TOPOLOGY OPTIMIZATION APPROACH OF DAMPING TREATMENT IN CABIN ACOUSTIC DESIGN

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1 TOPOLOGY OPTIMIZATION APPROACH OF DAMPING TREATMENT IN CABIN ACOUSTIC DESIGN Jianrun Zhang, Beibei Sun, Xi Lu Southeast University, School of Mechanical Engineering, Nanjing, Jiangsu, China The purpose of this study is to deal with acoustic design of damping layers by using topology optimization method. In the condition of small amplitude harmonic vibration, based on the acoustic radiation modes and direct boundary element method, analysis shows that the main factor affecting the acoustic radiation power and radiated sound pressure of plate is the normal velocity amplitude of the structure surface. Therefore, the objective function is defined as a combination of several root mean square values of normal velocities in the structure surfaces to find effective optimal damping layers. Vibro-acoustic model of cabin is established to analyze the coupling between structure vibration and sound radiation. Then, solid isotropic material with penalization (SIMP) method is applied to optimize the topology of damping layers on the structure surface of excavator cabin. The results show that the optimization method has a great significance for diminishing the interior noise of cabin. Keywords: Acoustic radiation, Normal vibration velocity, Damping treatment,multiple response peaks,topology optimization 1. Introduction Interior noise is one of the important factors affecting the comfort of the cabin. The cabin interior noise, apart from a small part of which is the airborne noise transmitted through the holes and cracks from outside sources, the main part is the low-frequency structure-borne noise radiated by the vibrating body panels [1-2]. Therefore, attaching the damping material on the structure surface to suppress the panel vibration is one of the most direct and effective way to reduce the cabin interior noise [3]. In recent years, a lot of research and application-oriented explorations have been carried out for the damping treatment of structure. For example, Duhring et al. [4] used the squared of sound pressure amplitude as the objective function to study the optimal placement of damping panels on walls of acoustic cavities. They demonstrated that the sound level can be effectively reduced by using topology optimizing the distribution of absorbing and reflecting materials. Kim et al. [5] compared the modal loss factors obtained by three different approaches: mode shape, the SED and topology optimization, in order to determine which approach provided a better damping treatment. They found that topology optimization could provide a higher modal loss factors. Lee [6] and Jung et al. [7] proposed a fractional-derivative model including the intrinsic nonlinearities of viscoelastic damping materials with respect to frequency and temperature. And the optimal layout of the constrained viscoelastic layer damping on the structure was identified using a gradient-based numerical search algorithm. Zheng et al. [8] introduced interface finite element that can directly couple together the base structure plate elements, the viscoelastic layer and the constraining layer plate elements. By using the SIMP model and MMA optimizer, the topology of the PCLD was optimized to minimize the sound power of plates at a certain frequency. In order to achieve a more clear optimal topology, Kang et al. [9] and Zhang et al. [10] proposed the artificial penalized damping model de- 1

2 pending on elemental stiffness and mass properties. Despite the fact that topology optimization has been an intensively studied topic, only a limited number of works have been devoted to find the optimal layout design of damping layers. Therefore, this paper presents a topology optimization formulation and numerical techniques for layout design of damping layers on the structure surfaces to minimize the sound pressure at a specific field point. In this paper, the root mean square value of normal velocities is introduced to design the layout of damping layers in the objective function. And it is applied to damping treatment design in excavator cabin, and the numerical implementation of the topology optimization procedures is proposed to find the optimum distribution of damping layers. 2. Acoustic Radiation of the Structure The acoustic radiation power has a strong correlation with vibration velocity shape and vibration velocity amplitude of the structure surface[11-15]. And the radiated sound pressure of an arbitrary field point in space mainly depends on the normal velocity of structure surface[16-19]. 2.1 Acoustic Radiation Mode Theory Consider a rectangular plate in an infinitely baffled plane subject to harmonic excitation and radiating sound to semi-infinite space which filled with a homogeneous isotropic acoustic medium. The plate is divided into N small elements with equal area and the maximal size is much smaller than the acoustic wavelength. Therefore, each element can be regarded as an independent lumped sound source. Then the acoustic power radiated from the ith element at the frequency of is given by 2 (1) where is the area of the ith element; is complex conjugate and Re is real; is the surface normal velocity of the ith element; is the surface sound pressure of the ith element. The surface sound pressure and surface normal velocity of N elements of the plate are composited of N order column vector, respectively expressed as, (2), (3) Therefore, the total acoustic power radiated from the plate at the frequency of is expressed as 2 (4) in which H is the complex conjugate transpose. Surface sound pressure and surface normal velocity have the following relationship Z (5) where Z is a N N impedance matrix, and its element pairs (i,j) is the ratio of surface sound pressure of the jth element and normal vibration velocity of the ith element. 2 (6) in which is the medium density; is the distance between the ith and jth element; is wave number and stands for the speed of sound in the acoustic medium. Substituting Equation (5) back into Equation (4) leads directly to a simple and compact expression of the acoustic radiation power 2 Z R (7) where R 2 (8) Introduce U ω as the amplitude of vibration velocity and V ω as the vector of vibration velocity, the surface normal velocity can be expressed by 2 ICSV24, London, July 2017

3 U ω V ω (9) in which U ω (10) V ω U ω (11) where V ω represents the vibration velocity mode. Substituting Equation (9) back into Equation (7), acoustic radiation power can be stated as V ω V ω U w (12) It can be seen from Equation (12) that, vibration velocity mode V (ω) and vibration velocity amplitude U (ω) of the structure surface are the two main factors affecting the acoustic radiation power for the plate structure which vibrating at the frequency of ω. 2.2 Direct Boundary Element Method The radiated sound pressure in semi-infinite space satisfies the following Helmholtz integral equation which governs the acoustic wave problems. P M M,Q, (13) where P M indicates the sound pressure of any field point M in semi-infinite space; is the sound pressure on the structure surface; is the normal velocity of point Q on the structure surface. The free space Green s function, for Helmholtz integral equation is given by, (14) where r is the distance between field point M and Q on the structure surface; is wave number and is the sound speed. Divide plate structure into N small elements which can be regarded as the lumped sound source. Then Equation (13) can be rewritten as (15) where is the sound pressure of an arbitrary field point in semi-infinite space; and obtained from the right side of Equation (13) are the vectors of acoustic coefficient; is the column vector of surface sound pressure of discrete elements; is the column vector of surface normal velocity of discrete elements. Substituting Equation (5) into Equation (15), the radiated sound pressure can be expressed as Z (16) where is the acoustic impedance vector of field point, which can be defined as Z (17) Substituting Equation (9) into Equation (16), yields V ω U ω (18) 2.3 Analysis of Acoustic Radiation Power and Radiated Sound Pressure Substituting Equation (12) into Equation (18), the radiated sound pressure can be written as V ω V ω V ω (19) In the case of small amplitude harmonic vibration, Equation (19) shows that acoustic radiation power is the main factor affecting, while Equation (12) indicates that the amplitude of normal vibration velocity U ω contribute the most in acoustic radiation power. Consequently, we can see from the above analysis that the acoustic radiation power can be reduced by controlling the amplitude of normal vibration velocity. 2.4 Analysis of Vibration Energy of Plate Structure Under normal circumstances, flexural wave causes much greater amplitude than the shear and longitudinal wave in the plate structure, so the radiated noise of plate structure is mainly caused by the flexural wave. ICSV24, London, July

ICSV24, London, 23-27 July 2017 Divide the plate structure into N elements and the vibration energy of plate structure can be expressed as (20) E where ω is the surface normal velocity of the ith

From the view of modal strain energy (MSE) method [20-21], the energy dissipation of damp- ing material on the structure surface can be written as E η (21) where is the material loss factor; is the

4 ICSV24, London, July 2017 Divide the plate structure into N elements and the vibration energy of plate structure can be expressed as (20) E where ω is the surface normal velocity of the ith element at the frequency of ; is flexurall rigidity of the ith element; is the wave number. From the view of modal strain energy (MSE) method [20-21], the energy dissipation of damp- ing material on the structure surface can be written as E η (21) where is the material loss factor; is the real eigenvector; is the stiffness matrix of damp- ing layer elements. 3. Objective function of Damping Layers Topology Optimization The purpose of topology optimization is to determine the optimal distribution of a given amount of material within the prescribed design domain and constraints. The Objective function of the topology optimization can be defined as Minimize f x, ] (22) Subject to 0 1 j 1,2 in which, f x is the multi-frequency optimization function; n is the number of peak frequencies to be considered; k is the kth peak frequency; l is the number of panel regions of structure to be con- sidered at the kth peak frequency; is the sound power contribution coefficient of the ith panel which is obtained from panel acoustic power contribution analysis;, is the root mean square value of the ith panel normal velocities; is the relative density of the damping material element; denotes the total number of damping layer elements in the design domain; is the volume of damping material after optimization while is the volume of damping material beforee optimiza- tion; denotes the volume fraction ratio and can be chosen depending on the maximumm allowable amount of damping material attached on the structure surfaces; is the damping material volume of the jth element; is the lower bound limit of the design variables, which is set to be in this study. The purpose of setting lower bound limit of the design variables is to avoid the possible problem of numerical singularity. 4. Numerical Analysiss of Cabin Sound Field Optimization 4.1 Finite Element Model of Cabin Structure In this study, the cabin structure is very complex and mainly composed of weldedd panels, as shown in Figure 1(a). The finite element model of cabin is shown in Figure 1( (b). (a)3d model (b)fem model Figure 1. Cabin structure model 4 ICSV24, London, July 2017

The structure dynamic analysis for obtaining the velocity boundary conditions is performed in the frequency range from 20Hz to 200Hz with an interval of 3Hz.

Through the structure dynamic analysis, the vibration velocities are obtained at every finite element node and then used as acoustic boundary conditions in the acoustic response analysis. 4.

5 The structure dynamic analysis for obtaining the velocity boundary conditions is performed in the frequency range from 20Hz to 200Hz with an interval of 3Hz. And the unit harmonic force excitation vertical to the cabin is applied in the lower left side of the mounted position of the cabin. Through the structure dynamic analysis, the vibration velocities are obtained at every finite element node and then used as acoustic boundary conditions in the acoustic response analysis. 4.2 Acoustic Boundary Element Model of Cabin The cabin seat has a great influence on the acoustic characteristics of interior sound field, because the seat occupies the larger internal space and has a certain sound absorption characteristics [1,24]. Therefore, the acoustic boundary element model of cabin including the seat is established in this paper. As shown in Figure 2. (a) (b) Figure 2. Acoustic boundary element model of cabin The acoustic response analysis for obtaining the sound pressure at the driver s right ear is performed in the same frequency range (20~200Hz) with an interval of 3Hz, and therein the reference pressure is 2 10 Pa. The A-weighted sound pressure level at the driver s right ear is shown in Figure Sound Pressure Level (db(a)) Frequency (Hz) Figure 3. Sound pressure level at the driver s right ear It is shown in Figure 3 that, there are a large number of sound pressure peaks on the curve of sound pressure level at the driver s right ear. The reasons for this phenomenon is that the cabin is composed of many steel panels which are grouped together to form dense partial modes. When the excitation frequency is accord with the natural frequency of panels, the sound pressure at the driver s right ear is easy to produce the response peak. And the acoustic optimization of multiple response peaks is used to reduce the sound pressure at the driver s right ear. 4.3 Topology Optimization of Damping Material Distribution Based on the SIMP method, the topology optimization of multiple response peaks is used to optimize the damping layers distribution on the panel regions which contribute most to the sound power. As mentioned before, the constraint condition of the optimization is that the volume fraction ratio of optimal material distribution and full damping layers (full damping treatment on the considered six panels) is not more than 60%, the design variable is the relative density of damping mate- ICSV24, London, July

6 ICSV24, London, July 2017 rial elements, and the optimization objectivee is to minimize the weighted value of several root mean square values of normal velocities on the prominent contribution regions. The optimal distribution of damping material is shown in Figure 4. (a)topology optimization results 6 ICSV24, London, July 2017

range of 0~1. the optimal partial damping layers attached on the panels are shown in Figure 4(b). At last, the added mass of damping material is reduced about 40%, from 14.84kg to 8.9kg.

7 (b)optimal partial damping layers Figure 4. Damping material distribution It can be seen from Figure 4(a), the relative densities of black regions and gray regions are respectively 1 and 0, and the rest regions of relative densities are in the range of 0~1. the optimal partial damping layers attached on the panels are shown in Figure 4(b). At last, the added mass of damping material is reduced about 40%, from 14.84kg to 8.9kg. The acoustic response analysis is respectively performed on the structural model of full damping layers and optimal partial damping layers attached on the considered panels. And the A- weighted sound pressure level at the driver s right ear is obtained, as shown in Figure 5. Sound Pressure Level (db(a)) 90 no damping layers full damping layers 80 optimal partial damping layers Frequency (Hz) Figure 5. Sound pressure level at the driver s right ear after optimization Figure 5 shows that, for the structural model of full damping layers attached on the considered panels, the response peak of sound pressure declines by 21.63dB at the frequency of 98Hz, and the rest four sound pressure peaks almost disappear respectively at the frequency of 116Hz, 137Hz, 161HZ, 185Hz. The total sound pressure level in the frequency range of 20~200Hz reduces from 85.14dB to 57.85dB. It is notable that the manner of optimal partial damping treatment enhances the efficiency of the damping layers so that the dual requirements of sound field improvement and lightweight cabin can be met simultaneously. subsubsections. 5. Conclusion This paper has presented a topology optimization approach to optimally design the damping layers, in order to improve the interior sound field of the cabin. In this study, we build a vibro-acoustic model including the seat of the excavator cabin using a FEM BEM approach. The model is utilized to predict the sound pressure level inside the cabin in the frequency range of Hz, and also to determine the contribution of each radiating panel to the sound power at the main response peaks. The free layer damping treatment is implemented on the prominent contribution panel regions based on the analysis of results. Then, the minimal weighted value of several root mean square values of surface normal velocities is used as the optimization objective to find the effective optimal damping layers. The results show that, the optimization method proposed in this paper has practical significance for improving the interior sound field and decreasing the added mass of damping layers in order to accord with lightweight trend. Acknowledgments ICSV24, London, July

8 The authors acknowledge 2014 Science and Technology Support Plan of Jiangsu China(Approval No. BE and No. BY ). REFERENCES 1 Mohanty A R, St Pierre B D, Suruli N P. Structure-borne noise reduction in a truck cab interior using numerical techniques[j]. Applied Acoustics, 2000, 59(1): Yuksel E, Kamci G, Basdogan I. Vibro-acoustic design optimization study to improve the sound pressure level inside the passenger cabin[j]. Journal of Vibration and Acoustics-Transactions of the ASME, 2012, 134(6): Danti M, Vige D E, Nierop G V. Modal methodology for the simulation and optimization of the free-layer damping treatment of a car body[j]. Journal of Vibration and Acoustics-Transactions of the ASME, 2010, 132(2): Duhring M B, Jensen J S, Sigmund O. Acoustic design by topology optimization[j]. Journal of Sound and Vibration, 2008, 317(3-5): Kim S Y, Mechefske C K, Kim I Y. Optimal damping layout in a shell structure using topology optimization[j]. Journal of Sound and Vibration, 2013, 332(12): Lee D H. Optimal placement of constrained-layer damping for reduction of interior noise[j]. AIAA Journal, 2008, 46(1): Jung B C, Lee D H, Youn B D. Optimal design of constrained-layer damping structures considering material and operational condition variability[j]. AIAA Journal, 2009, 47(12): Zheng W G, Lei Y F, Li S D, et al. Topology optimization of passive constrained layer damping with partial coverage on plate[j]. Shock and Vibration, 2013, 20(2): Kang Z, Zhang X P, Jiang S G, et al. On topology optimization of damping layer in shell structures under harmonic excitations[j]. Structural and multidisciplinary optimization, 2012, 46(1): Zhang X P, Kang Z. Topology optimization of damping layers for minimizing sound radiation of shell structures[j]. Journal of Sound and Vibration, 2013, 332(10): Cunefare K A, Currey M N, Johnson M E, et al. The radiation efficiency grouping of free-space acoustic radiation modes[j]. Journal of the Acoustical Society of America, 2001, 109(1): Hill S G, Snyder S D, Tanaka N. Practical implementation of an acoustic-based modal filtering sensing technique for active noise control[j]. Applied Acoustics, 2007, 68(11-12): Peters H, Kessissoglou N, Marburg S. Modal decomposition of exterior acoustic-structure interaction[j]. Journal of the Acoustical Society of America, 2013, 133(5): Marburg S, Losche E, Peters H, et al. Surface contributions to radiated sound power[j]. Journal of the Acoustical Society of America, 2013, 133(6): Wu H J, Jiang W K, Liu Y J. Analyzing acoustic radiation modes of baffled plates with a fast multipole boundary element method[j]. Journal of Vibration and Acoustics-Transactions of the ASME, 2013, 135(1): Sedaghatjoo Z, Adibi H. Calculation of domain integrals of two dimensional boundary element method[j]. Engineering Analysis with Boundary Elements, 2012, 36(12): Tomioka S, Nishiyama S. Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method[j]. Engineering Analysis with Boundary Elements, 2010, 34(4): Shao W, Mechefske C K. Acoustic analysis of a finite cylindrical duct based on Green's functions[j]. Journal of Sound and Vibration, 2005, 287(4-5): Gumerov N A, Duraiswami R. A broadband fast multipole accelerated boundary element method for the three dimensional Helmholtz equation[j]. Journal of the Acoustical Society of America, 2009, 125(1): Cura F, Mura A, Scarpa F. Modal strain energy based methods for the analysis of complex patterned free layer damped plates[j]. Journal of Vibration and Control, 2012, 18(9): Pan L J, Zhang B M. A new method for the determination of damping in cocured composite laminates with embedded viscoelastic layer[j]. Journal of Sound and Vibration, 2009, 319(3-5): Wu S F, Natarajan L K. Panel acoustic contribution analysis[j]. Journal of the Acoustical Society of America, 2013, 133(2): Han X, Guo Y J, Yu H D, et al. Interior sound field refinement of a passenger car using modified panel acoustic contribution analysis[j]. International Journal of Automotive Technology, 2009, 10(1): Wang Y, Zhang J R, Liu X B, et al. Interior acoustic field analysis of hydraulic excavator's cabin based on BEM[C]//2nd International Conference on Functional Manufacturing and Mechanical Dynamics. Hangzhou, China, 2012: ICSV24, London, July 2017

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