齿轮副啮合传动的动力学特性研究
|
摘要
齿轮系统作为各种工业设备中应用最为广泛的动力和运动传递装置,其工作时产生的振动是造成机器设备动态性能恶化的最主要因素。齿轮系统非线性振动理论是当前齿轮系统振动控制技术领域的热点,是目前振动噪声领域的前沿课题和发展方向。本文在研究罗茨风机齿轮系统结构基础上,对齿轮系统间隙非线性振动特性进行了研究,提出了相应的振动控制方法,进一步研究了系统的动力学特性。对于抑制结构振动噪声的传播,设计和评价低噪声风机齿轮传动系统结构提供了重要的理论依据,同时对于分析非线性齿轮系统的振动,合理设计齿轮参数具有十分广泛的理论意义和工程应用价值。
本文分别对齿轮系统非线性振动理论、判断周期解稳定性的Floquet理论以及振动特性评估方法进行了介绍,通过对罗茨风机结构的分析,建立了考虑时变啮合刚度和齿侧间隙的直齿轮副系统单自由度非线性动力学模型,推导出系统运动微分方程,进而分析了周期解的稳定性条件。在研究方法上,本文利用谐波平衡法及牛顿迭代法对系统的振动方程进行迭代求解,建立此类解的一般形式。在上述工作的基础上,研究系统参数对振动幅频特性影响规律,为合理设计齿轮参数以有效控制齿轮啮合过程的系统动力学行为和合理预测罗茨风机的机械噪声提供理论依据。
通过分析齿轮系统参数变化对振动响应中主谐波与二次谐波的影响,得出了以下重要结论:通过增加阻尼,减小外部激振力幅值,减小啮合刚度等方法均能有效减小主谐共振与二次谐波共振幅值;同时,幅频曲线对于ε的变化很敏感,随着ε增大,幅频曲线明显呈现不规律性,从而进入混沌状态;当减小齿形误差时,主谐共振幅值明显减小,而二次谐波共振幅值却呈增大趋势,但是由于主谐波占主要成分,二次谐波共振幅值跟主谐共振幅值比较而言要小的多,所以提高齿轮加工精度,减小制造误差也能有效控制主谐共振幅值,从而控制齿轮系统振动幅值。然而,随着齿形误差的减小,二次谐波振动幅值呈增大趋势,所以一味的改善齿形误差,当达到某一定值时,效果不再明显。
As the most widely used motivity and movement delivery device in various industrial equipments, gear system may produce vibration when it works, which is the key factor to make dynamic performances of the machinery equipment get worse. The nonlinear vibration theory of gear system is currently the hotspot of gear system vibration control area, as well as one of the most challenging and prospective work in the field of vibration and noise control. This dissertation, based on the investigation of the gear system structure of Roots blower, carried out a research work on backlash nonlinear vibration characteristics of gear system, putting forward relevant vibration control methods, then did a further study on the dynamic characteristics of system. Those provid an important theoretical basis on controlling the transmission of vibration and noise, designing and evaluating the structure of gear system for blowers of low noise.Meanwhile,there are far-ranging theoretical significance and engineering value on analyzing nonlinear vibration of gear system and designing rational gear parameters.
In this paper, nonlinear vibration of gear system, the Floquet theory for judging the stability of periodic solutions and the methods to evaluate vibration characteristics were introduced. Through the analysis of the structure of Roots blower, a nonlinear dynamics model of spur gears system as a single degree of freedom was derived, based on considering a time-varying mesh stiffness and backlash. Differential equation of the system was presented, then the stability conditions of periodic solutions was analysed. On the research methods, this dissertation uncoiled the vibration equation with Harmonic Balance Method and Newton-Raphson Method, then established a general form of this solution. Based on the work mentioned above, the influences that system parameters caused on the vibration amplitude and frequency characteristics were studied, which provided a theoretical basis on reasonably designing gear parameters to effectively control dynamics of gear system when meshing and reasonably forecasting mechanical noise of Roots blower.
By analyzing the influence that parameters' changement of gear system gives to primary harmonic and second harmonic in the vibration response, the following conclusions were drawn: Through increasing damping, reducing the external excitation amplitude and reducing meshing stiffness, the amplitude of primary harmonic resonance and second harmonic resonance could effectively be reduced; Meanwhile, amplitude and frequency curve was sensitive with the changement ofε.With increasingε,the curves showed no obvious law, which entered the chaotic state; When reducing profile errors, the amplitude of primary harmonic resonance significantly reduced, at the same time the amplitude of second harmonic resonance increased. Because primary harmonic was the major component, meanwhile the amplitude of second harmonic resonance was much less than primary harmonic's, increasing gear precision and reducing manufactural errors could effectively control the amplitude of primary harmonic resonance, and thus the vibration amplitude of gear system could be controUed.However, with the reduction of profile errors, the second harmonic vibration amplitude was increasing,so the effect wouldn't be evident any more, when profile errors reached a certain value.
本文分别对齿轮系统非线性振动理论、判断周期解稳定性的Floquet理论以及振动特性评估方法进行了介绍,通过对罗茨风机结构的分析,建立了考虑时变啮合刚度和齿侧间隙的直齿轮副系统单自由度非线性动力学模型,推导出系统运动微分方程,进而分析了周期解的稳定性条件。在研究方法上,本文利用谐波平衡法及牛顿迭代法对系统的振动方程进行迭代求解,建立此类解的一般形式。在上述工作的基础上,研究系统参数对振动幅频特性影响规律,为合理设计齿轮参数以有效控制齿轮啮合过程的系统动力学行为和合理预测罗茨风机的机械噪声提供理论依据。
通过分析齿轮系统参数变化对振动响应中主谐波与二次谐波的影响,得出了以下重要结论:通过增加阻尼,减小外部激振力幅值,减小啮合刚度等方法均能有效减小主谐共振与二次谐波共振幅值;同时,幅频曲线对于ε的变化很敏感,随着ε增大,幅频曲线明显呈现不规律性,从而进入混沌状态;当减小齿形误差时,主谐共振幅值明显减小,而二次谐波共振幅值却呈增大趋势,但是由于主谐波占主要成分,二次谐波共振幅值跟主谐共振幅值比较而言要小的多,所以提高齿轮加工精度,减小制造误差也能有效控制主谐共振幅值,从而控制齿轮系统振动幅值。然而,随着齿形误差的减小,二次谐波振动幅值呈增大趋势,所以一味的改善齿形误差,当达到某一定值时,效果不再明显。
As the most widely used motivity and movement delivery device in various industrial equipments, gear system may produce vibration when it works, which is the key factor to make dynamic performances of the machinery equipment get worse. The nonlinear vibration theory of gear system is currently the hotspot of gear system vibration control area, as well as one of the most challenging and prospective work in the field of vibration and noise control. This dissertation, based on the investigation of the gear system structure of Roots blower, carried out a research work on backlash nonlinear vibration characteristics of gear system, putting forward relevant vibration control methods, then did a further study on the dynamic characteristics of system. Those provid an important theoretical basis on controlling the transmission of vibration and noise, designing and evaluating the structure of gear system for blowers of low noise.Meanwhile,there are far-ranging theoretical significance and engineering value on analyzing nonlinear vibration of gear system and designing rational gear parameters.
In this paper, nonlinear vibration of gear system, the Floquet theory for judging the stability of periodic solutions and the methods to evaluate vibration characteristics were introduced. Through the analysis of the structure of Roots blower, a nonlinear dynamics model of spur gears system as a single degree of freedom was derived, based on considering a time-varying mesh stiffness and backlash. Differential equation of the system was presented, then the stability conditions of periodic solutions was analysed. On the research methods, this dissertation uncoiled the vibration equation with Harmonic Balance Method and Newton-Raphson Method, then established a general form of this solution. Based on the work mentioned above, the influences that system parameters caused on the vibration amplitude and frequency characteristics were studied, which provided a theoretical basis on reasonably designing gear parameters to effectively control dynamics of gear system when meshing and reasonably forecasting mechanical noise of Roots blower.
By analyzing the influence that parameters' changement of gear system gives to primary harmonic and second harmonic in the vibration response, the following conclusions were drawn: Through increasing damping, reducing the external excitation amplitude and reducing meshing stiffness, the amplitude of primary harmonic resonance and second harmonic resonance could effectively be reduced; Meanwhile, amplitude and frequency curve was sensitive with the changement ofε.With increasingε,the curves showed no obvious law, which entered the chaotic state; When reducing profile errors, the amplitude of primary harmonic resonance significantly reduced, at the same time the amplitude of second harmonic resonance increased. Because primary harmonic was the major component, meanwhile the amplitude of second harmonic resonance was much less than primary harmonic's, increasing gear precision and reducing manufactural errors could effectively control the amplitude of primary harmonic resonance, and thus the vibration amplitude of gear system could be controUed.However, with the reduction of profile errors, the second harmonic vibration amplitude was increasing,so the effect wouldn't be evident any more, when profile errors reached a certain value.
引文
[1]王玉松,谭达明.“主动隔振控制系统的研究”,微机发展,1996,2
[2]李禄胜.噪声污染的危害及控制[J],教学参考,1995,4:38-39
[3]王孚懋.罗茨风机噪声源识别与控制研究[J],现代制造技术与装备,2006,41(1)
[4]周亨达.工程流体力学,冶金工业出版社,1988
[5]蔡业彬,陈再良,方子严.罗茨风机噪声产生的机理及降噪措施[J],化工装备技术,2000,21(3):40-44
[6]熊安然,熊滨生,杨国良,冯力等.大流量三叶罗茨风机的流量特性及噪声分析,中原工学院学报,2006,Vol.17,No.2:35-38,45
[7]邱炳正.引风机振动原因及其处理,江西煤炭科技,2001,1
[8]彭学院,何志龙,张怀亮等.罗茨风机渐开线型转子型线的改进设计[J],风机技术,2003,3:3-5
[9]叶仲和,蔡海毅.扭叶转子罗茨鼓风机的设计、仿真与制造[D],福州大学,2002
[10]彭玲生编.工厂噪声控制技术与应用,北京:中国铁道出版社,1987:183-189
[11]闻邦椿,李以农,韩清凯编著.非线性振动理论中的解析方法及工程应用,东北大学出版社,2001.10
[12]刘延柱,陈立群.“非线性振动”,高等教育出版社,2001
[13]王海期.非线性振动,北京:高等教育出版社,1992
[14]陈予恕.非线性振动,高等教育出版社,2002
[15]韩幼祥.齿轮传动系统动力学的研究方向,煤矿机械,2005,7:61-63
[16]陈予恕,唐云,陆启韶,郑兆昌等.非线性动力学中的现代分析方法,科学出版社,1992
[17]刘继岩,孙涛,戚厚军.RV减速器的动力学模型与固有频率研究,中国机械工程,1999,10(5):381-384
[18]刘更等.斜齿圆柱齿轮轮齿动态响应的研究(一),齿轮,1988,12(6):8-10
[19]邵长健,孙涛,沈允文.基于结构动力学修改重分析的齿轮系统减振研究, 航空动力学报,1998,13(3):305-309
[20]K.Ichimarru and F.Hirano,Dynamic Behavior of Heavilly-loaded Spur Gears,Journal of Engineering for Industry,1974,96:373-381
[21]R.Kasuba and J.W Ewans.An Extended Model for Determining DynamicLoads in Spur Gearing,Journal of Mechanical Design,1981,103:398-409
[22]R.W.Comell and W.W.Westervelt.Dynamic Tooth Loads and Stressing for High Contact Ratio Spur Gears,Journal of Mechanical Design,1978,100:69-76
[23]G.W.Blankenship and A.Kahraman,Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Non-Linearity,Journal of Sound and Vibration,1995,185(5):743-765
[24]A.Raghothama and S.Narayanan,Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method,Journal of Sound and Vibration,1999,226(3):469-492
[25]Y.B.Kim and S.T.Noah,Stability and Bifurcation Analysis if Oscillators with Piecewise-Linear Characteristic:A General Approach,Journal of Applied Mechanics,1991,58:545-553
[26]李润方,王建军.齿轮系统动力学—振动、冲击、噪声,科学出版社,1997
[27]王建军,李润方.齿轮系统间隙非线性振动研究综述,非线性动力学学报,1995,2(3)
[28]孙智民.功率分流齿轮传动系统非线性动力学研究,西北工业大学[博士论文],2001
[29]Bapat C N,Popplewell N,Mclachlan K,Periodic motions of an impacr pairs,J Sound Vib,1983,87:41-59
[30]Whiston G S,Global dynamics of a vibro-impacting linear oscillator,J Sound and Vib,1987,118(3):395-429
[31]Smith C E,Liu P P,Coefficients of resultation,ASME Appl.Mech,1992,59:663-669
[32]Dubowsky S,Freudenstein F,Dynamic analysis of mechanical systems with clearances,part 1,formulation of dynamic model,ASME Eng Ind,1971,93:310-316
[33]Tomlinson G R,Lam J,Frequency response characteristics of structure with single and multiple clearance type nonlinearities,J Sound Vib,1984,96:111-125
[34]Hunt K H,Grossley F R E,Coefficient of restitution interpreted as damping in vibro-impact,A SME Eng Ind,1975,42:440-445
[35]周纪卿,朱因远.非线性振动,西安交通大学出版社,1998,9
[36]王建军,李其汉,李润方等.齿轮系统非线性振动研究进展,力学进展,2005,35(1):37-51
[37]李华.逆算符法及其在机械非线性动力分析中的应用,西安电子科技大学[博士论文],1999年6月
[38]Ozguven H N,Houser D R,Dynamic analysis of high speed gears by using loaded ststic transmission error,J Sound Vib,1988,125:71-83
[39]Kahraman A,Singh R,Non-linear dynamics of a spur gear pair,J Sound Vib,1990,142(1):49-75
[40]马云海,文成秀等.分段线性-非线性振动机械周期运动关于激励频率的分叉,上海理工大学学报,Vol.22,No.4,2000
[41]Christopher K.Halse,R.Eddie Wilson,Mario di Bernardo,Martin E.Homer,Coexisting solutions and bifurcations in mechanical oscillators with backlash,Journal of Sound and Vibration,305(2007):854-885
[42]王立华,李润方,林腾蛟等.齿轮系统时变刚度和间隙非线性振动特性研究,中国机械工程,2003,14(13):1143-1146
[43]郜志英,沈允文,李素有等.间隙非线性齿轮系统周期解结构及其稳定性研究,机械工程学报,2004,40(5):17-22
[44]杨绍普,申永军,刘献栋等.基于增量谐波平衡法的齿轮系统非线性动力学,振动与冲击,2005,24(3):40-42
[45]挪威委里塔斯船舶集团编著,马运义等编译.船舶振动控制,大连海事大学出版社,1992.2:201-203
[46]凌复华.非线性振动系统周期运动及其稳定性的数值研究,力学进展,西北工业大学硕士学位论文参考文献,Vol.6,No.1,198
[47]诸德超,邢誉峰主编.工程振动基础,北京:北京航空航天大学出版社,2004.9:242-244
[48]黄安基.非线性振动,成都:西南交通大学出版社,1993
[49]闻邦榕等.非线性振动理论中的解析方法及工程应用,沈阳:东北大学出版 社,2001.10:170-171
[50]刘延柱,陈立群.非线性振动.高等教育出版社,2001
[51]李骊等.强非线性振动系统的定性理论与定量方法,科学出版社,1997
[52]朱因远,周纪卿.非线性振动和运动稳定性,西安交通大学出版,1992
[53]薛定宇.控制系统计算机辅助设计:MATLAB语言及应用,清华大学出版社,1996.7
[54]精锐创作组编著.MATLAB 6.0科学运算完整解决方案,人民邮电出版社,2001.7
[2]李禄胜.噪声污染的危害及控制[J],教学参考,1995,4:38-39
[3]王孚懋.罗茨风机噪声源识别与控制研究[J],现代制造技术与装备,2006,41(1)
[4]周亨达.工程流体力学,冶金工业出版社,1988
[5]蔡业彬,陈再良,方子严.罗茨风机噪声产生的机理及降噪措施[J],化工装备技术,2000,21(3):40-44
[6]熊安然,熊滨生,杨国良,冯力等.大流量三叶罗茨风机的流量特性及噪声分析,中原工学院学报,2006,Vol.17,No.2:35-38,45
[7]邱炳正.引风机振动原因及其处理,江西煤炭科技,2001,1
[8]彭学院,何志龙,张怀亮等.罗茨风机渐开线型转子型线的改进设计[J],风机技术,2003,3:3-5
[9]叶仲和,蔡海毅.扭叶转子罗茨鼓风机的设计、仿真与制造[D],福州大学,2002
[10]彭玲生编.工厂噪声控制技术与应用,北京:中国铁道出版社,1987:183-189
[11]闻邦椿,李以农,韩清凯编著.非线性振动理论中的解析方法及工程应用,东北大学出版社,2001.10
[12]刘延柱,陈立群.“非线性振动”,高等教育出版社,2001
[13]王海期.非线性振动,北京:高等教育出版社,1992
[14]陈予恕.非线性振动,高等教育出版社,2002
[15]韩幼祥.齿轮传动系统动力学的研究方向,煤矿机械,2005,7:61-63
[16]陈予恕,唐云,陆启韶,郑兆昌等.非线性动力学中的现代分析方法,科学出版社,1992
[17]刘继岩,孙涛,戚厚军.RV减速器的动力学模型与固有频率研究,中国机械工程,1999,10(5):381-384
[18]刘更等.斜齿圆柱齿轮轮齿动态响应的研究(一),齿轮,1988,12(6):8-10
[19]邵长健,孙涛,沈允文.基于结构动力学修改重分析的齿轮系统减振研究, 航空动力学报,1998,13(3):305-309
[20]K.Ichimarru and F.Hirano,Dynamic Behavior of Heavilly-loaded Spur Gears,Journal of Engineering for Industry,1974,96:373-381
[21]R.Kasuba and J.W Ewans.An Extended Model for Determining DynamicLoads in Spur Gearing,Journal of Mechanical Design,1981,103:398-409
[22]R.W.Comell and W.W.Westervelt.Dynamic Tooth Loads and Stressing for High Contact Ratio Spur Gears,Journal of Mechanical Design,1978,100:69-76
[23]G.W.Blankenship and A.Kahraman,Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Non-Linearity,Journal of Sound and Vibration,1995,185(5):743-765
[24]A.Raghothama and S.Narayanan,Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method,Journal of Sound and Vibration,1999,226(3):469-492
[25]Y.B.Kim and S.T.Noah,Stability and Bifurcation Analysis if Oscillators with Piecewise-Linear Characteristic:A General Approach,Journal of Applied Mechanics,1991,58:545-553
[26]李润方,王建军.齿轮系统动力学—振动、冲击、噪声,科学出版社,1997
[27]王建军,李润方.齿轮系统间隙非线性振动研究综述,非线性动力学学报,1995,2(3)
[28]孙智民.功率分流齿轮传动系统非线性动力学研究,西北工业大学[博士论文],2001
[29]Bapat C N,Popplewell N,Mclachlan K,Periodic motions of an impacr pairs,J Sound Vib,1983,87:41-59
[30]Whiston G S,Global dynamics of a vibro-impacting linear oscillator,J Sound and Vib,1987,118(3):395-429
[31]Smith C E,Liu P P,Coefficients of resultation,ASME Appl.Mech,1992,59:663-669
[32]Dubowsky S,Freudenstein F,Dynamic analysis of mechanical systems with clearances,part 1,formulation of dynamic model,ASME Eng Ind,1971,93:310-316
[33]Tomlinson G R,Lam J,Frequency response characteristics of structure with single and multiple clearance type nonlinearities,J Sound Vib,1984,96:111-125
[34]Hunt K H,Grossley F R E,Coefficient of restitution interpreted as damping in vibro-impact,A SME Eng Ind,1975,42:440-445
[35]周纪卿,朱因远.非线性振动,西安交通大学出版社,1998,9
[36]王建军,李其汉,李润方等.齿轮系统非线性振动研究进展,力学进展,2005,35(1):37-51
[37]李华.逆算符法及其在机械非线性动力分析中的应用,西安电子科技大学[博士论文],1999年6月
[38]Ozguven H N,Houser D R,Dynamic analysis of high speed gears by using loaded ststic transmission error,J Sound Vib,1988,125:71-83
[39]Kahraman A,Singh R,Non-linear dynamics of a spur gear pair,J Sound Vib,1990,142(1):49-75
[40]马云海,文成秀等.分段线性-非线性振动机械周期运动关于激励频率的分叉,上海理工大学学报,Vol.22,No.4,2000
[41]Christopher K.Halse,R.Eddie Wilson,Mario di Bernardo,Martin E.Homer,Coexisting solutions and bifurcations in mechanical oscillators with backlash,Journal of Sound and Vibration,305(2007):854-885
[42]王立华,李润方,林腾蛟等.齿轮系统时变刚度和间隙非线性振动特性研究,中国机械工程,2003,14(13):1143-1146
[43]郜志英,沈允文,李素有等.间隙非线性齿轮系统周期解结构及其稳定性研究,机械工程学报,2004,40(5):17-22
[44]杨绍普,申永军,刘献栋等.基于增量谐波平衡法的齿轮系统非线性动力学,振动与冲击,2005,24(3):40-42
[45]挪威委里塔斯船舶集团编著,马运义等编译.船舶振动控制,大连海事大学出版社,1992.2:201-203
[46]凌复华.非线性振动系统周期运动及其稳定性的数值研究,力学进展,西北工业大学硕士学位论文参考文献,Vol.6,No.1,198
[47]诸德超,邢誉峰主编.工程振动基础,北京:北京航空航天大学出版社,2004.9:242-244
[48]黄安基.非线性振动,成都:西南交通大学出版社,1993
[49]闻邦榕等.非线性振动理论中的解析方法及工程应用,沈阳:东北大学出版 社,2001.10:170-171
[50]刘延柱,陈立群.非线性振动.高等教育出版社,2001
[51]李骊等.强非线性振动系统的定性理论与定量方法,科学出版社,1997
[52]朱因远,周纪卿.非线性振动和运动稳定性,西安交通大学出版,1992
[53]薛定宇.控制系统计算机辅助设计:MATLAB语言及应用,清华大学出版社,1996.7
[54]精锐创作组编著.MATLAB 6.0科学运算完整解决方案,人民邮电出版社,2001.7