- Email: [email protected]
Optimal damping layout in a shell structure using topology optimization
Journal of Sound and Vibration 332 (2013) 2873–2883
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Rapid Communications
Optimal damping layout in a shell structure using topology optimization Sun Yong Kim n, Chris K. Mechefske, Il Yong Kim Department of Mechanical and Materials Engineering, Queen’s University, Kingston, Ontario, Canada
a r t i c l e i n f o
abstract
Article history: Received 1 June 2011 Received in revised form 3 November 2012 Accepted 10 January 2013 Handling Editor: I. Trendafilova Available online 1 March 2013
Viscoelastic damping material attached on the surface of a structure is widely used to suppress the resonance vibration in aerospace, automobiles, and various other applications. A full treatment of damping material is not an effective method because the damping effect is not significantly increased compared to that obtained by an effective partial damping treatment. In addition, the weight of the structure is increased significantly, which can cause poor system performance. Topology optimization is recently implemented in order to find an effective optimal damping treatment. The objective function is maximization of the damping effect (i.e. the modal loss factor) and the constraint is a maximum allowable volume of damping material. In this paper we compare the modal loss factors obtained by topology optimization to the ones obtained by other approaches, in order to determine which approach provides a better damping treatment (i.e. higher value of the modal loss factor). As a result, topology optimization provides about up to 61.14 per cent higher modal loss factor, as confirmed by numerical example. The numerical model for finite element analysis and topology optimization is also experimentally validated by comparing the numerical results to the experimental modal loss factors. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Control of noise and vibration has become a fundamental concern in many applications in order to improve the performance, durability, and comfort of products. There are many techniques that can reduce noise and vibration such as active, passive, and semi-active methods. Among them, the passive method, which is to attach viscoelastic damping material on the surface of a structure, is widely used as it is relatively simple and economical in implementation and it provides high damping effects over a wide range of frequency and temperature [1]. One important factor in reducing vibration with damping material is an effective partial placement of damping material since it is well known that the whole damping treatment does not provide a significantly improved damping effect. In the literature, many researchers have proposed a variety of approaches in order to find an effective partial placement of damping treatment within a given amount of damping material. Parathasarathy et al. [2] and Roy and Ganesan [3] made ‘‘manually’’ the configurations of partial damping layer treatments to analyze the damping effectiveness. Yildiz and Stevens [4] and Huang et al. [5] considered the thickness of the damping layer as a design variable so an optimization is used to vary the individual thickness of each damping layer. Zaiwei and Xinhua [6] adopted the Response Surface Method
n
Corresponding author. E-mail address: [email protected] (S.Y. Kim).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.01.029
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(RSM) to minimize the sound radiation from a vibrating panel by changing the thickness of the base panel and damping layer. Zheng et al. [7] used the Genetic Algorithm (GA) to find optimal locations of damping patches for minimizing vibration response of a cylindrical shell structure. Mohammed et al. [8] also used the GA as an optimization strategy for optimum design of the constrained-layer damping treatment. Alvelid [9] proposed a modified gradient method to find ‘‘permissible positions’’ of damping materials to mitigate noise and vibration. Zheng et al. [10] recently used topology optimization with MMA approach to optimally design damping treatment by maximizing damping effect. Several researchers, without implementing mathematical optimization techniques, sought the optimal design of damping treatment using the strain energy distribution (SED) method [11]. The basic idea of the SED method is that the damping material is distributed over regions of a base structure that experience high strain energy. More recently, an evolutionary structural optimization (ESO) was adopted to find an optimal damping layout [12]. However, the concept of this method in the unconstrained-layer problem is to apply the damping material where the regions are greatly deflected, which is based on the mode shape approach [13]. The various methods in the literature, which seek an optimal layout, may be classified into four primary approaches, as follows. An optimal damping material treatment is sought 1. 2. 3. 4.
based on an engineer’s experience and intuition, based on the mode shape (ESO approach), based on the strain energy distribution (SED method), or by using a mathematical optimization algorithm.
Since many different approaches are proposed in order to enhance the energy dissipation of vibrating structures, it may be necessary to determine which method provides a higher damping effect. Among them, we have chosen three different approaches: mode shape, the SED, and a mathematical optimization algorithm. There are also many different approaches in a mathematical optimization algorithm. Topology optimization [14] is selected in terms of mathematically rigorous optimal solutions, computational efficiency, and low manufacturing costs. Therefore, the objective of this paper is to provide a holistic study of damping material design by comparing the modal loss factor (i.e. damping effect) with respect to three widely used methods, and by conducting experimental validation of damping layouts obtained by topology optimization. To determine a damping treatment by mode shape and the SED, a sorting method (i.e. high deflection area will be applied by damping material using the mode shape approach, and higher strain energy area will be treated damping material using the SED method) is adopted. For the topology optimization approach, the objective function is maximization of damping effect at a specific mode, and the constraint is a maximum allowable damping material. The remainder of this paper is organized as follows. First, the linear viscoelastic constitutive behavior is reviewed, and then the topology optimization equations are developed. Second, a practical example is presented to demonstrate the effectiveness of topology optimization by comparing the modal loss factors. Then, the numerical analysis and optimization model is validated experimentally, and conclusions are presented.
2. Modeling of damping structure 2.1. Modeling of viscoelastic damping materials Viscoelastic damping material behavior occurs in a wide variety of materials and can be characterized by liquid-like elastic behavior. Materials that experience viscoelastic behavior include acrylics, rubber, and adhesives. The characteristics of viscoelastic materials depend on temperature and excitation frequency. For these materials, a linear elastic constitutive relationship using Hooke’s law is not an accurate representation. Instead, the complex modulus is extensively used to describe the dynamic characteristics of viscoelastic materials. The stress–strain relationship of a viscoelastic damping material subjected to steady-state oscillatory conditions can be represented by the structural damping model considering the complex modulus, En, as follows:
s~ ðoÞ ¼ En ðoÞe~ ðoÞ
(1)
where ðÞn denotes a complex quantity and s~ and e~ are the Fourier transforms of stress and strain, respectively. The complex modulus En is given by En ðoÞ ¼ E0 ðoÞ þ iE00 ðoÞ ¼ E0 ðoÞð1 þiZðoÞÞ
(2)
where E0 ðoÞ, E00 ðoÞ, and ZðoÞ are the frequency-dependent storage modulus, loss modulus, and material loss factor, respectively. To accurately describe the characteristic of viscoelastic damping material, the mathematical model has been described in a variety of ways, such as the ‘‘Maxwell model’’, the ‘‘Voigt model’’, and the standard ‘‘classical’’ model [1]. Recently, a great simplification in modeling viscoelastic damping material has been developed, particularly with respect to the
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frequency domain, by using fractional derivative models with four-parameters [15]: En ¼ Eð1 þ iZÞ ¼
a1 þ b1 ðioÞb 1 þ c1 ðioÞ
b
¼
a1 þ b1 ði2pf aðTÞÞb 1þ c1 ði2pf aðTÞÞb
(3)
damping material parameters, where En is a complex modulus, Z is a damping material loss factor, a1, b1, c1, and b are pffiffiffiffiffiffiffi which must be determined from a suitable curve-fit process applied to test data, and i ¼ 1. Furthermore, in accordance with the frequency–temperature equivalence principle, o can be expressed by the reduced frequency 2pf aðTÞ, where f is the frequency in Hertz, and aðTÞ depends on the temperature. T is an absolute temperature. The fractional derivative model has been widely used in past years since the model is able to predict the characteristics of viscoelastic damping material with only a few parameters. 3. Topology optimization 3.1. The formulation of topology optimization The aim of topology optimization is to determine an optimal layout within a given volume constraint. In topology optimization, the design domain is typically discretized by finite elements. Each element in the damping layer is assigned a design variable ðrd Þ whose value indicates the composition of the discretized domain. There is a variety of models such as the homogenization method, an artificial material design method, and a level set method. One of the models is the rational approximation for material properties (RAMP) [16]: Eðrd Þ ¼ Emin þ
rd 1 þ qð1rd Þ
ðE0 Emin Þ
(4)
E0
(5)
Adopting Emin ¼ 0, the equation can be simplified as Eðrd Þ ¼
rd 1 þ qð1rd Þ
where Eðrd Þ is intermediate storage modulus related to its density rd and E0 is a reference storage modulus. The value of q is chosen as 3, in this study. Note that rd is the design variable that represents the existence of material in each design element and it is different from the physical density of material. To find an optimal damping treatment using the RAMP method, the objective function is to maximize the modal loss factor at a specific mode, and the constraint is a maximum allowable volume of damping material. The optimization problem statement is maximize subject to
ND X
jk
rd ðxÞvd rV max
d¼1
rmin r rd ðxÞ r1
(6)
where jk is the modal loss factor at kth mode, and rd ðxÞ is the normalized density of the dth design variable at the damping layer. vd is the volume of the dth design variable, and V max is an allowable maximum volume which is expressed as a percentage of the entire design domain in this study. The lower bound, rmin , should be chosen to avoid the problem of numerical singularity. In this study, the lower bound is provided as rmin ¼ 0:01. To update design variables, sensitivities of the objective function with respect to design variables are necessary. The sensitivity of the objective function (the kth modal loss factor) with respect to the ith design variable, ri , can be determined by direct differentiation: qjk ¼ qri
Zk
P quk,d PNB quk,d PND ½ e ¼ 1 uk,e þ ND d ¼ 1 uk,d d ¼ 1 Zk uk,d qrd qrd PNB PND ½ e ¼ 1 uk,e þ d ¼ 1 uk,d 2 qu P Zk k,d NB e ¼ 1 uk,e qrd ¼ PNB PND ½ e ¼ 1 uk,e þ d ¼ 1 uk,d 2
(7)
where qjk =qri is the sensitivity of the kth mode with respect to the ith design variable. The sensitivity of the strain energy at the kth mode with respect to the dth element, quk,d =qrd , can be readily determined as follows: 0 E0 1 Ed quk,d 1 ¼ UTk,d k0d Uk,d ¼ d UTk,d kd Uk,d ¼ ðstrain energyÞ (8) 2 qrd Ed 2 Ed dth element where (0 ) denotes the derivative with respect to rd . Differentiating (5) with respect to the dth design variable becomes E0d ¼
1þ q ½1 þqð1rd Þ2
E0
(9)
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By substituting (9) into the (8), and finally obtaining the sensitivity calculation of quk,d =qrd as follows: 9 8 1 þq > > E > > = < 2 0 quk,d ½1 þ qð1rd Þ ¼ ðstrain energyÞ r > > qrd d > > E0 ; : 1 þ qð1rd Þ dth element 1 þq ðstrain energyÞ ¼ rd f1 þqð1rd Þg dth element
(10)
In order to verify the accuracy of (10), we compared the results to the finite difference method with 0.1 per cent of design variable perturbation. The results showed a good agreement. Although topology optimization is widely used, there still remains shortcomings such as checkerboard patterns and mesh-dependency. To overcome these shortcomings, the sensitivity filtering technique [14] is adopted. The scheme of sensitivity filtering is to modify the original sensitivity value as follows: ND X qd jk 1 qjk H^ j rj ¼ ðri Þ1 PND ^j qri qrj H j¼1 j¼1
(11)
where qjd k =qri and qjk =qrj are a modified element sensitivity value of the fth design variable and an original element sensitivity value of the jth design variable, respectively. The original sensitivity values are modified based on a weighted average of the design variable sensitivities in a fixed neighborhood and the convolution operator H^ j is written as H^ j ¼ r min distðd,jÞ,
fj 2 NDV9distðd,jÞ r r min g, d ¼ 1, . . . ,ND
(12)
where the operator distðd,jÞ is defined as the distance between the center of element d and the center of element j. The convolution operator H^ j is zero outside the filter size, r min . The convolution operator for element j decays gradually with the distance from element e. It is also necessary to define a ‘‘stop condition’’ of the optimization procedure. Even though there is a variety of convergence criteria, the following equation is mostly used in topology optimization: max9rd,l1 rd,l 9 r e, d
d ¼ 1, . . . ,ND
(13)
where rd,l1 and rd,l are the density of dth element at (l 1)th iteration and at lth iteration, respectively. For the tolerance value, e, we used 0.01 in this study. To update design variables in topology optimization, the Optimality Criteria (OC) method [14] is adopted. 4. Examples and physical validation The aim of an example is to show the effectiveness of topology optimization in order to find an optimal damping layout. It is known that an optimal damping treatment can be differently determined depending upon the approach so it is necessary to compare the modal loss factors. By comparing the modal loss factors obtained by three different approaches: mode shape, the SED, and topology optimization, it can be determined which method provides better optimal damping treatment. 4.1. Numerical example Table 1 shows the dimensions and material properties of a quarter-cylindrical shell structure. A clamped–clamped boundary condition is specified: all degrees-of-freedom at the two ends perpendicular to the z-direction are restrained. In order to examine how optimal damping layout changes according to the three different approaches, the shell problem is solved with 50 per cent damping material (Figs. 1–3). Quantitative performance comparisons are given in Table 2. This example shows that topology optimization usually provides better damping effect than the other two
Table 1 The material properties and dimensions of the base shell structure. Quarter-cylindrical shell structure Radius (R) (m) Length (L) (m) Thickness (t) (m) Theta (y) (deg) Young’s modulus (GPa) Poisson’s ratio Density (kg/m3)
0.19 0.45 0.002 90 200 0.3 7860
S.Y. Kim et al. / Journal of Sound and Vibration 332 (2013) 2873–2883
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Fig. 1. Optimal damping layouts obtained by mode shape approach with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
approaches. Specifically, the modal loss factor obtained by topology optimization at the 4th mode is 23.27 and 61.14 per cent higher than ones obtained by mode shape and the SED, respectively. 4.2. Experimental validation 4.2.1. Complex modulus of styrene–butadiene rubber (SBR) damping material It is necessary to measure the complex modulus of our SBR damping material (solid rubber type) [17] in our lab in order to accurately obtain the modal loss factors. Table 3 shows the obtained complex modulus of the SBR material. Our SBR material has a density of 1385.1 kg/m3; the density of usual SBR material is 930–1200 kg/m3 [18]. The thickness of our SBR damping material is 0.003 m. 4.2.2. Damped semi-cylindrical shell structure Fig. 4 shows a semi-cylindrical shell structure that is used in this study, and Table 4 shows the material properties and dimensions of the base shell structure. As shown in the figure, two supporting thick wooden plates are glued at the ends of the semi-cylindrical shell so that the following boundary conditions are satisfied: ux ¼ uy ¼ 0:0
at z ¼ 0 and z ¼ 0:9
(14)
where ux and uy represent displacements in x- and y-directions, respectively, and z is the coordinate in the longitudinal direction along the axis of revolution of the semi-cylinder. These boundary conditions state that the shell structure is clamped at both ends. The objective function for topology optimization is to maximize the modal loss factor at a specific mode and the maximum allowable volume constraint is 40 per cent. In topology optimization, the filtering radius is
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Fig. 2. Optimal damping layouts obtained by the SED approach with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
0.0161 m, and the OC method is used to update the design variables. The optimal damping layouts by topology optimization at three different modes are shown in Fig. 5. Then, the optimal SBR damping shapes obtained by topology optimization are applied to the semi-shell structure. Figs. 6–8 compare the experimental frequency response function of the base structure to that of the damped structure at the three target modes. It is noted that the frequency response function (H1,input ) is obtained for target mode 1 and target mode 3, but the frequency response function (H2,input ) is obtained for target mode 2. The subscriptions 1 and 2 of the frequency response function is measured at a different location. As shown in Fig. 6, the frequency response function of the damped structure is not significantly decreased (only 1 dB lower) compared to the one of the bare structure. The decrease at target mode 2, however, is significantly decreased (5.2 dB lower) as shown in Fig. 7. Since ‘‘the optimally shaped SBR damping treatments’’ are simplified according to the target modes (64, 142, and 175 Hz), the numerical modal loss factors are also recalculated based on the shape of the SBR damping material. The assumption in this study is the temperature which is constant at 23 1C, while, the frequency is considered to be a variable. It is important to obtain the complex modulus of the SBR damping material at the actual target modes (64, 142, and 175 Hz), but, the complex modulus is presented only at 23, 146, and 410.5 Hz as shown in Table 3. Based on our experience, it is assumed that there is only a minor difference in the complex modulus. Thus, the complex modulus of the SBR damping material at the 1st mode (23 Hz) is used for the calculation of the modal loss factor at target mode 1 (64 Hz) and the complex modulus of the SBR material at the 2nd mode (146 Hz) is used for calculation of the modal loss factors at
S.Y. Kim et al. / Journal of Sound and Vibration 332 (2013) 2873–2883
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Fig. 3. Optimal damping layouts obtained by topology optimization with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
Table 2 The comparison of the modal loss factors with 50 per cent damping material: clamped–clamped boundary condition. Mode
Mode shape
SED
Topology optimization
1st 2nd 3rd 4th
0.03207 0.04036 0.04201 0.03962
0.04215 0.05285 0.05507 0.03006
0.04518 0.05162 0.05509 0.04844
Table 3 The complex modulus of the composite beam. Mode
Damped natural frequencies (Hz)
Young’s modulus (GPa)
Loss factor
1st 2nd 3rd
23.5 146.0 410.5
0.287 0.162 0.167
0.101 0.210 0.469
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Fig. 4. Experimental set-up. Table 4 Material properties of the semi-shell cylindrical structure. Semi-cylindrical shell structure Radius (R) (m) Length (L) (m) Thickness (t) (m) Young’s modulus (GPa) Structural damping coefficient (%) Poisson’s ratio Density (kg/m3)
0.19 0.9 0.002 197 0.1 0.3 7850
Fig. 5. Optimal damping layouts of semi-cylindrical structure using topology optimization. (a) Target mode 1, (b) target mode 2 and (c) target mode 3.
target mode 2 (142 Hz) and target mode 3 (175 Hz). It is also noted that the reason that the frequency response function at target mode 2 decreases (5.2 dB lower) more significantly than at target mode 1 (1.0 dB lower) is that the loss factor of the damping material at the 2nd mode (0.210) is higher than at the 1st mode (0.101).
S.Y. Kim et al. / Journal of Sound and Vibration 332 (2013) 2873–2883
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Fig. 6. Comparison of frequency response functions between damped and bare structures: target mode 1.
Fig. 7. Comparison of frequency response functions between damped and bare structures: target mode 2.
Table 5 compares the modal loss factors obtained by numerical calculations and by experimental measurements. The maximum discrepancy between numerical simulation and experimental testing is only 6.29 per cent so topology optimization can be widely adopted to real-world problems in order to effectively determine an optimal damping layout. Among these three modes, the modal loss factor at target mode 2 is most accurately measured in terms of the complex modulus of the SBR damping material. This is because the resonant mode at target mode 2 is 142 Hz, and the complex modulus at the 2nd mode is obtained at 146 Hz.
5. Conclusion Since several approaches have been proposed in order to effectively design an optimal damping layout, it is necessary to investigate which approach provides a better result. The mode shape, the SED, and the topology optimization methods have been chosen to design an optimal damping layout of an unconstrained-layer shell structure. To determine an optimal damping layout using the mode shape and the SED approaches, a sorting method is adopted. For topology optimization, the objective function is to maximize the modal loss factor at a specific mode, and the constraint is an allowable maximum damping material. It is noted that the modal strain energy method is used to numerically calculate the modal loss factors.
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Fig. 8. Comparison of frequency response functions between damped and bare structures: target mode 3. Table 5 The comparison of the modal loss factors between numerical simulation and experimental testing. Target mode
Numerical simulation
Experimental testing
Ratio (%)
1 2 3
0.01184 0.02358 0.01384
0.01145 0.02237 0.01297
96.70 94.87 93.71
By quantitatively comparing the modal loss factors with respect to the different approaches, it was found that topology optimization usually provides a higher modal loss factors. Because it has been confirmed that topology optimization is the most effective, damping layouts determined by topology optimization are experimentally validated. For the validation of the numerical models, frequency response functions are measured for the damped structure in order to measure the modal loss factors. The results show that the numerical results have a high correlation with the testing data. References [1] C.T. Sun, Y.P. Lu, Vibration Damping of Structural Elements, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1995. [2] G. Parathasarathy, C.V.R. Reddy, N. Ganesan, Partial coverage of rectangular plates by unconstrained layer damping treatments, Journal of Sound and Vibration 102 (1985) 203–216, http://dx.doi.org/10.1016/S0022-460X(85)80053-5. [3] R.K. Roy, N. Ganesan, Dynamic studies on plates with unconstrained layer treatment, Computers & Structures 49 (1993) 473–480, http://dx.doi.org/ 10.1016/0045-7949(93)90048-I. [4] A. Yildiz, K. Stevens, Optimum thickness distribution of unconstrained viscoelastic layer treatment for plates, Journal of Sound and Vibration 103 (1985) 183–199, http://dx.doi.org/10.1016/0022-460X(85)90232-9. [5] S.C. Huang, D.J. Inman, E.M. Austin, Some design consideration for active and passive constrained layer damping patches, Smart Materials and Structures 5 (1996) 301–313, http://dx.doi.org/10.1088/0964-1726/5/3/008. [6] L. Zaiwei, L. Xinhua, Vibro-acoustic analysis and optimization of damping structure with response surface method, Materials and Design 28 (2007) 1999–2007, http://dx.doi.org/10.1016/j.matdes.2006.07.006. [7] H. Zheng, C. Cai, G.S.H. Pau, G.R. Liu, Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments, Journal of Sound and Vibration 279 (2005) 739–756, http://dx.doi.org/10.1016/j.jsv.2003.11.020. [8] A. Mohammed, A. Al, I.B. Raed, Optimum design of segmented passive-constrained layer damping treatment through genetic algorithms, Mechanics of Advanced Materials and Structures 15 (2008) 250–257, http://dx.doi.org/10.1080/15376490801907772. [9] M. Alvelid, Optimal position and shape of applied damping material, Journal of Sound and Vibration 310 (2008) 947–965, http://dx.doi.org/10.1016/ j.jsv.2007.08.024. [10] L. Zheng, R. Xie, Y. Wang, E.S. Adel, Topology optimization of constrained layer damping on plates using Method of Moving Asymptote MMA approach, Shock and Vibration 18 (2011) 221–244, http://dx.doi.org/10.3233/SAV-2010-0583. [11] M.G. Sainbury, R.S. Masti, Vibration damping of cylindrical shells using strain-energy-based distribution of an add-on viscoelastic treatment, Finite elements in Analysis and Design 43 (2007) 175–192, http://dx.doi.org/10.1016/j.finel.2006.09.003. [12] Z.Z. Guo, Y.Z. Chen, Q. Hou, Topology optimization in damping structure based on ESO, Journal of China Ordnance 4 (2008) 293–298. [13] X. Zang, D. Yu, L. Yao, R. Yan, Optimization of thickness distribution of unconstrained damping layer based on mode shapes, Zhongguo Jixie Gongcheng (China Mechanical Engineering) 21 (2010) 515–518. [14] M.P. Bendose, O. Sigmund, Topology Optimization, Theory, Methods and Applications, 2nd edition, Springer, Berlin, Heidelberg, Germany, 2003. [15] S.Y. Kim, D.H. Lee, Identification of fractional-derivative-model parameters of viscoelastic materials from measured FRFs, Journal of Sound and Vibration 324 (2009) 570–586, http://dx.doi.org/10.1016/j.jsv.2009.02.040.
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[16] M. Stolpe, K. Svanberg, An alternative interpolation scheme for minimum compliance topology optimization, Structural and Multidisciplinary Optimization 22 (2001) 116–124, http://dx.doi.org/10.1007/s001580100129. [17] /http://www.hannarubbercompany.com/S. [18] Y. Liao, V. Wells, Estimation of complex Young’s modulus of non-stiff materials using a modified Oberst beam technique, Journal of Sound and Vibration 316 (2008) 87–100, http://dx.doi.org/10.1016/j.jsv.2008.02.028.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Rapid Communications
Optimal damping layout in a shell structure using topology optimization Sun Yong Kim n, Chris K. Mechefske, Il Yong Kim Department of Mechanical and Materials Engineering, Queen’s University, Kingston, Ontario, Canada
a r t i c l e i n f o
abstract
Article history: Received 1 June 2011 Received in revised form 3 November 2012 Accepted 10 January 2013 Handling Editor: I. Trendafilova Available online 1 March 2013
Viscoelastic damping material attached on the surface of a structure is widely used to suppress the resonance vibration in aerospace, automobiles, and various other applications. A full treatment of damping material is not an effective method because the damping effect is not significantly increased compared to that obtained by an effective partial damping treatment. In addition, the weight of the structure is increased significantly, which can cause poor system performance. Topology optimization is recently implemented in order to find an effective optimal damping treatment. The objective function is maximization of the damping effect (i.e. the modal loss factor) and the constraint is a maximum allowable volume of damping material. In this paper we compare the modal loss factors obtained by topology optimization to the ones obtained by other approaches, in order to determine which approach provides a better damping treatment (i.e. higher value of the modal loss factor). As a result, topology optimization provides about up to 61.14 per cent higher modal loss factor, as confirmed by numerical example. The numerical model for finite element analysis and topology optimization is also experimentally validated by comparing the numerical results to the experimental modal loss factors. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Control of noise and vibration has become a fundamental concern in many applications in order to improve the performance, durability, and comfort of products. There are many techniques that can reduce noise and vibration such as active, passive, and semi-active methods. Among them, the passive method, which is to attach viscoelastic damping material on the surface of a structure, is widely used as it is relatively simple and economical in implementation and it provides high damping effects over a wide range of frequency and temperature [1]. One important factor in reducing vibration with damping material is an effective partial placement of damping material since it is well known that the whole damping treatment does not provide a significantly improved damping effect. In the literature, many researchers have proposed a variety of approaches in order to find an effective partial placement of damping treatment within a given amount of damping material. Parathasarathy et al. [2] and Roy and Ganesan [3] made ‘‘manually’’ the configurations of partial damping layer treatments to analyze the damping effectiveness. Yildiz and Stevens [4] and Huang et al. [5] considered the thickness of the damping layer as a design variable so an optimization is used to vary the individual thickness of each damping layer. Zaiwei and Xinhua [6] adopted the Response Surface Method
n
Corresponding author. E-mail address: [email protected] (S.Y. Kim).
0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.01.029
2874
S.Y. Kim et al. / Journal of Sound and Vibration 332 (2013) 2873–2883
(RSM) to minimize the sound radiation from a vibrating panel by changing the thickness of the base panel and damping layer. Zheng et al. [7] used the Genetic Algorithm (GA) to find optimal locations of damping patches for minimizing vibration response of a cylindrical shell structure. Mohammed et al. [8] also used the GA as an optimization strategy for optimum design of the constrained-layer damping treatment. Alvelid [9] proposed a modified gradient method to find ‘‘permissible positions’’ of damping materials to mitigate noise and vibration. Zheng et al. [10] recently used topology optimization with MMA approach to optimally design damping treatment by maximizing damping effect. Several researchers, without implementing mathematical optimization techniques, sought the optimal design of damping treatment using the strain energy distribution (SED) method [11]. The basic idea of the SED method is that the damping material is distributed over regions of a base structure that experience high strain energy. More recently, an evolutionary structural optimization (ESO) was adopted to find an optimal damping layout [12]. However, the concept of this method in the unconstrained-layer problem is to apply the damping material where the regions are greatly deflected, which is based on the mode shape approach [13]. The various methods in the literature, which seek an optimal layout, may be classified into four primary approaches, as follows. An optimal damping material treatment is sought 1. 2. 3. 4.
based on an engineer’s experience and intuition, based on the mode shape (ESO approach), based on the strain energy distribution (SED method), or by using a mathematical optimization algorithm.
Since many different approaches are proposed in order to enhance the energy dissipation of vibrating structures, it may be necessary to determine which method provides a higher damping effect. Among them, we have chosen three different approaches: mode shape, the SED, and a mathematical optimization algorithm. There are also many different approaches in a mathematical optimization algorithm. Topology optimization [14] is selected in terms of mathematically rigorous optimal solutions, computational efficiency, and low manufacturing costs. Therefore, the objective of this paper is to provide a holistic study of damping material design by comparing the modal loss factor (i.e. damping effect) with respect to three widely used methods, and by conducting experimental validation of damping layouts obtained by topology optimization. To determine a damping treatment by mode shape and the SED, a sorting method (i.e. high deflection area will be applied by damping material using the mode shape approach, and higher strain energy area will be treated damping material using the SED method) is adopted. For the topology optimization approach, the objective function is maximization of damping effect at a specific mode, and the constraint is a maximum allowable damping material. The remainder of this paper is organized as follows. First, the linear viscoelastic constitutive behavior is reviewed, and then the topology optimization equations are developed. Second, a practical example is presented to demonstrate the effectiveness of topology optimization by comparing the modal loss factors. Then, the numerical analysis and optimization model is validated experimentally, and conclusions are presented.
2. Modeling of damping structure 2.1. Modeling of viscoelastic damping materials Viscoelastic damping material behavior occurs in a wide variety of materials and can be characterized by liquid-like elastic behavior. Materials that experience viscoelastic behavior include acrylics, rubber, and adhesives. The characteristics of viscoelastic materials depend on temperature and excitation frequency. For these materials, a linear elastic constitutive relationship using Hooke’s law is not an accurate representation. Instead, the complex modulus is extensively used to describe the dynamic characteristics of viscoelastic materials. The stress–strain relationship of a viscoelastic damping material subjected to steady-state oscillatory conditions can be represented by the structural damping model considering the complex modulus, En, as follows:
s~ ðoÞ ¼ En ðoÞe~ ðoÞ
(1)
where ðÞn denotes a complex quantity and s~ and e~ are the Fourier transforms of stress and strain, respectively. The complex modulus En is given by En ðoÞ ¼ E0 ðoÞ þ iE00 ðoÞ ¼ E0 ðoÞð1 þiZðoÞÞ
(2)
where E0 ðoÞ, E00 ðoÞ, and ZðoÞ are the frequency-dependent storage modulus, loss modulus, and material loss factor, respectively. To accurately describe the characteristic of viscoelastic damping material, the mathematical model has been described in a variety of ways, such as the ‘‘Maxwell model’’, the ‘‘Voigt model’’, and the standard ‘‘classical’’ model [1]. Recently, a great simplification in modeling viscoelastic damping material has been developed, particularly with respect to the
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frequency domain, by using fractional derivative models with four-parameters [15]: En ¼ Eð1 þ iZÞ ¼
a1 þ b1 ðioÞb 1 þ c1 ðioÞ
b
¼
a1 þ b1 ði2pf aðTÞÞb 1þ c1 ði2pf aðTÞÞb
(3)
damping material parameters, where En is a complex modulus, Z is a damping material loss factor, a1, b1, c1, and b are pffiffiffiffiffiffiffi which must be determined from a suitable curve-fit process applied to test data, and i ¼ 1. Furthermore, in accordance with the frequency–temperature equivalence principle, o can be expressed by the reduced frequency 2pf aðTÞ, where f is the frequency in Hertz, and aðTÞ depends on the temperature. T is an absolute temperature. The fractional derivative model has been widely used in past years since the model is able to predict the characteristics of viscoelastic damping material with only a few parameters. 3. Topology optimization 3.1. The formulation of topology optimization The aim of topology optimization is to determine an optimal layout within a given volume constraint. In topology optimization, the design domain is typically discretized by finite elements. Each element in the damping layer is assigned a design variable ðrd Þ whose value indicates the composition of the discretized domain. There is a variety of models such as the homogenization method, an artificial material design method, and a level set method. One of the models is the rational approximation for material properties (RAMP) [16]: Eðrd Þ ¼ Emin þ
rd 1 þ qð1rd Þ
ðE0 Emin Þ
(4)
E0
(5)
Adopting Emin ¼ 0, the equation can be simplified as Eðrd Þ ¼
rd 1 þ qð1rd Þ
where Eðrd Þ is intermediate storage modulus related to its density rd and E0 is a reference storage modulus. The value of q is chosen as 3, in this study. Note that rd is the design variable that represents the existence of material in each design element and it is different from the physical density of material. To find an optimal damping treatment using the RAMP method, the objective function is to maximize the modal loss factor at a specific mode, and the constraint is a maximum allowable volume of damping material. The optimization problem statement is maximize subject to
ND X
jk
rd ðxÞvd rV max
d¼1
rmin r rd ðxÞ r1
(6)
where jk is the modal loss factor at kth mode, and rd ðxÞ is the normalized density of the dth design variable at the damping layer. vd is the volume of the dth design variable, and V max is an allowable maximum volume which is expressed as a percentage of the entire design domain in this study. The lower bound, rmin , should be chosen to avoid the problem of numerical singularity. In this study, the lower bound is provided as rmin ¼ 0:01. To update design variables, sensitivities of the objective function with respect to design variables are necessary. The sensitivity of the objective function (the kth modal loss factor) with respect to the ith design variable, ri , can be determined by direct differentiation: qjk ¼ qri
Zk
P quk,d PNB quk,d PND ½ e ¼ 1 uk,e þ ND d ¼ 1 uk,d d ¼ 1 Zk uk,d qrd qrd PNB PND ½ e ¼ 1 uk,e þ d ¼ 1 uk,d 2 qu P Zk k,d NB e ¼ 1 uk,e qrd ¼ PNB PND ½ e ¼ 1 uk,e þ d ¼ 1 uk,d 2
(7)
where qjk =qri is the sensitivity of the kth mode with respect to the ith design variable. The sensitivity of the strain energy at the kth mode with respect to the dth element, quk,d =qrd , can be readily determined as follows: 0 E0 1 Ed quk,d 1 ¼ UTk,d k0d Uk,d ¼ d UTk,d kd Uk,d ¼ ðstrain energyÞ (8) 2 qrd Ed 2 Ed dth element where (0 ) denotes the derivative with respect to rd . Differentiating (5) with respect to the dth design variable becomes E0d ¼
1þ q ½1 þqð1rd Þ2
E0
(9)
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By substituting (9) into the (8), and finally obtaining the sensitivity calculation of quk,d =qrd as follows: 9 8 1 þq > > E > > = < 2 0 quk,d ½1 þ qð1rd Þ ¼ ðstrain energyÞ r > > qrd d > > E0 ; : 1 þ qð1rd Þ dth element 1 þq ðstrain energyÞ ¼ rd f1 þqð1rd Þg dth element
(10)
In order to verify the accuracy of (10), we compared the results to the finite difference method with 0.1 per cent of design variable perturbation. The results showed a good agreement. Although topology optimization is widely used, there still remains shortcomings such as checkerboard patterns and mesh-dependency. To overcome these shortcomings, the sensitivity filtering technique [14] is adopted. The scheme of sensitivity filtering is to modify the original sensitivity value as follows: ND X qd jk 1 qjk H^ j rj ¼ ðri Þ1 PND ^j qri qrj H j¼1 j¼1
(11)
where qjd k =qri and qjk =qrj are a modified element sensitivity value of the fth design variable and an original element sensitivity value of the jth design variable, respectively. The original sensitivity values are modified based on a weighted average of the design variable sensitivities in a fixed neighborhood and the convolution operator H^ j is written as H^ j ¼ r min distðd,jÞ,
fj 2 NDV9distðd,jÞ r r min g, d ¼ 1, . . . ,ND
(12)
where the operator distðd,jÞ is defined as the distance between the center of element d and the center of element j. The convolution operator H^ j is zero outside the filter size, r min . The convolution operator for element j decays gradually with the distance from element e. It is also necessary to define a ‘‘stop condition’’ of the optimization procedure. Even though there is a variety of convergence criteria, the following equation is mostly used in topology optimization: max9rd,l1 rd,l 9 r e, d
d ¼ 1, . . . ,ND
(13)
where rd,l1 and rd,l are the density of dth element at (l 1)th iteration and at lth iteration, respectively. For the tolerance value, e, we used 0.01 in this study. To update design variables in topology optimization, the Optimality Criteria (OC) method [14] is adopted. 4. Examples and physical validation The aim of an example is to show the effectiveness of topology optimization in order to find an optimal damping layout. It is known that an optimal damping treatment can be differently determined depending upon the approach so it is necessary to compare the modal loss factors. By comparing the modal loss factors obtained by three different approaches: mode shape, the SED, and topology optimization, it can be determined which method provides better optimal damping treatment. 4.1. Numerical example Table 1 shows the dimensions and material properties of a quarter-cylindrical shell structure. A clamped–clamped boundary condition is specified: all degrees-of-freedom at the two ends perpendicular to the z-direction are restrained. In order to examine how optimal damping layout changes according to the three different approaches, the shell problem is solved with 50 per cent damping material (Figs. 1–3). Quantitative performance comparisons are given in Table 2. This example shows that topology optimization usually provides better damping effect than the other two
Table 1 The material properties and dimensions of the base shell structure. Quarter-cylindrical shell structure Radius (R) (m) Length (L) (m) Thickness (t) (m) Theta (y) (deg) Young’s modulus (GPa) Poisson’s ratio Density (kg/m3)
0.19 0.45 0.002 90 200 0.3 7860
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Fig. 1. Optimal damping layouts obtained by mode shape approach with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
approaches. Specifically, the modal loss factor obtained by topology optimization at the 4th mode is 23.27 and 61.14 per cent higher than ones obtained by mode shape and the SED, respectively. 4.2. Experimental validation 4.2.1. Complex modulus of styrene–butadiene rubber (SBR) damping material It is necessary to measure the complex modulus of our SBR damping material (solid rubber type) [17] in our lab in order to accurately obtain the modal loss factors. Table 3 shows the obtained complex modulus of the SBR material. Our SBR material has a density of 1385.1 kg/m3; the density of usual SBR material is 930–1200 kg/m3 [18]. The thickness of our SBR damping material is 0.003 m. 4.2.2. Damped semi-cylindrical shell structure Fig. 4 shows a semi-cylindrical shell structure that is used in this study, and Table 4 shows the material properties and dimensions of the base shell structure. As shown in the figure, two supporting thick wooden plates are glued at the ends of the semi-cylindrical shell so that the following boundary conditions are satisfied: ux ¼ uy ¼ 0:0
at z ¼ 0 and z ¼ 0:9
(14)
where ux and uy represent displacements in x- and y-directions, respectively, and z is the coordinate in the longitudinal direction along the axis of revolution of the semi-cylinder. These boundary conditions state that the shell structure is clamped at both ends. The objective function for topology optimization is to maximize the modal loss factor at a specific mode and the maximum allowable volume constraint is 40 per cent. In topology optimization, the filtering radius is
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Fig. 2. Optimal damping layouts obtained by the SED approach with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
0.0161 m, and the OC method is used to update the design variables. The optimal damping layouts by topology optimization at three different modes are shown in Fig. 5. Then, the optimal SBR damping shapes obtained by topology optimization are applied to the semi-shell structure. Figs. 6–8 compare the experimental frequency response function of the base structure to that of the damped structure at the three target modes. It is noted that the frequency response function (H1,input ) is obtained for target mode 1 and target mode 3, but the frequency response function (H2,input ) is obtained for target mode 2. The subscriptions 1 and 2 of the frequency response function is measured at a different location. As shown in Fig. 6, the frequency response function of the damped structure is not significantly decreased (only 1 dB lower) compared to the one of the bare structure. The decrease at target mode 2, however, is significantly decreased (5.2 dB lower) as shown in Fig. 7. Since ‘‘the optimally shaped SBR damping treatments’’ are simplified according to the target modes (64, 142, and 175 Hz), the numerical modal loss factors are also recalculated based on the shape of the SBR damping material. The assumption in this study is the temperature which is constant at 23 1C, while, the frequency is considered to be a variable. It is important to obtain the complex modulus of the SBR damping material at the actual target modes (64, 142, and 175 Hz), but, the complex modulus is presented only at 23, 146, and 410.5 Hz as shown in Table 3. Based on our experience, it is assumed that there is only a minor difference in the complex modulus. Thus, the complex modulus of the SBR damping material at the 1st mode (23 Hz) is used for the calculation of the modal loss factor at target mode 1 (64 Hz) and the complex modulus of the SBR material at the 2nd mode (146 Hz) is used for calculation of the modal loss factors at
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Fig. 3. Optimal damping layouts obtained by topology optimization with clamped–clamped boundary condition and 50 per cent damping material: (a) 1st mode, (b) 2nd mode, (c) 3rd mode and (d) 4th mode.
Table 2 The comparison of the modal loss factors with 50 per cent damping material: clamped–clamped boundary condition. Mode
Mode shape
SED
Topology optimization
1st 2nd 3rd 4th
0.03207 0.04036 0.04201 0.03962
0.04215 0.05285 0.05507 0.03006
0.04518 0.05162 0.05509 0.04844
Table 3 The complex modulus of the composite beam. Mode
Damped natural frequencies (Hz)
Young’s modulus (GPa)
Loss factor
1st 2nd 3rd
23.5 146.0 410.5
0.287 0.162 0.167
0.101 0.210 0.469
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Fig. 4. Experimental set-up. Table 4 Material properties of the semi-shell cylindrical structure. Semi-cylindrical shell structure Radius (R) (m) Length (L) (m) Thickness (t) (m) Young’s modulus (GPa) Structural damping coefficient (%) Poisson’s ratio Density (kg/m3)
0.19 0.9 0.002 197 0.1 0.3 7850
Fig. 5. Optimal damping layouts of semi-cylindrical structure using topology optimization. (a) Target mode 1, (b) target mode 2 and (c) target mode 3.
target mode 2 (142 Hz) and target mode 3 (175 Hz). It is also noted that the reason that the frequency response function at target mode 2 decreases (5.2 dB lower) more significantly than at target mode 1 (1.0 dB lower) is that the loss factor of the damping material at the 2nd mode (0.210) is higher than at the 1st mode (0.101).
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Fig. 6. Comparison of frequency response functions between damped and bare structures: target mode 1.
Fig. 7. Comparison of frequency response functions between damped and bare structures: target mode 2.
Table 5 compares the modal loss factors obtained by numerical calculations and by experimental measurements. The maximum discrepancy between numerical simulation and experimental testing is only 6.29 per cent so topology optimization can be widely adopted to real-world problems in order to effectively determine an optimal damping layout. Among these three modes, the modal loss factor at target mode 2 is most accurately measured in terms of the complex modulus of the SBR damping material. This is because the resonant mode at target mode 2 is 142 Hz, and the complex modulus at the 2nd mode is obtained at 146 Hz.
5. Conclusion Since several approaches have been proposed in order to effectively design an optimal damping layout, it is necessary to investigate which approach provides a better result. The mode shape, the SED, and the topology optimization methods have been chosen to design an optimal damping layout of an unconstrained-layer shell structure. To determine an optimal damping layout using the mode shape and the SED approaches, a sorting method is adopted. For topology optimization, the objective function is to maximize the modal loss factor at a specific mode, and the constraint is an allowable maximum damping material. It is noted that the modal strain energy method is used to numerically calculate the modal loss factors.
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Fig. 8. Comparison of frequency response functions between damped and bare structures: target mode 3. Table 5 The comparison of the modal loss factors between numerical simulation and experimental testing. Target mode
Numerical simulation
Experimental testing
Ratio (%)
1 2 3
0.01184 0.02358 0.01384
0.01145 0.02237 0.01297
96.70 94.87 93.71
By quantitatively comparing the modal loss factors with respect to the different approaches, it was found that topology optimization usually provides a higher modal loss factors. Because it has been confirmed that topology optimization is the most effective, damping layouts determined by topology optimization are experimentally validated. For the validation of the numerical models, frequency response functions are measured for the damped structure in order to measure the modal loss factors. The results show that the numerical results have a high correlation with the testing data. References [1] C.T. Sun, Y.P. Lu, Vibration Damping of Structural Elements, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1995. [2] G. Parathasarathy, C.V.R. Reddy, N. Ganesan, Partial coverage of rectangular plates by unconstrained layer damping treatments, Journal of Sound and Vibration 102 (1985) 203–216, http://dx.doi.org/10.1016/S0022-460X(85)80053-5. [3] R.K. Roy, N. Ganesan, Dynamic studies on plates with unconstrained layer treatment, Computers & Structures 49 (1993) 473–480, http://dx.doi.org/ 10.1016/0045-7949(93)90048-I. [4] A. Yildiz, K. Stevens, Optimum thickness distribution of unconstrained viscoelastic layer treatment for plates, Journal of Sound and Vibration 103 (1985) 183–199, http://dx.doi.org/10.1016/0022-460X(85)90232-9. [5] S.C. Huang, D.J. Inman, E.M. Austin, Some design consideration for active and passive constrained layer damping patches, Smart Materials and Structures 5 (1996) 301–313, http://dx.doi.org/10.1088/0964-1726/5/3/008. [6] L. Zaiwei, L. Xinhua, Vibro-acoustic analysis and optimization of damping structure with response surface method, Materials and Design 28 (2007) 1999–2007, http://dx.doi.org/10.1016/j.matdes.2006.07.006. [7] H. Zheng, C. Cai, G.S.H. Pau, G.R. Liu, Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments, Journal of Sound and Vibration 279 (2005) 739–756, http://dx.doi.org/10.1016/j.jsv.2003.11.020. [8] A. Mohammed, A. Al, I.B. Raed, Optimum design of segmented passive-constrained layer damping treatment through genetic algorithms, Mechanics of Advanced Materials and Structures 15 (2008) 250–257, http://dx.doi.org/10.1080/15376490801907772. [9] M. Alvelid, Optimal position and shape of applied damping material, Journal of Sound and Vibration 310 (2008) 947–965, http://dx.doi.org/10.1016/ j.jsv.2007.08.024. [10] L. Zheng, R. Xie, Y. Wang, E.S. Adel, Topology optimization of constrained layer damping on plates using Method of Moving Asymptote MMA approach, Shock and Vibration 18 (2011) 221–244, http://dx.doi.org/10.3233/SAV-2010-0583. [11] M.G. Sainbury, R.S. Masti, Vibration damping of cylindrical shells using strain-energy-based distribution of an add-on viscoelastic treatment, Finite elements in Analysis and Design 43 (2007) 175–192, http://dx.doi.org/10.1016/j.finel.2006.09.003. [12] Z.Z. Guo, Y.Z. Chen, Q. Hou, Topology optimization in damping structure based on ESO, Journal of China Ordnance 4 (2008) 293–298. [13] X. Zang, D. Yu, L. Yao, R. Yan, Optimization of thickness distribution of unconstrained damping layer based on mode shapes, Zhongguo Jixie Gongcheng (China Mechanical Engineering) 21 (2010) 515–518. [14] M.P. Bendose, O. Sigmund, Topology Optimization, Theory, Methods and Applications, 2nd edition, Springer, Berlin, Heidelberg, Germany, 2003. [15] S.Y. Kim, D.H. Lee, Identification of fractional-derivative-model parameters of viscoelastic materials from measured FRFs, Journal of Sound and Vibration 324 (2009) 570–586, http://dx.doi.org/10.1016/j.jsv.2009.02.040.
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