工程科学与技术   2017, Vol. 49 Issue (2): 209-216

1. 四川大学 制造科学与工程学院, 四川 成都 610065;
2. 四川建安工业有限责任公司, 四川 雅安 625000

Analysis of Nonlinear Vibration on a Cylindrical Gear with Variational Hyperbola and Circular-arc-tooth-trace
CHEN Zhongmin1,2, HOU Li1, DUAN Yang1, ZHAO Fei1, PENG Wenhua2, LUO Lan1
1. School of Manufacturing Sci. and Eng., Sichuan Univ., Chengdu 610065, China;
2. Sichuan Jian'an Industrial Co., LTD., Yaan 625000, China
Abstract: In order to get the vibration rule of a cylindrical gear with variational hyperbola and circular-arc-tooth-trace (CHATT) at work and design a stable and efficient gear, it's necessary to analyze its nonlinear vibration characteristics.The time-varying stiffness of meshing curve and axial error excitation were calculated by load tooth contact analysis (LTCA), and meshing impact excitation was received according to meshing impact model.The twelve-degree of bending-torsion-shaft multi-factor coupling dynamical model of CHATT was established based on the theory of concentrated parameter, and vibration differential equations were built on the basis of Newton's second law.Then, equations were solved by adopting fourth-order Runge-Kutta algorithm with variable step lengths.Vibration displacement, vibration velocity, vibration acceleration, dynamic load of bearing, vibration displacement & velocity phase and Poincaré section of driving gear and driven gear were calculated, and response amplitude spectrums of above signals were obtained with FFT method.The comparison of numerical solutions of driving gear and driven gear in verticality, torsion, and axial direction showed that the vibration law of driving gear always keeps consistent with driven gear, quasi-periodic motion occurs in verticality and torsion direction, and vibration in axial direction shows a nearly chaotic state of steady state response.Moreover, the influence rule on system vibration characteristics was researched by changing tooth-trace radius, loading torque and inputing speed, and results of vibration displacement & velocity phase indicated that vibration regularity of driving gear and driven gear in axial direction gradually becomes poor, which evolutes from quasi-periodic motion to nearly chaotic motion.The regularity of axial direction vibration is more susceptible by changing these three parameters.The establishment and solution of CHATT vibration model and parameter influencing analysis provide theoretical foundation for the later dynamic design, prediction of vibration response under different parameters, and noise reduction.
Key words: variational hyperbola and circular-arc-tooth-trace    load tooth contact analysis (LTCA)    nonlinear vibration    dynamic design

1 变双曲圆弧齿线圆柱齿轮系统内部激励

 图1 内部激励计算流程 Fig. 1 Calculation flow chart of internal excitations

1.1 计算时变刚度激励

 图2 啮合综合刚度曲线 Fig. 2 Synthetical meshing stiffness curve

 图3 单对齿啮合刚度曲线 Fig. 3 Meshing stiffness curve of single-pair teeth

1.2 计算误差激励

 图4 轴向位移曲线 Fig. 4 Axial displacement

1.3 计算冲击激励

 ${F_{\rm{s}}} = \Delta v\sqrt {\frac{{b{J_1}{J_2}}}{{({J_1}r_{_{b2}}^{^2} + {J_2}r_{_{b1}}^{^2}){q_{\rm{s}}}}}}$ (1)

 图5 啮入冲击力曲线 Fig. 5 Meshing impact

2 变双曲圆弧齿线圆柱齿轮振动模型

 图6 CHATT非线性振动模型 Fig. 6 Nonlinear vibration model of CHATT

 $\left\{ \alpha \right\} = {\{ {y_{p1}}, {z_{p1}}, {\psi _{p1}}, {y_{g1}}, {z_{g1}}, {\psi _{g1}}, {y_{p2}}, {z_{p2}}, {\psi _{p2}}, {y_{g2}}, {z_{g2}}, {\psi _{g2}}\} ^{\rm{T}}}。$

 $\left\{ \begin{array}{l} {m_{p1}}{{\ddot y}_{p1}} + {c_{p1y}}{{\dot y}_{p1}} + {k_{p1y}}{y_{p1}} + \\ \;\;\;\;\;\;\;{c_{py}}({{\dot y}_{p1}}-{{\dot y}_{p2}}) + {\rm{ }}{k_{py}}({y_{p1}}-{y_{p2}}) =-{F_{yp1}}, \\ {m_{p1}}{{\ddot z}_{p1}} + {c_{p12z}}({{\ddot z}_{p1}} + {{\dot z}_{p2}}) + {k_{p12z}}({z_{p1}} + {z_{p2}}) = - {F_{z1}}, \\ {I_{p1}}{{\ddot \psi }_{p1}} = - {F_{yp1}}{R_{p1}} - {F_{s1}}{R_{p1}} + {T_{p1}} + {c_1}{{\ddot \psi }_{p1}} + {k_1}{\psi _{p1}} \end{array} \right.$ (2)
 $\left\{ \begin{array}{l} {m_{g1}}{{\ddot y}_{g1}} + {c_{g1y}}{{\dot y}_{g1}} + {k_{g1y}}{y_{g1}} + {c_{gy}}({{\dot y}_{g1}}-{{\dot y}_{g2}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_{gy}}({y_{g1}}-{y_{g2}}) = {F_{yg1}}, \\ {m_{g1}}{{\ddot z}_{g1}} + {c_{g1z}}{{\dot z}_{g1}} + {k_{g1z}}{z_{g1}} + {c_{g12z}}({{\dot z}_{g1}} + {{\dot z}_{g2}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_{g12z}}({z_{g1}} + {z_{g2}}) = {F_{z1}}, \\ {I_{g1}}{{\ddot \psi }_{g1}} = {F_{yg1}}{R_{g1}} + {F_{s1}}{R_{g1}}-{T_{g1}} - {c_1}{{\ddot \psi }_{g1}} - {k_1}{\psi _{g1}} \end{array} \right.$ (3)
 $\left\{ \begin{array}{l} {m_{p2}}{{\ddot y}_{p2}} + {c_{p2y}}{{\dot y}_{p2}} + {k_{p2y}}{y_{p2}} + {\rm{ }}{c_{py}}({{\dot y}_{p2}}-{{\dot y}_{p1}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_{py}}({y_{p2}}-{y_{p1}}) =-{F_{yp2}}, \\ {m_{p2}}{{\ddot z}_{p2}} + {c_{p12z}}({{\dot z}_{p1}} + {{\dot z}_{p2}}) + {k_{p12z}}({z_{p1}} + {z_{p2}}) = - {F_{z2}}, \\ {I_{p2}}{{\ddot \psi }_{p2}} = - {F_{yp2}}{R_{p2}} - {F_{s2}}{R_{p2}} + {T_{p2}} + {c_2}{{\ddot \psi }_{p2}} + {k_2}{\psi _{p2}} \end{array} \right.$ (4)
 $\left\{ \begin{array}{l} {m_{g2}}{{\ddot y}_{g2}} + {c_{g2y}}{{\dot y}_{g2}} + {k_{g2y}}{y_{g2}} + {c_{gy}}({{\dot y}_{g2}}-{{\dot y}_{g1}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_{gy}}({y_{g2}}-{y_{g1}}) = {F_{yg2}}, \\ {m_{g2}}{{\ddot z}_{g2}} + {c_{g2z}}{{\dot z}_{g2}} + {k_{g2z}}{z_{g2}} + {c_{g12z}}({{\dot z}_{g1}} + {{\dot z}_{g2}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{k_{g12z}}({z_{g1}} + {z_{g2}}) = {F_{z2}}, \\ {I_{g2}}{{\ddot \psi }_{g2}} = {F_{yg2}}{R_{g2}} + {F_{s2}}{R_{g2}}-{T_{g2}} - {c_2}{{\ddot \psi }_{g2}} - {k_2}{\psi _{g2}} \end{array} \right.$ (5)

 \begin{align} & {{F}_{yij}}={{k}_{yj}}[{{{\bar{y}}}_{pj}}-{{{\bar{y}}}_{gj}}-{{e}_{yj}}]+{{c}_{yj}}[{{{\dot{\bar{y}}}}_{pj}}-{{{\dot{\bar{y}}}}_{gj}}-{{{\dot{\bar{e}}}}_{yj}}]+ \\ & {{\left( -1 \right)}^{s+1}}\left[ {{k}_{ix}}\left( {{\psi }_{i1}}-{{\psi }_{i2}} \right)+{{c}_{ix}}\left( {{{\dot{\psi }}}_{i1}}-{{{\dot{\psi }}}_{i2}} \right) \right] \\ \end{align} (6)

 \begin{align} & {{F}_{zj}}={{k}_{zj}}\left[ {{{\bar{z}}}_{pj}}-{{{\bar{z}}}_{gj}}-{{e}_{zj}} \right]+{{c}_{zj}}\left[ {{{\dot{\bar{z}}}}_{pj}}-{{{\dot{\bar{z}}}}_{gj}}-{{{\dot{\bar{e}}}}_{yj}} \right]+ \\ & \left( -1 \right)j\left[ {{k}_{zj}}{{z}_{e}}+{{c}_{zj}}{{{\dot{z}}}_{e}} \right] \\ \end{align} (7)

 $\left\{ \begin{array}{l} {{\bar y}_{pi}} = {y_{pi}} + {\psi _{pi}}{R_{pi}}, \\ {{\bar y}_{gi}} = {y_{gi}} + {\psi _{gi}}{R_{gi}}, \\ {{\bar z}_{pi}} = {z_{pi}}-{{\bar y}_{pi}}{\rm{tan}}\theta, \\ {{\bar z}_{gi}} = {z_{gi}}-{{\bar y}_{gi}}{\rm{tan}}\theta \end{array} \right.$ (8)

 ${\lambda _i} = {{\bar y}_{pi}}-{{\bar y}_{gi}}-{e_{yi}}$ (9)

 ${\gamma _j} = {\psi _{j1}}-{\psi _{j2}}$ (10)

3 振动模型的求解与分析

 图7 主动轮振动位移和速度相图及Poincaré截面 Fig. 7 Vibration displacement & velocity phase and Poincaré section of driving gear

 图8 主动轮轴承动载荷 Fig. 8 Dynamic loads of driving gear bearing

 图9 主动轮振动加速度及幅值谱 Fig. 9 Vibration accelerations and amplitude spectrums of driving gear

4 参数对耦合系统振动特性的影响分析

 图10 齿线半径和平均啮合刚度曲线 Fig. 10 Tooth-trace radius & average meshing stiffness curve

5 结论

1) 通过CHATT齿轮副承载接触分析，计算引起齿轮系统振动的主要内部激励。基于集中参数理论和牛顿第二定律建立CHATT的12自由度弯扭轴的多因素耦合振动模型以及振动微分方程组，对其系统的振动特性进行仿真分析。

2) 基于文中建立的振动模型，分析主动轮与从动轮的竖直、扭转和轴向3个方向的振动特性。仿真数据表明，主动轮和从动轮在竖直和扭转方向上的运动平稳性比轴向好，为后续动态设计和降噪提供理论依据。

3) 进一步研究齿线半径、负载转矩和输入转速等3个参数变化对系统振动特性的影响规律。结果表明，轴向振动的规律性更容易受到上述3个参数变化的影响，为预测不同参数下CHATT系统的振动响应趋势提供一定的理论依据。

4) 研究中忽略了摩擦、齿轮的偏心质量、支撑箱体的变形等因素，在后续建立更加精确的振动模型中应考虑这些参数的影响。

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