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Optimizing Transplanting Mechanism with Planetary Elliptic Gears Based on Multi-body Dynamic Analysis and Approximate Models
Available online at www.sciencedirect.com
ScienceDirect Available online at www.sciencedirect.com Procedia Manufacturing 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia
Procedia Manufacturing 35 (2019) 1356–1362
2nd International Conference on Sustainable Materials Processing and Manufacturing, (SMPM 2019)
Optimizing Transplanting Mechanism with Planetary Elliptic Gears Based on Multi-body Dynamic Analysis and Approximate Models Chu Tingtinga, Zhu Dequana*, Xiong Weib, Zhu Lina, Zhang Shuna, Jen Tien-Chienc*, Liao Juana aSchool
School of Engineering, Anhui Agricultural University, Hefei 230036, China; School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China c Depatment of Mechanical Engineering Science, University of Johannesburg, Johannesburg 2006, South Africa bSchool
Abstract The multidisciplinary design optimization (MDO) strategy of the transplanting mechanism was determined, which was decomposed into three disciplines of kinematics, dynamics, and structural mechanics. The multidisciplinary design collaborative optimization models of the transplanting mechanism were established. The Latin hypercube design method was used to generate the initial sample points and construct the kriging model between the system-level variables and the discipline-level optimization. The MDO platform on the planetary elliptic gears transplanting mechanism was established on the basis of ISIGHT software and calculated by using the hybrid algorithm of multi-island genetic algorithm and sequential quadratic programming method. Optimization results showed that the width of the trajectory dynamic hole of the seedling needle tip, the frame vibration peak force, and the overall quality of the transplanting mechanism decreased by 55.6%, 20.5%, and 9.33%, respectively. The optimum overall performance of the transplanting mechanism was obtained by using the MDO based on approximation technique to meet the agronomic requirements of rice transplanting under high-accuracy computation and low computation time. © 2016 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by B.V. committee of SMPM 2019. Peer-review under responsibility ofElsevier the organizing Peer-review under responsibility of the organizing committee of SMPM 2019. Keywords: collaborative
optimization; surrogate model; multidisciplinary design; transplanting mechanism
* Corresponding author. Tel.: +86-551-6578-6065; fax: +86-551-6578-6165 Prof Zhu and Prof Jen are co-corresponding authors E-mail address: [email protected]; [email protected] 2351-9789 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of SMPM 2019.
2351-9789 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of SMPM 2019. 10.1016/j.promfg.2019.09.003
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1. Introduction Many factors influence the working performance of the transplanting mechanism of a rice transplanter, which involve kinematics, dynamics, structural mechanics, and other disciplines [1-2]. The single-subject design method has difficulty in coordinating the design and optimization among different disciplines. The multidisciplinary design optimization (MDO) method fully considers multiple disciplines and their coupling relationship and uses the coordination among disciplines to optimize the overall system performance. It has been used extensively in aerospace, shipping, automobile, and other fields [3-6]. Collaborative optimization can eliminate a large number of system analysis in MDO, and all disciplines can be analyzed and optimized in parallel [7-10]. Approximate models can effectively reduce the computational complexity and improve the efficiency of analysis and optimization without reducing the accuracy [11-12]. Therefore, in the study, the collaborative optimization method based on approximate model is applied to the MDO of transplanting mechanism with planetary elliptic gears, and the multidisciplinary model is established. A subject-level optimizer uses the hybrid algorithm of multi-island genetic algorithm and sequential quadratic programming method for optimization. An integrated optimization platform using ISIGHT software was built to perform the approximate solution of MDO for transplanting mechanism. The optimal combination of the design parameters and the overall performance of the product is optimal. 2. Structure and working principle of the transplanting mechanism The transplanting mechanism with planetary elliptic gears is composed of two planting arms and one gear box (Fig. 1), which has five equal elliptic gears. Its center of rotation is on one focal point of the elliptic gears. The five elliptic gears are initially installed at the same position to ensure normal operation. The central gear is a sun wheel fixed with the power input shaft. The two gears at both ends of the gearbox are planetary gears fixed with the planting arm. The gears between the sun and planetary gears are the middle gears called transition gears [13]. During the operation of the transplanting mechanism, the sun wheel is fixed and the power input shaft drives the gear box to rotate around the sun wheel. At the same time, the planting arm revolves along the gear box around the sun wheel while accompanying the star wheel rotation to complete the compound movement. The rotation of the planetary gear and the planting arm changes periodically because of the elliptical gears. Moreover, the movement track of the seedling needle tip becomes a closed curve of a non-circle (similar to an ellipse); it is divided into static and dynamic tracks. The static track is the track curve of the seedling needle tip when the transplanter stops and the transplanting mechanism works; whereas the dynamic track is the track curve of the seedling needle tip.
1. Cam; 2. Shifting block; 3. Seedling push spring; 4. Spring supporting beam; 5. Transplanting arm; 6. Seedling push stick; 7. Seedling needle; 8. Gear box; 9. Planetary gear; 10. Middle gear; 11. Sun gear Note: α0 is the initial position angle of the seedling needle tip; φ0 is the initial position angle of the gear box; S is the distance from the center of rotation of the planting arm to the seedling needle tip; and l is the distance between two adjacent gear rotating centers. Fig. 1. Transplanting mechanism with planetary elliptic gears
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3. Multidisciplinary design optimization
3.1. Disciplinary analysis The multidisciplinary problem of the transplanting mechanism with planetary elliptic gears is aimed at the structural characteristics of the mechanism and the requirements of MDO and is divided into multi-body dynamic and structural mechanic performance. 3.1.1 Analysis of multi-body dynamics The dynamic equation for the transplanting mechanism can be expressed as follows:
= FS f1 ( z, k , 0 , 0 ,S) + 14.3 f 2 ( z, k , 0 , 0 ,S)
(1)
where Fs is the vibration peak force of the support on rack; and f1 and f2 are the peak forces of the X- and Y-direction of the support, respectively, N. By combining the analysis with the actual work situation, the optimization objective of multi-body dynamics is to minimize the cave length of the dynamic track d of the seedling needle tip point and the vibration peak force of the support on rack Fs. The optimization parameters are the elliptical gear teeth z, ratio of the short and long axes of the elliptic gears k, the initial installation angles of the planting arm α0 and gear box φ0, the distance from the needle tip to the axis of the planet wheel S, the rotation center distance of the adjacent elliptic gears l, the angular velocity of the power input shaft ω, and the forward velocity of the rice transplanter vm. Meanwhile, the constraint parameters are the height of the needle tip trajectory H, the picking seedling angle β0, the pushing seedling angle β1, and the difference between the picking and pushing seedling angles δ. 3.1.2 Analysis of structural mechanic performance The structural design of the transplanting mechanism involves the stiffness and deviation of the bar, the natural frequency, and the damping characteristics. In this study, only static factors, stress distribution, and bar deviation are considered in the design of the transplanting mechanism to minimize the deformation of the end under the condition of satisfying the deformation and strength requirements. The optimization objective of the structural mechanic performance is to reduce the mass of the transplanting mechanism. The optimization parameters are the teeth of the elliptical gear z; the ratio of the short and long axes of the elliptic gears k; the angular velocity of the power input shaft ω; and the length l0, width b0, and thickness t0 of the gearbox case. The constraint parameters are the crosssection angle yB, deflection θB, and yield strength σ of the shaft. 3.2. MDO model The mathematical model of the transplanting mechanism with planetary elliptic gears based on collaborative optimization is obtained on the basis of the analyses of multi-body dynamics and structural mechanic performance. The overall MDO block diagram is shown in Fig. 2. Z*sys-1 Z 1*
System level Find Z min F ={f1, f2, f3} s.t R 1sys 0.001 R2sys 0.001 R3sys 0.001 Z*sys-2
Kinematic Find Z1 min R1 s.t
Dynamics Find Z2 min R2 s.t
Z 2*
Z*sys-3 Z 3*
Structural performance mechanics Find Z3 min R3 s.t
Fig. 2. Overall MDO diagram of the transplanting mechanism with planetary elliptic gears
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3.3. Approximate model of MDO The approximate model is constructed between the system-level variables and the subject-level optimization objectives, in which the system-level variables correspond with one another in each discipline. Therefore, the number of sample points N should be determined by the number of variables m in each discipline. The number of variables in kinematics, dynamics, and structural mechanics is 6, 5, and 6 respectively, and the number of sample points is 10 times of the number of discipline variables, that is, 60, 50, and 60 respectively. The kriging model between the system-level variables and the optimization objectives of disciplines is established on the basis of the analysis above, and the approximate model between the system design variables and the optimization objectives of the three disciplines is constructed using an optimizing Latin hypercube design to generate the sample points. The accuracy of the three approximate models is evaluated to ensure that they meet the engineering requirements. The accuracy evaluation criteria of the approximate models are the deterministic coefficient R2, average absolute error MAE, root mean square error RMSE, and maximum absolute error MAX. The error analysis results of the kriging model are obtained, as shown in Table 1. Table 1. Evaluating the parameters of the kriging model of disciplines Coefficient
R2
MAE
RMSE
MAX
R1
0.97625
0.03261
0.02115
0.04157
R2
0.98211
0.01779
0.01726
0.03624
R3
0.98516
0.00915
0.01157
0.15058
The statistical data in Table 1 show that the deterministic coefficient R2 of the kriging model constructed by the system variables and optimization objectives is greater than 0.9 in all three disciplines, which indicates that the overall accuracy of the kriging model is extremely high in all three disciplines. Furthermore, the values of MAE and RMAE are less than 0.1, and the value of MAX is less than 0.2. Thus, the global and local errors of the kriging model of the three disciplines are minimal, which meets the design requirements. 4. Construction and result analysis of the optimization platform
4.1. Construction of the optimization platform ISIGHT software has three main functions, namely, automation, integration, and optimization. The updating of design parameters is automatically reflected in all integrated software, which can considerably shorten product development cycles and remarkably reduce costs [14-16]. ISIGHT software is selected as the integrated platform for the MDO of transplanting mechanism. In view of the serialization of data flow in the analysis process, SolidWorks software is used for three-dimensional modeling, ANSYS/Workbench software is used for static analysis, and Adams software is used for kinematics and dynamics analysis. On the basis of the established system- and subjectlevel optimization model and approximate model construction method, the corresponding selection is conducted in the optimization component of ISIGHT platform, and the corresponding settings are performed in the calculator component. An integrated platform of the MDO of transplanting mechanism with planetary elliptic gears based on the approximate model is constructed. 4.2. Analysis of optimization result After the integrated platform of the MDO of transplanting mechanism is constructed, the multidisciplinary calculation can be performed and the global optimal solution can be obtained. The optimization results are shown in Table 2. From Table 2, the cave width of the dynamic track of the seedling needle tip point of the transplanting mechanism decreased from 0.569 mm to 0.252 mm (55.7%). The transplanting stability of the mechanism and the survival rate of the seedlings were high. During the operation of the transplanting mechanism, the peak of the
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vibration force on the frame decreased by 20.5%, and the overall force distribution and fluctuation amplitude were small. Under the premise of satisfying the stiffness and strength, the overall mass of the transplanting mechanism decreased from 3054.78 g to 2785.45 g (9.33%), thereby making the overall structure compact and saving material and manufacturing costs. Further analysis showed that under the condition of satisfying various constraints, the hybrid optimization algorithm could obtain satisfactory optimization solution on the basis of the cooperative optimization framework. This result verifies the correctness of the MDO and proxy models of the elliptic gear planetary system. Table 2. MDO results of the transplanting mechanism with planetary elliptic gears Model elements
Optimal parameters
Objectives
Cave width of the dynamic track of the seedling needle tip point, d/mm Vibration peak force of the support on rack, Fs/N Mass of the transplanting mechanism, m/kg
Variables
Constraints
Teeth of the elliptical gear, z Ratio of the short and long axes of the elliptic gears, k Initial installation angle of the planting arm, α0/° Initial installation angle of the gear box, φ0/° Rotation center distance of the adjacent elliptic gears, l/mm Distance from the needle tip to the axis of the planet wheel, S/mm Angular velocity of the power input shaft, ω/rad·s−1 Forward velocity of the rice transplanter, vm/mm·s−1 Length of the gearbox case, l0/mm Width of the gearbox case, b0/mm Thickness of the gearbox case, t0/mm Cross-section angle of the shaft, yB/mm Deflection of the shaft, θB/° Yield strength of the shaft σ/MPa Height of the needle tip trajectory, H/mm Picking seedling angle, β0/° Pushing seedling angle, β1/° Difference between the picking and pushing seedling angles, δ /°
Initial values
First step optimization with MIGA Optimization Relative results increment/%
Second step optimization with NLPQ Optimization Relative results increment/%
Total increment/%
0.569
0.252
−55.7
0.252
0
−55.7
2842.3
2817.2
−0.88
2259.9
−19.78
−20.5
3054.78
2863.12
−6.77
2785.45
−2.74
−9.33
19
21
/
21
/
/
0.9873
0.9903
0.30
0.9911
0.08
0.38
−65
−61.16
−5.91
−60.15
−1.65
−7.56
50
48.44
−3.12
45.12
−6.85
−9.97
43.5
44.67
2.69
44.81
0.31
3.00
140
145.88
4.20
149.46
2.45
6.65
6.28
12.56
/
12.56
/
/
495
593
19.8
593
0
19.8
242.63
235.04
−3.13
234.97
−0.03
−3.16
64.52
63.25
−1.97
62.95
−0.47
−2.44
14.8
13.06
−11.76
13.02
−0.31
−12.06
4.77E-08
−9.49
4.77E-08
−0.06
−9.55
0.636
−16.75
0.627
−1.42
−18.17
8.93E-05
−12.82
8.92E-05
−0.02
−12.84
295.1
289.67
−1.84
298.14
2.92
1.08
77.53
76.79
−0.95
74.66
−2.77
−3.7
22.35
21.06
−5.77
19.71
−6.41
−11.8
55.18
55.73
1
54.95
−1.4
−0.42
5.27E08 0.764 1.02E04
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4.3. Test verification The optimization model is re-established on the basis of the optimized design parameters. Table 3 shows the simulation results. As shown in Table 3, the optimized cave length of the dynamic track of the seedling needle tip point improved, the peak of the vibration force on the frame caused by the support decreased, and the mass of the transplanting mechanism increased; however, the relative error was within 5%. Thus, the optimized results are consistent with the verified results, which further explains the reliability of the multidisciplinary collaborative optimization method based on approximate model. Table 3 Validating results of the simulation Model elements
Optimal parameters
Initial values
Optimizatio n results
Verificatio n results
Relative increment/%
Objectives
Cave width of the dynamic track of the seedling needle tip point, d/mm
0.569
0.252
0.263
4.37
Vibration peak force of the support on rack, Fs/N
2842.3
2259.9
2161.72
-4.34
Mass of the transplanting mechanism, m/kg
3054.78
2785.45
2867.65
2.95
5. Conclusion
MDO mathematical model of the transplanting mechanism with planetary elliptic gears is established on the basis of the cooperative optimization method. The kriging model is used as the approximate model. The evaluation of the approximate model shows that the accuracy is satisfied with the design requirement. Based on ISIGHT software, the MDO integration platform of the transplanting mechanism with planetary elliptic gears is constructed. Optimization results show that the width is reduced from 0.569 mm to 0.252 mm (55.7%), thereby making the transplanting stability of the mechanism and the survival rate of the seedlings high. During the operation of the transplanting mechanism, the peak vibration force on the frame is reduced by 20.5%. The entire force is uniformly distributed, and the fluctuation amplitude is small, thereby making the performance of the mechanism stable and reliable. The overall quality of the insertion mechanism is reduced from 3054.78 g to 2785.45 g (9.33%), which makes the overall structure compact and saves material and manufacturing costs. References [1]
B.L. Ye, W.M. Yi, G.H. Yu, Y. Gao, X. Zhao, Optimization design and test of rice plug seedling transplanting mechanism of planetary gear train with incomplete eccentric circular gear and non-circular gears, International Journal of Agricultural and Biological Engineering, 10(2017)43-55. [2] L. Sun, X. Chen, C.Y. Wu, G.F. Zhang, Y.D Xu, Synthesis and design of rice pot seedling transplanting mechanism based on labeled graph theory, Computers and Electronics in Agriculture, 143(2017)249-261. [3] M.I. Alam, R.S. Pant, Multi-objective multidisciplinary design analyses and optimization of high altitude airships, Aerospace science and technology, 78(2018)248-259. [4] T. Ashuri, M.B. Zaaijer, J.R.R.A. Martins, G. J. W. van Bussel, G. A. M. van Kuik, Multidisciplinary design optimization of offshore wind turbines for minimum levelized cost of energy, Renewable Energy, 68(2014)893-905. [5] J. T. Allison, D. R. Herber, Multidisciplinary design optimization of dynamic engineering systems, AIAA Journal, 52(2014)691-710. [6] T. Ashuri, M.B. Zaaijer, J.R.R.A. Martins, J. Zhang, Multidisciplinary design optimization of large wind turbines-Technical, economic, and design challenges, Energy Conversion and Management, 123(2016)56-70. [7] Y.C. Hu, S.S. Chen, S.P. Li, Research on multidisciplinary optimization of sugarcane harvester based on global coordination, China mechanical engineering, 18(2007)1355-1358. [8] C.Q. Wang, D.F. Wang, S. Zhang, Design and application of lightweight multi-objective collaborative optimization for a parametric bodyin-white structure, Proceedings of the Institution of Mechanical Engineers Part D-Journal of Automobile Engineering, 230(2016)273-288. [9] D. Temple, M. Colette, A goal-programming enhanced collaborative optimization approach to reducing lifecyle costs for naval vessels, Structural and Multidisciplinary Optimization, 53(2016)1261-1275. [10] S.G. Zuo, K.J. Wei, X.D. Wu, Multi-objective optimization of a multi-chamber perforated muffler using an approximate model and genetic algorithm, International Journal of Acoustics and Vibration, 21(2016)152-163. [11] Y. Feng, L.J. Zhang, Precisely deciding CSL formulas through approximate model checking for CTMCs, Journal of Computer and System Sciences, 89(2017)361-371.
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[12] K. Kwak, J.S. Kim, D.F. Huber, T. Kanade, Online approximate model representation based on scale-normalized and fronto-parallel appearance, International Journal of Computer Vision, 117(2016)48-69. [13] D.Q. Zhu, Study on optimization design method for mechanical-electrical-hydraulic integrated system based on collaborative simulation, University of Science and Technology of China, Hefei, 2012. [14] P.N. Koch, J.P. Evans, D. Powell, Interdigitation for effective design space exploration using iSIGHT, Structural and Multidisciplinary Optimization, 23(2002)111-126. [15] S. Nagendra, J.B. Staubach, A.J. Suydam, S.J. Ghunakikar, V.R. Akula, Optimal rapid multidisciplinary response networks: RAPIDDISK, Structural and Multidisciplinary Optimization, 29(2005)213-231. [16] J.J. Huang, L.X. Zheng, Q. Mei, Design and optimization method of a two-disk rotor system, International Journal of Turbo & Jet-Engines, 33(2016)1-8.
ScienceDirect Available online at www.sciencedirect.com Procedia Manufacturing 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia
Procedia Manufacturing 35 (2019) 1356–1362
2nd International Conference on Sustainable Materials Processing and Manufacturing, (SMPM 2019)
Optimizing Transplanting Mechanism with Planetary Elliptic Gears Based on Multi-body Dynamic Analysis and Approximate Models Chu Tingtinga, Zhu Dequana*, Xiong Weib, Zhu Lina, Zhang Shuna, Jen Tien-Chienc*, Liao Juana aSchool
School of Engineering, Anhui Agricultural University, Hefei 230036, China; School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China c Depatment of Mechanical Engineering Science, University of Johannesburg, Johannesburg 2006, South Africa bSchool
Abstract The multidisciplinary design optimization (MDO) strategy of the transplanting mechanism was determined, which was decomposed into three disciplines of kinematics, dynamics, and structural mechanics. The multidisciplinary design collaborative optimization models of the transplanting mechanism were established. The Latin hypercube design method was used to generate the initial sample points and construct the kriging model between the system-level variables and the discipline-level optimization. The MDO platform on the planetary elliptic gears transplanting mechanism was established on the basis of ISIGHT software and calculated by using the hybrid algorithm of multi-island genetic algorithm and sequential quadratic programming method. Optimization results showed that the width of the trajectory dynamic hole of the seedling needle tip, the frame vibration peak force, and the overall quality of the transplanting mechanism decreased by 55.6%, 20.5%, and 9.33%, respectively. The optimum overall performance of the transplanting mechanism was obtained by using the MDO based on approximation technique to meet the agronomic requirements of rice transplanting under high-accuracy computation and low computation time. © 2016 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by B.V. committee of SMPM 2019. Peer-review under responsibility ofElsevier the organizing Peer-review under responsibility of the organizing committee of SMPM 2019. Keywords: collaborative
optimization; surrogate model; multidisciplinary design; transplanting mechanism
* Corresponding author. Tel.: +86-551-6578-6065; fax: +86-551-6578-6165 Prof Zhu and Prof Jen are co-corresponding authors E-mail address: [email protected]; [email protected] 2351-9789 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of SMPM 2019.
2351-9789 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the organizing committee of SMPM 2019. 10.1016/j.promfg.2019.09.003
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1. Introduction Many factors influence the working performance of the transplanting mechanism of a rice transplanter, which involve kinematics, dynamics, structural mechanics, and other disciplines [1-2]. The single-subject design method has difficulty in coordinating the design and optimization among different disciplines. The multidisciplinary design optimization (MDO) method fully considers multiple disciplines and their coupling relationship and uses the coordination among disciplines to optimize the overall system performance. It has been used extensively in aerospace, shipping, automobile, and other fields [3-6]. Collaborative optimization can eliminate a large number of system analysis in MDO, and all disciplines can be analyzed and optimized in parallel [7-10]. Approximate models can effectively reduce the computational complexity and improve the efficiency of analysis and optimization without reducing the accuracy [11-12]. Therefore, in the study, the collaborative optimization method based on approximate model is applied to the MDO of transplanting mechanism with planetary elliptic gears, and the multidisciplinary model is established. A subject-level optimizer uses the hybrid algorithm of multi-island genetic algorithm and sequential quadratic programming method for optimization. An integrated optimization platform using ISIGHT software was built to perform the approximate solution of MDO for transplanting mechanism. The optimal combination of the design parameters and the overall performance of the product is optimal. 2. Structure and working principle of the transplanting mechanism The transplanting mechanism with planetary elliptic gears is composed of two planting arms and one gear box (Fig. 1), which has five equal elliptic gears. Its center of rotation is on one focal point of the elliptic gears. The five elliptic gears are initially installed at the same position to ensure normal operation. The central gear is a sun wheel fixed with the power input shaft. The two gears at both ends of the gearbox are planetary gears fixed with the planting arm. The gears between the sun and planetary gears are the middle gears called transition gears [13]. During the operation of the transplanting mechanism, the sun wheel is fixed and the power input shaft drives the gear box to rotate around the sun wheel. At the same time, the planting arm revolves along the gear box around the sun wheel while accompanying the star wheel rotation to complete the compound movement. The rotation of the planetary gear and the planting arm changes periodically because of the elliptical gears. Moreover, the movement track of the seedling needle tip becomes a closed curve of a non-circle (similar to an ellipse); it is divided into static and dynamic tracks. The static track is the track curve of the seedling needle tip when the transplanter stops and the transplanting mechanism works; whereas the dynamic track is the track curve of the seedling needle tip.
1. Cam; 2. Shifting block; 3. Seedling push spring; 4. Spring supporting beam; 5. Transplanting arm; 6. Seedling push stick; 7. Seedling needle; 8. Gear box; 9. Planetary gear; 10. Middle gear; 11. Sun gear Note: α0 is the initial position angle of the seedling needle tip; φ0 is the initial position angle of the gear box; S is the distance from the center of rotation of the planting arm to the seedling needle tip; and l is the distance between two adjacent gear rotating centers. Fig. 1. Transplanting mechanism with planetary elliptic gears
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3. Multidisciplinary design optimization
3.1. Disciplinary analysis The multidisciplinary problem of the transplanting mechanism with planetary elliptic gears is aimed at the structural characteristics of the mechanism and the requirements of MDO and is divided into multi-body dynamic and structural mechanic performance. 3.1.1 Analysis of multi-body dynamics The dynamic equation for the transplanting mechanism can be expressed as follows:
= FS f1 ( z, k , 0 , 0 ,S) + 14.3 f 2 ( z, k , 0 , 0 ,S)
(1)
where Fs is the vibration peak force of the support on rack; and f1 and f2 are the peak forces of the X- and Y-direction of the support, respectively, N. By combining the analysis with the actual work situation, the optimization objective of multi-body dynamics is to minimize the cave length of the dynamic track d of the seedling needle tip point and the vibration peak force of the support on rack Fs. The optimization parameters are the elliptical gear teeth z, ratio of the short and long axes of the elliptic gears k, the initial installation angles of the planting arm α0 and gear box φ0, the distance from the needle tip to the axis of the planet wheel S, the rotation center distance of the adjacent elliptic gears l, the angular velocity of the power input shaft ω, and the forward velocity of the rice transplanter vm. Meanwhile, the constraint parameters are the height of the needle tip trajectory H, the picking seedling angle β0, the pushing seedling angle β1, and the difference between the picking and pushing seedling angles δ. 3.1.2 Analysis of structural mechanic performance The structural design of the transplanting mechanism involves the stiffness and deviation of the bar, the natural frequency, and the damping characteristics. In this study, only static factors, stress distribution, and bar deviation are considered in the design of the transplanting mechanism to minimize the deformation of the end under the condition of satisfying the deformation and strength requirements. The optimization objective of the structural mechanic performance is to reduce the mass of the transplanting mechanism. The optimization parameters are the teeth of the elliptical gear z; the ratio of the short and long axes of the elliptic gears k; the angular velocity of the power input shaft ω; and the length l0, width b0, and thickness t0 of the gearbox case. The constraint parameters are the crosssection angle yB, deflection θB, and yield strength σ of the shaft. 3.2. MDO model The mathematical model of the transplanting mechanism with planetary elliptic gears based on collaborative optimization is obtained on the basis of the analyses of multi-body dynamics and structural mechanic performance. The overall MDO block diagram is shown in Fig. 2. Z*sys-1 Z 1*
System level Find Z min F ={f1, f2, f3} s.t R 1sys 0.001 R2sys 0.001 R3sys 0.001 Z*sys-2
Kinematic Find Z1 min R1 s.t
Dynamics Find Z2 min R2 s.t
Z 2*
Z*sys-3 Z 3*
Structural performance mechanics Find Z3 min R3 s.t
Fig. 2. Overall MDO diagram of the transplanting mechanism with planetary elliptic gears
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3.3. Approximate model of MDO The approximate model is constructed between the system-level variables and the subject-level optimization objectives, in which the system-level variables correspond with one another in each discipline. Therefore, the number of sample points N should be determined by the number of variables m in each discipline. The number of variables in kinematics, dynamics, and structural mechanics is 6, 5, and 6 respectively, and the number of sample points is 10 times of the number of discipline variables, that is, 60, 50, and 60 respectively. The kriging model between the system-level variables and the optimization objectives of disciplines is established on the basis of the analysis above, and the approximate model between the system design variables and the optimization objectives of the three disciplines is constructed using an optimizing Latin hypercube design to generate the sample points. The accuracy of the three approximate models is evaluated to ensure that they meet the engineering requirements. The accuracy evaluation criteria of the approximate models are the deterministic coefficient R2, average absolute error MAE, root mean square error RMSE, and maximum absolute error MAX. The error analysis results of the kriging model are obtained, as shown in Table 1. Table 1. Evaluating the parameters of the kriging model of disciplines Coefficient
R2
MAE
RMSE
MAX
R1
0.97625
0.03261
0.02115
0.04157
R2
0.98211
0.01779
0.01726
0.03624
R3
0.98516
0.00915
0.01157
0.15058
The statistical data in Table 1 show that the deterministic coefficient R2 of the kriging model constructed by the system variables and optimization objectives is greater than 0.9 in all three disciplines, which indicates that the overall accuracy of the kriging model is extremely high in all three disciplines. Furthermore, the values of MAE and RMAE are less than 0.1, and the value of MAX is less than 0.2. Thus, the global and local errors of the kriging model of the three disciplines are minimal, which meets the design requirements. 4. Construction and result analysis of the optimization platform
4.1. Construction of the optimization platform ISIGHT software has three main functions, namely, automation, integration, and optimization. The updating of design parameters is automatically reflected in all integrated software, which can considerably shorten product development cycles and remarkably reduce costs [14-16]. ISIGHT software is selected as the integrated platform for the MDO of transplanting mechanism. In view of the serialization of data flow in the analysis process, SolidWorks software is used for three-dimensional modeling, ANSYS/Workbench software is used for static analysis, and Adams software is used for kinematics and dynamics analysis. On the basis of the established system- and subjectlevel optimization model and approximate model construction method, the corresponding selection is conducted in the optimization component of ISIGHT platform, and the corresponding settings are performed in the calculator component. An integrated platform of the MDO of transplanting mechanism with planetary elliptic gears based on the approximate model is constructed. 4.2. Analysis of optimization result After the integrated platform of the MDO of transplanting mechanism is constructed, the multidisciplinary calculation can be performed and the global optimal solution can be obtained. The optimization results are shown in Table 2. From Table 2, the cave width of the dynamic track of the seedling needle tip point of the transplanting mechanism decreased from 0.569 mm to 0.252 mm (55.7%). The transplanting stability of the mechanism and the survival rate of the seedlings were high. During the operation of the transplanting mechanism, the peak of the
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vibration force on the frame decreased by 20.5%, and the overall force distribution and fluctuation amplitude were small. Under the premise of satisfying the stiffness and strength, the overall mass of the transplanting mechanism decreased from 3054.78 g to 2785.45 g (9.33%), thereby making the overall structure compact and saving material and manufacturing costs. Further analysis showed that under the condition of satisfying various constraints, the hybrid optimization algorithm could obtain satisfactory optimization solution on the basis of the cooperative optimization framework. This result verifies the correctness of the MDO and proxy models of the elliptic gear planetary system. Table 2. MDO results of the transplanting mechanism with planetary elliptic gears Model elements
Optimal parameters
Objectives
Cave width of the dynamic track of the seedling needle tip point, d/mm Vibration peak force of the support on rack, Fs/N Mass of the transplanting mechanism, m/kg
Variables
Constraints
Teeth of the elliptical gear, z Ratio of the short and long axes of the elliptic gears, k Initial installation angle of the planting arm, α0/° Initial installation angle of the gear box, φ0/° Rotation center distance of the adjacent elliptic gears, l/mm Distance from the needle tip to the axis of the planet wheel, S/mm Angular velocity of the power input shaft, ω/rad·s−1 Forward velocity of the rice transplanter, vm/mm·s−1 Length of the gearbox case, l0/mm Width of the gearbox case, b0/mm Thickness of the gearbox case, t0/mm Cross-section angle of the shaft, yB/mm Deflection of the shaft, θB/° Yield strength of the shaft σ/MPa Height of the needle tip trajectory, H/mm Picking seedling angle, β0/° Pushing seedling angle, β1/° Difference between the picking and pushing seedling angles, δ /°
Initial values
First step optimization with MIGA Optimization Relative results increment/%
Second step optimization with NLPQ Optimization Relative results increment/%
Total increment/%
0.569
0.252
−55.7
0.252
0
−55.7
2842.3
2817.2
−0.88
2259.9
−19.78
−20.5
3054.78
2863.12
−6.77
2785.45
−2.74
−9.33
19
21
/
21
/
/
0.9873
0.9903
0.30
0.9911
0.08
0.38
−65
−61.16
−5.91
−60.15
−1.65
−7.56
50
48.44
−3.12
45.12
−6.85
−9.97
43.5
44.67
2.69
44.81
0.31
3.00
140
145.88
4.20
149.46
2.45
6.65
6.28
12.56
/
12.56
/
/
495
593
19.8
593
0
19.8
242.63
235.04
−3.13
234.97
−0.03
−3.16
64.52
63.25
−1.97
62.95
−0.47
−2.44
14.8
13.06
−11.76
13.02
−0.31
−12.06
4.77E-08
−9.49
4.77E-08
−0.06
−9.55
0.636
−16.75
0.627
−1.42
−18.17
8.93E-05
−12.82
8.92E-05
−0.02
−12.84
295.1
289.67
−1.84
298.14
2.92
1.08
77.53
76.79
−0.95
74.66
−2.77
−3.7
22.35
21.06
−5.77
19.71
−6.41
−11.8
55.18
55.73
1
54.95
−1.4
−0.42
5.27E08 0.764 1.02E04
6
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4.3. Test verification The optimization model is re-established on the basis of the optimized design parameters. Table 3 shows the simulation results. As shown in Table 3, the optimized cave length of the dynamic track of the seedling needle tip point improved, the peak of the vibration force on the frame caused by the support decreased, and the mass of the transplanting mechanism increased; however, the relative error was within 5%. Thus, the optimized results are consistent with the verified results, which further explains the reliability of the multidisciplinary collaborative optimization method based on approximate model. Table 3 Validating results of the simulation Model elements
Optimal parameters
Initial values
Optimizatio n results
Verificatio n results
Relative increment/%
Objectives
Cave width of the dynamic track of the seedling needle tip point, d/mm
0.569
0.252
0.263
4.37
Vibration peak force of the support on rack, Fs/N
2842.3
2259.9
2161.72
-4.34
Mass of the transplanting mechanism, m/kg
3054.78
2785.45
2867.65
2.95
5. Conclusion
MDO mathematical model of the transplanting mechanism with planetary elliptic gears is established on the basis of the cooperative optimization method. The kriging model is used as the approximate model. The evaluation of the approximate model shows that the accuracy is satisfied with the design requirement. Based on ISIGHT software, the MDO integration platform of the transplanting mechanism with planetary elliptic gears is constructed. Optimization results show that the width is reduced from 0.569 mm to 0.252 mm (55.7%), thereby making the transplanting stability of the mechanism and the survival rate of the seedlings high. During the operation of the transplanting mechanism, the peak vibration force on the frame is reduced by 20.5%. The entire force is uniformly distributed, and the fluctuation amplitude is small, thereby making the performance of the mechanism stable and reliable. The overall quality of the insertion mechanism is reduced from 3054.78 g to 2785.45 g (9.33%), which makes the overall structure compact and saves material and manufacturing costs. References [1]
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