面齿轮传动啮合刚度分析与修形减振优化
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摘要
为准确求解面齿轮传动的啮合刚度,基于齿轮的承载接触分析(LTCA)技术,综合考虑变位、小轮偏置、齿面修形以及安装误差,提出了面齿轮传动啮合刚度计算方法,并验证了该方法的精确性。分析了负载、变位、小轮偏置和安装误差对面齿轮传动综合啮合刚度均值和波动幅值的影响,并将LTCA技术与遗传算法相结合,建立了面齿轮传动修形减振优化模型。研究结果表明:面齿轮传动综合啮合刚度幅值波动较大,存在阶跃突变现象,但波动幅值对负载、变位、小轮偏置和安装误差并不敏感,而综合啮合刚度均值受负载、小轮偏置和安装误差影响较大,且在3类安装误差中,轴夹角误差对综合啮合刚度均值影响最大;优化小轮修形参数后使综合啮合刚度的波动幅值大幅下降,从而可有效减小面齿轮传动的振动和噪声。
To accurately calculate meshing stiffness of face-gear drives, comprehensively considering effects of profile-shift, pinion offset, tooth surface modification and misalignment, a method for calculating meshing stiffness of face-gear drives was proposed based on the loaded tooth contact analysis(LTCA) technique of gears. The accuracy of the proposed method was verified. The effects of load, profile-shift, pinion offset and misalignment on the mean value and amplitude fluctuation of comprehensive meshing stiffness of face-gear drives were analyzed. Combining the LTCA technique with the genetic algorithm, an optimization model of vibration reduction and modification for face-gear drives was established. The results showed that the amplitude fluctuation of the comprehensive meshing stiffness of face-gear drives is larger, and there is a step abrupt change phenomenon; the amplitude fluctuation is not sensitive to load, pinion offset and misalignment, but these 3 factors significantly affect the mean value of the comprehensive meshing stiffness; in 3 types of misalignment, shaft angle error has the largest effect on the mean value of the comprehensive meshing stiffness; after optimizing pinion modification parameters, the amplitude fluctuation of the comprehensive meshing stiffness decreases by 91% to effectively reduce vibration and noise of face-gear drives.
To accurately calculate meshing stiffness of face-gear drives, comprehensively considering effects of profile-shift, pinion offset, tooth surface modification and misalignment, a method for calculating meshing stiffness of face-gear drives was proposed based on the loaded tooth contact analysis(LTCA) technique of gears. The accuracy of the proposed method was verified. The effects of load, profile-shift, pinion offset and misalignment on the mean value and amplitude fluctuation of comprehensive meshing stiffness of face-gear drives were analyzed. Combining the LTCA technique with the genetic algorithm, an optimization model of vibration reduction and modification for face-gear drives was established. The results showed that the amplitude fluctuation of the comprehensive meshing stiffness of face-gear drives is larger, and there is a step abrupt change phenomenon; the amplitude fluctuation is not sensitive to load, pinion offset and misalignment, but these 3 factors significantly affect the mean value of the comprehensive meshing stiffness; in 3 types of misalignment, shaft angle error has the largest effect on the mean value of the comprehensive meshing stiffness; after optimizing pinion modification parameters, the amplitude fluctuation of the comprehensive meshing stiffness decreases by 91% to effectively reduce vibration and noise of face-gear drives.
引文
[1] PIMSARN M, KAZEROUNIAN K. Efficient evaluation of spur gear tooth mesh load using pseudo-interference stiffness estimation method[J]. Mechanism and Machine Theory, 2002, 37(8): 769-786.
[2] MA H, ZENG J, FENG R, et al. An improved analytical method for mesh stiffness calculation of spur gears with tip relief[J]. Mechanism & Machine Theory, 2016, 98: 64-80.
[3] PEDERSEN N L, J?RGENSEN M F. On gear tooth stiffness evaluation[J]. Computers & Structures, 2014, 135(3): 109-117.
[4] YESILYURT I, GU F, BALL A D. Gear tooth stiffness reduction measurement using modal analysis and its use in wear fault severity assessment of spur gears[J]. NDT & E International, 2003, 36(5): 357-372.
[5] 李明, 孙涛, 胡海岩. 齿轮传动转子-轴承系统动力学的研究进展[J]. 振动工程学报, 2002, 15(3): 249-256. LI Ming, SUN Tao, HU Haiyan. Review on dynamics of geared rotor-bearing systems[J]. Journal of Vibration Engineering, 2002, 15(3): 249-256.
[6] 李亚鹏. 齿轮时变啮合刚度改进算法及刚度激励研究[D]. 大连: 大连理工大学, 2009.
[7] TERAUCHI Y, NAGAMURA K. Study on deflection of spur gear teeth: 1st report, calculation, of tooth deflection by two-dimensional elastic theory[J]. Bulletin of JSME, 1980, 23(184): 1682-1688.
[8] TERAUCHI Y, NAGAMURA K. Study on deflection of spur gear teeth: 2nd report, calculation of tooth deflection for spur gears with various tooth profiles[J]. Bulletin of JSME, 1981, 24(188): 447-452.
[9] 常乐浩, 贺朝霞, 刘岚, 等. 一种确定斜齿轮传递误差和啮合刚度的快速有效方法[J]. 振动与冲击, 2017, 36(6): 157-162. CHANG Lehao, HE Zhaoxia, LIU Lan, et al. Express method for determining the transmission error and mesh stiffness of helical gears[J]. Journal of Vibration and Shock, 2017, 36(6): 157-162.
[10] WU Y J, WANG J J, HAN Q K. Contact finite element method for dynamic meshing characteristics analysis of continuous engaged gear drives[J]. Journal of Mechanical Science and Technology, 2012, 26(6): 1671-1685.
[11] RINCON A F D, VIADERO F, IGLESIAS M, et al. A model for the study of meshing stiffness in spur gear transmissions[J]. Mechanism & Machine Theory, 2013, 61: 30-58.
[12] 唐进元, 蒲太平. 基于有限元法的螺旋锥齿轮啮合刚度计算[J]. 机械工程学报, 2011, 47(11):23-29. TANG Jinyuan, PU Taiping. Spiral bevel gear meshing stiffness calculations based on the finite element method[J]. Journal of Mechanical Engineering, 2011, 47(11): 23-29.
[13] HEATH G F, SLAUGHTER S C, FISHER D J, et al. Helical face gear development under the enhanced rotorcraft drive system program[R]. Cleveland, OH: NASA Technical Reports Server, 2011: 1-20.
[14] LEWICKI D G, HEATH G F, FILLER R R, et al. Face gear surface durability investigations[J]. Journal of the American Helicopter Society, 2008, 53(3): 282-289.
[15] 李政民卿, 黄鹏, 李晓贞. 面齿轮轮齿刚度的计算方法及其影响因素分析[J]. 重庆大学学报, 2014, 37(1): 26-30. LI Zhengminqing, HUANG Peng, LI Xiaozhen. A calculation method of the stiffness of face gear tooth and analysis of its influence factors[J]. Journal of Chongqing University, 2014, 37(1): 26-30.
[16] LI Z M Q, WANG J, ZHU R P. Influence predictions of contact effects on mesh stiffness of face gear drives with spur gear[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2015, 32(5): 566-570.
[17] 雷敦财, 唐进元. 一种面齿轮传动时变啮合刚度数值计算方法[J]. 中国机械工程, 2014, 25(17): 2300-2304. LEI Duncai, TANG Jinyuan. A calculation method of mesh stiffness for face gear transmission system[J]. China Mechanical Engineering, 2014, 25(17): 2300-2304.
[18] ZHANG Yi, FANG Zongde. Analysis of tooth contact and load distribution of helical gears with crossed axes[J]. Mechanism and Machine Theory, 1999, 34: 41-57.
[19] 方宗德. 齿轮轮齿承载接触分析(LTCA)的模型和方法[J]. 机械传动, 1998, 22(2): 1-3. FANG Zongde. Model and approach for loaded tooth contact analysis (LTCA) of gear drives[J]. Journal of Mechanical Transmission, 1998, 22(2): 1-3.
[20] SHUTING L I. Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear[J]. Journal of Mechanical Design, 2002, 124(3): 511-517.
[21] 蒋进科. 高速渐开线圆柱齿轮齿面设计及数控加工技术研究[D]. 西安: 西北工业大学, 2015.
[22] 付学中, 方宗德, 侯祥颖,等. 变位面齿轮副承载特性分析及变位系数优化[J]. 华中科技大学学报(自然科学版), 2017, 45(6):57-62. FU Xuezhong, FANG Zongde, HOU Xiangying, et al. Bearing characteristics and optimal modification coefficient of modified face gear pair[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2017, 45(6): 57-62.
[23] 付学中, 方宗德, 李建华, 等. 偏置面齿轮的碟形砂轮磨齿及啮合性能[J]. 华南理工大学学报(自然科学版), 2016, 44(7): 77-82. FU Xuezhong, FANG Zongde, LI Jianhua, et al. Grinding and meshing performance of offset face gear modified with disk wheel[J]. Journal of South China University of Technology (Natural Science Edition), 2016, 44(7): 77-82.
[24] ROSEN J B. The gradient projection method for nonlinear programming. Part I. Linear Constraints[J]. Journal of the Society for Industrial and Applied Mathematics, 1961, 4(4): 514-532.
[25] SMITH J D. Gear noise and vibration[M]. New York: Marcel Dekker, 2003.
[26] MAATAR M, VELEX P. An analytical expression for the time-varying contact length in perfect cylindrical gears: some possible applications in gear dynamics[J]. Journal of Mechanical Design, 1996, 118(4): 586-589.
[27] 王峰. 人字齿轮传动系统振动特性分析与试验研究[D]. 西安: 西北工业大学, 2014.
[28] 常山, 徐振忠, 霍肇波, 等. 斜齿圆柱齿轮瞬时啮合刚度及齿廓修形的研究[J]. 热能动力工程, 1997, 12(4): 270-274. CHANG Shan, XU Zhenzhong, HUO Zhaobo, et al. A study on the momentary meshing rigidity and tooth profile modification of helical cylindrical gears[J]. Journal of Engineering for Thermal Energy and Power, 1997, 12(4): 270-274.
[29] LITVIN F L, VECCHIATO D, YUKISHIMA K, et al. Reduction of noise of loaded and unloaded misaligned gear drives[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 5523-5536.
[30] 付学中, 方宗德, 李建华, 等. 面齿轮传动小轮拓扑修形设计及啮合性能分析[J]. 哈尔滨工程大学学报, 2016, 37(9): 1281-1286. FU Xuezhong, FANG Zongde, LI Jianhua, et al. Design of topological shape modification of auxiliary pinion of face gears and analysis of meshing performance[J]. Journal of Harbin Engineering University, 2016, 37(9): 1281-1286.
[31] DEB K, MEMBER A, PRATAP A, et al. A Fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
[2] MA H, ZENG J, FENG R, et al. An improved analytical method for mesh stiffness calculation of spur gears with tip relief[J]. Mechanism & Machine Theory, 2016, 98: 64-80.
[3] PEDERSEN N L, J?RGENSEN M F. On gear tooth stiffness evaluation[J]. Computers & Structures, 2014, 135(3): 109-117.
[4] YESILYURT I, GU F, BALL A D. Gear tooth stiffness reduction measurement using modal analysis and its use in wear fault severity assessment of spur gears[J]. NDT & E International, 2003, 36(5): 357-372.
[5] 李明, 孙涛, 胡海岩. 齿轮传动转子-轴承系统动力学的研究进展[J]. 振动工程学报, 2002, 15(3): 249-256. LI Ming, SUN Tao, HU Haiyan. Review on dynamics of geared rotor-bearing systems[J]. Journal of Vibration Engineering, 2002, 15(3): 249-256.
[6] 李亚鹏. 齿轮时变啮合刚度改进算法及刚度激励研究[D]. 大连: 大连理工大学, 2009.
[7] TERAUCHI Y, NAGAMURA K. Study on deflection of spur gear teeth: 1st report, calculation, of tooth deflection by two-dimensional elastic theory[J]. Bulletin of JSME, 1980, 23(184): 1682-1688.
[8] TERAUCHI Y, NAGAMURA K. Study on deflection of spur gear teeth: 2nd report, calculation of tooth deflection for spur gears with various tooth profiles[J]. Bulletin of JSME, 1981, 24(188): 447-452.
[9] 常乐浩, 贺朝霞, 刘岚, 等. 一种确定斜齿轮传递误差和啮合刚度的快速有效方法[J]. 振动与冲击, 2017, 36(6): 157-162. CHANG Lehao, HE Zhaoxia, LIU Lan, et al. Express method for determining the transmission error and mesh stiffness of helical gears[J]. Journal of Vibration and Shock, 2017, 36(6): 157-162.
[10] WU Y J, WANG J J, HAN Q K. Contact finite element method for dynamic meshing characteristics analysis of continuous engaged gear drives[J]. Journal of Mechanical Science and Technology, 2012, 26(6): 1671-1685.
[11] RINCON A F D, VIADERO F, IGLESIAS M, et al. A model for the study of meshing stiffness in spur gear transmissions[J]. Mechanism & Machine Theory, 2013, 61: 30-58.
[12] 唐进元, 蒲太平. 基于有限元法的螺旋锥齿轮啮合刚度计算[J]. 机械工程学报, 2011, 47(11):23-29. TANG Jinyuan, PU Taiping. Spiral bevel gear meshing stiffness calculations based on the finite element method[J]. Journal of Mechanical Engineering, 2011, 47(11): 23-29.
[13] HEATH G F, SLAUGHTER S C, FISHER D J, et al. Helical face gear development under the enhanced rotorcraft drive system program[R]. Cleveland, OH: NASA Technical Reports Server, 2011: 1-20.
[14] LEWICKI D G, HEATH G F, FILLER R R, et al. Face gear surface durability investigations[J]. Journal of the American Helicopter Society, 2008, 53(3): 282-289.
[15] 李政民卿, 黄鹏, 李晓贞. 面齿轮轮齿刚度的计算方法及其影响因素分析[J]. 重庆大学学报, 2014, 37(1): 26-30. LI Zhengminqing, HUANG Peng, LI Xiaozhen. A calculation method of the stiffness of face gear tooth and analysis of its influence factors[J]. Journal of Chongqing University, 2014, 37(1): 26-30.
[16] LI Z M Q, WANG J, ZHU R P. Influence predictions of contact effects on mesh stiffness of face gear drives with spur gear[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2015, 32(5): 566-570.
[17] 雷敦财, 唐进元. 一种面齿轮传动时变啮合刚度数值计算方法[J]. 中国机械工程, 2014, 25(17): 2300-2304. LEI Duncai, TANG Jinyuan. A calculation method of mesh stiffness for face gear transmission system[J]. China Mechanical Engineering, 2014, 25(17): 2300-2304.
[18] ZHANG Yi, FANG Zongde. Analysis of tooth contact and load distribution of helical gears with crossed axes[J]. Mechanism and Machine Theory, 1999, 34: 41-57.
[19] 方宗德. 齿轮轮齿承载接触分析(LTCA)的模型和方法[J]. 机械传动, 1998, 22(2): 1-3. FANG Zongde. Model and approach for loaded tooth contact analysis (LTCA) of gear drives[J]. Journal of Mechanical Transmission, 1998, 22(2): 1-3.
[20] SHUTING L I. Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear[J]. Journal of Mechanical Design, 2002, 124(3): 511-517.
[21] 蒋进科. 高速渐开线圆柱齿轮齿面设计及数控加工技术研究[D]. 西安: 西北工业大学, 2015.
[22] 付学中, 方宗德, 侯祥颖,等. 变位面齿轮副承载特性分析及变位系数优化[J]. 华中科技大学学报(自然科学版), 2017, 45(6):57-62. FU Xuezhong, FANG Zongde, HOU Xiangying, et al. Bearing characteristics and optimal modification coefficient of modified face gear pair[J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2017, 45(6): 57-62.
[23] 付学中, 方宗德, 李建华, 等. 偏置面齿轮的碟形砂轮磨齿及啮合性能[J]. 华南理工大学学报(自然科学版), 2016, 44(7): 77-82. FU Xuezhong, FANG Zongde, LI Jianhua, et al. Grinding and meshing performance of offset face gear modified with disk wheel[J]. Journal of South China University of Technology (Natural Science Edition), 2016, 44(7): 77-82.
[24] ROSEN J B. The gradient projection method for nonlinear programming. Part I. Linear Constraints[J]. Journal of the Society for Industrial and Applied Mathematics, 1961, 4(4): 514-532.
[25] SMITH J D. Gear noise and vibration[M]. New York: Marcel Dekker, 2003.
[26] MAATAR M, VELEX P. An analytical expression for the time-varying contact length in perfect cylindrical gears: some possible applications in gear dynamics[J]. Journal of Mechanical Design, 1996, 118(4): 586-589.
[27] 王峰. 人字齿轮传动系统振动特性分析与试验研究[D]. 西安: 西北工业大学, 2014.
[28] 常山, 徐振忠, 霍肇波, 等. 斜齿圆柱齿轮瞬时啮合刚度及齿廓修形的研究[J]. 热能动力工程, 1997, 12(4): 270-274. CHANG Shan, XU Zhenzhong, HUO Zhaobo, et al. A study on the momentary meshing rigidity and tooth profile modification of helical cylindrical gears[J]. Journal of Engineering for Thermal Energy and Power, 1997, 12(4): 270-274.
[29] LITVIN F L, VECCHIATO D, YUKISHIMA K, et al. Reduction of noise of loaded and unloaded misaligned gear drives[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 5523-5536.
[30] 付学中, 方宗德, 李建华, 等. 面齿轮传动小轮拓扑修形设计及啮合性能分析[J]. 哈尔滨工程大学学报, 2016, 37(9): 1281-1286. FU Xuezhong, FANG Zongde, LI Jianhua, et al. Design of topological shape modification of auxiliary pinion of face gears and analysis of meshing performance[J]. Journal of Harbin Engineering University, 2016, 37(9): 1281-1286.
[31] DEB K, MEMBER A, PRATAP A, et al. A Fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ[J]. Transactions on Evolutionary Computation, 2002, 6(2): 182-197.