圆柱齿轮传动齿面形状与非线性特性关联规律分析及实验研究
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摘要
齿轮传动系统在航空、航天、交通、机械和仪表制造等各个工业部门获得广泛的应用。齿轮系统是由齿轮副、传动轴、轴承、箱体等组成的弹性机械系统。齿轮啮合时轮齿的弹性变形、啮入啮出冲击、齿轮间隙、制造误差、修形、装配误差等都对轮齿静、动力接触特性、系统动态性能、系统传动精度等有很大影响。为了满足各类机械装备性能不断提高的要求,必须考虑这些非线性因素,对齿轮系统的非线性动力学问题进行研究,因此从设计角度,将设计-静态分析-动态分析-实验分析进行系统地研究是齿轮传动的一个重要基础性研究课题。
论文根据齿轮修形加工原理,推导了标准直齿轮及修形直齿轮的参数化坐标方程,基于TCA理论,给出了考虑三种安装误差时,修形直齿轮的接触轨迹,为齿轮传动几何参数的分析、动态性能分析及实验研究提供分析方法。
基于Labview和LMS测量系统,搭建了齿轮动(静)态传递误差和振动噪音测量试验平台,给出了基于光栅盘的齿轮动(静)态传递误差测量、计算方法,并测量得到不同安装误差时的静态传递误差及轮齿间隙,分析了安装误差、转速、载荷等对动态传递误差及齿轮系统的冲击振动的影响。
针对齿轮冲击的能量守恒问题,建立了一种新的考虑时变啮合刚度、摩擦齿轮冲击动力学模型。该模型刚度、摩擦均根据啮合点逐点求解,然后积分动力学方程,得到系统的动态响应。应用数值仿真的方法研究了不同载荷、摩擦对系统的瞬态冲击行为规律。研究表明：(1)轮齿存在脱啮冲击时,传递误差主要是以冲击频率为主,但冲击频率与啮合频率、轴频有差异；(2)载荷的波动对低速时系统的动态传递误差、转速波动等的影响有限,高速时齿频载荷波动使传递误差的啮合频率部分急剧增大,冲击频率基本保持不变；(3)轴频载荷波动对系统的影响更加显著,会引起系统出现脱啮-冲击,甚至双边冲击,系统响应以低频冲击振动为主。(4)低速时摩擦对系统的影响非常明显,摩擦可以遏制轮齿脱啮,但是其对低频振动的影响不可忽视。高速时摩擦对高频分量的影响比较小,但是对低频分量影响比较大。以上现象,在经典动力学模型分析中不可能得到,所以在研究轮齿的拍击特征时,有必要从冲击动力学的角度去分析轮齿的冲击行为,该研究对理解齿轮的冲击特征、齿轮结构优化、可靠性分析提供了一个新的思路。
论文研究了冲击阻尼参数对系统非线性动力学响应的影响,研究表明：在重载时,参数n不会改变间隙引起的跳跃现象,但是会影响跳跃的长度,同时加剧系统的振动。在轻载时,参数n会使间隙引起的跳跃特征消失,从而使系统具有线性特征。冲击阻尼可以遏制响应的次谐分量的幅值,但是对主要啮合频率及其倍频分量的影响比较小。啮合频率部分的幅值受阻尼的影响比较小,但是次谐分量随着冲击阻尼的增加而明显减小。轮齿直接冲击的速度也不尽然相同,对应的恢复系数也有很大的差别,所以低速的轮齿拍击、启停比较频繁时的轮齿冲击分析必须要考虑恢复系数对系统响应的影响。在低速时,动载荷主要与传递误差的高频分量(啮合频率及其倍频分量)有关,提高齿面的精度、改善齿廓偏差及粗糙度可以有效的遏制传递误差的高频分量幅值从而降低系统的动载荷。在高速时,动载荷主要受传递误差轴频分量的影响。虽然齿轮可能处于啮合频率的共振区域,但是动态传递误差仍然以传递误差的轴频分量为主。
从工程角度来说,单纯的从提高齿轮材料性能远不及对轮齿进行修形得以改善啮合特征的效果好,虽然目前有学者建议齿轮冲击采用n=1.5的形式,但是精确的齿轮冲击实验验证还没有见到报道,这也是基于参数n的动态修形设计的一个基础条件,必须在实验的基础上准确确定参数n。该研究表明我们除了关心系统的共振区域,避免齿轮传动系统的共振外,还应该考虑在低转速时冲击引起过高的冲击动载荷。
针对本研究提出的含动态域的非光滑动力学系统,本文首先给出了基于多重打靶方法周期解求解方法,并结合Floquet理论,给出了该类系统余维一和余维二分岔的数值计算求解格式,以经典的倍周期分岔、鞍结分岔为基础,分析了该类系统的切分岔特征,及转速参数ω、间隙参数b0和载荷波动系数f1对系统余维一和余维二分岔的影响。
在齿轮传动系统的振动噪音研究方面,一般人们把轴频的调制,归结为系统安装误差、轴承误差等因素引起的。动力学分析往往忽略静态传递误差的轴频部分,从而得不到含轴频调制的响应。根据本文提出的建立动力学系统的方法,在动力学方程中,将静态传递误差仍然表示为位移激励,从而避免了位移激励在二次求导时,轴频分量的相对值变小,以致从数学意义上被忽略。本文得到的动态响应物理意义明确,其与实际的齿轮系统的振动信号分析具有可比性。
Gear transmission systems are broadly adopted in the engineering field if aircraft and space, transportation, mechanical engineering and instruments as well. The gear transmission systems are elastic mechanical system included gear pair, shaft, bearing and box. The static and/or dynamics contact characteristic, transmission behaviors and dynamics transmission error are affected by gear mesh elastic deformation, tooth impact, backlash, manufacturing error, tooth modification and assembling error in the mesh process. Therefore, nonlinear dynamics analysis of gear transmission system with these nonlinear factors is an imperative request to improve performance of the mechanical system. From the perspectives of dynamic, it's an important research direction to systematically analyze the gear design, static analysis, dynamics analysis and experimental investigation
Firstly, parametric equations of standard and modified spur gear are deduced based on gear mesh theory. The contact patterns of modified spur gears are studied with the effect of three kinds of assembling error based on TCA.
A test rig for measuring the static and/or dynamics transmission error, vibration and noise is built based on Ni.Labview and LMS.test systems. With the help of high precision optical encoder, effects of gear misalignment on unloaded and lightly loaded dynamic transmission errors, which are relative to gear rattle, are investigated. The gear mesh misalignment is introduced by eccentric sleeve assembled on the output shaft. Effects of modification and misalignment on the dynamic transmission error, are studied at different load and driving velocity conditions.
Based on nonlinear Hertz impact damping model, a new impact model of gear transmission system combining with time varying stiffness and friction is developed. The time varying stiffness and friction force functions are obtained at each contact point, which are functions of rotation angles of gear and pinion. The numerical program is constructed to analyze the transient impact dynamics behaviors with different load and friction conditions. The numerical results shown that,(1) the main parts of dynamics transmission error is impact frequency component, which is determined by initial impact velocity.(2) Fluctuation of load is less effect to the dynamics transmission error and output velocity with lower input velocity. But in higher input velocity condition, the fluctuation of load made the meshing frequency component of dynamics transmission error increasing rapidly. And the impact frequency change slightly. However,(3) when the fluctuation of load is formularized with shaft frequency, the system occurs loss contact motions moreover double side impact motion.(4) The friction force can reduce the loss contact impact motion at low input shaft condition. The low frequency vibrations are affected by friction obviously but it has less effect on the high frequency vibration. The main results gotten in this paper is appeared in previous research. The transient impact dynamics behavior analysis is essential step to disclose the gear rattling.
Effects of impact damping on impact dynamic characteristics of crowned gear transmission system are investigated numerically. The results show that jump induced by backlash are not changed with parameter n, but the length of jump region changes obviously and the system vibration is aggravating at heavy load. Under lightly loaded condition, the jump deduced by backlash disappears with the effect of parameter n. The flank modification applied to gear pairs can vary the parameter n then the relation between modification and vibration behavior can be analyzed directly, which may be a new stratagem for dynamic characteristic design. Moreover, impact damping can suppress the amplitude of sub-harmonic component and slightly affect the main and second meshing frequency component. And the effect of coefficient of restitution, depended on initial impact velocity, should be involved into the analyses of gear impact when the system undergoes gear rattling or intermittent motion. And the process of collision process described by the constant coefficient of constitution method is inappropriate.
The static transmission error (TE) can be combined with backlash parameter with comparative analysis of experimental dynamic transmission error directly. At low speed condition, the dynamic load is mainly connected with high frequency component of transmission error. The high frequency amplitude of transmission error can be suppressed with the increase of tooth surface accuracy, improvement of tooth profile error and roughness effectively. At high speed condition, the shaft frequency component of transmission error has a dominate influence on dynamic load, though the gear may be in the resonance area. Parameter n is decreasing and dynamic load coefficient is increasing especially in resonance area. When n is1.0, loss contact impact area exists in the system, and the amplitudes of shaft frequency component in dynamic transmission error are almost equal, which indicates that dynamic characteristics are little affected by load and parameter n when there is no loss contact impact condition. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. When n is1.5, loss contact impact do not exist in system, and the ratios of prior three harmonic components are approximate to the linear increase of load.
The dynamics model considered in this paper are reduced to a new non-smooth dynamics model with varying domain. A multiple shooting method and Floquet theory are developed to get the stable periodic solution. The numerical strategy is performed to analyze the co-dimension one and co-dimension two bifurcations. And the effects of frequency parameter ω and backlash b0and fluctuation load f1on co-dimension two bifurcations are investigated.
论文根据齿轮修形加工原理,推导了标准直齿轮及修形直齿轮的参数化坐标方程,基于TCA理论,给出了考虑三种安装误差时,修形直齿轮的接触轨迹,为齿轮传动几何参数的分析、动态性能分析及实验研究提供分析方法。
基于Labview和LMS测量系统,搭建了齿轮动(静)态传递误差和振动噪音测量试验平台,给出了基于光栅盘的齿轮动(静)态传递误差测量、计算方法,并测量得到不同安装误差时的静态传递误差及轮齿间隙,分析了安装误差、转速、载荷等对动态传递误差及齿轮系统的冲击振动的影响。
针对齿轮冲击的能量守恒问题,建立了一种新的考虑时变啮合刚度、摩擦齿轮冲击动力学模型。该模型刚度、摩擦均根据啮合点逐点求解,然后积分动力学方程,得到系统的动态响应。应用数值仿真的方法研究了不同载荷、摩擦对系统的瞬态冲击行为规律。研究表明：(1)轮齿存在脱啮冲击时,传递误差主要是以冲击频率为主,但冲击频率与啮合频率、轴频有差异；(2)载荷的波动对低速时系统的动态传递误差、转速波动等的影响有限,高速时齿频载荷波动使传递误差的啮合频率部分急剧增大,冲击频率基本保持不变；(3)轴频载荷波动对系统的影响更加显著,会引起系统出现脱啮-冲击,甚至双边冲击,系统响应以低频冲击振动为主。(4)低速时摩擦对系统的影响非常明显,摩擦可以遏制轮齿脱啮,但是其对低频振动的影响不可忽视。高速时摩擦对高频分量的影响比较小,但是对低频分量影响比较大。以上现象,在经典动力学模型分析中不可能得到,所以在研究轮齿的拍击特征时,有必要从冲击动力学的角度去分析轮齿的冲击行为,该研究对理解齿轮的冲击特征、齿轮结构优化、可靠性分析提供了一个新的思路。
论文研究了冲击阻尼参数对系统非线性动力学响应的影响,研究表明：在重载时,参数n不会改变间隙引起的跳跃现象,但是会影响跳跃的长度,同时加剧系统的振动。在轻载时,参数n会使间隙引起的跳跃特征消失,从而使系统具有线性特征。冲击阻尼可以遏制响应的次谐分量的幅值,但是对主要啮合频率及其倍频分量的影响比较小。啮合频率部分的幅值受阻尼的影响比较小,但是次谐分量随着冲击阻尼的增加而明显减小。轮齿直接冲击的速度也不尽然相同,对应的恢复系数也有很大的差别,所以低速的轮齿拍击、启停比较频繁时的轮齿冲击分析必须要考虑恢复系数对系统响应的影响。在低速时,动载荷主要与传递误差的高频分量(啮合频率及其倍频分量)有关,提高齿面的精度、改善齿廓偏差及粗糙度可以有效的遏制传递误差的高频分量幅值从而降低系统的动载荷。在高速时,动载荷主要受传递误差轴频分量的影响。虽然齿轮可能处于啮合频率的共振区域,但是动态传递误差仍然以传递误差的轴频分量为主。
从工程角度来说,单纯的从提高齿轮材料性能远不及对轮齿进行修形得以改善啮合特征的效果好,虽然目前有学者建议齿轮冲击采用n=1.5的形式,但是精确的齿轮冲击实验验证还没有见到报道,这也是基于参数n的动态修形设计的一个基础条件,必须在实验的基础上准确确定参数n。该研究表明我们除了关心系统的共振区域,避免齿轮传动系统的共振外,还应该考虑在低转速时冲击引起过高的冲击动载荷。
针对本研究提出的含动态域的非光滑动力学系统,本文首先给出了基于多重打靶方法周期解求解方法,并结合Floquet理论,给出了该类系统余维一和余维二分岔的数值计算求解格式,以经典的倍周期分岔、鞍结分岔为基础,分析了该类系统的切分岔特征,及转速参数ω、间隙参数b0和载荷波动系数f1对系统余维一和余维二分岔的影响。
在齿轮传动系统的振动噪音研究方面,一般人们把轴频的调制,归结为系统安装误差、轴承误差等因素引起的。动力学分析往往忽略静态传递误差的轴频部分,从而得不到含轴频调制的响应。根据本文提出的建立动力学系统的方法,在动力学方程中,将静态传递误差仍然表示为位移激励,从而避免了位移激励在二次求导时,轴频分量的相对值变小,以致从数学意义上被忽略。本文得到的动态响应物理意义明确,其与实际的齿轮系统的振动信号分析具有可比性。
Gear transmission systems are broadly adopted in the engineering field if aircraft and space, transportation, mechanical engineering and instruments as well. The gear transmission systems are elastic mechanical system included gear pair, shaft, bearing and box. The static and/or dynamics contact characteristic, transmission behaviors and dynamics transmission error are affected by gear mesh elastic deformation, tooth impact, backlash, manufacturing error, tooth modification and assembling error in the mesh process. Therefore, nonlinear dynamics analysis of gear transmission system with these nonlinear factors is an imperative request to improve performance of the mechanical system. From the perspectives of dynamic, it's an important research direction to systematically analyze the gear design, static analysis, dynamics analysis and experimental investigation
Firstly, parametric equations of standard and modified spur gear are deduced based on gear mesh theory. The contact patterns of modified spur gears are studied with the effect of three kinds of assembling error based on TCA.
A test rig for measuring the static and/or dynamics transmission error, vibration and noise is built based on Ni.Labview and LMS.test systems. With the help of high precision optical encoder, effects of gear misalignment on unloaded and lightly loaded dynamic transmission errors, which are relative to gear rattle, are investigated. The gear mesh misalignment is introduced by eccentric sleeve assembled on the output shaft. Effects of modification and misalignment on the dynamic transmission error, are studied at different load and driving velocity conditions.
Based on nonlinear Hertz impact damping model, a new impact model of gear transmission system combining with time varying stiffness and friction is developed. The time varying stiffness and friction force functions are obtained at each contact point, which are functions of rotation angles of gear and pinion. The numerical program is constructed to analyze the transient impact dynamics behaviors with different load and friction conditions. The numerical results shown that,(1) the main parts of dynamics transmission error is impact frequency component, which is determined by initial impact velocity.(2) Fluctuation of load is less effect to the dynamics transmission error and output velocity with lower input velocity. But in higher input velocity condition, the fluctuation of load made the meshing frequency component of dynamics transmission error increasing rapidly. And the impact frequency change slightly. However,(3) when the fluctuation of load is formularized with shaft frequency, the system occurs loss contact motions moreover double side impact motion.(4) The friction force can reduce the loss contact impact motion at low input shaft condition. The low frequency vibrations are affected by friction obviously but it has less effect on the high frequency vibration. The main results gotten in this paper is appeared in previous research. The transient impact dynamics behavior analysis is essential step to disclose the gear rattling.
Effects of impact damping on impact dynamic characteristics of crowned gear transmission system are investigated numerically. The results show that jump induced by backlash are not changed with parameter n, but the length of jump region changes obviously and the system vibration is aggravating at heavy load. Under lightly loaded condition, the jump deduced by backlash disappears with the effect of parameter n. The flank modification applied to gear pairs can vary the parameter n then the relation between modification and vibration behavior can be analyzed directly, which may be a new stratagem for dynamic characteristic design. Moreover, impact damping can suppress the amplitude of sub-harmonic component and slightly affect the main and second meshing frequency component. And the effect of coefficient of restitution, depended on initial impact velocity, should be involved into the analyses of gear impact when the system undergoes gear rattling or intermittent motion. And the process of collision process described by the constant coefficient of constitution method is inappropriate.
The static transmission error (TE) can be combined with backlash parameter with comparative analysis of experimental dynamic transmission error directly. At low speed condition, the dynamic load is mainly connected with high frequency component of transmission error. The high frequency amplitude of transmission error can be suppressed with the increase of tooth surface accuracy, improvement of tooth profile error and roughness effectively. At high speed condition, the shaft frequency component of transmission error has a dominate influence on dynamic load, though the gear may be in the resonance area. Parameter n is decreasing and dynamic load coefficient is increasing especially in resonance area. When n is1.0, loss contact impact area exists in the system, and the amplitudes of shaft frequency component in dynamic transmission error are almost equal, which indicates that dynamic characteristics are little affected by load and parameter n when there is no loss contact impact condition. It proves that the nonlinear impact model is sufficient to study the rattling process of gear pairs. When n is1.5, loss contact impact do not exist in system, and the ratios of prior three harmonic components are approximate to the linear increase of load.
The dynamics model considered in this paper are reduced to a new non-smooth dynamics model with varying domain. A multiple shooting method and Floquet theory are developed to get the stable periodic solution. The numerical strategy is performed to analyze the co-dimension one and co-dimension two bifurcations. And the effects of frequency parameter ω and backlash b0and fluctuation load f1on co-dimension two bifurcations are investigated.
引文
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