Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

Le Qi; Jianxing Zhou; Huachao Xu

doi:10.1177/14613484221091075

Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

Qi, Le; Zhou, Jianxing; Xu, Huachao 2022-04-11 00:00:00 Aiming at noise problem of two-stage gearbox, this paper uses the panel acoustic contribution analysis (PACA) and response surface methodology to conduct structural-borne acoustics analysis and multi-objective optimization of gearbox. Using this optimization, optimal layout and parameters of the added ribs can be determined. The dynamic model of a spur gear-shaft-bearing system is built in consideration of the time-varying mesh stiffness, the time-varying bearing stiffness, and the ﬂexibility of the shaft, while dynamic load of the bearing is obtained through solving the model. The dynamic load of bearings is considered as excitation to analyze the vibration and noise radiation characteristics of housing. Furthermore, acoustic transfer vector, modal acoustic contribution and PACA are used to select the region with the most acoustic contribution. The variable ribs structure model (VRSM) is established in this region. Finally, the VRSM are adopted to optimize the housing. The optimization model based on the peak sound pressure level (PSPL) and the ribs mass (R ) is established by using response surface method. Through multi-objective optimization, the variable in VRSM can be modiﬁed or deleted to reduce the radiated noise of the gearbox. The optimization results demonstrate that the PSPL after optimization is decreased by 4.88 dB at frequencyof3130Hz. At thesamemass, thenoise reductioneffect of VRSM is 12% higher than that of standard rib. Therefore, the optimization effect of the housing structure is obvious. PACA-VRSM-RSM- NSGA-II technology provides a new research scheme for reducing the vibration and noise of the gearbox. Keywords gearbox, radiated noise, panel acoustic participation analysis, response surface methodology, multi-objective optimization Introduction Gear system was widely used in automotive, aerospace, and power generation industries due to its compact structure andhightransmission efﬁciency. The vibration of gearbox is mostly caused by the meshing action of gear teeth, in which it passes through gears, axles and bearings, and ﬁnally transfers to housing, which causing forced vibration of housing and radiating noise into the surrounding environment. The ﬁnite element/boundary element method (FEM/ 1–3 4 BEM) is the prevailing method applied to the numerical analysis for the gearbox vibration and noise. Kubur et al. proposed a dynamic model of a multi-shaft helical gear reduction unit formed by N ﬂexible shafts, and studied the inﬂuence of bearing stiffness, shaft size, and other parameters on the vibration of the gearbox. Later, Wei et al. established the coupled nonlinear dynamics model of a wind turbine planetary gear transmissionsystem, andproposed a lightweight and reliability constrained optimization design method according to the sequential quadratic pro- gramming algorithm. Subsequently, Chang et al. proposed a comprehensive coupled gearbox system dynamics model of a gear-shaft-bearing-shell system, which improved the accuracy of bearing response and thus more accurately School of Mechanical Engineering, Xinjiang University, Urumqi, China Corresponding author: *Jianxing Zhou, School of Mechanical Engineering, Xinjiang University, Urumqi 830047, China. Email: jianzhou82923@163.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access pages (https://us.sagepub.com/ en-us/nam/open-access-at-sage). 2 Journal of Low Frequency Noise, Vibration and Active Control 0(0) estimated the radiation noise of the gearbox. In order to solve the noise problem generated by operating gearboxes, Han et al. proposed a computational method for predicting gearbox noise from steady dynamic response to acoustic radiation calculation. Based on the gear dynamics model, many scholars have provided different schemes to reduce the radiated noise of the gearbox. To reduce the periodic change of meshing stiffness, Dogruer et al. proposed a nonlinear controller which adjusts the torque acting on the driving gear to minimize the adverse effect of time-varying mesh stiffness. And a spur gearbox was designed which does not emit much noise and vibration. Jolivet et al. addressed the issue of the impact tooth ﬂank ﬁnishingon noise andvibration. Ghosh et al. used the genetic algorithm to optimize the geometrical shape of the gear, reducing the noise of the tractor gearbox. In addition, the surface ribbing of the shell is one of the main methods of noise reduction. Many of structures contain an important account of ribs. Moyne et al. discussed the inﬂuence of adding ribs to an automobile gearbox shell on the radiation noise. The inﬂuence of ribs was theoretically discussed and three different physical effects were distinguished: the “vibrating,” the “obstacle,” and the “source” effects. Furthermore, panel acoustic contribution analysis (PACA) is an important method for optimizing the shell structure, which can determine the approximate position of the ribs plate distribution. Based on the dynamics model of a gear system, Zhou et al. added ribs using the acoustic contributions of the surface of the gearbox, and designed two gear stiffness improvement schemes to reduce the radiated noise of the gearbox. Although PACA can identify the panel area of the greatest acoustic contribution, it cannot determine the exact location of the ribs in this area, nor the geometry of the added ribs, which requires further calculations. Wang et al. proposed an optimal ribs layout design method to reduce radiation noise based on topology optimization and acoustic contribution analysis, veriﬁed the accuracy of the result by modal test and acoustic measurement. However, the results of topology optimization are complex, diverse and difﬁcult to achieve. Compared to topology optimization, traditional structure optimization is often costly and inefﬁcient. Artiﬁcial intelligence algorithms can signiﬁcantly reduce calculation time and material cost. Wang et al. proposed a multi-objective optimization design based on the response surface method and PACA, and pasted a damping layer on the panel surface to reduce radiation noise. Zhou et al. adopted the free damping structure and new stiffener structure to optimize the rear end cover of a double planetary gear housing. The opti- mization model based on the vibration acceleration of the rear end cover surface was established by applying K-S function and response surface method. However, there has been no serious attempt to determine the optimum layout and parameters of ribs. To address this, our investigation adds acoustic transfer vector (ATV) and modal acoustic contribution (MAC) to help identify the regions with the greatest acoustic contribution, enriching the screening process of the PACA method. Based on ATV-MAC-PACA, the variable ribs structure model (VRSM) as design variables in which utilized ribs width, thickness, number, and direction is proposed. Through multi-objective optimization, the variable in VRSM can be modiﬁed or deleted to reduce the radiated noise of the housing. PACA-VRSM-RSM- NSGA-II technology provides a new research scheme for the layout of gearbox ribs. Design methods and processes This section analyzes the vibration and noise calculation method of a two-stage gearbox system. And the PACA-VRSM- RSM-NSGA-II method for the gearbox structure optimization is as shown in Figure 1. The entire analysis process is composed of ﬁve components, namely the (a) gearbox dynamic excitation calculation, (b) gearbox dynamic response analysis, (c) gearbox radiated noise calculation, (d) determine ribs addition area and VRSM, and (e) Multi-objective optimization. (a) Gearbox dynamic excitation calculation. Analytical model is established by using lumped mass method. In the model, the inﬂuence of the TVMS, TVBS, and shaft ﬂexibility is considered. The dynamic load of the bearings is calculated by the Newmark integral method. (b) Gearbox dynamic response analysis. Modal analysis of the shell ﬁnite element model with boundary conditions is carried out. The dynamic load of the system is then added to the corresponding position of the shell ﬁnite element model. The modal superposition method is used to solve the dynamic response of the gearbox structure, and the vibration response and surface acceleration of the housing are also obtained. (c) Gearbox radiated noise calculation. The surface element of the ﬁnite element model is extracted, and the shell element is used to ﬁll the holes on the surface of the structure. Finally, the gearbox boundary element model is obtained by remeshing the surface elements of gearbox. And the obtained boundary element model and ATV are used to solve the radiated noise of the gearbox. Qi et al. 3 Figure 1. Multi-objective optimization calculation process of ribs layout of the two-stage gearbox. (d) Determine ribs addition region and VRSM. Three screening methods, ATV, MAC, and PACA, were used to de- termine the region with the greatest acoustic contribution. And the variable ribs model is added in this region. (e) Multi-objective optimization. Eleven structural parameters of the VRSM are used as input variables. PSPL and R are used as output variables. Response surface models of PSPL and R are then constructed by central composite design (CCD), and R is used to test the accuracy of the model. When the model accuracy meets the requirements, the obtained response surface objective function obtained is exported to NSGA-II, and the optimal ideal point is determined by analyzing the Pareto solution set. Finally, an experimental comparison is conducted to obtain the ideal VRSM. Gearbox dynamic excitation calculation The object of analysis in this paper is a two-stage gear reducer, whose model is as shown in Figure 2, which is a two-stage straight toothed cylinder gear reducer, and the minor chamfering and feature in structure is simpliﬁed in the model. The speciﬁc parameters of the gear are shown in Table 1, and the bearing parameters are shown in Table 2. Gear transmission system dynamic model The spur gear transmission system model has six degrees of freedom (DOF), namely rotation DOF of driving and driven wheels, the translation DOF along x and y directions (see Figure 3). Then the relative total displacement of meshing elements of spur gear pair δ is deﬁned as δ ¼ x sin α þ y cos α r θ x sin α y cos α r θ eðtÞ (1) p p p p g g g g where α is the pressure angle, r is the base circle radius of the driving gear, and r is the base circle radius of the driven gear. p g 4 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 2. (a) 3D model of two-stage gearbox, (b) Internal structure of the gearbox. Table 1. Gear system parameters. st st nd nd Parameters 1 stage driving gear 1 stage driven gear 2 stage driving gear 2 stage driven gear Number of teeth 36 90 29 100 Modulus (mm) 1.5 1.5 1.5 1.5 Pressure angle ()20 20 20 20 Tooth width (mm) 12 12 12 12 Table 2. Bearing parameters. Diameter of inner race (mm) Diameter of outer race (mm) Ball number Ball diameter (mm) 28.7 46.6 8 8.7 Pitch circle diameter (mm) Radial internal clearance Curvature radius (mm) Goodness ﬁt 37.65 0.5 4.5 0.5172 Figure 3. Dynamics model of gearbox transmission system. Qi et al. 5 Considering the inﬂuence of time-varying meshing stiffness and the gear meshing error, the motion differential equation of the meshing element of the spur gear can be expressed as > _ > m x€ þ c δ sin α þ k δ sin α þ f sin α ¼ 0 > p p m m s > _ > m y€ þ c δ cos α þ k δ cos α þ f cos α ¼ 0 > p m m s > € _ I θ c δr k δr f r ¼ 0 p p m p m p s p (2) > _ > m x€ c δ sin α k δ sin α f sin α ¼ 0 > g g m m s > _ > m y€ c δ cos α k δ cos α f cos α ¼ 0 > g m m s > € _ I θ c δr k δr f r ¼ 0 > g g m g m g s g where m is the mass of gear, k is the meshing stiffness, c is the meshing damping, I is the moment of inertia of gear and f m m s is the normal impact force. During the operation of the gear transmission system, the bearing supports the shaft system and transmits vibration from the gears to the housing. The bearing stiffness matrix can be expressed as K ¼ diag k ,k ,0 (3) bx by The ﬁnite element model of the gearbox (see Figure 4(a)) is established according to the gear meshing, bearing, and shaft elements, with a total of 29 elements and 32 nodes. The stiffness matrix of each element is superimposed according to the position of the element nodes to obtain the overall stiffness matrix, as shown in Figure 4(b). The generalized displacement vector of the system is fxg ¼ x , x , x , x , x , x ……x , x , x (4) 1x 1y 1θ 2x 2y 2θ nx ny nθ where n denotes the number of nodes. The dynamics equation of the gear system can be expressed as n o n o ½m x€ðtÞ þ½c x_ðtÞ þ½kðtÞfxðtÞg ¼ FðtÞ (5) where {x(t)} is the generalized nodal displacement vector, [m], [c], and [k] are the matrix of global mass, damping, and stiffness, respectively, and {[F(t)]} is the external force vector received by the system. Bearing dynamic load The input speed is 5400 r/min, and the output torque is 100 Nm. The time domain and frequency domain diagrams of dynamic loads of each bearing are obtained by Newmark integral method, as shown in Figure 5. where different colors Figure 4. Solution of overall stiffness of gearbox. (a) Finite element model of gearbox, (b) Overall stiffness matrix of gearbox. 6 Journal of Low Frequency Noise, Vibration and Active Control 0(0) represent different bearings. f (3240 Hz) and f (1044 Hz) represent the meshing frequency of the ﬁrst gear pair and the m1 m2 second gear pair, respectively. Experimental veriﬁcation The platform used in this experiment is the SQI Wind turbine transmission system test bench (shown in Figure 6(a)). The main devices include the test bed, drive motor, controller, gearbox, and data collector, which can be used to study the dynamic analysis of the gear transmission in this paper. The driving motor is used to adjust the speed automatically, increasing 240 rpm each time, and the acceleration value after each speed change is recorded by the acceleration sensor (shown in Figure 6(b)). The experimental results are compared with the simulation results (see Figure 7). It can be seen that the simulation results are very close to the trend of the experiment, indicating that the simulation results can replace the experiment. Gearbox dynamic response analysis Figure 8 is the ﬁnite element model of the gearbox. The material of the plate and ribs are both steels having Young’s 11 3 modulus E = 2.07 × 10 Pa, Poisson’s ratio μ = 0.3, and mass density ρ = 7800 kg/m . In order to apply dynamic load to the gearbox, a mass node is created in the center of the bearing hole and the node is coupled with the node on the inside surface of the bearing hole. Then the dynamic load is applied on the central node. A displacement constraint is applied at the bottom of the gearbox. The ﬁnite element model of housing and the ﬁrst eight natural vibration modes of the gearbox are shown in Figure 8. When the bottom of the gearbox is restrained, the vibration of the lower part of the gearbox is weak, while the vibration amplitude of the top part is obvious. Figure 5. The bearing dynamic load of the gear system. (a) Time response of loads from bearing 1 to bearing 6, (b) Spectrum of loads from bearing 1 to bearing 6. Figure 6. Transmission system test bench. (a) System equipment, (b) Sensor measurement position. Qi et al. 7 Gearbox radiated noise calculation The BEM is obtained by extracting surface elements from the gearbox ﬁnite element model and repairing the bearing holes with shell elements. The acoustic ﬂuid is air having mass density ρ0 = 1.2 kg/m3 and sound speed c = 343.4 m/s. Based on the boundary element model, the hemispherical shape acoustic ﬁeld is established. Four points with the same radius are Figure 7. Comparison between simulation and experiment. Figure 8. Finite element model of housing, (b) Mode of vibration of gearbox. Figure 9. (a) Gearbox ﬁeld point distribution, (b) ATV. 8 Journal of Low Frequency Noise, Vibration and Active Control 0(0) selected on the acoustic ﬁeld as the measured ﬁeld points, namely, ﬁeld points A, B, C, and D, as shown in Figure 9(a). In the case of small perturbations, the acoustic equation can be considered linear. Thus, a linear relationship can be established between the input (the normal vibration of the structure surface) and the output (the sound pressure at a point in the sound ﬁeld). According to the relationship shown in Figure 9(b), the sound pressure at a certain ﬁeld point can be expressed as pðωÞ¼ ATV ðωÞ v ðωÞ (6) where ATV is the acoustic transfer vector calculation, and ω is the angular frequency. The boundary element method is used to solve the noise radiation of the gearbox by taking the normal vibration acceleration of the shell’s outer surface as the boundary condition. Figure 10 shows the noise spectrum when the input speed is 5400 rpm, and the torque is 100 Nm. Due to the consideration of shaft ﬂexibility, the overall noise increases ﬁrst and then decreases. From the sound pressure level curves, it is well noticed that peak is generated on the position of 3 × fm2. The corresponding frequency of the maximum radiated noise is 3 × f (3132 Hz), which is close to the eighth order natural m2 frequency of housing (3098.35 Hz). At 3 × f (3132 Hz), the SPL at ﬁeld point A is the largest, so ﬁeld point A is selected m2 as the target ﬁeld point. The radiated noise at point A can be reduced by suppressing the PSPL at 3 × f . m2 Determine ribs addition region and VRSM In this study, the regions with larger ATV can be found by the analysis of ATV. Then, the regions with larger MAC can be obtained by MAC. Finally, panels can be divided on the regions with larger MAC and ATV concurrently and the acoustic contribution of each panel can be obtained by PACA. This process can be found the region with maximum acoustic contribution. Based on ATV-MAC-PACA, the VRSM was established. Acoustic transfer vector (ATV) Figure 11 demonstrates the amplitude of the acoustic transfer vector at 3 × f (3132 Hz) to ﬁeld point A. It can be seen that m2 the amplitude on the upper part of the gearbox is signiﬁcantly greater than that on the bottom, indicating that the normal vibration on the upper part of the gearbox has a more obvious impact on the PSPL. In summary, the ﬁrst screening region is the upper part of the gearbox. Figure 10. Noise spectrum of gearbox. Qi et al. 9 Figure 11. Amplitude of acoustic transfer vector at ﬁeld point A at 3 × f . m2 Modal acoustic contribution analysis (MACA) Using the modal superposition method, the displacement response of the multi-free system can be obtained as f ðωÞ φ xðωÞ¼ φ (7) a ðiω λ Þ k k k¼0 f ðωÞ φ where k is the order of the mode, q ðωÞ¼ is the structural modal participation factor, λ is the k-th eigenvalue of the k k a ðiωλ Þ k k system, {φ} denotes the mode shape of the k-th order; {f (ω)} is the load vector, and α denotes a constant related to the k k characteristics of the system. By projecting {x(ω)} onto the normal direction of the box surface and taking the derivative, the vibration velocity in the normal direction of the gear box surface can be obtained as v ðωÞ¼ iω q ðωÞφ (8) n k jn k¼1 where {φ} is the component of the mode shape on the normal line of the gearbox surface. jn By substituting equations (9) into (10), the sound pressure at any point in the sound ﬁeld can be obtained as n n n X X X T T pðωÞ¼ ATV ðωÞ iω q ðωÞφ ¼ iωq ðωÞATV ðωÞ φ ¼ P (9) k k sk nk nk k¼1 k¼1 k¼1 where P ¼ iωq ðωÞATV ðωÞ φ denotes the sound pressure generated by mode K. The projection of P in the sk k sk nk k¼1 direction of the total sound pressure P(ω) is called the modal acoustic contribution. The acoustic contribution of each mode at ﬁeld point A is obtained by MACA, as shown in Figure 12. It can be seen that the eighth mode has a large acoustic contribution to the PSPL at ﬁeld point A. The corresponding main mode shape of this mode is shown in Figure 12. The eighth main mode vibrates obviously at the top of the gearbox. Therefore, the result of the second screening is the top region of the gearbox. 10 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 12. The modal acoustic contributions at the ﬁeld point A at the meshing frequency. Panel acoustic contribution analysis (PACA) The main panels corresponding to the ﬁrst eight modes are divided, and PACA analysis was carried out. The acoustical contribution of each panel to the ﬁeld point A at 3×f (3132 Hz) was obtained, as shown in Figure 13. It can be seen from m2 the ﬁgure that panel 1 and panel 2 have the largest contribution, more than 25 percent, so the screening results through PACA are panel 1 and panel 2. In summary, Using the ATV-MAC-PACA method, the central region of the top of the gearbox has the largest acoustic contribution to the ﬁeld point A at 3×f (3132 Hz). Therefore, the VRSM can be added at this region to reduce the radiated m2 noise of the gearbox. Variable ribs structure model (VRSM) Although the general region of the added ribs can be obtained from the previous screening, the research on the distribution and number of ribs and the speciﬁc parameters of each rib requires further design. The ribs layout method adopted in this paper refers to literature. Figure 14(a) is the rectangular region determined according to the screening results. To study the effect of the layout and parameters of the ribs on reducing the radiated noise of the shell, the ribs model is added to the top center of the shell (note: all model ribs are added to the inner surface of the top of the box body to prevent disturbance of the ATV and improve calculation efﬁciency). The characteristic of VRSM is characterized by the fact that there is no deﬁnite ribs layout and size, and variables in the model can be changed as required. Thus, different VRSMs can be realized according to different optimization results. As shown in Figure 14(b), the model contains 11 design variables; namely, the number of horizontal and vertical ribs (n and n ), the width of the ribs (d , d , d , and d ), and the thickness of the ribs (t , t , t , and t ). φ is the 1 2 1 2 3 4 1 2 3 4 rotation angle of the ribs relative to the x-axis (shown in Figure 14(c)). In addition, the spacing y and y between the ribs are 1 2 determined by the number of horizontal and vertical ribs, respectively; y and y reﬂect variation the density of rib 1 2 distribution in the gearbox. It can be seen that through the selection of value range, the rib height and thickness variables can be set to zero, and then the redundant ribs can be deleted in the corresponding optimization. Multi-objective optimization There is a great relationship between the mass of ribs and peak value of radiated noise of the gearbox, and the two are in conﬂict in essence, so it is very necessary to establish a multi-objective optimization model. Response surface model (RSM) Based on statistical and mathematical methods, RSM can build and evaluate a variety of complex problems. In this method, the output variable (system response) is approximately ﬁtted to the speciﬁed design space to replace the real response Qi et al. 11 Figure 13. Analysis results of acoustical contribution of plate surface at meshing frequency. Figure 14. 3D model of ribs layout design. (a) the position where the ribs are added in the gearbox, (b)VRSM, and (c) variable φ: the rotation Angle of the ribs relative to the x-axis. surface. In order to give consideration to the ﬁtting accuracy and optimization calculation time, the central composite design (CCD) was adopted. In this design, alpha = 0.5, the center point is 1 and the factorial core uses the min-res V method. The relation between the independent variable in the response surface and response Z can be expressed as Z ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞþ ε (10) 1 2 1 2 3 4 1 2 3 4 where ε is the calculation error. RSM usually adopts quadratic polynomial model to ﬁt the parameter model of the response. Its approximate function can be expressed as n n X X Z ¼ c þ c x þ c ðx Þ þ i ¼ 1nj > icijxixj (11) 0 i i ii i i¼1 i¼1 where Z denote the estimated response; c denote the constant term coefﬁcient; c denotes the coefﬁcients of the input factor 0 i x ; c denotes the quadratic coefﬁcients on the independent variable x and c denotes the coefﬁcients of the interaction term i ii i ij between the input variable x and x . i j 12 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Multi-objective genetic algorithm Genetic Algorithms is a heuristic algorithm developed on the basis of biological evolutionary theory and Genetic thought. Compared with single objective optimization, multi-objective optimization can better solve problems with multiple objective requirements. NSGA-II proposed by Deb et al. is a multi-objective optimization algorithm based on Pareto optimal solution, that is, a fast non-dominated multi-objective optimization algorithm with elite retention strategy. The three major characteristics of NSGA-II are the proposed congestion degree and congestion comparison operator, and the in- troduction of non-dominated ranking and elite strategy. In this paper, the optimal VRSM is solved while considering multiple targets. The objective functions of Rm and PSPL are obtained by a large number of experimental data and RSM. The optimization problem model is as follows > Minimize R ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞ > m 1 1 2 1 2 3 4 1 2 3 4 > Minimize PSPL ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞ > 2 1 2 1 2 3 4 1 2 3 4 > subject to constraints 2 ≤ n ,n ≤ 6 (12) 1 2 0 ≤ d ,d ,d ,d ≤ 5 1 2 3 4 0 ≤ t ,t ,t ,t ≤ 6 1 2 3 4 0 ≤ φ ≤ 90 where f ()and f () denote objective functions for R and PSPL, respectively. 1 2 m Results and discussion As per CCD design, 91 experiments are performed for VRSM. See Appendix 1 for speciﬁc simulation experiment data. The regression analysis is an assessment of the accuracy and predictive power of response surface models. When the response surface model is more accurate, the numerical model will be closer to the physical model, and the predicted value will be 2 2 more accurate. Generally, R and adjusted R are used to characterize the accuracy of response surface model, and the mathematical formulations are given by SE R ¼ 1 (13) ST S ðN K 1Þ SE R ¼ (14) adj S ðN 1Þ ST S ¼ ðY y Þ (15) SE i i i¼1 N N X X S ¼ Y y N (16) ST i i¼1 i¼1 where S denotes the sum of squares of regression residuals, S denotes the sum of squares of total variation, N denotes the SE ST number of experimental sites, K denotes the number of unknown coefﬁcients, Y denotes the response vector of the test point, y denotes the RSM approximate estimate vector. Quadratic equation for PSPL A mathematical model for prediction of PSPL using VRSM as the desiccant material is described in equation (17) Qi et al. 13 PSPL ¼ 93:11 1:77n þ 0:86n 0:18d þ 0:14d 0:20d þ 0:45d þ 0:05t þ 0:14t 1 2 1 2 3 4 1 2 þ0:15t þ 0:07t þ 0:01φ 0:02n d þ 0:01n t 0:04n t 0:03n d 0:04 × n d 3 4 1 3 1 2 1 3 2 1 2 4 0:03n t 0:01d d þ 0:02d d 0:04d t 0:01d t 0:01d t 0:04 × d t þ 0:02d t 2 2 1 2 1 3 1 1 1 3 2 1 2 2 2 3 (17) 0:01d t þ 0:001d φ þ 0:01d d þ 0:01d t 0:08d t þ 0:02d t 0:02d t 0:02d t 2 4 2 3 4 3 1 3 3 3 4 4 1 4 2 2 2 þ0:04d t 0:07d t þ 0:01t t þ 0:01t t þ 0:02t t 0:0006t φ þ 0:22n 0:09n 4 3 4 4 1 2 2 3 3 4 4 1 2 2 2 2 2 2 2 2 2 2 þ0:06d 0:03d þ 0:03d 0:06d 0:003t 0:02t 0:03t 0:01t 0:0001φ 1 2 3 4 1 2 3 4 The response surface model of PSPL is established through the above calculation formula. Before using the mathematical 2 2 model, the goodness of developed models is assessed by evaluating the coefﬁcient of determination (R ). The values of R = 0.9832 and adjusted R = 0.9632 are very close to 1, and the difference between the predicted value and the adjusted value is less than 0.2. Therefore, the mathematical model established for the optimization objective meets the accuracy re- quirements. As shown in Table 3, when the resulting p-value is less than 0.0001, the model can be considered signiﬁcantly important. The items exceeding the recognized standard value of 0.05 can then be deleted to improve the model accuracy. Nevertheless, the purpose of this study is to investigate the effects of all parameters on response variables, and as many parameters as possible can be retained for research when the accuracy is satisﬁed. According to the sum of squares, the linear term has a contribution of 44.22%. The contribution of the cross-interaction term is 36.45%, and the contribution of self-interaction term was 0.11%. Figure 15 shows the two-dimensional contour plot of part of the response surface, in which the contour plot of the PSPL is in good agreement with the experimental data. The contour lines in the ﬁgure are divided into Table 3. Factors and levels of the central composition design. PSPL R PSPL R m m Sum of p-value prob p-value prob Sum of p-value prob Sum of p-value prob Source squares > f Sum of squares >f Source squares >f squares >f Model 176.62 <0.0001 1,564,182 <0.0001 d t 1.13 0.0003 2576 0.0002 2 3 n 4.10 <0.0001 43,291 <0.0001 d d 0.34 0.0363 1605 0.0019 1 3 4 n 2.86 <0.0001 28,604 <0.0001 d t 0.66 0.0047 924 0.0143 2 3 1 d 5.63 <0.0001 46,528 <0.0001 d t 17.62 <0.0001 114,466 <0.0001 1 3 3 d 3.07 <0.0001 28,738 <0.0001 d t 0.79 0.0021 4168 <0.0001 2 3 4 d 13.36 <0.0001 93,237 <0.0001 d t 1.48 0.0001 4263 <0.0001 3 4 2 d 15.23 <0.0001 94,319 <0.0001 d t 3.43 <0.0001 9530 <0.0001 4 4 3 t 2.47 <0.0001 12,380 <0.0001 d t 14.21 <0.0001 90,296 <0.0001 1 4 4 t 5.44 <0.0001 49,436 <0.0001 t t 0.38 0.0286 3470 <0.0001 2 1 2 t 16.10 <0.0001 117,150 <0.0001 t t 0.46 0.0162 1018 0.0105 3 2 3 t 9.83 <0.0001 69,887 <0.0001 t t 1.24 0.0002 3643 <0.0001 4 3 4 φ 0.03 0.5288 721 0.0283 t φ 0.36 0.0334 2410 0.0003 n d 0.77 0.0024 5360 <0.0001 n 0.11 0.2308 23.76 0.6763 1 3 1 n t 2.90 0.0358 20,953 <0.0001 n 0.02 0.6147 4.06 0.8627 1 3 2 n d 0.86 0.0000 2904 0.0001 d 0.02 0.6163 0.43 0.9550 2 1 1 n d 2.58 0.0014 682 0.0326 d 0.004 0.8164 0.72 0.9419 2 4 2 n t 0.99 <0.0001 5197 <0.0001 d 0.01 0.7762 0.60 0.9469 2 2 3 d d 0.22 0.0007 2434 <0.0001 d 0.02 0.6339 0.64 0.9455 1 2 4 d d 0.52 0.0885 1591 <0.0001 t 0.001 0.9666 3.30 0.8762 1 3 1 d t 5.66 0.0113 38,861 <0.0001 t 0.002 0.8589 4.19 0.8607 1 1 2 d t 0.26 <0.0001 2501 <0.0001 t 0.01 0.6928 13.85 0.7498 1 3 3 d t 0.58 0.0682 7048 <0.0001 t 0.002 0.8803 14.97 0.7403 2 1 4 d t 4.50 0.0074 30,060 <0.0001 φ 0.01 0.7333 1.85 0.9071 2 2 Model statistics R 0.9832 0.9979 Adjusted R 0.9632 0.9924 Predicted R 0.9120 0.9445 Adeq precision 29.5684 61.1448 14 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 15. Contour plots for PSPL in terms of (a) n vs. d , (b) n vs. d , (c) n vs. t , (d) d vs. d , (e) d vs. t , (f) d vs. t , (g) d vs. t , (h) d vs. 1 3 2 4 2 2 1 3 2 1 2 2 2 3 3 d , (i) d vs. t , (j) d vs. t , (k) d vs. t , (l) d vs. t , (m) t vs. t , (n) t vs. t , (o) t vs. t , and (p) t vs. φ. 4 3 1 4 2 4 3 4 4 1 2 2 3 3 4 4 ﬁve levels, from red to blue, indicating gradual changes from high to low. It can be observed that when the other nine variables are ﬁxed, and the horizontal coordinate and vertical coordinate variables change, PSPL will also change ac- cordingly (Figure 15). Qi et al. 15 Figure 16. Contour plots for R in terms of (a) n vs. d , (b) n vs. d , (c) n vs. t , (d) d vs. d , (e) d vs. t , (f) d vs. t , (g) d vs. t , (h) d vs. m 1 3 2 4 2 2 1 3 2 1 2 2 2 3 3 d , (i) d vs. t , (j) d vs. t , (k) d vs. t , (l) d vs. t , (m) t vs. t , (n) t vs. t , (o) t vs. t , and (p) t vs. φ. 4 3 1 4 2 4 3 4 4 1 2 2 3 3 4 4 16 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Quadratic equations for R . Regression function concerning R can be expressed as follows m m R ¼ 65:13 þ 8:56n 4:92n 20:41d 2:79d 1:69d 9:42d 8:56t 10:66t m 1 2 1 2 3 4 1 2 19:93t 8:89t þ 0:46φ þ 2:88n d þ 2:33n d þ 1:25n d þ 0:61n t þ 3:63n t þ 1:20n t 3 4 1 1 1 3 1 4 1 1 1 3 1 4 1:8n d þ 1:36n d þ 0:79n d þ 2:43n d þ 2:14n t þ 2:08n t 0:65n t þ 0:07n φ 2 1 2 2 2 3 2 4 2 2 2 3 2 4 2 þ1:07d d 1:07d d 0:42d d þ 4:01d t þ 1:10d t þ 0:34d t þ 0:42d d 0:82d d 1 2 1 3 1 4 1 1 1 3 1 4 2 3 2 4 (18) þ1:67d t þ 3:69d t 1:10d t 0:09d φ 0:94d d 0:67d t 0:62d t þ 6:81d t 2 1 2 2 2 3 2 3 4 3 1 3 2 3 3 1:27d t þ 1:44d t 2:04d t þ 7:08d t 0:95t t 0:37t t 0:37t t 0:62t t 3 4 4 2 4 3 4 4 1 2 1 3 1 3 1 4 2 2 2 þ0:04t φ 0:67t t þ 0:07t φ 1:10t t þ 0:05t φ þ 0:06t φ 3:29n 1:36n 0:28d 1 2 3 2 3 4 3 4 1 2 1 2 2 2 2 2 2 2 2 0:37d 0:34d 0:34d þ 0:54t þ 0:61t þ 1:12t þ 1:16t 0:002φ 2 3 4 1 2 3 4 The above regression function is mathematically veriﬁed by the ANOVA test, and the obtained results are shown in Table 3. Based on the sum of squares, it can be seen that the contribution ratio of the linear term is 37.35%, and the contribution ratio of the cross-action term is 22.76%. According to the corresponding calculated statistical error value R = 0.9979 and adjusted R = 0.9924. The results show that the experimental data are well-adapted to the established model. Figure 16 depicts the contour map of part of R , showing the inﬂuence of each parameter and the interaction between the parameters on the corresponding response. Optimization results Statistical analysis shows that the results obtained by RSM correspond well with the experimental data. Therefore, the regression equation obtained by the response surface method can be used as the objective function of the subsequent multi- objective algorithm. NSGA-II is used to calculate the optimal solution set of PSPL and Rm. NSGA-II has been im- plemented in this study for a population size of 50, a crossover fraction of 0.8, a Pareto front population fraction of 0.35. Due to the conﬂict and competition among optimization objectives, multi-objective optimization has no unique solution, and its solution is a set of optimal non-inferior solutions. It is noteworthy that in the present study, NSGA-II has been run 20 times for the case of optimization, and corresponding ﬁrst ranked Pareto fronts have been selected. Figure 17 shows the Pareto solution set for PSPL and Rm solved by NSGA-II. The coordinates of the points in the Pareto solution set in the ﬁgure are derived from Table 4. Table 4 shows the optimal values of the two objective functions and the corresponding values of the 11 parameters. The values of variables in the table are rounded results. As seen in Table 4, 4 and 6 occur most frequently among all optimization results for n1 and n2. The highest number of occurrences in d1, d2, d3, and d4 are 1, 4, 1, and 5, respectively. The highest number of occurrences in t1, t2, t3, and t4 are 5, 6, 6, and 1, respectively. The optimal value of φ is close to 90°. Among all variables, such as the number, width, and thickness of ribs and φ, the thickness of the ribs has the highest value and the most obvious effect on noise reduction. Therefore, the primary consideration should be to increase the thickness of the ribs. Figure 17 clearly illustrates that at point Figure 17. Pareto solution set calculated with NSGA-II. Qi et al. 17 A(n =4, n =5, d = 5 mm, d = 3 mm, d = 5 mm, d = 5 mm, t = 6 mm, t = 4 mm, t = 6 mm, t = 6 mm, φ = 78°), PSPL is 1 2 1 2 3 4 1 2 3 4 reduced to the minimum, so point A can be regarded as the optimal layout condition for the most obvious noise reduction of the gearbox after adding variable ribs. R drops to its lowest at point B (n =4, n =2, d = 1 mm, d = 4 mm, d = 0 mm, m 1 2 1 2 3 d = 0 mm, t = 1 mm, t = 1 mm, t = 2 mm, t = 4 mm, φ = 88°). Thus, point B is considered to be the optimal layout 4 1 2 3 4 condition for variable ribs with the lowest mass. In order to verify the rationality of the theoretical points, the parameters of the variable ribs model corresponding to points A and B in the multi-objective optimization results are substituted into the calculation ﬂow of the gearbox radiation noise, and the obtained results are compared with points A and B. The comparison results are provided in Table 5. In the corresponding output variables of the point A, the PSPL and R optimization results and simulation values errors are 0.52% and +1.8 g, respectively. In the output variables corresponding to point B, the Table 4. The corresponding parameter variables and corresponding values of the output variables in the Pareto solution set. Variables Objective function n1 n2 d1 d2 d3 d4 t1 t2 t3 t4 φ FSPL R 4 633255 6 1 3 11 87.68 280.24 4 553556 4 6 6 78 86.49 521.49 5 543356 4 6 6 80 86.71 499.77 5 544356 4 6 6 86 86.79 472.89 4 635545 6 6 1 38 86.95 426.18 4 635445 6 6 1 43 86.99 422.70 4 635455 6 6 1 41 87.29 376.95 4 621454 6 5 5 33 87.44 345.47 4 614253 6 3 6 17 87.61 330.93 4 615253 6 3 4 13 87.63 313.50 3 654154 5 1 3 21 87.68 303.13 4 640155 6 0 2 10 88.40 175.97 4 630155 5 1 3 6 88.50 173.18 5 213303 0 6 2 90 88.79 156.86 4 640355 4 0 2 8 88.82 138.81 4 640355 3 0 1 5 89.15 128.64 3 631156 3 0 1 1 89.20 125.56 4 640355 3 0 1 5 89.31 114.61 4 640155 1 0 1 1 89.39 105.53 4 640156 0 0 1 3 89.42 97.15 4 640156 1 0 1 3 89.54 93.40 3 224201 0 5 3 84 90.22 45.18 4 213101 1 6 3 74 90.32 41.68 4 215111 1 5 5 89 90.42 34.21 4 214101 0 5 3 88 90.44 23.00 4 214101 0 4 3 80 90.69 13.43 5 233201 0 1 3 81 90.75 10.85 4 214101 0 4 3 85 90.78 8.55 4 214001 1 2 4 88 90.96 6.28 Table 5. The calculated ideal points are compared with the corresponding experimental data. Design variable point A, B Output variable Estimated Experimental Deviation A PSPL 86.49 86.95 0.53% R 521.49 523.29 +1.8 g B PSPL 90.96 91.58 0.67% R 6.28 11.1 +4.82 g m 18 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 18. radiation noise spectrum of the original structure and each scheme model at ﬁeld point A. Table 6. The PSPL at ﬁeld point A. Standard rib, dB VRSM, dB PSPL 87.48 86.95 theoretical and calculated error values of PSPL and R are 0.67% and +4.82 g, respectively. Figure 18 shows the spectral comparison of sound pressure levels of the improved front and rear gearboxes, where the noise reduction amplitude of the ideal point A is very obvious compared with that before adding ribs, while the noise reduction amplitude of ideal point B is very small. Researchers can choose the appropriate variable ribs layout according to the speciﬁc requirements of the R .For this purpose, the relationship between PSPL and R is deduced according to the Pareto front as follows R ¼ 177873:68 3890:95ðPSPLÞþ 22:28ðPSPLÞ (19) The region obtained by PACA method is used as the base area of the rib to establish the standard rib. Adjust the thickness of the standard rib so that the mass of the standard rib is the same as the mass of the VRSM at the optimal point A. The PSPL of the standard rib and VRSM at the ﬁeld point A is shown in Table 6. At the same mass, the noise reduction effect of VRSM is 12% higher than that of standard rib. Conclusions In this paper, a two-stage gearbox radiated noise model is established, and the response surface method and NSGA-II can modify the variables in the VRSM to reduce the noise of the gearbox. The conclusions of this paper are as follows: 1. ATV and MAC improve the accuracy of the PACA method. The ATV-MAC-PACA method can also be used to add damping materials, shock absorbers, and other mechanisms. 2. The thickness of the ribs is the key variable for noise reduction. The PSPL after optimization is decreased by 4.88 dB at frequency of 3130 Hz. At the same mass, the noise reduction effect of VRSM is 12% higher than that of standard rib. Qi et al. 19 3. The VRSM should be determined according to different ﬁeld points and different excitation frequencies. A dis- advantage of this method is that acoustic contribution analysis must be performed again after changing the working conditions and considering the ﬁeld points to achieve a good noise reduction effect. Declaration of conﬂicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study was funded by the Chinese National Natural Science Foundation (51665054), Natural Science Foundation of Xinjiang Province (No. 2021D01C050) and Xinjiang Province key research and development (2021B01003-1). ORCID iDs Le Qi  https://orcid.org/0000-0002-7345-2636 Jianxing Zhou https://orcid.org/0000-0002-0764-1172 Huachao Xu https://orcid.org/0000-0003-4471-805X References 1. Attia AY. Noise of involute helical gears. J Eng Ind 1969; 2: 165–171. 2. Neriya S, Bhat R, and Sankar T. Coupled torsional-ﬂexural vibration of a geared shaft system using ﬁnite element analysis. Shock Vib Bull 1985; 55: 13–25. 3. Abbes M, Bouaziz S, Chaari F, et al. An acoustic-structural interaction modelling for the evaluation of a gearbox-radiated noise. Int J Mech Sci 2008; 50: 569–577. 4. Kubur M, Kahraman A, and Zinid M. Dynamic analysis of a multi-shaft helical gear transmission by ﬁnite elements model and experiment. J Vib Acoust 2004; 126: 398–406. 5. Wei J, Lv C, Sun W, et al. A study on optimum design method of gear transmission system for wind turbine. Int J Precis Eng Manuf 2013; 14: 767–778. 6. Chang L, He C, and Liu G. Dynamic modeling of parallel shaft gear transmissions using ﬁnite element method. Shock Vib 2016; 35: 47–53. 7. Han J, Liu Y, Yu S, et al. Acoustic-vibration analysis of the gear-bearing-housing coupled system. Appl Acoust 2021; 178: 108024. 8. Dogruer C and Pirsoltan A. Active vibration control of a single-stage spur gearbox. Mech Syst Signal Process 2017; 85: 429–444. 9. Jolivet S, Mezghani S, Mansori M, et al. Dependence of tooth ﬂank ﬁnishing on powertrain gear noise. J Manuf Syst 2015; 37: 467–471. 10. Ghosh S and Chakraborty G. On optimal tooth proﬁle modiﬁcation for reduction of vibration and noise in spur gear pairs. Mech Mach Theory 2016; 105: 145–163. 11. Moyne S and Tebec J. Ribs effects in acoustic radiation of a gearbox - their modelling in a boundary element method. Appl Acoust 2002; 63: 223–233. 12. Zhou J, Sun W, and Tao Q. Gearbox low-noise design method based on panel acoustic contribution. Math Probl Eng 2014; 2014: 144–151. 13. Wang J, Chang S, Liu G, et al. Optimal rib layout design for noise reduction based on topology optimization and acoustic contribution analysis. Struct Multidiscipl Optim 2017; 56: 1093–1108. 14. Wang Y, Qin X, Huang S, et al. Structural-borne acoustics analysis and multi-objective optimization by using panel acoustic participation and response surface methodology. Appl Acoust 2017; 116: 139–151. 15. Zhou W, Zuo Y, and Zheng M. Analysis and optimization of the vibration and noise of a double planetary gear power coupling mechanism. Shock Vib 2018; 2018: 9048695. 16. Wang S, Xieeryazidan A, Zhang X, et al. An improved computational method for vibration response and radiation noise analysis of two-Stage gearbox. IEEE Access 2020; 8: 85973–85988. 17. Qiao Z, Zhou J, Sun W, et al. A coupling dynamics analysis method for two-stage spur gear under multisource time-varying excitation. Shock Vib 2019; 2019: 1–18. 18. Ehsani A and Dalir H. Multi-objective design optimization of variable ribs composite grid plates. Struct Multidiscipl Optim 2020; 63: 407–418. 20 Journal of Low Frequency Noise, Vibration and Active Control 0(0) 19. Box G and Wilson K. On the experimental attainment of optimum conditions. J R Stat Soc 1951; 13: 1–45. 20. Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 2002; 6: 182–197. Appendix Notation [m], [c], [k] Matrix of global mass, damping, and stiffness ATV Acoustic transfer vector c Coefﬁcients of the input factor x i i c Quadratic coefﬁcients on the independent variable x ii i c Coefﬁcients of the interaction term between the input variable x and x ij i j c Meshing damping d Width of the horizontal ribs d Width of the vertical ribs d Width of the diagonal ribs d Width of the diagonal ribs FEM/BEM ﬁnite element/boundary element method f Normal impact force I Moment of inertia of gear k Order of the mode K Number of unknown coefﬁcients k Meshing stiffness m Mass of gear MAC Modal acoustic contribution n Number of nodes N Number of experimental sites n Number of the horizontal ribs n Number of the vertical ribs NSGA II Non-dominated sorted genetic algorithm II PACA Panel acoustic contribution analysis PSPL Peak sound pressure level R Determination coefﬁcient r Base circle radius of the driven gear R Ribs mass r Base circle radius of the driving gear RSM Response surface methodology S Sum of squares of regression residuals SE S Sum of squares of total variation ST t Thickness of the horizontal ribs t The thickness of the vertical ribs t Thickness of the diagonal ribs t The thickness of the diagonal ribs VRSM Variable ribs structure model Y Response vector of the test point y RSM approximate estimate vector α Pressure angle ω Angular frequency Z Estimated response φ Rotation angle λ K-th eigenvalue of the system {[F(t)]} External force vector received by the system {f (ω)} Load vector Qi et al. 21 {x(t)} Generalized nodal displacement vector {φ} Mode shape of the k-th order Appendix 1 Results of faced central composite design for establishing RS functions. Variables Run n1 n2 d1(mm) d2(mm) d3(mm) d4(mm) t1(mm) t2(mm) t3(mm) t4(mm) φ(°) PSPL(dB) Rm(g) 1 6 2 0 5 0 0 6 6 0 6 0 90.97 70.65 2 2 2 5 0 5 0 6 6 6 0 0 89.55 173.85 3 4 4 2.5 2.5 2.5 2.5 3 3 3 4.5 45 90.14 179.06 4 2 2 0 0 0 0 0 0 6 6 90 91.83 0.00 5 4 3 2.5 2.5 2.5 2.5 3 3 3 3 45 90.56 133.81 6 6 6 0 5 5 0 0 0 0 6 0 91.83 0.00 7 6 2 0 0 5 5 0 0 0 6 0 90 209.36 8 4 4 2.5 3.75 2.5 2.5 3 3 3 3 45 90.24 163.35 9 4 4 2.5 2.5 2.5 2.5 4.5 3 3 3 45 90.17 166.19 10 2 6 0 0 5 5 6 6 6 0 90 89.13 236.31 11 2 2 5 0 0 0 0 6 6 6 90 91.83 0.00 12 6 2 5 0 0 5 0 0 6 0 90 91.83 0.00 13 4 5 2.5 2.5 2.5 2.5 3 3 3 3 45 90.11 166.94 14 2 6 0 0 0 0 6 0 6 0 0 91.83 0.00 15 3 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.66 135.90 16 2 6 0 0 5 5 6 6 6 6 0 88.87 237.59 17 4 4 2.5 2.5 2.5 1.25 3 3 3 3 45 90.45 129.76 18 4 4 2.5 2.5 3.75 2.5 3 3 3 3 45 90.14 172.43 19 6 6 5 0 0 5 6 6 0 0 90 89.37 196.81 20 6 6 0 5 0 5 0 0 6 0 0 91.83 0.00 21 2 6 0 5 0 5 0 6 0 6 0 87.07 399.43 22 2 6 5 0 0 5 6 0 6 6 90 88.43 284.74 23 4 4 2.5 1.25 2.5 2.5 3 3 3 3 45 90.53 138.98 24 6 6 5 0 5 5 6 0 6 0 0 87.34 400.12 25 6 2 5 5 5 5 6 0 0 6 0 88.72 322.78 26 6 2 5 0 5 5 6 6 6 6 90 87.09 481.30 27 6 6 5 5 0 0 6 0 0 0 0 90.58 155.43 28 4 4 2.5 2.5 2.5 2.5 3 4.5 3 3 45 90.26 167.70 29 6 2 0 5 5 5 6 6 6 0 0 88.51 260.91 30 4 4 2.5 2.5 2.5 2.5 3 3 3 3 67.5 90.61 146.41 31 6 2 0 0 0 0 0 0 0 0 0 91.83 0.00 32 4 4 2.5 2.5 2.5 2.5 1.5 3 3 3 45 90.67 139.73 33 6 6 5 0 0 5 0 6 0 6 0 88.63 302.66 34 6 6 5 5 0 0 0 6 0 6 90 90.59 155.43 35 6 2 5 0 5 5 0 6 0 0 0 91.83 0.00 36 6 6 0 5 0 0 6 0 6 6 0 91.83 0.00 37 6 6 0 0 5 5 0 0 6 6 90 87 481.23 (continued) 22 Journal of Low Frequency Noise, Vibration and Active Control 0(0) (continued) Variables Run n1 n2 d1(mm) d2(mm) d3(mm) d4(mm) t1(mm) t2(mm) t3(mm) t4(mm) φ(°) PSPL(dB) Rm(g) 38 2 6 0 0 0 5 0 0 0 0 90 91.83 0.00 39 2 2 0 5 0 5 6 6 6 6 90 89.76 172.00 40 6 2 5 5 0 5 6 6 6 0 0 89.65 211.95 41 2 6 5 0 0 0 0 6 0 0 0 91.83 0.00 42 4 4 2.5 2.5 2.5 2.5 3 1.5 3 3 45 90.53 138.53 43 6 2 0 0 0 5 0 6 6 6 0 90 209.36 44 4 4 2.5 2.5 2.5 2.5 3 3 3 1.5 45 90.66 129.63 45 2 6 5 5 5 0 6 0 6 6 90 89.15 284.74 46 2 2 0 5 0 0 0 6 6 0 0 90.8 70.65 47 6 6 5 5 5 5 0 0 0 0 90 91.83 0.00 48 2 2 5 0 5 5 0 0 6 6 0 89.23 240.61 49 6 6 0 5 5 0 0 6 6 6 90 88.2 394.51 50 6 2 0 0 0 5 6 0 0 6 90 89.4 209.91 51 4 4 1.25 2.5 2.5 2.5 3 3 3 3 45 90.65 140.05 52 6 6 5 0 5 0 6 6 6 6 90 87.89 420.63 53 5 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.64 161.00 54 2 6 5 5 0 0 6 6 0 6 0 88.65 249.63 55 2 2 0 5 5 5 0 0 6 0 90 91.47 51.81 56 6 6 0 0 5 0 0 6 0 0 0 91.83 0.00 57 6 2 0 5 5 0 6 0 0 0 90 91.83 0.00 58 2 2 5 5 5 0 0 0 0 0 0 91.83 0.00 59 6 6 0 5 5 5 6 6 0 6 90 87.47 389.07 60 2 6 0 5 5 0 0 6 0 0 90 90.61 155.43 61 2 2 5 0 0 0 6 0 0 6 0 91.44 51.81 62 2 6 5 5 0 5 0 6 6 0 90 90.61 155.43 63 2 2 5 5 5 5 0 6 6 6 0 88.47 294.50 64 4 4 2.5 2.5 2.5 3.75 3 3 3 3 45 90.23 172.64 65 6 2 5 5 0 0 6 0 6 6 90 89.34 196.81 66 2 6 5 5 0 0 0 0 6 6 0 91.83 0.00 67 4 4 2.5 2.5 2.5 2.5 3 3 3 3 22.5 90.12 155.22 68 2 2 5 5 0 5 0 0 0 6 90 90.09 128.15 69 6 6 5 5 5 0 6 6 6 0 0 86.34 585.42 70 2 6 0 0 5 0 0 0 6 0 0 88.53 258.17 71 2 6 5 0 5 0 6 0 0 0 0 91.6 51.81 72 2 2 5 0 0 0 6 0 0 0 90 90.85 70.65 73 2 6 5 0 5 5 0 6 0 6 90 89.13 236.31 74 6 2 5 0 5 0 0 0 0 6 90 91.83 0.00 75 2 2 0 5 5 5 6 0 0 6 90 90.09 128.15 76 6 2 0 5 0 5 0 6 0 0 90 91.52 51.81 77 4 4 3.75 2.5 2.5 2.5 3 3 3 3 45 90.39 162.54 78 2 2 0 0 5 0 0 6 0 6 0 91.83 0.00 79 2 2 5 5 5 5 6 6 0 0 90 90.47 117.75 80 6 2 0 0 0 0 6 6 6 0 90 91.83 0.00 81 6 6 0 5 0 0 0 0 6 0 90 91.83 0.00 82 6 2 0 0 5 0 6 0 6 6 0 88.68 209.36 83 4 4 2.5 2.5 2.5 2.5 3 3 1.5 3 45 90.66 129.63 84 2 6 0 0 0 0 6 6 0 6 90 91.83 0.00 85 4 4 2.5 2.5 1.25 2.5 3 3 3 3 45 90.82 129.99 86 2 2 0 0 0 5 6 6 0 0 0 91.83 0.00 (continued) Qi et al. 23 (continued) 87 2 6 0 5 0 5 6 0 0 0 90 91.83 0.00 88 4 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.38 151.61 89 6 2 5 5 5 0 0 6 6 0 90 88.03 357.96 90 2 6 5 5 5 5 6 0 6 0 0 88.57 292.70 91 4 4 2.5 2.5 2.5 2.5 3 3 4.5 3 45 90.05 178.86 PSPL = peak sound pressure level, Rm = ribs mass http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE http://www.deepdyve.com/lp/sage/multi-objective-optimization-of-gearbox-based-on-panel-acoustic-kFxI25chrX

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References (20)

A. Attia (1969)
Noise of Involute Helical Gears
Journal of Engineering for Industry, 91
Zizhen Qiao, Jianxing Zhou, W. Sun, Xiangfeng Zhang (2019)
A Coupling Dynamics Analysis Method for Two-Stage Spur Gear under Multisource Time-Varying Excitation
Shock and Vibration
Jing-Siou Wei, C. Lv, Wei Sun, Xiang Li, Yang Wang (2013)
A study on optimum design method of gear transmission system for wind turbine
International Journal of Precision Engineering and Manufacturing, 14
Dynamic modeling of parallel shaft gear transmissions using finite element method
Sabyasachi Ghosh, G. Chakraborty (2016)
On optimal tooth profile modification for reduction of vibration and noise in spur gear pairs
Mechanism and Machine Theory, 105
Jinpeng Wang, Shangchow Chang, Geng Liu, Lan Liu, Liyan Wu (2017)
Optimal rib layout design for noise reduction based on topology optimization and acoustic contribution analysis
Structural and Multidisciplinary Optimization, 56
S. Jolivet, S. Mezghani, M. Mansori, Benoit Jourdain (2015)
Dependence of tooth flank finishing on powertrain gear noise
Journal of Manufacturing Systems, 37
M. Abbes, S. Bouaziz, F. Chaari, M. Maatar, M. Haddar (2008)
An acoustic–structural interaction modelling for the evaluation of a gearbox-radiated noise
International Journal of Mechanical Sciences, 50
Shengnan Wang, Adayi Xieeryazidan, Xiangfeng Zhang, Jianxing Zhou (2020)
An Improved Computational Method for Vibration Response and Radiation Noise Analysis of Two-Stage Gearbox
IEEE Access, 8
(1985)
Coupled torsional-flexural vibration of a geared shaft system using finite element analysis
M. Kubur, A. Kahraman, D. Zini, K. Kienzle (2004)
Dynamic Analysis of a Multi-Shaft Helical Gear Transmission by Finite Elements: Model and Experiment
Journal of Vibration and Acoustics, 126
Yongliang Wang, X. Qin, Song Huang, L. Lu, Qingkai Zhang, Jianfeng Feng (2017)
Structural-borne acoustics analysis and multi-objective optimization by using panel acoustic participation and response surface methodology
Applied Acoustics, 116
G. Box, K. Wilson (1951)
On the Experimental Attainment of Optimum Conditions
Journal of the royal statistical society series b-methodological, 13
A. Ehsani, H. Dalir (2021)
Multi-objective design optimization of variable ribs composite grid plates
Structural and Multidisciplinary Optimization, 63
Weijian Zhou, Y. Zuo, M. Zheng (2018)
Analysis and Optimization of the Vibration and Noise of a Double Planetary Gear Power Coupling Mechanism
Shock and Vibration
C. Dogruer, A. Pirsoltan (2017)
Active vibration control of a single-stage spur gearbox
Mechanical Systems and Signal Processing, 85
Jianxing Zhou, W. Sun, Qing Tao (2014)
Gearbox Low-Noise Design Method Based on Panel Acoustic Contribution
Mathematical Problems in Engineering, 2014
Jiyuan Han, Yang Liu, Shuanghe Yu, Siyao Zhao, Hui Ma (2021)
Acoustic-vibration analysis of the gear-bearing-housing coupled system
Applied Acoustics
S. Moyne, J. Tébec (2002)
Ribs effects in acoustic radiation of a gearbox — their modelling in a boundary element method
Applied Acoustics, 63
K. Deb, S. Agrawal, Amrit Pratap, T. Meyarivan (2002)
A fast and elitist multiobjective genetic algorithm: NSGA-II
IEEE Trans. Evol. Comput., 6

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DOI: 10.1177/14613484221091075
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Abstract

Aiming at noise problem of two-stage gearbox, this paper uses the panel acoustic contribution analysis (PACA) and response surface methodology to conduct structural-borne acoustics analysis and multi-objective optimization of gearbox. Using this optimization, optimal layout and parameters of the added ribs can be determined. The dynamic model of a spur gear-shaft-bearing system is built in consideration of the time-varying mesh stiffness, the time-varying bearing stiffness, and the ﬂexibility of the shaft, while dynamic load of the bearing is obtained through solving the model. The dynamic load of bearings is considered as excitation to analyze the vibration and noise radiation characteristics of housing. Furthermore, acoustic transfer vector, modal acoustic contribution and PACA are used to select the region with the most acoustic contribution. The variable ribs structure model (VRSM) is established in this region. Finally, the VRSM are adopted to optimize the housing. The optimization model based on the peak sound pressure level (PSPL) and the ribs mass (R ) is established by using response surface method. Through multi-objective optimization, the variable in VRSM can be modiﬁed or deleted to reduce the radiated noise of the gearbox. The optimization results demonstrate that the PSPL after optimization is decreased by 4.88 dB at frequencyof3130Hz. At thesamemass, thenoise reductioneffect of VRSM is 12% higher than that of standard rib. Therefore, the optimization effect of the housing structure is obvious. PACA-VRSM-RSM- NSGA-II technology provides a new research scheme for reducing the vibration and noise of the gearbox. Keywords gearbox, radiated noise, panel acoustic participation analysis, response surface methodology, multi-objective optimization Introduction Gear system was widely used in automotive, aerospace, and power generation industries due to its compact structure andhightransmission efﬁciency. The vibration of gearbox is mostly caused by the meshing action of gear teeth, in which it passes through gears, axles and bearings, and ﬁnally transfers to housing, which causing forced vibration of housing and radiating noise into the surrounding environment. The ﬁnite element/boundary element method (FEM/ 1–3 4 BEM) is the prevailing method applied to the numerical analysis for the gearbox vibration and noise. Kubur et al. proposed a dynamic model of a multi-shaft helical gear reduction unit formed by N ﬂexible shafts, and studied the inﬂuence of bearing stiffness, shaft size, and other parameters on the vibration of the gearbox. Later, Wei et al. established the coupled nonlinear dynamics model of a wind turbine planetary gear transmissionsystem, andproposed a lightweight and reliability constrained optimization design method according to the sequential quadratic pro- gramming algorithm. Subsequently, Chang et al. proposed a comprehensive coupled gearbox system dynamics model of a gear-shaft-bearing-shell system, which improved the accuracy of bearing response and thus more accurately School of Mechanical Engineering, Xinjiang University, Urumqi, China Corresponding author: *Jianxing Zhou, School of Mechanical Engineering, Xinjiang University, Urumqi 830047, China. Email: jianzhou82923@163.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access pages (https://us.sagepub.com/ en-us/nam/open-access-at-sage). 2 Journal of Low Frequency Noise, Vibration and Active Control 0(0) estimated the radiation noise of the gearbox. In order to solve the noise problem generated by operating gearboxes, Han et al. proposed a computational method for predicting gearbox noise from steady dynamic response to acoustic radiation calculation. Based on the gear dynamics model, many scholars have provided different schemes to reduce the radiated noise of the gearbox. To reduce the periodic change of meshing stiffness, Dogruer et al. proposed a nonlinear controller which adjusts the torque acting on the driving gear to minimize the adverse effect of time-varying mesh stiffness. And a spur gearbox was designed which does not emit much noise and vibration. Jolivet et al. addressed the issue of the impact tooth ﬂank ﬁnishingon noise andvibration. Ghosh et al. used the genetic algorithm to optimize the geometrical shape of the gear, reducing the noise of the tractor gearbox. In addition, the surface ribbing of the shell is one of the main methods of noise reduction. Many of structures contain an important account of ribs. Moyne et al. discussed the inﬂuence of adding ribs to an automobile gearbox shell on the radiation noise. The inﬂuence of ribs was theoretically discussed and three different physical effects were distinguished: the “vibrating,” the “obstacle,” and the “source” effects. Furthermore, panel acoustic contribution analysis (PACA) is an important method for optimizing the shell structure, which can determine the approximate position of the ribs plate distribution. Based on the dynamics model of a gear system, Zhou et al. added ribs using the acoustic contributions of the surface of the gearbox, and designed two gear stiffness improvement schemes to reduce the radiated noise of the gearbox. Although PACA can identify the panel area of the greatest acoustic contribution, it cannot determine the exact location of the ribs in this area, nor the geometry of the added ribs, which requires further calculations. Wang et al. proposed an optimal ribs layout design method to reduce radiation noise based on topology optimization and acoustic contribution analysis, veriﬁed the accuracy of the result by modal test and acoustic measurement. However, the results of topology optimization are complex, diverse and difﬁcult to achieve. Compared to topology optimization, traditional structure optimization is often costly and inefﬁcient. Artiﬁcial intelligence algorithms can signiﬁcantly reduce calculation time and material cost. Wang et al. proposed a multi-objective optimization design based on the response surface method and PACA, and pasted a damping layer on the panel surface to reduce radiation noise. Zhou et al. adopted the free damping structure and new stiffener structure to optimize the rear end cover of a double planetary gear housing. The opti- mization model based on the vibration acceleration of the rear end cover surface was established by applying K-S function and response surface method. However, there has been no serious attempt to determine the optimum layout and parameters of ribs. To address this, our investigation adds acoustic transfer vector (ATV) and modal acoustic contribution (MAC) to help identify the regions with the greatest acoustic contribution, enriching the screening process of the PACA method. Based on ATV-MAC-PACA, the variable ribs structure model (VRSM) as design variables in which utilized ribs width, thickness, number, and direction is proposed. Through multi-objective optimization, the variable in VRSM can be modiﬁed or deleted to reduce the radiated noise of the housing. PACA-VRSM-RSM- NSGA-II technology provides a new research scheme for the layout of gearbox ribs. Design methods and processes This section analyzes the vibration and noise calculation method of a two-stage gearbox system. And the PACA-VRSM- RSM-NSGA-II method for the gearbox structure optimization is as shown in Figure 1. The entire analysis process is composed of ﬁve components, namely the (a) gearbox dynamic excitation calculation, (b) gearbox dynamic response analysis, (c) gearbox radiated noise calculation, (d) determine ribs addition area and VRSM, and (e) Multi-objective optimization. (a) Gearbox dynamic excitation calculation. Analytical model is established by using lumped mass method. In the model, the inﬂuence of the TVMS, TVBS, and shaft ﬂexibility is considered. The dynamic load of the bearings is calculated by the Newmark integral method. (b) Gearbox dynamic response analysis. Modal analysis of the shell ﬁnite element model with boundary conditions is carried out. The dynamic load of the system is then added to the corresponding position of the shell ﬁnite element model. The modal superposition method is used to solve the dynamic response of the gearbox structure, and the vibration response and surface acceleration of the housing are also obtained. (c) Gearbox radiated noise calculation. The surface element of the ﬁnite element model is extracted, and the shell element is used to ﬁll the holes on the surface of the structure. Finally, the gearbox boundary element model is obtained by remeshing the surface elements of gearbox. And the obtained boundary element model and ATV are used to solve the radiated noise of the gearbox. Qi et al. 3 Figure 1. Multi-objective optimization calculation process of ribs layout of the two-stage gearbox. (d) Determine ribs addition region and VRSM. Three screening methods, ATV, MAC, and PACA, were used to de- termine the region with the greatest acoustic contribution. And the variable ribs model is added in this region. (e) Multi-objective optimization. Eleven structural parameters of the VRSM are used as input variables. PSPL and R are used as output variables. Response surface models of PSPL and R are then constructed by central composite design (CCD), and R is used to test the accuracy of the model. When the model accuracy meets the requirements, the obtained response surface objective function obtained is exported to NSGA-II, and the optimal ideal point is determined by analyzing the Pareto solution set. Finally, an experimental comparison is conducted to obtain the ideal VRSM. Gearbox dynamic excitation calculation The object of analysis in this paper is a two-stage gear reducer, whose model is as shown in Figure 2, which is a two-stage straight toothed cylinder gear reducer, and the minor chamfering and feature in structure is simpliﬁed in the model. The speciﬁc parameters of the gear are shown in Table 1, and the bearing parameters are shown in Table 2. Gear transmission system dynamic model The spur gear transmission system model has six degrees of freedom (DOF), namely rotation DOF of driving and driven wheels, the translation DOF along x and y directions (see Figure 3). Then the relative total displacement of meshing elements of spur gear pair δ is deﬁned as δ ¼ x sin α þ y cos α r θ x sin α y cos α r θ eðtÞ (1) p p p p g g g g where α is the pressure angle, r is the base circle radius of the driving gear, and r is the base circle radius of the driven gear. p g 4 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 2. (a) 3D model of two-stage gearbox, (b) Internal structure of the gearbox. Table 1. Gear system parameters. st st nd nd Parameters 1 stage driving gear 1 stage driven gear 2 stage driving gear 2 stage driven gear Number of teeth 36 90 29 100 Modulus (mm) 1.5 1.5 1.5 1.5 Pressure angle ()20 20 20 20 Tooth width (mm) 12 12 12 12 Table 2. Bearing parameters. Diameter of inner race (mm) Diameter of outer race (mm) Ball number Ball diameter (mm) 28.7 46.6 8 8.7 Pitch circle diameter (mm) Radial internal clearance Curvature radius (mm) Goodness ﬁt 37.65 0.5 4.5 0.5172 Figure 3. Dynamics model of gearbox transmission system. Qi et al. 5 Considering the inﬂuence of time-varying meshing stiffness and the gear meshing error, the motion differential equation of the meshing element of the spur gear can be expressed as > _ > m x€ þ c δ sin α þ k δ sin α þ f sin α ¼ 0 > p p m m s > _ > m y€ þ c δ cos α þ k δ cos α þ f cos α ¼ 0 > p m m s > € _ I θ c δr k δr f r ¼ 0 p p m p m p s p (2) > _ > m x€ c δ sin α k δ sin α f sin α ¼ 0 > g g m m s > _ > m y€ c δ cos α k δ cos α f cos α ¼ 0 > g m m s > € _ I θ c δr k δr f r ¼ 0 > g g m g m g s g where m is the mass of gear, k is the meshing stiffness, c is the meshing damping, I is the moment of inertia of gear and f m m s is the normal impact force. During the operation of the gear transmission system, the bearing supports the shaft system and transmits vibration from the gears to the housing. The bearing stiffness matrix can be expressed as K ¼ diag k ,k ,0 (3) bx by The ﬁnite element model of the gearbox (see Figure 4(a)) is established according to the gear meshing, bearing, and shaft elements, with a total of 29 elements and 32 nodes. The stiffness matrix of each element is superimposed according to the position of the element nodes to obtain the overall stiffness matrix, as shown in Figure 4(b). The generalized displacement vector of the system is fxg ¼ x , x , x , x , x , x ……x , x , x (4) 1x 1y 1θ 2x 2y 2θ nx ny nθ where n denotes the number of nodes. The dynamics equation of the gear system can be expressed as n o n o ½m x€ðtÞ þ½c x_ðtÞ þ½kðtÞfxðtÞg ¼ FðtÞ (5) where {x(t)} is the generalized nodal displacement vector, [m], [c], and [k] are the matrix of global mass, damping, and stiffness, respectively, and {[F(t)]} is the external force vector received by the system. Bearing dynamic load The input speed is 5400 r/min, and the output torque is 100 Nm. The time domain and frequency domain diagrams of dynamic loads of each bearing are obtained by Newmark integral method, as shown in Figure 5. where different colors Figure 4. Solution of overall stiffness of gearbox. (a) Finite element model of gearbox, (b) Overall stiffness matrix of gearbox. 6 Journal of Low Frequency Noise, Vibration and Active Control 0(0) represent different bearings. f (3240 Hz) and f (1044 Hz) represent the meshing frequency of the ﬁrst gear pair and the m1 m2 second gear pair, respectively. Experimental veriﬁcation The platform used in this experiment is the SQI Wind turbine transmission system test bench (shown in Figure 6(a)). The main devices include the test bed, drive motor, controller, gearbox, and data collector, which can be used to study the dynamic analysis of the gear transmission in this paper. The driving motor is used to adjust the speed automatically, increasing 240 rpm each time, and the acceleration value after each speed change is recorded by the acceleration sensor (shown in Figure 6(b)). The experimental results are compared with the simulation results (see Figure 7). It can be seen that the simulation results are very close to the trend of the experiment, indicating that the simulation results can replace the experiment. Gearbox dynamic response analysis Figure 8 is the ﬁnite element model of the gearbox. The material of the plate and ribs are both steels having Young’s 11 3 modulus E = 2.07 × 10 Pa, Poisson’s ratio μ = 0.3, and mass density ρ = 7800 kg/m . In order to apply dynamic load to the gearbox, a mass node is created in the center of the bearing hole and the node is coupled with the node on the inside surface of the bearing hole. Then the dynamic load is applied on the central node. A displacement constraint is applied at the bottom of the gearbox. The ﬁnite element model of housing and the ﬁrst eight natural vibration modes of the gearbox are shown in Figure 8. When the bottom of the gearbox is restrained, the vibration of the lower part of the gearbox is weak, while the vibration amplitude of the top part is obvious. Figure 5. The bearing dynamic load of the gear system. (a) Time response of loads from bearing 1 to bearing 6, (b) Spectrum of loads from bearing 1 to bearing 6. Figure 6. Transmission system test bench. (a) System equipment, (b) Sensor measurement position. Qi et al. 7 Gearbox radiated noise calculation The BEM is obtained by extracting surface elements from the gearbox ﬁnite element model and repairing the bearing holes with shell elements. The acoustic ﬂuid is air having mass density ρ0 = 1.2 kg/m3 and sound speed c = 343.4 m/s. Based on the boundary element model, the hemispherical shape acoustic ﬁeld is established. Four points with the same radius are Figure 7. Comparison between simulation and experiment. Figure 8. Finite element model of housing, (b) Mode of vibration of gearbox. Figure 9. (a) Gearbox ﬁeld point distribution, (b) ATV. 8 Journal of Low Frequency Noise, Vibration and Active Control 0(0) selected on the acoustic ﬁeld as the measured ﬁeld points, namely, ﬁeld points A, B, C, and D, as shown in Figure 9(a). In the case of small perturbations, the acoustic equation can be considered linear. Thus, a linear relationship can be established between the input (the normal vibration of the structure surface) and the output (the sound pressure at a point in the sound ﬁeld). According to the relationship shown in Figure 9(b), the sound pressure at a certain ﬁeld point can be expressed as pðωÞ¼ ATV ðωÞ v ðωÞ (6) where ATV is the acoustic transfer vector calculation, and ω is the angular frequency. The boundary element method is used to solve the noise radiation of the gearbox by taking the normal vibration acceleration of the shell’s outer surface as the boundary condition. Figure 10 shows the noise spectrum when the input speed is 5400 rpm, and the torque is 100 Nm. Due to the consideration of shaft ﬂexibility, the overall noise increases ﬁrst and then decreases. From the sound pressure level curves, it is well noticed that peak is generated on the position of 3 × fm2. The corresponding frequency of the maximum radiated noise is 3 × f (3132 Hz), which is close to the eighth order natural m2 frequency of housing (3098.35 Hz). At 3 × f (3132 Hz), the SPL at ﬁeld point A is the largest, so ﬁeld point A is selected m2 as the target ﬁeld point. The radiated noise at point A can be reduced by suppressing the PSPL at 3 × f . m2 Determine ribs addition region and VRSM In this study, the regions with larger ATV can be found by the analysis of ATV. Then, the regions with larger MAC can be obtained by MAC. Finally, panels can be divided on the regions with larger MAC and ATV concurrently and the acoustic contribution of each panel can be obtained by PACA. This process can be found the region with maximum acoustic contribution. Based on ATV-MAC-PACA, the VRSM was established. Acoustic transfer vector (ATV) Figure 11 demonstrates the amplitude of the acoustic transfer vector at 3 × f (3132 Hz) to ﬁeld point A. It can be seen that m2 the amplitude on the upper part of the gearbox is signiﬁcantly greater than that on the bottom, indicating that the normal vibration on the upper part of the gearbox has a more obvious impact on the PSPL. In summary, the ﬁrst screening region is the upper part of the gearbox. Figure 10. Noise spectrum of gearbox. Qi et al. 9 Figure 11. Amplitude of acoustic transfer vector at ﬁeld point A at 3 × f . m2 Modal acoustic contribution analysis (MACA) Using the modal superposition method, the displacement response of the multi-free system can be obtained as f ðωÞ φ xðωÞ¼ φ (7) a ðiω λ Þ k k k¼0 f ðωÞ φ where k is the order of the mode, q ðωÞ¼ is the structural modal participation factor, λ is the k-th eigenvalue of the k k a ðiωλ Þ k k system, {φ} denotes the mode shape of the k-th order; {f (ω)} is the load vector, and α denotes a constant related to the k k characteristics of the system. By projecting {x(ω)} onto the normal direction of the box surface and taking the derivative, the vibration velocity in the normal direction of the gear box surface can be obtained as v ðωÞ¼ iω q ðωÞφ (8) n k jn k¼1 where {φ} is the component of the mode shape on the normal line of the gearbox surface. jn By substituting equations (9) into (10), the sound pressure at any point in the sound ﬁeld can be obtained as n n n X X X T T pðωÞ¼ ATV ðωÞ iω q ðωÞφ ¼ iωq ðωÞATV ðωÞ φ ¼ P (9) k k sk nk nk k¼1 k¼1 k¼1 where P ¼ iωq ðωÞATV ðωÞ φ denotes the sound pressure generated by mode K. The projection of P in the sk k sk nk k¼1 direction of the total sound pressure P(ω) is called the modal acoustic contribution. The acoustic contribution of each mode at ﬁeld point A is obtained by MACA, as shown in Figure 12. It can be seen that the eighth mode has a large acoustic contribution to the PSPL at ﬁeld point A. The corresponding main mode shape of this mode is shown in Figure 12. The eighth main mode vibrates obviously at the top of the gearbox. Therefore, the result of the second screening is the top region of the gearbox. 10 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 12. The modal acoustic contributions at the ﬁeld point A at the meshing frequency. Panel acoustic contribution analysis (PACA) The main panels corresponding to the ﬁrst eight modes are divided, and PACA analysis was carried out. The acoustical contribution of each panel to the ﬁeld point A at 3×f (3132 Hz) was obtained, as shown in Figure 13. It can be seen from m2 the ﬁgure that panel 1 and panel 2 have the largest contribution, more than 25 percent, so the screening results through PACA are panel 1 and panel 2. In summary, Using the ATV-MAC-PACA method, the central region of the top of the gearbox has the largest acoustic contribution to the ﬁeld point A at 3×f (3132 Hz). Therefore, the VRSM can be added at this region to reduce the radiated m2 noise of the gearbox. Variable ribs structure model (VRSM) Although the general region of the added ribs can be obtained from the previous screening, the research on the distribution and number of ribs and the speciﬁc parameters of each rib requires further design. The ribs layout method adopted in this paper refers to literature. Figure 14(a) is the rectangular region determined according to the screening results. To study the effect of the layout and parameters of the ribs on reducing the radiated noise of the shell, the ribs model is added to the top center of the shell (note: all model ribs are added to the inner surface of the top of the box body to prevent disturbance of the ATV and improve calculation efﬁciency). The characteristic of VRSM is characterized by the fact that there is no deﬁnite ribs layout and size, and variables in the model can be changed as required. Thus, different VRSMs can be realized according to different optimization results. As shown in Figure 14(b), the model contains 11 design variables; namely, the number of horizontal and vertical ribs (n and n ), the width of the ribs (d , d , d , and d ), and the thickness of the ribs (t , t , t , and t ). φ is the 1 2 1 2 3 4 1 2 3 4 rotation angle of the ribs relative to the x-axis (shown in Figure 14(c)). In addition, the spacing y and y between the ribs are 1 2 determined by the number of horizontal and vertical ribs, respectively; y and y reﬂect variation the density of rib 1 2 distribution in the gearbox. It can be seen that through the selection of value range, the rib height and thickness variables can be set to zero, and then the redundant ribs can be deleted in the corresponding optimization. Multi-objective optimization There is a great relationship between the mass of ribs and peak value of radiated noise of the gearbox, and the two are in conﬂict in essence, so it is very necessary to establish a multi-objective optimization model. Response surface model (RSM) Based on statistical and mathematical methods, RSM can build and evaluate a variety of complex problems. In this method, the output variable (system response) is approximately ﬁtted to the speciﬁed design space to replace the real response Qi et al. 11 Figure 13. Analysis results of acoustical contribution of plate surface at meshing frequency. Figure 14. 3D model of ribs layout design. (a) the position where the ribs are added in the gearbox, (b)VRSM, and (c) variable φ: the rotation Angle of the ribs relative to the x-axis. surface. In order to give consideration to the ﬁtting accuracy and optimization calculation time, the central composite design (CCD) was adopted. In this design, alpha = 0.5, the center point is 1 and the factorial core uses the min-res V method. The relation between the independent variable in the response surface and response Z can be expressed as Z ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞþ ε (10) 1 2 1 2 3 4 1 2 3 4 where ε is the calculation error. RSM usually adopts quadratic polynomial model to ﬁt the parameter model of the response. Its approximate function can be expressed as n n X X Z ¼ c þ c x þ c ðx Þ þ i ¼ 1nj > icijxixj (11) 0 i i ii i i¼1 i¼1 where Z denote the estimated response; c denote the constant term coefﬁcient; c denotes the coefﬁcients of the input factor 0 i x ; c denotes the quadratic coefﬁcients on the independent variable x and c denotes the coefﬁcients of the interaction term i ii i ij between the input variable x and x . i j 12 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Multi-objective genetic algorithm Genetic Algorithms is a heuristic algorithm developed on the basis of biological evolutionary theory and Genetic thought. Compared with single objective optimization, multi-objective optimization can better solve problems with multiple objective requirements. NSGA-II proposed by Deb et al. is a multi-objective optimization algorithm based on Pareto optimal solution, that is, a fast non-dominated multi-objective optimization algorithm with elite retention strategy. The three major characteristics of NSGA-II are the proposed congestion degree and congestion comparison operator, and the in- troduction of non-dominated ranking and elite strategy. In this paper, the optimal VRSM is solved while considering multiple targets. The objective functions of Rm and PSPL are obtained by a large number of experimental data and RSM. The optimization problem model is as follows > Minimize R ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞ > m 1 1 2 1 2 3 4 1 2 3 4 > Minimize PSPL ¼ f ðn ,n ,d ,d ,d ,d ,t ,t ,t ,t ,φÞ > 2 1 2 1 2 3 4 1 2 3 4 > subject to constraints 2 ≤ n ,n ≤ 6 (12) 1 2 0 ≤ d ,d ,d ,d ≤ 5 1 2 3 4 0 ≤ t ,t ,t ,t ≤ 6 1 2 3 4 0 ≤ φ ≤ 90 where f ()and f () denote objective functions for R and PSPL, respectively. 1 2 m Results and discussion As per CCD design, 91 experiments are performed for VRSM. See Appendix 1 for speciﬁc simulation experiment data. The regression analysis is an assessment of the accuracy and predictive power of response surface models. When the response surface model is more accurate, the numerical model will be closer to the physical model, and the predicted value will be 2 2 more accurate. Generally, R and adjusted R are used to characterize the accuracy of response surface model, and the mathematical formulations are given by SE R ¼ 1 (13) ST S ðN K 1Þ SE R ¼ (14) adj S ðN 1Þ ST S ¼ ðY y Þ (15) SE i i i¼1 N N X X S ¼ Y y N (16) ST i i¼1 i¼1 where S denotes the sum of squares of regression residuals, S denotes the sum of squares of total variation, N denotes the SE ST number of experimental sites, K denotes the number of unknown coefﬁcients, Y denotes the response vector of the test point, y denotes the RSM approximate estimate vector. Quadratic equation for PSPL A mathematical model for prediction of PSPL using VRSM as the desiccant material is described in equation (17) Qi et al. 13 PSPL ¼ 93:11 1:77n þ 0:86n 0:18d þ 0:14d 0:20d þ 0:45d þ 0:05t þ 0:14t 1 2 1 2 3 4 1 2 þ0:15t þ 0:07t þ 0:01φ 0:02n d þ 0:01n t 0:04n t 0:03n d 0:04 × n d 3 4 1 3 1 2 1 3 2 1 2 4 0:03n t 0:01d d þ 0:02d d 0:04d t 0:01d t 0:01d t 0:04 × d t þ 0:02d t 2 2 1 2 1 3 1 1 1 3 2 1 2 2 2 3 (17) 0:01d t þ 0:001d φ þ 0:01d d þ 0:01d t 0:08d t þ 0:02d t 0:02d t 0:02d t 2 4 2 3 4 3 1 3 3 3 4 4 1 4 2 2 2 þ0:04d t 0:07d t þ 0:01t t þ 0:01t t þ 0:02t t 0:0006t φ þ 0:22n 0:09n 4 3 4 4 1 2 2 3 3 4 4 1 2 2 2 2 2 2 2 2 2 2 þ0:06d 0:03d þ 0:03d 0:06d 0:003t 0:02t 0:03t 0:01t 0:0001φ 1 2 3 4 1 2 3 4 The response surface model of PSPL is established through the above calculation formula. Before using the mathematical 2 2 model, the goodness of developed models is assessed by evaluating the coefﬁcient of determination (R ). The values of R = 0.9832 and adjusted R = 0.9632 are very close to 1, and the difference between the predicted value and the adjusted value is less than 0.2. Therefore, the mathematical model established for the optimization objective meets the accuracy re- quirements. As shown in Table 3, when the resulting p-value is less than 0.0001, the model can be considered signiﬁcantly important. The items exceeding the recognized standard value of 0.05 can then be deleted to improve the model accuracy. Nevertheless, the purpose of this study is to investigate the effects of all parameters on response variables, and as many parameters as possible can be retained for research when the accuracy is satisﬁed. According to the sum of squares, the linear term has a contribution of 44.22%. The contribution of the cross-interaction term is 36.45%, and the contribution of self-interaction term was 0.11%. Figure 15 shows the two-dimensional contour plot of part of the response surface, in which the contour plot of the PSPL is in good agreement with the experimental data. The contour lines in the ﬁgure are divided into Table 3. Factors and levels of the central composition design. PSPL R PSPL R m m Sum of p-value prob p-value prob Sum of p-value prob Sum of p-value prob Source squares > f Sum of squares >f Source squares >f squares >f Model 176.62 <0.0001 1,564,182 <0.0001 d t 1.13 0.0003 2576 0.0002 2 3 n 4.10 <0.0001 43,291 <0.0001 d d 0.34 0.0363 1605 0.0019 1 3 4 n 2.86 <0.0001 28,604 <0.0001 d t 0.66 0.0047 924 0.0143 2 3 1 d 5.63 <0.0001 46,528 <0.0001 d t 17.62 <0.0001 114,466 <0.0001 1 3 3 d 3.07 <0.0001 28,738 <0.0001 d t 0.79 0.0021 4168 <0.0001 2 3 4 d 13.36 <0.0001 93,237 <0.0001 d t 1.48 0.0001 4263 <0.0001 3 4 2 d 15.23 <0.0001 94,319 <0.0001 d t 3.43 <0.0001 9530 <0.0001 4 4 3 t 2.47 <0.0001 12,380 <0.0001 d t 14.21 <0.0001 90,296 <0.0001 1 4 4 t 5.44 <0.0001 49,436 <0.0001 t t 0.38 0.0286 3470 <0.0001 2 1 2 t 16.10 <0.0001 117,150 <0.0001 t t 0.46 0.0162 1018 0.0105 3 2 3 t 9.83 <0.0001 69,887 <0.0001 t t 1.24 0.0002 3643 <0.0001 4 3 4 φ 0.03 0.5288 721 0.0283 t φ 0.36 0.0334 2410 0.0003 n d 0.77 0.0024 5360 <0.0001 n 0.11 0.2308 23.76 0.6763 1 3 1 n t 2.90 0.0358 20,953 <0.0001 n 0.02 0.6147 4.06 0.8627 1 3 2 n d 0.86 0.0000 2904 0.0001 d 0.02 0.6163 0.43 0.9550 2 1 1 n d 2.58 0.0014 682 0.0326 d 0.004 0.8164 0.72 0.9419 2 4 2 n t 0.99 <0.0001 5197 <0.0001 d 0.01 0.7762 0.60 0.9469 2 2 3 d d 0.22 0.0007 2434 <0.0001 d 0.02 0.6339 0.64 0.9455 1 2 4 d d 0.52 0.0885 1591 <0.0001 t 0.001 0.9666 3.30 0.8762 1 3 1 d t 5.66 0.0113 38,861 <0.0001 t 0.002 0.8589 4.19 0.8607 1 1 2 d t 0.26 <0.0001 2501 <0.0001 t 0.01 0.6928 13.85 0.7498 1 3 3 d t 0.58 0.0682 7048 <0.0001 t 0.002 0.8803 14.97 0.7403 2 1 4 d t 4.50 0.0074 30,060 <0.0001 φ 0.01 0.7333 1.85 0.9071 2 2 Model statistics R 0.9832 0.9979 Adjusted R 0.9632 0.9924 Predicted R 0.9120 0.9445 Adeq precision 29.5684 61.1448 14 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 15. Contour plots for PSPL in terms of (a) n vs. d , (b) n vs. d , (c) n vs. t , (d) d vs. d , (e) d vs. t , (f) d vs. t , (g) d vs. t , (h) d vs. 1 3 2 4 2 2 1 3 2 1 2 2 2 3 3 d , (i) d vs. t , (j) d vs. t , (k) d vs. t , (l) d vs. t , (m) t vs. t , (n) t vs. t , (o) t vs. t , and (p) t vs. φ. 4 3 1 4 2 4 3 4 4 1 2 2 3 3 4 4 ﬁve levels, from red to blue, indicating gradual changes from high to low. It can be observed that when the other nine variables are ﬁxed, and the horizontal coordinate and vertical coordinate variables change, PSPL will also change ac- cordingly (Figure 15). Qi et al. 15 Figure 16. Contour plots for R in terms of (a) n vs. d , (b) n vs. d , (c) n vs. t , (d) d vs. d , (e) d vs. t , (f) d vs. t , (g) d vs. t , (h) d vs. m 1 3 2 4 2 2 1 3 2 1 2 2 2 3 3 d , (i) d vs. t , (j) d vs. t , (k) d vs. t , (l) d vs. t , (m) t vs. t , (n) t vs. t , (o) t vs. t , and (p) t vs. φ. 4 3 1 4 2 4 3 4 4 1 2 2 3 3 4 4 16 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Quadratic equations for R . Regression function concerning R can be expressed as follows m m R ¼ 65:13 þ 8:56n 4:92n 20:41d 2:79d 1:69d 9:42d 8:56t 10:66t m 1 2 1 2 3 4 1 2 19:93t 8:89t þ 0:46φ þ 2:88n d þ 2:33n d þ 1:25n d þ 0:61n t þ 3:63n t þ 1:20n t 3 4 1 1 1 3 1 4 1 1 1 3 1 4 1:8n d þ 1:36n d þ 0:79n d þ 2:43n d þ 2:14n t þ 2:08n t 0:65n t þ 0:07n φ 2 1 2 2 2 3 2 4 2 2 2 3 2 4 2 þ1:07d d 1:07d d 0:42d d þ 4:01d t þ 1:10d t þ 0:34d t þ 0:42d d 0:82d d 1 2 1 3 1 4 1 1 1 3 1 4 2 3 2 4 (18) þ1:67d t þ 3:69d t 1:10d t 0:09d φ 0:94d d 0:67d t 0:62d t þ 6:81d t 2 1 2 2 2 3 2 3 4 3 1 3 2 3 3 1:27d t þ 1:44d t 2:04d t þ 7:08d t 0:95t t 0:37t t 0:37t t 0:62t t 3 4 4 2 4 3 4 4 1 2 1 3 1 3 1 4 2 2 2 þ0:04t φ 0:67t t þ 0:07t φ 1:10t t þ 0:05t φ þ 0:06t φ 3:29n 1:36n 0:28d 1 2 3 2 3 4 3 4 1 2 1 2 2 2 2 2 2 2 2 0:37d 0:34d 0:34d þ 0:54t þ 0:61t þ 1:12t þ 1:16t 0:002φ 2 3 4 1 2 3 4 The above regression function is mathematically veriﬁed by the ANOVA test, and the obtained results are shown in Table 3. Based on the sum of squares, it can be seen that the contribution ratio of the linear term is 37.35%, and the contribution ratio of the cross-action term is 22.76%. According to the corresponding calculated statistical error value R = 0.9979 and adjusted R = 0.9924. The results show that the experimental data are well-adapted to the established model. Figure 16 depicts the contour map of part of R , showing the inﬂuence of each parameter and the interaction between the parameters on the corresponding response. Optimization results Statistical analysis shows that the results obtained by RSM correspond well with the experimental data. Therefore, the regression equation obtained by the response surface method can be used as the objective function of the subsequent multi- objective algorithm. NSGA-II is used to calculate the optimal solution set of PSPL and Rm. NSGA-II has been im- plemented in this study for a population size of 50, a crossover fraction of 0.8, a Pareto front population fraction of 0.35. Due to the conﬂict and competition among optimization objectives, multi-objective optimization has no unique solution, and its solution is a set of optimal non-inferior solutions. It is noteworthy that in the present study, NSGA-II has been run 20 times for the case of optimization, and corresponding ﬁrst ranked Pareto fronts have been selected. Figure 17 shows the Pareto solution set for PSPL and Rm solved by NSGA-II. The coordinates of the points in the Pareto solution set in the ﬁgure are derived from Table 4. Table 4 shows the optimal values of the two objective functions and the corresponding values of the 11 parameters. The values of variables in the table are rounded results. As seen in Table 4, 4 and 6 occur most frequently among all optimization results for n1 and n2. The highest number of occurrences in d1, d2, d3, and d4 are 1, 4, 1, and 5, respectively. The highest number of occurrences in t1, t2, t3, and t4 are 5, 6, 6, and 1, respectively. The optimal value of φ is close to 90°. Among all variables, such as the number, width, and thickness of ribs and φ, the thickness of the ribs has the highest value and the most obvious effect on noise reduction. Therefore, the primary consideration should be to increase the thickness of the ribs. Figure 17 clearly illustrates that at point Figure 17. Pareto solution set calculated with NSGA-II. Qi et al. 17 A(n =4, n =5, d = 5 mm, d = 3 mm, d = 5 mm, d = 5 mm, t = 6 mm, t = 4 mm, t = 6 mm, t = 6 mm, φ = 78°), PSPL is 1 2 1 2 3 4 1 2 3 4 reduced to the minimum, so point A can be regarded as the optimal layout condition for the most obvious noise reduction of the gearbox after adding variable ribs. R drops to its lowest at point B (n =4, n =2, d = 1 mm, d = 4 mm, d = 0 mm, m 1 2 1 2 3 d = 0 mm, t = 1 mm, t = 1 mm, t = 2 mm, t = 4 mm, φ = 88°). Thus, point B is considered to be the optimal layout 4 1 2 3 4 condition for variable ribs with the lowest mass. In order to verify the rationality of the theoretical points, the parameters of the variable ribs model corresponding to points A and B in the multi-objective optimization results are substituted into the calculation ﬂow of the gearbox radiation noise, and the obtained results are compared with points A and B. The comparison results are provided in Table 5. In the corresponding output variables of the point A, the PSPL and R optimization results and simulation values errors are 0.52% and +1.8 g, respectively. In the output variables corresponding to point B, the Table 4. The corresponding parameter variables and corresponding values of the output variables in the Pareto solution set. Variables Objective function n1 n2 d1 d2 d3 d4 t1 t2 t3 t4 φ FSPL R 4 633255 6 1 3 11 87.68 280.24 4 553556 4 6 6 78 86.49 521.49 5 543356 4 6 6 80 86.71 499.77 5 544356 4 6 6 86 86.79 472.89 4 635545 6 6 1 38 86.95 426.18 4 635445 6 6 1 43 86.99 422.70 4 635455 6 6 1 41 87.29 376.95 4 621454 6 5 5 33 87.44 345.47 4 614253 6 3 6 17 87.61 330.93 4 615253 6 3 4 13 87.63 313.50 3 654154 5 1 3 21 87.68 303.13 4 640155 6 0 2 10 88.40 175.97 4 630155 5 1 3 6 88.50 173.18 5 213303 0 6 2 90 88.79 156.86 4 640355 4 0 2 8 88.82 138.81 4 640355 3 0 1 5 89.15 128.64 3 631156 3 0 1 1 89.20 125.56 4 640355 3 0 1 5 89.31 114.61 4 640155 1 0 1 1 89.39 105.53 4 640156 0 0 1 3 89.42 97.15 4 640156 1 0 1 3 89.54 93.40 3 224201 0 5 3 84 90.22 45.18 4 213101 1 6 3 74 90.32 41.68 4 215111 1 5 5 89 90.42 34.21 4 214101 0 5 3 88 90.44 23.00 4 214101 0 4 3 80 90.69 13.43 5 233201 0 1 3 81 90.75 10.85 4 214101 0 4 3 85 90.78 8.55 4 214001 1 2 4 88 90.96 6.28 Table 5. The calculated ideal points are compared with the corresponding experimental data. Design variable point A, B Output variable Estimated Experimental Deviation A PSPL 86.49 86.95 0.53% R 521.49 523.29 +1.8 g B PSPL 90.96 91.58 0.67% R 6.28 11.1 +4.82 g m 18 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 18. radiation noise spectrum of the original structure and each scheme model at ﬁeld point A. Table 6. The PSPL at ﬁeld point A. Standard rib, dB VRSM, dB PSPL 87.48 86.95 theoretical and calculated error values of PSPL and R are 0.67% and +4.82 g, respectively. Figure 18 shows the spectral comparison of sound pressure levels of the improved front and rear gearboxes, where the noise reduction amplitude of the ideal point A is very obvious compared with that before adding ribs, while the noise reduction amplitude of ideal point B is very small. Researchers can choose the appropriate variable ribs layout according to the speciﬁc requirements of the R .For this purpose, the relationship between PSPL and R is deduced according to the Pareto front as follows R ¼ 177873:68 3890:95ðPSPLÞþ 22:28ðPSPLÞ (19) The region obtained by PACA method is used as the base area of the rib to establish the standard rib. Adjust the thickness of the standard rib so that the mass of the standard rib is the same as the mass of the VRSM at the optimal point A. The PSPL of the standard rib and VRSM at the ﬁeld point A is shown in Table 6. At the same mass, the noise reduction effect of VRSM is 12% higher than that of standard rib. Conclusions In this paper, a two-stage gearbox radiated noise model is established, and the response surface method and NSGA-II can modify the variables in the VRSM to reduce the noise of the gearbox. The conclusions of this paper are as follows: 1. ATV and MAC improve the accuracy of the PACA method. The ATV-MAC-PACA method can also be used to add damping materials, shock absorbers, and other mechanisms. 2. The thickness of the ribs is the key variable for noise reduction. The PSPL after optimization is decreased by 4.88 dB at frequency of 3130 Hz. At the same mass, the noise reduction effect of VRSM is 12% higher than that of standard rib. Qi et al. 19 3. The VRSM should be determined according to different ﬁeld points and different excitation frequencies. A dis- advantage of this method is that acoustic contribution analysis must be performed again after changing the working conditions and considering the ﬁeld points to achieve a good noise reduction effect. Declaration of conﬂicting interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following ﬁnancial support for the research, authorship, and/or publication of this article: This study was funded by the Chinese National Natural Science Foundation (51665054), Natural Science Foundation of Xinjiang Province (No. 2021D01C050) and Xinjiang Province key research and development (2021B01003-1). ORCID iDs Le Qi  https://orcid.org/0000-0002-7345-2636 Jianxing Zhou https://orcid.org/0000-0002-0764-1172 Huachao Xu https://orcid.org/0000-0003-4471-805X References 1. Attia AY. Noise of involute helical gears. J Eng Ind 1969; 2: 165–171. 2. Neriya S, Bhat R, and Sankar T. Coupled torsional-ﬂexural vibration of a geared shaft system using ﬁnite element analysis. Shock Vib Bull 1985; 55: 13–25. 3. Abbes M, Bouaziz S, Chaari F, et al. An acoustic-structural interaction modelling for the evaluation of a gearbox-radiated noise. Int J Mech Sci 2008; 50: 569–577. 4. Kubur M, Kahraman A, and Zinid M. Dynamic analysis of a multi-shaft helical gear transmission by ﬁnite elements model and experiment. J Vib Acoust 2004; 126: 398–406. 5. Wei J, Lv C, Sun W, et al. A study on optimum design method of gear transmission system for wind turbine. Int J Precis Eng Manuf 2013; 14: 767–778. 6. Chang L, He C, and Liu G. Dynamic modeling of parallel shaft gear transmissions using ﬁnite element method. Shock Vib 2016; 35: 47–53. 7. Han J, Liu Y, Yu S, et al. Acoustic-vibration analysis of the gear-bearing-housing coupled system. Appl Acoust 2021; 178: 108024. 8. Dogruer C and Pirsoltan A. Active vibration control of a single-stage spur gearbox. Mech Syst Signal Process 2017; 85: 429–444. 9. Jolivet S, Mezghani S, Mansori M, et al. Dependence of tooth ﬂank ﬁnishing on powertrain gear noise. J Manuf Syst 2015; 37: 467–471. 10. Ghosh S and Chakraborty G. On optimal tooth proﬁle modiﬁcation for reduction of vibration and noise in spur gear pairs. Mech Mach Theory 2016; 105: 145–163. 11. Moyne S and Tebec J. Ribs effects in acoustic radiation of a gearbox - their modelling in a boundary element method. Appl Acoust 2002; 63: 223–233. 12. Zhou J, Sun W, and Tao Q. Gearbox low-noise design method based on panel acoustic contribution. Math Probl Eng 2014; 2014: 144–151. 13. Wang J, Chang S, Liu G, et al. Optimal rib layout design for noise reduction based on topology optimization and acoustic contribution analysis. Struct Multidiscipl Optim 2017; 56: 1093–1108. 14. Wang Y, Qin X, Huang S, et al. Structural-borne acoustics analysis and multi-objective optimization by using panel acoustic participation and response surface methodology. Appl Acoust 2017; 116: 139–151. 15. Zhou W, Zuo Y, and Zheng M. Analysis and optimization of the vibration and noise of a double planetary gear power coupling mechanism. Shock Vib 2018; 2018: 9048695. 16. Wang S, Xieeryazidan A, Zhang X, et al. An improved computational method for vibration response and radiation noise analysis of two-Stage gearbox. IEEE Access 2020; 8: 85973–85988. 17. Qiao Z, Zhou J, Sun W, et al. A coupling dynamics analysis method for two-stage spur gear under multisource time-varying excitation. Shock Vib 2019; 2019: 1–18. 18. Ehsani A and Dalir H. Multi-objective design optimization of variable ribs composite grid plates. Struct Multidiscipl Optim 2020; 63: 407–418. 20 Journal of Low Frequency Noise, Vibration and Active Control 0(0) 19. Box G and Wilson K. On the experimental attainment of optimum conditions. J R Stat Soc 1951; 13: 1–45. 20. Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 2002; 6: 182–197. Appendix Notation [m], [c], [k] Matrix of global mass, damping, and stiffness ATV Acoustic transfer vector c Coefﬁcients of the input factor x i i c Quadratic coefﬁcients on the independent variable x ii i c Coefﬁcients of the interaction term between the input variable x and x ij i j c Meshing damping d Width of the horizontal ribs d Width of the vertical ribs d Width of the diagonal ribs d Width of the diagonal ribs FEM/BEM ﬁnite element/boundary element method f Normal impact force I Moment of inertia of gear k Order of the mode K Number of unknown coefﬁcients k Meshing stiffness m Mass of gear MAC Modal acoustic contribution n Number of nodes N Number of experimental sites n Number of the horizontal ribs n Number of the vertical ribs NSGA II Non-dominated sorted genetic algorithm II PACA Panel acoustic contribution analysis PSPL Peak sound pressure level R Determination coefﬁcient r Base circle radius of the driven gear R Ribs mass r Base circle radius of the driving gear RSM Response surface methodology S Sum of squares of regression residuals SE S Sum of squares of total variation ST t Thickness of the horizontal ribs t The thickness of the vertical ribs t Thickness of the diagonal ribs t The thickness of the diagonal ribs VRSM Variable ribs structure model Y Response vector of the test point y RSM approximate estimate vector α Pressure angle ω Angular frequency Z Estimated response φ Rotation angle λ K-th eigenvalue of the system {[F(t)]} External force vector received by the system {f (ω)} Load vector Qi et al. 21 {x(t)} Generalized nodal displacement vector {φ} Mode shape of the k-th order Appendix 1 Results of faced central composite design for establishing RS functions. Variables Run n1 n2 d1(mm) d2(mm) d3(mm) d4(mm) t1(mm) t2(mm) t3(mm) t4(mm) φ(°) PSPL(dB) Rm(g) 1 6 2 0 5 0 0 6 6 0 6 0 90.97 70.65 2 2 2 5 0 5 0 6 6 6 0 0 89.55 173.85 3 4 4 2.5 2.5 2.5 2.5 3 3 3 4.5 45 90.14 179.06 4 2 2 0 0 0 0 0 0 6 6 90 91.83 0.00 5 4 3 2.5 2.5 2.5 2.5 3 3 3 3 45 90.56 133.81 6 6 6 0 5 5 0 0 0 0 6 0 91.83 0.00 7 6 2 0 0 5 5 0 0 0 6 0 90 209.36 8 4 4 2.5 3.75 2.5 2.5 3 3 3 3 45 90.24 163.35 9 4 4 2.5 2.5 2.5 2.5 4.5 3 3 3 45 90.17 166.19 10 2 6 0 0 5 5 6 6 6 0 90 89.13 236.31 11 2 2 5 0 0 0 0 6 6 6 90 91.83 0.00 12 6 2 5 0 0 5 0 0 6 0 90 91.83 0.00 13 4 5 2.5 2.5 2.5 2.5 3 3 3 3 45 90.11 166.94 14 2 6 0 0 0 0 6 0 6 0 0 91.83 0.00 15 3 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.66 135.90 16 2 6 0 0 5 5 6 6 6 6 0 88.87 237.59 17 4 4 2.5 2.5 2.5 1.25 3 3 3 3 45 90.45 129.76 18 4 4 2.5 2.5 3.75 2.5 3 3 3 3 45 90.14 172.43 19 6 6 5 0 0 5 6 6 0 0 90 89.37 196.81 20 6 6 0 5 0 5 0 0 6 0 0 91.83 0.00 21 2 6 0 5 0 5 0 6 0 6 0 87.07 399.43 22 2 6 5 0 0 5 6 0 6 6 90 88.43 284.74 23 4 4 2.5 1.25 2.5 2.5 3 3 3 3 45 90.53 138.98 24 6 6 5 0 5 5 6 0 6 0 0 87.34 400.12 25 6 2 5 5 5 5 6 0 0 6 0 88.72 322.78 26 6 2 5 0 5 5 6 6 6 6 90 87.09 481.30 27 6 6 5 5 0 0 6 0 0 0 0 90.58 155.43 28 4 4 2.5 2.5 2.5 2.5 3 4.5 3 3 45 90.26 167.70 29 6 2 0 5 5 5 6 6 6 0 0 88.51 260.91 30 4 4 2.5 2.5 2.5 2.5 3 3 3 3 67.5 90.61 146.41 31 6 2 0 0 0 0 0 0 0 0 0 91.83 0.00 32 4 4 2.5 2.5 2.5 2.5 1.5 3 3 3 45 90.67 139.73 33 6 6 5 0 0 5 0 6 0 6 0 88.63 302.66 34 6 6 5 5 0 0 0 6 0 6 90 90.59 155.43 35 6 2 5 0 5 5 0 6 0 0 0 91.83 0.00 36 6 6 0 5 0 0 6 0 6 6 0 91.83 0.00 37 6 6 0 0 5 5 0 0 6 6 90 87 481.23 (continued) 22 Journal of Low Frequency Noise, Vibration and Active Control 0(0) (continued) Variables Run n1 n2 d1(mm) d2(mm) d3(mm) d4(mm) t1(mm) t2(mm) t3(mm) t4(mm) φ(°) PSPL(dB) Rm(g) 38 2 6 0 0 0 5 0 0 0 0 90 91.83 0.00 39 2 2 0 5 0 5 6 6 6 6 90 89.76 172.00 40 6 2 5 5 0 5 6 6 6 0 0 89.65 211.95 41 2 6 5 0 0 0 0 6 0 0 0 91.83 0.00 42 4 4 2.5 2.5 2.5 2.5 3 1.5 3 3 45 90.53 138.53 43 6 2 0 0 0 5 0 6 6 6 0 90 209.36 44 4 4 2.5 2.5 2.5 2.5 3 3 3 1.5 45 90.66 129.63 45 2 6 5 5 5 0 6 0 6 6 90 89.15 284.74 46 2 2 0 5 0 0 0 6 6 0 0 90.8 70.65 47 6 6 5 5 5 5 0 0 0 0 90 91.83 0.00 48 2 2 5 0 5 5 0 0 6 6 0 89.23 240.61 49 6 6 0 5 5 0 0 6 6 6 90 88.2 394.51 50 6 2 0 0 0 5 6 0 0 6 90 89.4 209.91 51 4 4 1.25 2.5 2.5 2.5 3 3 3 3 45 90.65 140.05 52 6 6 5 0 5 0 6 6 6 6 90 87.89 420.63 53 5 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.64 161.00 54 2 6 5 5 0 0 6 6 0 6 0 88.65 249.63 55 2 2 0 5 5 5 0 0 6 0 90 91.47 51.81 56 6 6 0 0 5 0 0 6 0 0 0 91.83 0.00 57 6 2 0 5 5 0 6 0 0 0 90 91.83 0.00 58 2 2 5 5 5 0 0 0 0 0 0 91.83 0.00 59 6 6 0 5 5 5 6 6 0 6 90 87.47 389.07 60 2 6 0 5 5 0 0 6 0 0 90 90.61 155.43 61 2 2 5 0 0 0 6 0 0 6 0 91.44 51.81 62 2 6 5 5 0 5 0 6 6 0 90 90.61 155.43 63 2 2 5 5 5 5 0 6 6 6 0 88.47 294.50 64 4 4 2.5 2.5 2.5 3.75 3 3 3 3 45 90.23 172.64 65 6 2 5 5 0 0 6 0 6 6 90 89.34 196.81 66 2 6 5 5 0 0 0 0 6 6 0 91.83 0.00 67 4 4 2.5 2.5 2.5 2.5 3 3 3 3 22.5 90.12 155.22 68 2 2 5 5 0 5 0 0 0 6 90 90.09 128.15 69 6 6 5 5 5 0 6 6 6 0 0 86.34 585.42 70 2 6 0 0 5 0 0 0 6 0 0 88.53 258.17 71 2 6 5 0 5 0 6 0 0 0 0 91.6 51.81 72 2 2 5 0 0 0 6 0 0 0 90 90.85 70.65 73 2 6 5 0 5 5 0 6 0 6 90 89.13 236.31 74 6 2 5 0 5 0 0 0 0 6 90 91.83 0.00 75 2 2 0 5 5 5 6 0 0 6 90 90.09 128.15 76 6 2 0 5 0 5 0 6 0 0 90 91.52 51.81 77 4 4 3.75 2.5 2.5 2.5 3 3 3 3 45 90.39 162.54 78 2 2 0 0 5 0 0 6 0 6 0 91.83 0.00 79 2 2 5 5 5 5 6 6 0 0 90 90.47 117.75 80 6 2 0 0 0 0 6 6 6 0 90 91.83 0.00 81 6 6 0 5 0 0 0 0 6 0 90 91.83 0.00 82 6 2 0 0 5 0 6 0 6 6 0 88.68 209.36 83 4 4 2.5 2.5 2.5 2.5 3 3 1.5 3 45 90.66 129.63 84 2 6 0 0 0 0 6 6 0 6 90 91.83 0.00 85 4 4 2.5 2.5 1.25 2.5 3 3 3 3 45 90.82 129.99 86 2 2 0 0 0 5 6 6 0 0 0 91.83 0.00 (continued) Qi et al. 23 (continued) 87 2 6 0 5 0 5 6 0 0 0 90 91.83 0.00 88 4 4 2.5 2.5 2.5 2.5 3 3 3 3 45 90.38 151.61 89 6 2 5 5 5 0 0 6 6 0 90 88.03 357.96 90 2 6 5 5 5 5 6 0 6 0 0 88.57 292.70 91 4 4 2.5 2.5 2.5 2.5 3 3 4.5 3 45 90.05 178.86 PSPL = peak sound pressure level, Rm = ribs mass

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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE

Published: Apr 11, 2022

Keywords: gearbox; radiated noise; panel acoustic participation analysis; response surface methodology; multi-objective optimization

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Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

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Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology:

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