NONLINEAR DYNAMIC ANALYSIS OF HIGH SPEED AND HEAVY LOAD HERRINGBONE GEAR TRANSMISSION
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摘要: 人字齿轮承载能力强, 重合度大, 可靠性高, 多于高速、重载工况下使用. 探究高速重载人字齿轮传动系统非线性动力学特性, 可为其设计提供参考. 首先, 计算齿轮副时变啮合刚度; 引入齿侧间隙、间隙非线性函数和综合传动误差, 计算时变啮合力; 引入轴承游隙, 计算轴承受力. 随后, 建立高速重载人字齿轮传动系统非线性动力学方程, 使用4阶Runge-Kutta法对方程求解. 最后, 探究不同因素对系统动态响应影响. 保持系统其他参数不变, 分别改变输入转矩、啮合阻尼、齿侧间隙、啮合刚度及激励频率, 绘制系统时间−位移图像、时间−速度图像、空间相图、空间频谱图及分岔图, 观察系统非线性动力学响应变化趋势, 判别系统运动状态. 结果表明: 在一定范围内, 系统稳定性与啮合阻尼、啮合刚度呈正相关关系, 与齿侧间隙、输入转矩呈负相关关系; 逐渐增大外部激励频率时, 系统运动从单周期运动逐渐变为混沌运动, 随后又回归单周期运动. 为保证系统平稳运行, 应合理选取外部激励频率.Abstract: Herringbone gears have strong bearing capacity, large contact ratio and high reliability, and are mostly used in high-speed and heavy-load working occasions. Exploring the nonlinear dynamic characteristics of high-speed and heavy-duty herringbone gear transmission system and finding out the stable operation interval of the system can provide reference for its design. Firstly, the time-varying meshing stiffness of gear pair is calculated, and the time-varying meshing force is calculated by introducing the backlash, the nonlinear function of backlash and the comprehensive transmission error. The bearing clearance is introduced to calculate the bearing force. Subsequently, the nonlinear dynamic equation of the high-speed and heavy-duty herringbone gear transmission system is established, and the fourth-order Runge-Kutta method is used to solve the equation. Finally, the influence of different factors on the stability of the system is explored. Keeping other parameters of the system unchanged, the meshing damping, backlash, meshing stiffness and excitation frequency are changed respectively. The time-displacement image, time-velocity image, spatial phase diagram, spatial frequency diagram and bifurcation diagram of the system are drawn. The change trend of nonlinear dynamic response of the system is observed and the motion state of the system is judged. The results show that within a certain range, the stability of the system is positively correlated with meshing damping and meshing stiffness, and negatively correlated with the backlash. When the external excitation frequency is gradually increased, the system motion gradually changes from single-cycle motion to chaotic motion, and then returns to stable single-cycle motion. Therefore, in order to ensure the smooth operation of the system, the external excitation frequency should be reasonably selected.
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引言
人字齿轮可消除斜齿轮啮合过程中附加轴向载荷, 增大重合度, 使啮合更平稳, 因此常用于武装直升机动力传输系统中. 对人字齿轮系统非线性动力学特性进行分析, 探究各因素对非线性行为影响, 找出稳定运行区间, 能为齿轮系统设计提供参考.
诸多学者深入探究不同因素对齿轮系统非线性行为影响[1-3]. 以上学者考虑时变啮合刚度、轴承游隙、齿侧间隙、间隙非线性函数和齿心偏差等因素影响, 建立系统非线性动力学方程并求解. 肖乾等[4]将齿间滑动及摩擦考虑在内, 建立列车齿轮传动系统模型; 莫帅等[5-6]将轮齿裂纹考虑在内, 建立小模数变位齿轮故障动力学模型; 袁冰等[7]将齿间累积误差和综合传递误差引入, 建立人字齿轮非线性动力学模型; Mo等[8-12]建立非正交面齿轮轴承系统模型, 研究在不同激励频率下系统动力学响应及非线性全局行为演化过程, 采用胞映射法研究非正交面齿轮轴承系统多自由度耦合振动模型动态行为演化过程. Chen等[13]综合考虑摩擦、时变刚度对系统冲击影响, 建立齿轮−转子−轴承系统多自由度非线性动力学模型; Miroslav等[14]考虑时变啮合刚度与滚动轴承轴颈和外壳之间的非线性接触力, 建立齿轮传动非线性动力学模型. 诸多学者[15-17]提出动力学模型建立方式, 对非线性动力学模型建立有推进作用.
王云等[18]建立了人字齿轮副平移−扭转振动分析模型, 结合混合弹流润滑理论计算了混合弹流润滑摩擦因数, 并探究其对系统动态特性影响; Yang等[19]提出一种由轴颈轴承支撑的人字齿轮转子系统耦合动力学模型, 用时间位移图像、相平面图和分岔图从波度类型和波度幅值两方面研究人字齿轮转子系统非线性动力学行为; Xu等[20]在摩擦动力学条件下, 通过迭代数值方案提出一种新型HPGS耦合动力学模型, 并对考虑齿面摩擦下该系统动力学特性进行研究, 得出齿面摩擦与动态啮合力关系. Wang等[21]等学者根据谐波平衡展开法和多尺度法分别推导单稳态能量收集器位移和电压频率响应函数, 并从实验测量中确定单稳态系统参数, 将其用于谐波平衡和多尺度方法理论解数值研究; Wang等[22]和Moradi等[23]分别建立铁路机车单自由度裂纹齿轮模型和直齿轮副非线性振动模型, 采用多尺度方法对共振进行参数化研究, 揭示不同因素对系统稳定性影响; 诸多学者[24-30]采用不同研究方式, 分别对考虑不同因素下系统非线性响应进行研究, 使非线性动力学研究方法更加完善.
本文旨在研究高速重载人字齿轮传动系统非线性振动特性, 综合考虑时变啮合刚度、传动误差、齿侧间隙和轴承间隙等因素, 建立系统动力学模型, 通过控制变量法, 探究在不同齿侧间隙、啮合阻尼、啮合刚度及外部激励频率下系统非线性动力学响应, 输出系统时间−位移图像、时间−速度图像、空间频谱图、空间相图、小波时频图及分岔图, 通过观测图像曲线变化趋势判别系统是否稳定.
1. 人字齿轮系统动力学模型
引入传递误差、时变啮合刚度和齿侧间隙等因素, 考虑轴承滚动体与内外滚道之间游隙及轴承非线性振动, 构建图1所示人字齿轮系统多自由度耦合振动模型.
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图 1 人字齿轮系统多自由度耦合振动模型Figure 1. Multi-degree-of-freedom coupling vibration model of herringbone gear system图中m1, m3, I1和I3分别为右、左两侧主动斜齿轮质量及右、左两侧主动斜齿轮转动惯量, m2, m4, I2和I4分别为右、左两侧从动斜齿轮质量及右、左两侧从动斜齿轮转动惯量; mA1和mA2分别为输入轴和输出轴右侧轴承质量, mA3和mA4分别为输入轴和输出轴左侧轴承质量; kA1i, kA2i, kA3i, kA4i, cA1i, cA2i, cA3i和cA4i (i = x, y)为各轴承内圈沿坐标轴x向和y向支撑刚度和支撑阻尼; k1i, k2i, k3i, k4i, c1i, c2i, c3i和c4i (i = x, y, z, θ)为齿轮与轴承连接轴段沿坐标轴x向、y向和z向支撑刚度、扭转刚度及相应阻尼; k13i, k24i, c13i和c24i (i = x, y, z, θ)为中间轴段沿坐标轴各向支撑刚度、扭转刚度及相应阻尼.
把轴承与旋转轴视作刚性连接, 齿轮、一侧轴承与旋转轴质量集中在其质心, 系统考虑24个自由度
$$ \begin{split} &\left\{x_{1},y_{1},z_{1},\theta_{1},x_{2},y_{2},z_{2},\theta_{2},x_{3},y_{3},z_{3},\theta_{3},x_{4},y_{4},z_{4},\theta_{4},\right.\\ &\qquad \left.x_{A1}, y_{A1},x_{A2},y_{A2},x_{A3},y_{A3},x_{A4},y_{A4}\right\} \end{split}$$ 其中, xi, yi, zi, θi (i = 1, 2, 3, 4)为各斜齿轮沿坐标轴各向振动与扭转位移, xAj, yAj (j = 1, 2, 3, 4)为各轴承内圈沿坐标轴相应方向位移.
在本研究所用齿轮参数条件下, 系统额定负载扭矩为200 N·m, 最大工作转速超过5000 r/min, 在高速重载工况下运行, 其中, 人字齿轮及轴承参数如表1及表2所示.
表 1 人字齿轮参数Table 1. Parameters of the herringbone gearsInput gear Output gear number of teeth 22 41 mess/kg 10.7 29.74 modulus/mm 5 5 pressure angle/(°) 20 20 helix angle/(°) 22 22 teeth width/mm 120 120 表 2 轴承参数Table 2. Bearing parametersBearing A1&A3 Bearing A2&A4 bearing type NJ209 6211 outer raceway diameter/mm 85 95 inner raceway diameter/mm 45 55 width/mm 19 21 系统振动微分方程如下所示
$$ \left. \begin{aligned} &{m_i}{{ x}_i} + {c_{xi}}({{ x}_i} - {{ x}_{Av}}) + {c_{xj}}({{ x}_i} - {{ x}_p}) + {k_{xi}}({x_i} - {x_{Av}}) +\\ &\qquad {k_{xj}}({x_i} - {x_p}) = ( - {k_{mi}}f({x_{nk}}) - {c_{mi}}{{ x}_{nk}})\sin {\alpha _n} \\ & {m_i}{{ y}_i} + {c_{yi}}({{ y}_i} - {{ y}_{Av}}) + {c_{yj}}({{ y}_i} - {{ y}_p}) + {k_{yi}}({y_i} - {y_{Av}})+ \\ &\qquad {k_{yj}}({y_i} - {y_p}) = ( - {k_{mi}}f({x_{nk}}) - {c_{mi}}{{ x}_{nk}})\cos {\alpha _n}\cos \beta - {f_g} \\ & {m_i} {{ z}_i} + {c_{zi}}({{ z}_i} - {{ z}_{Av}}) + {c_{zj}}({{ z}_i} - {{ z}_p}) + {k_{zi}}({z_i} - {z_{Av}})+ \\ & \qquad {k_{zj}}({z_i} - {z_p}) = ( - {k_{mi}}f({x_{nk}}) - {c_{mi}}{{ x}_{nk}})\cos {\alpha _n}\sin \beta \\ & {I_i}{{ \theta }_i} + {c_{tj}}({{ \theta }_i} - {{ \theta }_p}) + {c_{ti}}{{ \theta }_i} + {k_{tj}}({\theta _i} - {\theta _p}) + {k_{ti}}{\theta _i} = {T_i}- \\ & \qquad {F_{myk}}{R_{bi}} - {m_{qi}}g{R_{bi}} + {F_i}{R_{bi}} \end{aligned} \right\} $$ (1) $$ \left.\begin{aligned} & {m_i}{{ x}_i} + {c_{xj}}({{ x}_i} - {{ x}_j}) + {c_{xi}}{{ x}_i}+ \\ &\qquad {k_{xj}}({x_i} - {x_j}) + {k_{xi}}{x_i} = - {F_{bxi}} \\ & {m_i}{{ y}_i} + {c_{yj}}({{ y}_i} - {{ y}_j}) + {c_{yj}}{{ y}_i} + \\ &\qquad {k_{yj}}({y_i} - {y_j}) + {k_{yi}}{x_i} = - {F_{byi}} - {m_i}g \end{aligned} \right\} $$ (2) 为提升计算精度, 以齿轮啮合线位移替换系统中齿轮扭转位移, 将方程无量纲化, 如下所示
$$ \left.\begin{aligned} & {{ {\bar x}}_{n1}} = ({{ {\bar x}}_1} - {{ {\bar x}}_2})\sin {\alpha _n} + ({{ {\bar y}}_1} - {{ {\bar y}}_2})\cos {\alpha _n}\cos \beta + \\ &\qquad ({{ {\bar z}}_1} - {{ {\bar z}}_2})\cos {\alpha _n}\sin \beta - {\kappa _{m1}}f({{\bar x}_{n1}}) - {\xi _{m1}}{{ {\bar x}}_{n1}} + {f_1} + {f_{e1}} \\ & {{ {\bar x}}_{n2}} = ({{ {\bar x}}_3} - {{ {\bar x}}_4})\sin {\alpha _n} + ({{ {\bar y}}_3} - {{ {\bar y}}_4})\cos {\alpha _n}\cos \beta + \\ &\qquad ({{ {\bar z}}_3} - {{ {\bar z}}_4})\cos {\alpha _n}\sin \beta - {\kappa _{m2}}f({{\bar x}_{n2}}) - {\xi _{m2}}{{ {\bar x}}_{n2}} + {f_1} + {f_{e2}} \end{aligned}\right\} $$ (3) $$ \left.\begin{aligned} &{{ {\bar x}}_i} + {\xi _{xi}}({{ {\bar x}}_i} - {{ {\bar x}}_{qv}}) + {\xi _{xj}}({{ {\bar x}}_i} - {{ {\bar x}}_p}) + {\kappa _{xi}}({{\bar x}_i} - {{\bar x}_{qv}}) +\\ &\qquad {\kappa _{xj}}({{\bar x}_i} - {{\bar x}_p}) = ( - {\kappa _{mi}}f({{\bar x}_{nk}}) - {\xi _{mi}}{{ {\bar x}}_{nk}})\sin {\alpha _n} \\ & {{ {\bar y}}_i} + {\xi _{yi}}({{ {\bar y}}_i} - {{ {\bar y}}_{qv}}) + {\xi _{yj}}({{ {\bar y}}_i} - {{ {\bar y}}_p}) + {\kappa _{yi}}({{\bar y}_i} - {{\bar y}_{qv}})+ \\ &\qquad {\kappa _{yj}}({{\bar y}_i} - {{\bar y}_p}) = ( - {\kappa _{mi}}f({{\bar x}_{nk}}) - {\xi _{mi}}{{ {\bar x}}_{nk}})\cos {\alpha _n}\cos \beta - {f_g} \\ & {{ {\bar z}}_i} + {\xi _{zi}}({{ {\bar z}}_i} - {{ {\bar z}}_{qv}}) + {\xi _{zj}}({{ {\bar z}}_i} - {{ {\bar z}}_p}) + {\kappa _{zi}}({{\bar z}_i} - {{\bar z}_{qv}})+ \\ &\qquad {\kappa _{zj}}({{\bar z}_i} - {{\bar z}_p}) = ( - {\kappa _{mi}}f({{\bar x}_{nk}}) - {\xi _{mi}}{{ {\bar x}}_{nk}})\cos {\alpha _n}\sin \beta \end{aligned} \right\} $$ (4) $$ \left. \begin{aligned} & {{ x}_i} + {\xi _{xj}}({{ x}_i} - {{ x}_j}) + {\xi _{xi}}{{ x}_i}+ \\ &\qquad {\kappa _{xj}}({x_i} - {x_j}) + {\kappa _{xi}}{x_i} = - {f_{bxi}} \\ & {{ y}_i} + {\xi _{yj}}({{ y}_i} - {{ y}_j}) + {\xi _{yj}}{{ y}_i} + \\ &\qquad {\kappa _{yj}}({y_i} - {y_j}) + {\kappa _{yi}}{x_i} = - {f_{byi}} - {m_i}g \end{aligned} \right\} $$ (5) 式(1)和式(3)中, j取13, 当i = 1时, p = 3, k = 1, v = 1; 当i = 3时, p = 1, k = 2, v = 3. j取24, 当i = 2时, p = 1, k = 1, v = 2; 当i = 4时, p = 2, k = 2, v = 4; 式(2)和式(4)中, 当i = A1时, j = 1; 当i = A3时, j = 3; 当i = A2时, j = 2; 当i = A4时, j = 4.
无量纲化方式如下所示
$$ \begin{split} & {\text{ }}{{\bar x}_i} = {x_i}/{b_m},{\text{ }}{{\bar y}_i} = {y_i}/{b_m},{\text{ }}{{\bar z}_i} = {z_i}/{b_m},{\text{ }}\tau = {\omega _n}t \\ & {\omega _n} = \sqrt {\frac{{{K_m}}}{{{m_e}}}} ,{\text{ }}{\omega _{ji}} = \sqrt {\frac{{{K_{ji}}}}{{{m_{ji}}}}} ,{\text{ }}{\kappa _{ji}} = \frac{{\omega _{ji}^2}}{{\omega _n^2}},{\text{ }}{\kappa _{mi}} = \frac{{\omega _{xi}^2}}{{\omega _n^2}}\kappa (\tau ){\text{ }} \\ & \kappa (\tau ) = \frac{{{K_{mv}}(t)}}{{{K_a}}},{\text{ }}{\xi _{ji}} = \frac{{{C_{ji}}}}{{2{m_i}{\omega _n}}},{\text{ }}{\xi _{mv}} = \frac{{{C_{mv}}}}{{2{m_{ev}}{\omega _n}}}{\text{ }} \\ & {f_{bkAv}} = \frac{{{F_{bkAv}}\cos {\alpha _n}}}{{{m_{ev}}{b_m}\omega _n^2}},{\text{ }}{f_g} = \frac{g}{{{b_m}\omega _n^2}},{f_{ev}} = \frac{{{{\rm{d}}^2}{e_v}(\tau )/{\rm{d}}{\tau ^2}}}{{{b_m}}} \\ & (i = 1,2,3,4;v = 1,2;j = x,y,z,t;k = x,y) \end{split} $$ 其中, τ表示无量纲时间, ωn表示固有频率, Km表示啮合刚度均值, fg为无量纲重力.
2. 系统时变激励
2.1 时变啮合刚度及啮合力
定义右侧主动与从动斜齿轮啮合线为啮合线1, 左侧主动与从动斜齿轮啮合线为啮合线2, αn和β为法面压力角和螺旋角, e为静态传递误差均值, ωm为斜齿轮副啮合频率, e(t)为静态传递误差, 则右侧主动斜齿轮和右侧从动斜齿轮在啮合线1上投影位移可由下式计算
$$ \left. \begin{aligned} & {x_{n1}} = ({x_1} - {x_2})\sin {\alpha _n} + ({y_1} - {y_2})\cos {\alpha _n}\cos \beta+ \\ &\qquad ({z_1} - {z_2})\cos {\alpha _n}\sin \beta + ({R_{b1}}{\theta _1} - {R_{b2}}{\theta _2})\cos {\alpha _n}\\ &\qquad \cos \beta - e(t) \\ & e(t) = e\sin ({\omega _m}t) \end{aligned} \right\} $$ (6) 基于此, 啮合力计算方法如下
$$ \left.\begin{split} & {F_{m1}} = {K_m}(t)f({x_{n1}}) + {C_m}{{ x}_{n1}} \\ & f({x_{n1}}) = \left\{ \begin{aligned} & {x_{n1}} - {b_m},{\text{ }}{x_{n1}} \gt {b_m} \\ & 0,{\text{ }}\left| {{x_{n1}}} \right| \lt {b_m} \\ & {x_{n1}} + {b_m},{\text{ }}{x_{n1}} \lt - {b_m} \end{aligned} \right. \end{split} \right\} $$ (7) 式中, Km(t)为该斜齿轮副时变啮合刚度; Cm为该斜齿轮副啮合阻尼, bm为齿侧间隙一半.
根据动力学模型中所建立三维坐标系, 将啮合力分解至x, y, z方向, 如下式所示
$$ \left.\begin{split} & {F_{mx1}} = {F_{m1}}\sin {\alpha _n} \\ & {F_{my1}} = {F_{m1}}\cos {\alpha _n}\cos \beta \\ & {F_{mz1}} = {F_{m1}}\cos {\alpha _n}\sin \beta \end{split} \right\} $$ (8) 左侧斜齿轮副啮合力计算方式同理.
根据ISO standard 6336-1-2006计算右侧斜齿轮副时变啮合刚度, 如下式所示
$$ \left. \begin{split} & K(t) = {K_m} + {K_i}\cos ({\omega _m}t + {\phi _0}) \\ & {K_i} = \frac{{\sqrt {2 - 2\cos [2\text{π} i({\varepsilon _\alpha } - 1)}] }}{{\text{π} i}}{K_c} \\ & {K_m} = (0.75{\varepsilon _\alpha } + 0.25){K_c} \end{split} \right\} $$ (9) 式中, K(t)为斜齿啮合时变刚度, Km为平均啮合刚度, Ki为刚度变化幅值, ωm为啮合频率, 其值等于输入轴转速频率乘以输入齿轮齿数, $\phi $为初始相位角, εα为端面重合系数, Kc为单对齿轮的啮合刚度.
左侧斜齿轮副时变啮合刚度可用相同方式计算. 取相同参数, 啮合线1与啮合线2时变啮合刚度曲线变化趋势相同, 如图2所示.
2.2 时变轴承支撑力
轴承用于支撑旋转构件, 其内部轴承力会对系统非线性响应产生影响. 人字齿轮系统模型中, 轴承载荷计算方式如下所示.
以ωbi和ωbo分别为轴承内、外圈绕旋转轴轴线转动角速度, 由于外圈固定于轴承座, 故ωbo = 0. 滚动体与轴承内、外圈接触点处线速度vbi和vbo, 滚动体个数为Zb, 轴承内、外圈半径分别为rbi和rbo, 滚动体公转角速度ωbn及位置角θi(t)计算方式如下所示
$$ \left. \begin{aligned} & {\omega _{bn}} = \frac{{{v_{bi}} + {v_{bo}}}}{{{r_{bi}} + {r_{bo}}}} = \frac{{{\omega _{bi}} \cdot {r_{bi}}}}{{{r_{bi}} + {r_{bo}}}} \\ & {\theta _i}(t) = \frac{{2\text{π} (i - 1)}}{{{Z_b}}} + {\omega _{bn}}t{\text{ }}(i = 1,2,\cdots,{Z_b})\end{aligned} \right\} $$ (10) 轴承未因受力发生形变时, 内、外圈曲率中心分别为Obi和Obo, 曲率半径分别为rbi和rbo, 滚动体与内、外圈初始接触角均为γi; 轴承受力变形后, 内、外圈曲率中心变为$O'_{bi}$和$O'_{bo}$, 接触角变为$\gamma '_i $. 因轴承外圈固定于轴承座, 故受力时其曲率中心位置不变, 轴承内圈曲率中心由Obi变化到$O'_{bi} $. 以$l' $为外圈曲率中心中心距, δ为滚动体变形, c为轴承径向游隙, $\gamma '_i $为滚动体变形后与轴承内圈接触角, 上述各项参数计算公式为
$$ \left.\begin{split} & \delta = l' - l - c \\[-2pt] & l' = \sqrt {(l\sin {\gamma _i} + {r_{bo}}{\delta _\theta }\cos {\theta _i})^2 + {\left( {l\cos {\gamma _i} + x\cos {\theta _i} + y\sin {\theta _i}} \right)^2}} \\[-2pt] &\tan {{\gamma '_i}} = \frac{{l\sin {\gamma _i} + {r_{bo}}{\delta _\theta }\cos {\theta _i}}}{{ l\cos {\gamma _i} + x\cos {\theta _i} + y\sin {\theta _i}}} \end{split} \right\} $$ (11) 式中, x和y分别为轴承内圈沿坐标轴相应方向振动位移.
引入Heaviside函数, 以H(δ)表示, 忽略滚动体及保持架惯性, Frb为滚动体径向受力, Kb为滚动体与滚道接触刚度, 轴承内圈在坐标系各方向受力由下式计算
$$ \left. \begin{split} & H(\delta ) = \left\{\begin{aligned} & 1,{\text{ }} & \delta > 0 \\ & 0,{\text{ }} & \delta < 0 \\ \end{aligned} \right. \\ & {F_{rb}} = {K_b}{\delta ^n}\cos {{\gamma '}_i} \cdot H\left( \delta \right) \\[-3pt] & {F_{bx}} = \sum\limits_{i = 1}^{{Z_b}} {{F_{rb}}\cos {\theta _i}} \\[-3pt] & {F_{by}} = \sum\limits_{i = 1}^{{Z_b}} {{F_{rb}}\sin {\theta _i}} \end{split} \right\} $$ (12) 3. 人字齿轮系统非线性动力学响应
选取额定输入转矩为150 N·m, 探究改变不同系统参数时, 系统动力学响应变化趋势. 考虑到工程实际情况, 在模型建立过程中, 左、右斜齿轮副动态啮合误差激励波动幅值不同, 啮合线1和啮合线2动力学响应曲线变化趋势存在差异.
3.1 不同转矩下系统响应
本部分探究输入转速为6000 r/min时, 人字齿轮传动系统在不同输入转矩下非线性响应. 为更直观反映振动曲线数值, 将啮合线1和啮合线2无量纲时间−位移曲线各点数值乘以bm, 无量纲时间−速度曲线各点数值乘以ωnbm, 绘制不同阻尼系数下系统响应, 如图3所示. 通过分析图3(a)和图3(b)可以看出, 改变Tin值将改变振动位移曲轨迹变化趋势. 图3(c)和图3(d)表明系统运动速度随时间变化关系, 结合空间相图及庞加莱映射点可知, 当Tin缓慢增加时, 系统运动从稳定单周期运动逐渐变为双周期运动、多周期运动和混沌运动, 且振幅逐渐增大, 在此过程中, 啮合线1振动速度峰值为5.2 μm/s, 啮合线2振动速度峰值分别为0.27 μm/s, 0.33 μm/s, 4.5 μm/s和5.3 μm/s.
3.2 不同啮合阻尼下系统响应
本部分探究输入转速为6000 r/min、输入转矩为150 N·m时, 人字齿轮传动系统在不同啮合阻尼系数下非线性响应. 为更直观反映振动曲线数值, 将啮合线1和啮合线2无量纲时间−位移曲线各点数值乘以bm, 无量纲时间−速度曲线各点数值乘以ωnbm, 绘制不同阻尼系数下系统响应, 如图4所示. 通过分析图4(a)和图4(b)可以看出, 改变ξm值将改变振动位移曲轨迹变化趋势. 图4(c)和图4(d)表明系统运动速度随时间变化关系, 结合空间相图及庞加莱映射点可知, 当ξm缓慢增加时, 系统运动从稳定单周期运动逐渐变为双周期运动和多周期运动, 且振幅逐渐增大.
3.3 不同齿侧间隙下系统响应
保持系统其他参数不变, 改变bm值, 分别观察啮合线1和啮合线2非线性位移. 为更直观反映振动曲线数值, 将啮合线1和啮合线2无量纲时间−位移曲线各点数值乘以bm, 无量纲时间−速度曲线各点数值乘以ωnbm, 绘制不同阻尼系数下系统响应, 如图5所示. 从图5(a)和图5(b)可以看出, 改变bm值将改变振动位移曲线的轨迹趋势. 结合空间相图及庞加莱映射点可知, 在不同bm值下, 啮合线1和啮合线2运动状态总体变化趋势相同, 当bm缓慢增大时, 啮合线振动位移从稳定单周期运动逐渐变为双周期运动以及多周期运动, 波动幅度逐渐增大. 当齿侧间隙从1 μm变化至4 μm时, 啮合线1稳定单周期振动速度峰值为0.23 μm/s, 混沌运动及多周期运动振动速度峰值为0.25 μm/s和0.53 μm/s, 啮合线2稳定单周期振动速度峰值为2.5 μm/s.
3.4 不同啮合刚度下系统响应
本部分探究在不同啮合刚度下系统非线性响应. 为更直观反映振动曲线数值, 将啮合线1和啮合线2无量纲时间−位移曲线各点数值乘以bm, 无量纲时间−速度曲线各点数值乘以ωnbm, 绘制不同阻尼系数下系统响应, 如图6所示. 通过分析图6(a)和图6(b)可以看出, 改变Km值将改变振动位移曲线的轨迹趋势. 空间相图及庞加莱映射点表示, 当Km缓慢增加时, 庞加莱映射点集由散乱无序点变化为相轨迹线上两点, 最终变化为相轨迹线上一点, 表明啮合线上投影振动位移起初为混沌运动状态, 随后逐渐变为倍周期运动状态, 最后当啮合刚度突破一定阈值时, 系统趋于稳定, 做单周期运动. 如图6(c)和图6(d)所示, 在稳定运动状态下, 啮合线1稳定单周期振动速度峰值为0.43 μm/s, 混沌运动及多周期运动振动速度峰值为0.8 μm/s和0.33 μm/s, 啮合线2稳定单周期振动速度的峰值为0.45 μm/s.
3.5 不同外部激励频率下系统响应
改变外激励频率, 当输入转速逐渐增大至10000 r/min时, 啮合线1运动空间频谱如图7(a)所示. 由图可知, 当激励频率约为0.6时, 振动频率呈现单一峰值状态, 表明系统处于稳定单周期运动状态. 当激励频率约为1时, 振动频率出现两个峰值, 幅值分别为0.3和0.6, 表明系统处于倍周期或多周期运动状态. 同时, 当激励频率约为1.25和1.68时, 出现较多混沌频率, 表明系统处于混沌运动状态. 此外, 图7(b)为空间状态下相平面图和庞加莱截面图, 系统运动状态转变过程为: 从稳定单周期运动状态逐渐走向倍周期分岔运动状态和混沌运动状态, 最终重新回到稳定周期运动状态.
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图 7 啮合线1空间频谱图及空间相图Figure 7. Spatial spectrum and spatial phase diagram of meshing line 1通过小波变换方法分析啮合线上振动位移, 如图8所示. 当ωe = 0.7时, 啮合线上振动位移主要由0.1ωe组成, 幅值约为0.28. 当ωe = 1时, 啮合线位移主振频率为0.8ωe和分量1.6ωe, 系统处于倍周期运动状态. 当ωe = 1.55时, 除频率为0.25ωe外, 还出现较多混沌频率, 幅值集中在0.05左右, 此时系统在做混沌运动. 当ωe = 2.2时, 啮合位移主要由0.35ωe组成. 幅值约为0.12, 对应系统稳定单周期运动状态.
为进一步研究外部激励频率对系统动态特性影响, 表征系统混沌运动状态, 判别系统运动, 绘制分岔参数为ωe时系统分岔图与李雅普诺夫指数图, 如图9所示. 图中的数据表明, 当ωe < 0.7时, 最大李雅普诺夫指数小于0, 系统振动位移变化周期与啮合周期一致, 运动状态为稳定单周期运动; 当ωe在0.7 ~ 1.2之间时, 系统振动位移变化周期为啮合周期2倍, 系统做倍周期运动; 当ωe在1.2 ~ 1.3之间时, 系统振动位移变化周期为啮合周期4倍, 系统处于周期4运动状态; 当ωe在1.3 ~ 1.72之间时, 最大李雅普诺夫指数大于0, 系统振动位移变化不再具有周期性, 表明系统处于混沌运动状态; 当ωe在1.72 ~ 1.88之间时, 系统振动位移变化周期为啮合周期二倍, 系统做倍周期运动. 当ωe > 1.88时, 最大李雅普诺夫指数降到0线以下, 此时系统所处运动状态为稳定单周期运动.
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图 9 啮合线1系统位移分岔图及最大李雅普诺夫指数图Figure 9. Displacement bifurcation diagram and maximum Lyapunov exponent diagram of meshing line 1改变外激励频率, 当输入转速逐渐增大至10000 r/min时, 啮合线2运动空间频谱如图10(a)所示. 由图可知, 当激励频率小于0.7时, 频谱图上只出现单个主共振峰, 振动幅值最大值为0.7, 表明系统在做稳定单周期运动; 当无量纲激励频率约为1.21和1.75时, 系统振动频率出现多值现象, 表明系统处于混沌运动状态. 此外, 图10(b)为空间相平面图和庞加莱截面图. 图中信息表示, 当无量纲激励频率ωe从0开始增大至2时, 系统运动状态由稳定单周期运动状态过渡至多周期运动状态, 随后陷入混沌运动状态, 在经历倍周期运动状态后, 最终转变回单周期运动状态, 此时系统又趋于稳定.
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图 10 啮合线2空间频谱图及空间相图Figure 10. Spatial spectrum and spatial phase diagram of meshing line 2通过小波变换方法分析齿轮传动系统啮合线1上振动位移变化趋势, 如图11所示. 当ωe = 0.6时, 坐标轴各向投影至啮合线上振动位移主要由0.1ωe组成, 幅值约为0.27, 表明系统稳定. 当ωe = 1.45时, 除频率为0.23ωe外, 还出现较多混沌频率, 幅值集中在0.1左右. 当ωe = 2.1时, 啮合位移主要由0.35ωe组成, 幅值约为0.12, 对应系统稳定单周期运动状态.
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图 11 啮合线2小波变换时频图 (续)Figure 11. Meshing line 2 time-frequency diagram of wavelet transform (continued)为进一步研究外部激励频率对系统动态特性影响, 表征系统混沌运动状态, 判别系统运动, 绘制分岔参数为ωe时系统分岔图及李雅普诺夫指数图, 如图12所示. 图中的数据表明, 当ωe < 0.63时, 系统进行稳定单周期运动; 当ωe在0.63 ~ 1.13之间时, 最大李雅普诺夫指数小于0, 系统做倍周期运动; 当ωe在1.13 ~ 1.25之间时, 系统处于周期4运动状态; 当ωe在1.25 ~ 1.75之间时, 系统处于混沌运动状态, 此时, 最大李雅普诺夫指数上升, 突破0线; 当ωe在1.75 ~ 1.9之间时, 系统做倍周期运动. 当ωe > 1.9时, 最大李雅普诺夫指数降至负值, 表明系统已经脱离混沌运动, 进入稳定单周期运动.
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图 12 啮合线2系统位移分岔图及最大李雅普诺夫指数图Figure 12. Displacement bifurcation diagram and maximum Lyapunov exponent diagram of meshing line 24. 结论
通过对高速重载人字轮非线性动力学分析得到如下结论.
(1) 引入轴承游隙、齿侧间隙和动态啮合误差激励等因素, 建立人字齿轮系统非线性动力学模型, 通过控制变量法分析齿侧间隙对系统非线性特性影响, 结果表明, 当齿侧间隙取值范围为1 × 10−6 ~ 4 × 10−6 m时, 若增大齿侧间隙, 则系统周期运动稳定性降低.
(2) 系统其他参数选定为初始参数, 在150 ~ 300 N·m范围内逐渐增大输入转矩以及在区间(0.15, 0.3)内逐渐减小啮合阻尼比时, 系统非线性振动由单周期演变为多周期和混沌, 为使系统运动处于单周期运动状态, 系统输入转矩不宜过高, 啮合阻尼比不宜过低.
(3) 通过控制变量法探究啮合刚度及激励频率对系统影响. 在1.5 × 108 ~ 6 × 108 N/m范围内逐渐增大啮合刚度时, 系统运动状态由混沌状态经历多周期运动状态后, 最终变为稳定单周期运动状态. 逐渐增大激励频率时, 系统将经历周期、准周期、混沌及其他运动状态. 为保证系统稳定运行, 啮合刚度不宜过小, 激励频率不应大于3250 Hz.
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图 1 人字齿轮系统多自由度耦合振动模型
Figure 1. Multi-degree-of-freedom coupling vibration model of herringbone gear system
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图 7 啮合线1空间频谱图及空间相图
Figure 7. Spatial spectrum and spatial phase diagram of meshing line 1
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图 9 啮合线1系统位移分岔图及最大李雅普诺夫指数图
Figure 9. Displacement bifurcation diagram and maximum Lyapunov exponent diagram of meshing line 1
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图 10 啮合线2空间频谱图及空间相图
Figure 10. Spatial spectrum and spatial phase diagram of meshing line 2
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图 11 啮合线2小波变换时频图 (续)
Figure 11. Meshing line 2 time-frequency diagram of wavelet transform (continued)
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图 12 啮合线2系统位移分岔图及最大李雅普诺夫指数图
Figure 12. Displacement bifurcation diagram and maximum Lyapunov exponent diagram of meshing line 2
表 1 人字齿轮参数
Table 1 Parameters of the herringbone gears
Input gear Output gear number of teeth 22 41 mess/kg 10.7 29.74 modulus/mm 5 5 pressure angle/(°) 20 20 helix angle/(°) 22 22 teeth width/mm 120 120 
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表 2 轴承参数
Table 2 Bearing parameters
Bearing A1&A3 Bearing A2&A4 bearing type NJ209 6211 outer raceway diameter/mm 85 95 inner raceway diameter/mm 45 55 width/mm 19 21
下载: 导出CSV
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[1] 徐向阳, 李龙, 任博等. 空间轴交角人字行星齿轮系统动态特性研究. 机械工程学报, 2023, 59(1): 141-150 (Xu Xiangyang, Li Long, Ren Bo, et al. Study on the dynamic characteristics of herringbone planetary gear system with space shafts angle. Journal of Mechanical Engineering, 2023, 59(1): 141-150 (in Chinese) Xu Xiangyang, Li Long, Ren Bo, et al. Study on the Dynamic Characteristics of Herringbone Planetary Gear System with Space Shafts Angle. Journal of Mechanical Engineering, 2023, 59(01): 141-150 (in Chinese)
[2] 林腾蛟, 陈梦寒, 杨金. 齿廓修形人字齿轮副时变啮合刚度计算方法. 振动与冲击, 2021, 40(9): 175-183 (Lin Tengjiao, Chen Menghan, Yang Jin. Calculation method of time-varying meshing stiffness of herringbone gear pair with tooth profile modification. Journal of Vibration and Shock, 2021, 40(9): 175-183 (in Chinese) doi: 10.13465/j.cnki.jvs.2021.09.023 Lin Tengjiao, Chen Menghan, Yang Jin. Calculation method of time-varying meshing stiffness of herringbone gear pair with tooth profile modification. Journal of Vibration and Shock, 2021, 40(09): 175-183 (in Chinese) doi: 10.13465/j.cnki.jvs.2021.09.023
[3] 张佳雄, 魏静, 张春鹏等. 高阶调谐齿轮参数设计及动态响应研究. 振动工程学报, 2022, 35(2): 369-378 (Zhang Jiaxiong, Wei Jing, Zhang Chunpeng, et al. Parameter design and dynamic response study of a high-order tuning gear. Journal of Vibration Engineering, 2022, 35(2): 369-378 (in Chinese) Zhang Jiaxiong, Wei Jing, Zhang Chunpeng, et al. Parameter design and dynamic response study of a high-order tuning gear. Journal of Vibration Engineering, 2022, 35(02): 369-378 (in Chinese)
[4] 肖乾, 王丹红, 陈道云等. 考虑齿间滑动影响的高速列车传动齿轮动态接触特性分析. 机械工程学报, 2021, 57(10): 87-94 Xiao Qian, Wang Danhong, Chen Daoyun, et al. Analysis of dynamic contact characteristics of high-speed train transmission gear considering the influence of sliding between the teeth. Journal of Mechanical Engineering, 2021, 57(10): 87-94 (in Chinese)
[5] 莫帅, 周长鹏, 王檑等. 机器人关节裂纹传动系统机电耦合动态特性研究. 机械工程学报, 2022, 58(19): 57-67 Mo Shuai, Zhou Changpeng, Wang Lei, et al. Research on dynamic characteristic of electromechanical coupling of robot joint crack transmission system. Journal of Mechanical Engineering, 2022, 58(19): 57-67 (in Chinese)
[6] 莫帅, 周长鹏, 高瀚君等. 机器人智能关节机电耦合系统瞬态动力学研究. 华中科技大学学报(自然科学版), 2023, 51(2): 131-138 Mo Shuai, Zhou Changpeng, Gao Hanjun, et al. Research on transient dynamics of robot intelligent joint electromechanical coupling system. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2023, 51(2): 131-138 (in Chinese)
[7] 袁冰, 常山, 刘更等. 考虑齿距累积误差的人字齿轮系统动态特性分析. 振动与冲击, 2020, 39(3): 120-126 Yuan Bing, Chang Shan, Liu Geng, et al. Dynamic characteristics of a double helical gear system considering accumulation pitch error. Journal of Vibration and Shock, 2020, 39(3): 120-126 (in Chinese)
[8] Mo S, Zhang Y, Luo B, et al. The global behavior evolution of non-orthogonal face gear-bearing transmission system. Mechanism and Machine Theory, 2022, 175: 104969 doi: 10.1016/j.mechmachtheory.2022.104969
[9] Mo S, Zhang T, Jin GG, et al. Analytical investigation on load sharing characteristics of herringbone planetary gear train with flexible support and floating sun gear. Mechanism and Machine Theory, 2020, 144(2): 1-27
[10] Mo S, Zhang YD, Wu Q. Research on multiple-split load sharing of two-stage star gearing system in consideration of displacement compatibility. Mechanism and Machine Theory, 2015, 88(1): 1-15
[11] Mo S, Zhang YD, Wu Q, et al. Research on Natural Characteristics of double-helical star gearing system for GTF aero-engine. Mechanism and Machine Theory, 2016, 106(12): 166-189
[12] Mo S, Zhou CP, Dang HY. Lubrication effect of the nozzle layout for arc tooth cylindrical gears. Journal of Tribology, 2022, 144(3): 1-25
[13] Chen SY, Tang JY, Luo CW, et al. Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction. Mechanism and Machine Theory, 2011, 46(4): 466-478 doi: 10.1016/j.mechmachtheory.2010.11.016
[14] Miroslav B, Vladimír Z. On modeling and vibration of gear drives influenced by nonlinear couplings. Mechanism and Machine Theory, 2011, 46(3): 375-397
[15] Wang SY, Zhu RP. Modeling and theoretical investigation of nonlinear torsional characteristics for double-helical star gearing system in GTF gearbox. Journal of Vibration Engineering & Technologies, 2022, 10: 193-209
[16] Wang SY, Zhu RP. Nonlinear torsional dynamics of star gearing transmission system of GTF gearbox. Shock and Vibration, 2020, 2020(4): 1-15
[17] Tian G, Gao Z, Liu P, et al. Dynamic modeling and stability analysis for a spur gear system considering gear backlash and bearing clearance. Machines, 2022, 10(6): 43
[18] 王云, 杨为, 唐小林. 基于动态啮合力的人字齿轮摩擦因数分析. 振动与冲击, 2021, 40(18): 265-272 Wang Yun, Yang Wei, Tang Xiaolin. Friction coefficient analysis for herringbone gears based on dynamic meshing forces. Journal of Vibration and Shock, 2021, 40(18): 265-272 (in Chinese)
[19] Yang J, Zhu R, Yue Y, et al. Nonlinear analysis of herringbone gear rotor system based on the surface waviness excitation of journal bearing. Journal of the Brazilian Society of Mechanical Science and Engineering, 2022, 54: 44-52
[20] Xu X, Jiang G, Wang H, et al. Investigation on dynamic characteristics of herringbone planetary gear system considering tooth surface friction. Meccanica, 2022, 57: 1677-1699 doi: 10.1007/s11012-022-01526-4
[21] Wang W, Cao JY, Dhiman M, et al. Comparison of harmonic balance and multi-scale method in characterizing the response of monostable energy harvesters. Mechanical Systems and Signal Processing, 2018, 108: 252-261
[22] Wang JG, Bo L, Rui S, et al. Resonance and stability analysis of a cracked gear system for railway locomotive. Applied Mathematical Modelling, 2020, 77(1): 253-266
[23] Moradi H, Salarieh H. Analysis of nonlinear oscillations in spur gear pairs with approximated modelling of backlash nonlinearity. Mechanism and Machine Theory, 2012, 51: 14-31 doi: 10.1016/j.mechmachtheory.2011.12.005
[24] Yi Y, Huang K, Xiong YS, et al. Nonlinear dynamic modelling and analysis for a spur gear system with time-varying pressure angle and gear backlash. Mechanical Systems and Signal Processing, 2019, 132: 18-34
[25] Tatsuhito A, Kensho S. Theoretical analysis of nonlinear vibration characteristics of gear pair with shafts. Theoretical and Applied Mechanics Letters, 2022, 12(2): 100324
[26] Yang Y, Xia WK, Han JM, et al. Vibration analysis for tooth crack detection in a spur gear system with clearance nonlinearity. International Journal of Mechanical Sciences, 2019, 157: 648-661
[27] Celikay A, Donmez A, Kahraman A. An experimental and theoretical study of subharmonic resonances of a spur gear pair. Journal of Sound and Vibration, 2021, 515: 116421
[28] Kandil A, Hamed Y, Mohamed M, et al. Third-order superharmonic resonance analysis and control in a nonlinear dynamical system. Mathematics. 2022, 10(8): 1282
[29] Öztürk V, Cigeroglu E, Özgüven H. Ideal tooth profile modifications for improving nonlinear dynamic response of planetary gear trains. Journal of Sound and Vibration, 2021, 500: 116007
[30] 金花, 吕小红, 张子豪等. 齿轮传动系统共存吸引子的不连续分岔. 力学学报, 2023, 55(1): 203-212 Jing Hua, Lv Xiaohong, Zhang Zihao, et al. Discontinuous bifurcations of coexisting attractors for a gear transmission system. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 203-212 (in Chinese)
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