Optimization of Noise Transfer Path Based on the Composite Panel Acoustic and Modal Contribution Analysis

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Introduction
Noise, vibration, and harshness (NVH) is usually used to evaluate the drivers' comfort of a vehicle [1,2]. With the increasing academic attention to the environment and the technical improvement of vibration and noise standards of construction machinery, the NVH performance of construction machinery has been adopted by major manufacturers in recent years. Drawing on the achievement of automotive NVH, analysis methods and the design of construction machinery NVH have been gradually formed with the finite element and boundary element as the core [3,4].
For analysis and control of vibration and noise in a cab, continuous efforts have been made. Sheng and Tijs used the wavenumber finite and boundary element methods to obtain a numerical solution [5,6]. Li used the experimental method to obtain a structural modal of the body-in-white of a car, which provides an excellent way to solve similar problems [7,8]. Mohanty identified that the frequency of structural noise was less than 1 kHz, which belongs to the low and middle frequency range, and can be controlled by suppressing structural vibration [9]. Cheung performed finite element modeling of the vehicle cab after measuring the random excitation. e low-frequency structural noise of the cab was predicted when the contribution of the body panels was analyzed [10]. Hyunwoo and Siano calculated the acoustic response and predicted the noise by inputting excitation [11][12][13]. Guo and Cheng reduced the structure-borne noise since the automotive body panel vibration was controlled [14,15]. Bao analyzed and predicted the sound field inside the cab of the shovel loader using acoustic vibration coupling, and the design was optimized by confirming the panel with the largest acoustic contribution [16]. Long and Wolff analyzed acoustic response, panel contribution volume, and modal participation factor by applying ATV and MATV methods for the cab of a truck, and the panels were confirmed to be improved [17,18]. Wu established HELS-based near-field acoustic holography (NAH) formulations that can analyze and rank the panel acoustic contributions toward the acoustic field and reconstructed the acoustic field inside the vehicle passenger compartment [19]. At present, the traditional method for PACA aims at the maximum sound pressure level peak at a field point. en the panels which contribute the most to the maximum sound pressure level are recognized. Finally, the panels are ameliorated to reduce noise. However, in practice, there is often more than one sound pressure peak with small differences in each other. e contributions of a panel are also different at main frequencies. While the maximum peak of the sound pressure is reduced at a frequency, the other one may be increased at another frequency. erefore, the effectiveness of noise reduction is unsatisfactory. In this study, a method of composite panel acoustic contribution analysis is proposed, which takes the multiple frequencies into consideration.
After recognition of the regions that contribute most to the interior noise, it is necessary to optimize the panels. Han compared three different conditions to affirm the optimum proposal [20]. Liang used structural topology optimization to modify the panels [21]. Most researchers used experiments or topology optimization methods to optimize panels to reduce the interior noise of a cab but rarely used optimization algorithms in construction machinery. Optimization algorithms have been studied widely to solve multiobjective optimization problems, such as the global optimization algorithm [22,23], the local optimization algorithm without gradient [24][25][26], the local optimization algorithm based on gradient [27][28][29], and so on. e optimization of reducing noise in a cab is a constrained, multiobjective, and multivariable problem, which is suitable for the optimization algorithm without gradient. Chan and Kumar studied algorithms in the NLOPT library and selected appropriate algorithm to solve the multiobjective problem [30,31]. e optimization algorithm in the NLOPT library can meet the requirements.
In summary, the current research mostly focuses on the single-frequency PACA while rarely considers the composite contribution of the multiple frequencies when recognizing the optimized panels. Furthermore, most of the transfer path optimization methods are topology optimization or multischeme experimental comparison, while the ways of combining the optimization algorithm with simulation software are rarely used in a cab. In this paper, the theory and method of composite panel acoustic and modal contribution analysis are constructed. e composite panel acoustic and modal contribution are integrated to determine the improved panels and areas. en, the transfer path optimization coupling acoustic software, Python language, and algorithm are used to optimize the panels and areas. Finally, the effectiveness of the proposed method is verified in an excavator cab.

Composite Panel Acoustic Contribution Analysis Method.
Among the energy transferring from the vibration source to the cab panel, the bending wave is the main fluctuation form.
e periodic average power transmitted by a simple harmonic bending wave is expressed as where V max ,n is the maximum normal velocity of a thin panel structure, K s is the bending stiffness, ω is the angular frequency, and k is the wave number. e vibration energy per unit wavelength of a simple harmonic bending wave can be expressed as e transmission of vibration energy within a thin panel can be determined by the maximum normal velocity at different locations. e wave equation governing sound field is as follows: where p denotes the sound pressure, v 0 denotes the speed of sound, t denotes the time, and ∇ 2 denotes the Laplace operator. Refer to the panel theory, the vibration differential equation of the panel structure in the cab is as follows: where ρ s denotes the density of the thin panel, h s denotes the thickness of the thin panel, and f s denotes the normal load acting on the surface of the structure per unit area. e finite element representation is established by applying the structural dynamics equation of the panel with the wave equation and using the Galerkin method.
where u s denotes the displacement column vector of the node; p f denotes the sound pressure column vector of the node; M s , C s , and K s are the mass matrix, damping matrix, and stiffness matrix, respectively; F s denotes the column vector of external load acting on the structure or the sound cavity; the subscripts s and f denote the structure body and the sound cavity, respectively; and A denotes the coupling matrix of sound pressure and displacement. Neglecting the sound absorption materials and damping materials, the acoustic cavity damping matrix and structural damping matrix are zero. Applying Fourier transform, the above equation is transformed as follows: , U s , P f , and Q f are frequency-domain parameters, while u s , p f , and F f are the time-domain parameters. Since the acoustic cavity has no load other than the wall panel vibration, the external load Q f � 0. Let en equation (6) is transformed to It can be inferred that controlling the vibration speed of the thin cab panel can affect the vibration energy and acoustic radiation.
ATV illustrates the relation between sound pressure response at a field point and surface vibration velocity.
where P site denotes the sound pressure at a field point, A T ATV (ω) denotes the acoustic transfer vector, and V n (ω) denotes the normal vibration velocity of a panel. e acoustic contribution is the algebraic sum of the acoustic contribution of every element in the panel to the field points. e mathematical formula of the element acoustic contribution is as follows: where P i,j denotes the acoustic contribution of the i th element in the j th panel to the field point. Based on the structure-acoustic coupling equation, the synthetic sound pressure at the field point within the acoustic cavity is expressed as follows: where P panel N,j is the sound pressure generated by panel j at site N and n denotes the number of all relevant panels. To quantify the contribution of the panel vibration to the sound pressure level at the field point, the sound pressure component caused by any panel j at site N is projected in the direction of the complex vector of the combined sound pressure to obtain the conventional contribution of the panel to the synthetic sound pressure.
e composite panel acoustic field contribution analysis takes all sound pressure peaks into consideration, i.e., the same importance is given to all sound pressure peaks. According to the contributions to the overall sound pressure level of the field point, it is necessary to give different weighting to the sound pressure peaks, and the correction coefficient β is introduced and defined as follows: where P i is the peak sound pressure of interest frequency and P RMS site is the root mean square (RMS) value, and the composite panel acoustic contribution is obtained as follows:

Modal Contribution Analysis
Method. e modal acoustic transfer vector illustrates the relationship between the sound pressure at the field point and the modal participation factor, which takes into account the structural modal on the basis of ATV. In accordance with the vibration displacement vector of the surface, the normal vibration velocity vector is projected to the normal direction of the surface.
where Φ n denotes the matrix of structural modal and MRSP(ω) denotes the matrix of modal participation factors. e sound pressure at the field point can be calculated as where MATV(ω) denotes the modal acoustic transfer vector and can be calculated as e modal contribution of each frequency is calculated by extracting the modal matrix to recognize the optimized area.

Composite Panel Acoustic and Modal
Contribution Analysis Method is paper uses a composite panel acoustic and modal contribution analysis method to recognize the optimized area and then uses a noise transfer path optimization with vibroacoustic coupling response to determine the optimal area to reduce low-frequency noise.
Complexity Figure 1 shows the process of composite panel acoustic and modal contribution analysis. First, the FEM and the acoustic model of a cab are established and coupled. e structure model and the acoustic cavity model are calculated for the modal participation factor. Based on the vibroacoustic coupling model and external excitation, the SPL at the field point is calculated. e correctness of the vibroacoustic model is verified by comparison with experimental and simulation results. Second, based on the ATV method, the composite panel acoustic contribution analysis method is proposed to identify the main panels affecting the field point. Composite panel acoustic contribution analysis gives different weighting to the sound pressure peaks. Combined with the modal acoustic contribution of the MATV method, the areas which have the largest influence noise at the field point are confirmed in panels. Finally, the noise transfer path optimization with vibroacoustic coupling response analysis can determine the panel thickness and reduce low-frequency noise.
e optimization algorithm NLOPT is applied to design the areas.

Application and Verification
is paper takes an excavator cab as the application target of the new method. Excavators are the most widely applied equipment in construction. For the purpose of reducing the low-frequency noise caused by panel vibration in an excavator cab, the composite panel acoustic and modal contribution analysis and noise transfer path optimization are applied.
e optimization results are verified by an experiment.

e Cab Structural Model.
e excavator cab is mainly made of the welded panel and shell structure. Some structural parts such as small holes, chamfered corners, and tabs that have less impact on the cab model are applied to simplify the process. e material of the body in white is Q235 with 210 GPa elastic modulus, 7850 kg/m 3 density, and 0.3 Poisson's ratio. e glass is added around the FEM of the body in white to constitute an enclosed cab. e elastic modulus of the glass is 72 GPa, density is 2500 kg/m 3 , and the Poisson's ratio is 0.22. e cab is meshed with an average mesh cell size of 10 mm. Figure 2(a) shows the FEM of the cab with a cell number of 324072 and a node number of 319817. According to the procedure of solving the structural model of the cab, the first four structural modes of the enclosed cab are obtained, as shown in Figure 3. Most large displacements in modal shapes are concentrated in the glasses, which demonstrates that the glasses are prone to be excited vibration. e maximum length of the acoustic grid is 283.33 mm. e length of the acoustic grid is set as 40 mm. Moreover, the acoustic model is obtained with 9453 elements, as shown in Figure 2(b). Figure 4 shows the first four acoustic modes of the excavator cab. e structural modal and acoustic modal can be imported to the vibroacoustic model for acoustic response analysis.

e Cab Vibroacoustic Model.
e structural model of the enclosed cab is first inputted into the simulation software. en the acoustic cavity model also is imported. e field points are set up in the acoustic cavity elements. e excitation points are created under the cab. Finally, the structure model and acoustic cavity model have achieved mutual mapping by way of establishing the coupling face. e vibroacoustic coupling model of the excavator cab is established, as shown in Figure 5. Figure 6 shows the process of data acquisition in the excavator cab. e vibration acceleration in the Z direction of the cab suspension and the corresponding one-third octave right ear sound pressure of a medium excavator at idle speed (1050 rpm) are measured as input to the panel acoustic contribution analysis, as shown in Figures 7 and 8, respectively. A low-frequency band, mainly 20-200 Hz, possesses the main energy of sound pressure.

Baseline Measurements.
e frequencies corresponding to the sound pressure peaks are 50 Hz, 100 Hz, and so on.

Acoustic Response Analysis.
By acoustic response analysis, the sound pressure at the field point connects with the elements in FEM. e acceleration excitation of the suspension is imported to the vibroacoustic coupling model. e frequency range of the simulation is set as 20-200 Hz, and the step size is 1 Hz. After the simulation calculation, Figure 9 shows the comparison simulation results with experimental results about SPL of DRE.
In the frequency range of 20-200 Hz, the tendency of the simulation curve and the experiment curve is roughly accordant and the frequency corresponding to the peak also coincides. It shows that the simulation results can be reflected the acoustic response and verified the accuracy of the coupling model.

Composite Panel Acoustic Contribution Analysis.
e excavator cab is divided into 12 panels in the acoustic model, and the panel numbers and the corresponding panel names are shown in Table 1.
As an example, Figures 10 and 11 show the panel acoustic contribution at 50 Hz and 100 Hz, respectively. e panels have a great difference in contribution at diverse frequencies, or even opposite contributions. Hence, it is important to consider the composite contribution of the main frequencies. e composite panel acoustic contribution analysis for the partitioned panels is calculated in accordance with equations (13) and (14). e histogram is shown in Figure 12. From the figure, it can be seen that panels 1, 3, 4, 5, 6, 7, and 12 have a positive contribution to the overall SPL at the field point and the rest of the panels have a negative contribution. Among the panels contributing positively, panels with a contribution degree greater than 0.05 are the right glass, door glass, and bottom panel, which need to be optimized.

Modal Contribution Analysis.
e enclosed structure modal obtained from the MSC Nastran solution is imported into the acoustic software. Moreover, the calculation results of ATV are also imported to map the structure modal to the acoustic mesh, and the modal participation factor is calculated. en, the MATV matrix is calculated using ATV as the modal transfer matrix and combining the modal participation factors. Afterward, the MATV matrix is extracted by combining the frequency corresponding to sound pressure peaks in the DRE sound pressure response curve. According to Figure 8, the frequency of the higher peak value of sound pressure is 50 Hz and 100 Hz. Figures 13 and 14 show the value of the modal contribution of the cab modal at 50 Hz and 100 Hz, respectively. e top five-mode orders in modal contributions at 50 Hz are 3 rd , 4 th , 6 th , 36 th , and 40 th orders, while the top fivemode orders in modal contributions at 100 Hz are the 3 rd , 4 th , 6 th , 23 rd , and 25 th orders. e related mode vibration patterns are shown in Table 2, and the representative mode vibration patterns are shown in Figures 15, 16, and 17. From the representative modal vibration patterns, it can be concluded that the regions with greater influence on the peak sound pressure corresponding to 50 Hz and 100 Hz are mainly the center of the door glass, the center of the right glass, and the center of the bottom panel.

Structure Optimization.
e method of composite panel acoustic and modal contribution analysis identified the key locations for improvement i.e. the door glass, the right side glass, and the bottom panel. e optimization method of the noise transfer path is adopted to reduce cab noise. Since the upper part of the bottom panel is the seat and the bottom is the hydraulic pipeline, it is not easy to change the structure. us, the thickness optimization of the center of the bottom panel is not considered. e structure of the excavator cab is optimized Complexity using the NLOPT optimization program combining with the acoustic software. e NLOPT script is shown in Figure 18. e left side door glass center and the right glass center can be optimized. NLOPT involves various kinds of optimization algorithms with a uniform interface; the derivative-free optimization with LN_COBYLA function is suitable for the accuracy and speed of the optimization. Applying the same boundary conditions and excitation conditions to the optimized model, the optimization program is terminated after 20 iterations. Moreover, an optimal design is found that the total weight of the whole board is less than 35 kg, and the sound pressure level is low. After the optimization, the thickness of the glass on the right is 0.00566 m, and the thickness of the left side door glass is 0.0062 m. e optimized sound response in the cab is analyzed. Figure 19 shows the optimized DRE sound pressure curve and the original DRE sound pressure curve. It can be seen that the SPL at the frequency of 100 Hz is reduced from 53.06 dB to 51.36 dB. Although the SPL in the frequency range from 120 Hz to 200 Hz increased, the total sound pressure has been reduced by 2.3 dB, which indicates the direction of noise reduction for the cab.

Experimental Verification.
According to the results of the structure optimization, it can be concluded that the driver's right ear noise can be reduced by increasing the thickness of the right glass and left side door glass. e manual scheme is experimented by adding a mass block on the right glass and left side door glass, as shown in Figures 20 and 21.
When the right ear noise is collected under the 1050 rpm working condition, compared with baseline measurement results, as shown in Figure 22, the sound pressure level is reduced by 3.18 dB in the frequency band of 20 Hz-200 Hz, and the overall sound pressure level is reduced by 2.25 dB. e gap is because the sound pressure level of 315 Hz and 400 Hz and other medium and high-frequency noise increases, which leads to the noise level reduction of the whole frequency band. e experiment proves that the low-frequency noise can be reduced by thickening the panels.

Excitation points
Field point Acoustic cavity model

Conclusion
Reducing the vibration noise usually achieved by improving the structure, but how to quickly and accurately identify and optimize the structure with a large contribution, is a challenge. In this paper, a method of noise transfer path optimization based on the composite panel acoustic and modal contribution analysis is proposed. e acoustic transfer vector method is used to analyze the acoustic characteristics in the vibroacoustic model. To avoid the lack of considering a single frequency contribution in the traditional panel acoustic contribution analysis, the composite panel acoustic contribution analysis that takes the multiple frequencies into consideration is proposed. Synthesizing panel contribution and modal contribution results, the panels and areas that need to be optimized identified. e parameter optimization method coupling acoustic software, Python language, and algorithm is adopted for the target panel and areas. e method is verification in an excavator cab. e acoustic response of the excavator cab is consistent with the experimental sound pressure curve at the field point. By optimizing the thickness of the right glass and left side door glass, the overall sound pressure level at DRE is reduced by 2.25 dB. e experimental result verifies the feasibility of analysis and optimization of the noise based on the composite panel contribution and modal acoustic contribution methods, which provide a reference for passive noise control in a cab.

Data Availability
e data used to support the findings of this study have not been made available because the data are one of the research results of Sunward Intelligent Equipment Co., Ltd.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
Mass block e right glass  14 Complexity