10 15 20 25 30 35 400

0.02

0.04

0.06

Am

plitu

de (m

/N)

10 15 20 25 30 35 40−200

−150

−100

−50

0

Frequency (Hz)

Frequency (Hz)

Phas

e (de

g)

Full coverage—numericalImproved sensitivity analysis—numericalConventional sensitivity analysis—numericalFull coverage—experimentalImproved sensitivity analysis—experimentalConventional sensitivity analysis—experimental

Figure 17: Frequency response comparison between numerical and experimental results of the cantilever plates.


the optimal layout of CLD treatment on plates can signif-icantly reduce the displacement response for one or moreresonant frequencies in a frequency range by the proposedtopology optimization method. Moreover, by contrast, theimproved sensitivity analysis proposed in this paper canincrease the modal damping ratio of CLD system and supplymore accurate layout of CLD treatment. This results in moreeffective vibration suppression in plates with CLD treatment.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundationof China (Grant no. 50775225) and the State Key Laboratoryof Mechanical Transmission, Chongqing University (Grantno. 0301002109165). These financial supports are gratefullyacknowledged.

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