NON-LINEAR DYNAMICS OF AERONAUTICAL NON-ORTHOGONAL OFFSET GEAR SPLIT FLOW SYSTEM
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摘要: 在面齿轮传动系统研究领域, 研究对象多为单级面齿轮副, 针对面齿轮分汇流传动系统动态特性的研究内容相对较少. 聚焦航空非正交偏置面齿轮分汇流系统的非线性动力学特性, 建立轴承滚动体变形、齿侧间隙以及综合传递误差的数学模型, 推导出包含轴承在内的非正交偏置面齿轮传动多自由度耦合弯扭组合动力学模型. 利用数值方法求解航空非正交偏置面齿轮分汇流系统的振动微分方程, 结合时域图、频域图、相图、Poincaré映射、李雅普诺夫指数以及分岔图研究复杂系统非线性动态特性. 研究了激励频率和啮合阻尼比变化对分流级与汇流级齿轮副之间啮合线位移的影响, 结果表明: 系统分流级与汇流级在特定参数下表现出不同的非线性特性, 随激励频率增大, 分流级出现单周期、多周期以及混沌现象, 而汇流级无混沌现象产生; 随啮合阻尼比增大, 分流级状态变化近似于逆倍周期分岔过程, 而汇流级在较小的阻尼比也可能保持周期运动状态; 合理的激励频率和啮合阻尼比能使得系统跳出混沌并保持稳定.Abstract: In the research field of face gear transmission system, most of the research objects are single-stage face gear pairs, and there are relatively few research contents on the dynamic characteristics of face gear split-confluence transmission system. Focusing on the nonlinear dynamic characteristics of the aviation non-orthogonal offset face gear shunt system, the mathematical models of bearing rolling element deformation, tooth side clearance and comprehensive transmission error are established, and the multi-degree-of-freedom coupling bending-torsion combination dynamic model of non-orthogonal offset face gear transmission including bearing is derived. The vibration differential equation of the non-orthogonal offset face gear flow separation and confluence system is solved by numerical method. The nonlinear dynamic characteristics of the complex system are studied by combining the time domain diagram, frequency domain diagram, phase diagram, Poincaré map, Lyapunov exponent and bifurcation diagram. The influence of excitation frequency and meshing damping ratio on the displacement of the meshing line between the shunt stage and the confluence stage gear pair is studied. The results show that the shunt stage and the confluence stage of the system show different nonlinear characteristics under specific parameters. With the increase of excitation frequency, the shunt stage appears single-period, multi-period and chaotic phenomena, while the confluence stage does not have chaotic phenomena. With the increase of meshing damping ratio, the state change of the shunt stage is similar to the inverse period doubling bifurcation process, while the confluence stage may also maintain a periodic motion state at a smaller damping ratio. Therefore, reasonable excitation frequency and meshing damping ratio can make the system jump out of chaos and remain stable, which provides a theoretical basis for fault prevention and fault diagnosis of face gear split-confluence transmission system. At the same time, it provides a reference for the design of face gear split-confluence transmission system and the selection of lubrication conditions.
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Keywords:
- face gear /
- bearing deformation /
- nonlinear dynamics /
- dynamic response /
- bifurcation theory
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引 言
面齿轮传动系统通常由圆柱齿轮和共轭面齿轮构成, 至今已演变出多种构型, 例如非正交面齿轮、偏置面齿轮等, 具有重合度大、分流效果好、结构紧凑以及噪声低等多方面优点, 被应用于直升机传动等特殊领域中.
一些学者针对面齿轮设计理论、相关刀具设计和接触特性等方面进行了研究. Wang等[1-2]提出了一种蜗杆滚齿加工面齿轮的方法以及面齿轮的剃齿加工方法. Zhou等[3]提出一种新的几何分析方法, 用于缩小面齿轮CAD模型与数控铣削之间的差异. Zhou等[4]提出了一种盘形砂轮加工面齿轮的磨削方法. 高凌锋等[5]推导出了一种偏置正交弧线齿面齿轮, 计算了齿面根切位置. Wang等[6]提出一种面齿轮齿面的测量模型, 实现了数字化齿面接触分析. 王延忠等[7-8]针对面齿轮提出了一种偏置曲线修整方法; 考虑载荷作用下面齿轮传动系统内部结构及轮齿变形, 对其振动特性进行了研究.
在齿轮建模方面, 许多学者进行了深入研究. Li等[9]研究了润滑情况下齿轮的损耗以及动态传动误差的变化. Wang等[10]研究了铁路机车齿轮裂纹系统的参数共振和稳定性. 林腾蛟等[11]研究了锥−平行轴−行星多级齿轮传动系统非线性振动特性. Liu等[12]比较了点接触和线接触条件下面齿轮啮合刚度, 通过理论计算得出点接触条件下轮齿整体刚度低于线接触条件.
随着对齿轮齿面创成和啮合特性的一系列研究, 学者开始对齿轮传动的静、动力学行为进行研究. 靳广虎等[13]针对面齿轮−圆柱齿轮分流传动构型, 研究了输入轴和分扭轴刚度对均载特性的影响. 李晓贞等[14]考虑齿面摩擦, 研究了齿面摩擦力对面齿轮传动系统振动特性的影响. Wang等[15]基于频域特征识别对GTF齿轮箱的非线性动力学进行了分析. Xiong等[16]研究了齿侧间隙对啮合刚度以及非线性动力学的影响. Chen等[17]对齿轮转子轴承系统的非线性动态特性进行了研究. 常乐浩等[18]建立了齿轮承载接触分析修正模型, 提出了已知齿面误差分布时啮合刚度和综合啮合误差的确定方法. Yang等[19]提出一种改进的裂纹齿轮啮合刚度计算方法, 并使用有限元法进行了验证. 宁志远等[20]将齿轮磨损与动力学耦合, 对磨损情况下的行星齿轮动力学进行了研究. 石建飞等[21]对两空间耦合下的齿轮传动系统的多稳态特性进行了研究. Hu等[22]研究了平均载荷以及齿侧间隙对面齿轮传动系统动力学响应的影响. Chen等[23]研究了6自由度面齿轮传动系统的非线性动态特性. Mo等[24-25]研究了非正交面齿轮−轴承传动系统的全局动力学行为, 对双输入面齿轮传动系统的固有特性进行了分析. 林何等[26-27]针对由弧齿锥齿轮和行星轮系构成的传动系统, 研究系统参数对其均载特性的影响; 建立齿轮−轴承系统含间隙的非线性振动模型, 研究了其混沌响应减振控制问题. Peng等[28]引入SAM方法来抑制面齿轮传动的振动. Dong等[29-30]针对面齿轮−轴−轴承系统进行了动力学分析, 同时对同心面齿轮系统的振动特性以及负载分配特性进行了研究. 上述学者针对齿轮传动系统开展了一系列研究, 但是在面齿轮动态特性研究领域, 其研究对象多为单级面齿轮副, 针对面齿轮分汇流系统的动态特性研究则相对较少.
针对航空传动系统空间狭小、传动形式灵活多变等特点, 采用非正交偏置面齿轮构成的传动系统有望在狭小空间内实现动力传输, 因此对其传动系统的非线性动力学行为的研究有重大意义. 本文以航空非正交偏置面齿轮分汇流系统为研究对象, 建立含轴承游隙、齿侧间隙和综合传递误差等因素的航空非正交偏置面齿轮传动系统动力学模型, 研究不同激励频率与不同啮合阻尼比条件下系统非线性动力学行为.
1. 传动系统动力学模型
根据轴交角的不同, 可将面齿轮分为正交、非正交面齿轮, 其中正交面齿轮的轴交角为90°, 而非正交面齿轮的轴交角可根据需要进行设计, 使得传动方案的布置更加灵活多变; 根据圆柱齿轮轴线是否与面齿轮轴线相交可分为偏置和非偏置面齿轮, 应用偏置面齿轮可实现交错轴之间的动力传输.
本文研究对象是一种由直齿轮、非正交偏置面齿轮以及斜齿轮构成的服务于航空领域的特殊传动系统, 此外轴承也被考虑在内. 此航空非正交偏置面齿轮分汇流传动系统动力学模型如图1所示, 扭矩由直齿轮1输入, 经面齿轮2, 3分流, 后经斜齿轮4, 5汇流到斜齿轮6输出. 所涉及齿轮的部分参数如下: 齿轮质量分别为1.8, 18.2, 18.2, 3.5, 3.5和16.7 kg, 模数m1 = m2 = m3 = mn4 = mn5 = mn6 = 4 mm, z1 = 28, z2 = z3 = 85, z4 = z5 = 37, z6 = 87, 法向压力角均为25°, 斜齿轮螺旋角为10°, 杨氏模量为209 GPa, 泊松比为0.27. 所涉及轴承的部分参数如下, 轴承11和轴承12的内、外径分别为70 mm和125 mm, 宽度为20 mm; 轴承21, 22, 31, 32的内、外径分别为80 mm和140 mm, 宽度为26 mm; 轴承41和轴承42的内、外径分别为100 mm和180 mm, 宽度为34 mm.
1.1 滚动轴承动力学模型
滚动轴承由滚动体、保持架以及内、外圈构成. 假设在该传动系统中, 轴承外圈与轴承座固接, 其线速度为零; 内圈与传动轴刚性连接; 滚动体均匀分布在沟槽中, 工作时, 滚动体与滚道始终保持纯滚动状态; 则轴承中第i个滚动体在某一时刻的转动角度θi可表示为
$$ {\theta _i}(t) = \frac{{2{\text{π}} (i - 1)}}{{{Z_b}}} + \frac{{{r_{bn}}{\omega _{bn}}}}{{{r_{bn}} + {r_{bw}}}}t $$ (1) 式中, Zb为滚动体数量; rbn和rbw分别为内、外圈半径; ωbn为内圈角速度, 而外圈角速度为0. 轴承滚动体变形关系如图2所示. 变形后接触角γ1可表示为
$$ \left. \begin{split} &{l_1} = \sqrt {({l_0}\sin {\gamma _0} + z + {r_{bw}}{\delta _\theta }\cos {\theta _i})^2 + {({l_0}\cos {\gamma _0} + x\cos {\theta _i} + y\sin {\theta _i})^2}} \\ &{\gamma _1} = \arctan \dfrac{{{l_0}\sin {\gamma _0} + z + {r_{bw}}{\delta _\theta }\cos {\theta _i}}}{{{l_0}\cos {\gamma _0} + x\cos {\theta _i} + y\sin {\theta _i}}} \end{split} \right\} $$ (2) 根据Hertz接触理论, 单个滚动体与内外圈之间轴向和径向力表示为
$$ {F_{ab}} = {K_b}{\delta ^n}\sin {\gamma _1}H(\delta ){\text{ }},{\text{ }}{F_{rb}} = {K_b}{\delta ^n}\cos {\gamma _1}H(\delta ) $$ (3) 式中, Kb为轴承支承刚度; 对于球轴承n = 3/2, 滚子轴承n = 10/9; δ为滚动体变形量δ = l1−l0−c, c为轴承间隙; H(δ)为Heaviside函数. 根据上述结果, 可得到因滚动体变形而产生的作用力, 并将其分解到对应坐标系如下式
$$ \left. \begin{split} & {F_{bx}} = \sum\limits_{i = 1}^{{Z_{{b}}}} {{K_{{b}}}{\delta ^n}\cos {\gamma _1}H(\delta )\cos {\theta _i}} \\ &{F_{by}} = \sum\limits_{i = 1}^{{Z_{{b}}}} {{K_{{b}}}{\delta ^n}\cos {\gamma _1}H(\delta )\sin {\theta _i}} \\ & {F_{bz}} = \sum\limits_{i = 1}^{{Z_{{b}}}} {{K_{{b}}}{\delta ^n}\sin {\gamma _1}H(\delta )} \end{split} \right\} $$ (4) 1.2 齿轮副相对位移及动态啮合力
如图3所示, 以直齿轮1和面齿轮2为例, 推导面齿轮副的相对位移以及动态啮合力.
直齿轮为3自由度模型, 包括径向位移y1, z1以及绕轴线的旋转角位移θ1; 而面齿轮为4自由度模型, 包括径向位移x2, y2, z2以及绕轴线的旋转角位移θ2. 经推导得出, 直齿轮和面齿轮之间沿啮合线方向的相对位移xn12为
$$ \begin{split} {x_{n12}} = &({y_1} + {r_1}{\theta _1} - {y_2}\cos {\beta _m} - {r_2}{\theta _2}\cos {\beta _m} - \\ & {x_2}\sin {\beta _m}\sin {\gamma _m} - {z_2}\sin {\beta _m}\cos {\gamma _m})\cos {\alpha _n} + \\ & ({z_1} - {x_2}\cos {\gamma _m} - {z_2}\sin {\gamma _m})\sin {\alpha _n} + {e_{12}}(t) \end{split} $$ (5) 式中, r1为直齿轮分度圆半径; r2为面齿轮回转轴线到齿宽中点的垂直距离; βm为偏置角, 满足sinβm = ep/r2, ep为偏置距; γm为轴交角; e12(t)为综合传动误差.
齿侧间隙对齿轮动态特性有重要影响, 直齿轮1与面齿轮2之间齿侧间隙函数可表示为
$$ f({x_{n12}}) = \left\{ \begin{gathered} {x_{n12}} - {b_{m12}},{\text{ }}{x_{n12}} > {b_{m12}} \\ 0,{\text{ }}\left| {{x_{n12}}} \right| \leqslant {b_{m12}} \\ {x_{n12}} + {b_{m12}},{\text{ }}{x_{n12}} < - {b_{m12}} \\ \end{gathered} \right. $$ (6) 式中, bm12为半值齿侧间隙, 则沿啮合线方向的啮合力及其分量为
$$ \left. \begin{array}{l} {F_{n12}} = {K_{m12}}\left( t \right)f({x_{n12}}) + {C_{m12}}{{\dot x}_{n12}} \\ {F_{{y_1}}} = {F_{n12}}\cos {\alpha _n}\\ {F_{{z_1}}} = {F_{n12}}\sin {\alpha _n} \\ {F_{{x_2}}} = {F_{n12}}\sin {\alpha _n}\cos {\gamma _m} + {F_{n12}}\cos {\alpha _n}\sin {\beta _m}\sin {\gamma _m} \\ {F_{{y_2}}} = {F_{n12}}\cos {\alpha _n}\cos {\beta _m} \\ {F_{{z_2}}} = {F_{n12}}\sin {\alpha _n}\sin {\gamma _m} + {F_{n12}}\cos {\alpha _n}\sin {\beta _m}\cos {\gamma _m} \end{array} \right\} $$ (7) 式中, Km12(t)为直齿轮与非正交偏置面齿轮之间的时变啮合刚度; 而Cm12为二者之间的啮合阻尼; $ {C_{m12}} = 2{\zeta _m}\sqrt {{{{K_{a12}}{J_1}{J_2}} \mathord{\left/ {\vphantom {{{K_{a12}}{J_1}{J_2}} {({r_1}^2{J_1} + {r_2}^2{J_2})}}} \right. } {({r_1}^2{J_1} + {r_2}^2{J_2})}}} $; ζm为啮合阻尼比; J1, J2分别为直齿轮与面齿轮的转动惯量. 其余面齿轮副的相对位移以及各齿轮副之间啮合力可由相似的思路推导得到.
以斜齿轮4, 6为例推导斜齿轮副沿啮合线方向相对位移xn46
$$ \begin{split} {x_{n46}} = &({r_4}{\theta _4} - {r_6}{\theta _6} - {x_4} + {x_6})\cos \beta \cos {\alpha _{46}} + \\ & ({z_4} - {z_6})\sin \beta \cos {\alpha _{46}} - ({y_4} - {y_6})\sin {\alpha _{46}} + {e_{46}}(t) \end{split} $$ (8) 式中, r4和r6分别为斜齿轮4和6的分度圆半径; $ {\theta _4} $和$ {\theta _6} $为斜齿轮4, 6扭转位移; x4, x6, y4, y6, z4和z6为斜齿轮4, 6沿x, y, z轴的振动位移; α46为斜齿轮法面压力角; β为螺旋角; e46(t)为综合传动误差.
斜齿轮4, 6之间的动态啮合力及其分量为
$$ {F_{n46}} = {K_{m46}}\left( t \right)f({x_{n46}}) + {C_{m46}}{\dot x_{n46}} $$ (9) 式中, Km46(t)为斜齿轮副的时变啮合刚度; Cm46为啮合阻尼; $ {C_{m46}} = 2{\zeta _m}\sqrt {{{{K_{a46}}{J_4}{J_6}} \mathord{\left/ {\vphantom {{{K_{a46}}{J_4}{J_6}} {({r_4}^2{J_4} + {r_6}^2{J_6})}}} \right. } {({r_4}^2{J_4} + {r_6}^2{J_6})}}} $; ζm为啮合阻尼比; Ka46为啮合刚度均值; f(xn46)为齿侧间隙函数; J4和J6分别为斜齿轮4, 6的转动惯量. 同理, 其余斜齿轮副的相对位移以及动态啮合力可由相似的思路推导得到.
2. 非正交偏置面齿轮分汇流系统振动微分方程
基于集中参数法以及牛顿第二定律, 推导非正交偏置面齿轮分汇流传动系统的弯−扭耦合多自由度振动微分方程组.
视输入级部分轴承为2自由度模型, 忽略重力影响, 考虑其沿径向的振动位移, 由滚动体变形产生的轴承力由式(1) ~ 式(4)可得, 则输入级部分轴承的振动微分方程为
$$ \left. \begin{gathered} {m_{b1i}}{{\ddot y}_{b1i}} + {C_b}_{1iy}{{\dot y}_{b1i}} + {C_{s1y}}{{\dot y}_{b1j}} + {K_{s1y}}{y_{b1j}} = {F_{b1iy}} \\ {m_{b1i}}{{\ddot z}_{b1i}} + {C_b}_{1iz}{{\dot z}_{b1i}} + {C_{s1z}}{{\dot z}_{b1j}} + {K_{s1z}}{z_{b1j}} = {F_{b1iz}} \\ \end{gathered} \right\} $$ (10) 式中, mb为轴承质量; Cb为轴承支承阻尼; Ks, Cs分别为轴支承刚度和阻尼; 当i = 1, yb1j = yb11 − yb12; 当i = 2, yb1j = 2yb12 − yb11 − y1, zb1j = 2yb12 − yb11 − y1.
结合式(5) ~ 式(7)即可得到分流级齿轮振动微分方程组如下式
$$ \left. \begin{split} & {m_1}{{\ddot y}_1} + {C_{s1y}}({{\dot y}_1} - {{\dot y}_{b12}}) + {K_{s1y}}({y_1} - {y_{b12}}) = \\ & \qquad - {F_{n12}}\cos {\alpha _n} - {F_{n13}}\cos {\alpha _n} \\ & {m_1}{{\ddot z}_1} + {C_{s1y}}({{\dot z}_1} - {{\dot z}_{b12}}) + {K_{s1y}}({z_1} - {z_{b12}}) = \\ & \qquad - {F_{n12}}\sin {\alpha _n} - {F_{n13}}\sin {\alpha _n} \\ & {J_1}{{\ddot \theta }_1} = {T_1} - {r_1}({F_{n12}} + {F_{n13}})\cos {\alpha _n} \end{split} \right\} $$ (11) $$ \left. \begin{gathered} {m_2}{{\ddot x}_2} + {C_{s2x}}({{\dot x}_2} - {{\dot x}_{b21}}) + {K_{s2x}}({x_2} - {x_{b21}}) = {F_{{x_2}}} \\ {m_2}{{\ddot y}_2} + {C_{s2y}}({{\dot y}_2} - {{\dot y}_{b21}}) + {K_{s2y}}({y_2} - {y_{b21}}) = {F_{{y_2}}} \\ {m_2}{{\ddot z}_2} + {C_{s2z}}({{\dot z}_2} - {{\dot z}_{b21}}) + {K_{s2z}}({z_2} - {z_{b21}}) = {F_{{z_2}}} \\ {J_2}{{\ddot \theta }_2} + {C_{t2}}({{\dot \theta }_2} - {{\dot \theta }_4}) + {K_{t2}}({\theta _2} - {\theta _4}) = {r_2}{F_{n12}}\cos {\alpha _n} \end{gathered} \right\} $$ (12) $$ \left. \begin{array}{l} {m_3}{{\ddot x}_3} + {C_{s3x}}({{\dot x}_3} - {{\dot x}_{b31}}) + {K_{s3x}}({x_3} - {x_{b31}}) = {F_{3x}} \\ {m_3}{{\ddot y}_3} + {C_{s3y}}({{\dot y}_3} - {{\dot y}_{b31}}) + {K_{s3y}}({y_3} - {y_{b31}}) = {F_{3y}} \\ {m_3}{{\ddot z}_3} + {C_{s3z}}({{\dot z}_3} - {{\dot z}_{b31}}) + {K_{s3z}}({z_3} - {z_{b31}}) = {F_{3z}} \\ {J_3}{{\ddot \theta }_3} + {C_{t3}}({{\dot \theta }_3} - {{\dot \theta }_5}) + {K_{t3}}({\theta _3} - {\theta _5}) = {M_3} \end{array} \right\} $$ (13) $$ \left. \begin{gathered} {F_{3x}} = {F_{n13}}\sin {\alpha _n}\cos {\gamma _m} + {F_{n13}}\cos {\alpha _n}\sin {\beta _m}\sin {\gamma _m} \\ {F_{3z}} = {F_{n13}}\sin {\alpha _n}\sin {\gamma _m} + {F_{n13}}\cos {\alpha _n}\sin {\beta _m}\cos {\gamma _m} \\ {F_{3y}} = {F_{n13}}\cos {\alpha _n}\cos {\beta _m}{\text{ }},{\text{ }}{M_3} = {r_3}{F_{n13}}\cos {\alpha _n} \end{gathered} \right\} $$ (14) 式中, Kt和Ct分别为传动轴的扭转刚度和扭转阻尼; θ4, θ5分别为斜齿轮4和5绕各自回转轴的旋转角位移. 同样地, 可得到其余轴承以及输出级斜齿轮振动微分方程组如下
$$ \left. \begin{gathered} {m_{b21}}{{\ddot x}_{b21}} + {C_b}_{21x}{{\dot x}_{b21}} + {C_{s2x}}(2{{\dot x}_{b21}} - {{\dot x}_2} - {{\dot x}_{b22}}) + \\ \qquad {K_{s2x}}(2{x_{b21}} - {x_2} - {x_{b22}}) = {F_{b21y}} \\ {m_{b21}}{{\ddot y}_{b21}} + {C_b}_{21y}{{\dot y}_{b21}} + {C_{s2y}}(2{{\dot y}_{b21}} - {{\dot y}_2} - {{\dot y}_{b22}}) + \\ \qquad {K_{s2y}}(2{y_{b21}} - {y_2} - {y_{b22}}) = {F_{b21y}} \\ {m_{b21}}{{\ddot z}_{b21}} + {C_b}_{21z}{{\dot z}_{b21}} + {C_{s2z}}(2{{\dot z}_{b21}} - {{\dot z}_2} - {{\dot z}_{b22}}) + \\ \qquad {K_{s2x}}(2{z_{b21}} - {z_2} - {z_{b22}}) = {F_{b21z}} \\ \end{gathered} \right\} $$ (15) $$ \left. \begin{gathered} {m_{b22}}{{\ddot x}_{b22}} + {C_b}_{22x}{{\dot x}_{b22}} + {C_{s2x}}(2{{\dot x}_{b22}} - {{\dot x}_4} - {{\dot x}_{b21}}) + \\ \qquad {K_{s2x}}(2{x_{b22}} - {x_4} - {x_{b21}}) = {F_{b22y}} \\ {m_{b22}}{{\ddot y}_{b22}} + {C_b}_{22y}{{\dot y}_{b22}} + {C_{s2y}}(2{{\dot y}_{b22}} - {{\dot y}_4} - {{\dot y}_{b21}}) + \\ \qquad {K_{s2y}}(2{y_{b22}} - {y_4} - {y_{b21}}) = {F_{b22y}} \\ {m_{b22}}{{\ddot z}_{b22}} + {C_b}_{22z}{{\dot z}_{b22}} + {C_{s2z}}(2{{\dot z}_{b22}} - {{\dot z}_4} - {{\dot z}_{b21}}) + \\ \qquad {K_{s2x}}(2{z_{b22}} - {z_4} - {z_{b21}}) = {F_{b22z}} \\ \end{gathered} \right\} $$ (16) $$ \left. \begin{gathered} {m_{b31}}{{\ddot x}_{b31}} + {C_b}_{31x}{{\dot x}_{b31}} + {C_{s3x}}(2{{\dot x}_{b31}} - {{\dot x}_3} - {{\dot x}_{b32}}) + \\ \qquad {K_{s3x}}(2{x_{b31}} - {x_3} - {x_{b32}}) = {F_{b31y}} \\ {m_{b31}}{{\ddot y}_{b31}} + {C_b}_{31y}{{\dot y}_{b31}} + {C_{s3y}}(2{{\dot y}_{b31}} - {{\dot y}_3} - {{\dot y}_{b32}}) + \\ \qquad {K_{s3y}}(2{y_{b31}} - {y_3} - {y_{b32}}) = {F_{b31y}} \\ {m_{b31}}{{\ddot z}_{b31}} + {C_b}_{31z}{{\dot z}_{b31}} + {C_{s3z}}(2{{\dot z}_{b31}} - {{\dot z}_3} - {{\dot z}_{b32}}) + \\ \qquad {K_{s3x}}(2{z_{b31}} - {z_3} - {z_{b32}}) = {F_{b31z}} \\ \end{gathered} \right\} $$ (17) $$ \left. \begin{gathered} {m_{b32}}{{\ddot x}_{b32}} + {C_b}_{32x}{{\dot x}_{b32}} + {C_{s3x}}(2{{\dot x}_{b32}} - {{\dot x}_5} - {{\dot x}_{b31}}) + \\ \qquad {K_{s3x}}(2{x_{b32}} - {x_5} - {x_{b31}}) = {F_{b32y}} \\ {m_{b32}}{{\ddot y}_{b32}} + {C_b}_{32y}{{\dot y}_{b32}} + {C_{s3y}}(2{{\dot y}_{b32}} - {{\dot y}_5} - {{\dot y}_{b31}}) + \\ \qquad {K_{s3y}}(2{y_{b32}} - {y_5} - {y_{b31}}) = {F_{b32y}} \\ {m_{b32}}{{\ddot z}_{b32}} + {C_b}_{32z}{{\dot z}_{b32}} + {C_{s3z}}(2{{\dot z}_{b32}} - {{\dot z}_5} - {{\dot z}_{b31}}) + \\ \qquad {K_{s3x}}(2{z_{b32}} - {z_5} - {z_{b31}}) = {F_{b32z}} \\ \end{gathered} \right\} $$ (18) $$ \left. \begin{gathered} {m_4}{{\ddot x}_4} + {C_{s2x}}({{\dot x}_4} - {{\dot x}_{b22}}) + {K_{s2x}}({x_4} - {x_{b22}}) = \\[-4pt] \qquad - {F_{n46}}\cos {\alpha _{46}}\cos \beta \\[-4pt] {m_4}{{\ddot y}_4} + {C_{s2y}}({{\dot y}_4} - {{\dot y}_{b22}}) + {K_{s2y}}({y_4} - {y_{b22}}) = \\[-4pt] \qquad - {F_{n46}}\sin {\alpha _{46}} \\ {m_4}{{\ddot z}_4} + {C_{s2z}}({{\dot z}_4} - {{\dot z}_{b22}}) + {K_{s2z}}({z_4} - {z_{b22}}) = \\[-4pt] \qquad - {F_{n46}}\cos {\alpha _{46}}\sin \beta \\[-4pt] {J_4}{{\ddot \theta }_4} + {C_{t2}}({{\dot \theta }_4} - {{\dot \theta }_2}) + {K_{t2}}({\theta _4} - {\theta _2}) = \\[-4pt] \qquad - {r_4}\cos \beta {F_{n46}}\cos {\alpha _{46}} \\ \end{gathered} \right\} $$ (19) $$ \left. \begin{gathered} {m_5}{{\ddot x}_5} + {C_{s3x}}({{\dot x}_5} - {{\dot x}_{b32}}) + {K_{s3x}}({x_5} - {x_{b32}}) = \\[-4pt] \qquad - {F_{n56}}\cos {\alpha _{56}}\cos \beta \\[-4pt] {m_5}{{\ddot y}_5} + {C_{s3y}}({{\dot y}_5} - {{\dot y}_{b32}}) + {K_{s3y}}({y_5} - {y_{b32}}) = \\[-4pt] \qquad - {F_{n56}}\sin {\alpha _{56}} \\[-4pt] {m_5}{{\ddot z}_5} + {C_{s3z}}({{\dot z}_5} - {{\dot z}_{b32}}) + {K_{s3z}}({z_5} - {z_{b32}}) = \\[-4pt] \qquad - {F_{n56}}\cos {\alpha _{56}}\sin \beta \\[-4pt] {J_5}{{\ddot \theta }_5} + {C_{t3}}({{\dot \theta }_5} - {{\dot \theta }_3}) + {K_{t3}}({\theta _5} - {\theta _3}) = \\[-4pt] \qquad - {r_4}\cos \beta {F_{n56}}\cos {\alpha _{56}} \\ \end{gathered} \right\} $$ (20) $$ \left. \begin{gathered} {m_6}{{\ddot x}_6} + {C_{s4x}}(2{{\dot x}_6} - {{\dot x}_{b41}} - {{\dot x}_{b42}}) + {K_{s4y}}(2{x_6} - {x_{b41}}- \\[-2pt] \quad {x_{b41}}) = {F_{n46}}\cos {\alpha _{46}}\cos \beta + {F_{n56}}\cos {\alpha _{56}}\cos \beta \\[-2pt] {m_6}{{\ddot y}_6} + {C_{s4y}}(2{{\dot y}_6} - {{\dot y}_{b41}} - {{\dot y}_{b42}}) + {K_{s4y}}(2{y_6} - {y_{b41}} - \\[-2pt] \quad {y_{b42}}) = {F_{n46}}\sin {\alpha _{46}} + {F_{n56}}\sin {\alpha _{56}} \\[-2pt] {m_6}{{\ddot z}_6} + {C_{s4z}}(2{{\dot z}_6} - {{\dot z}_{b41}} - {{\dot z}_{b42}}) + {K_{s4z}}(2{z_6} - {z_{b41}} - \\[-2pt] \quad {z_{b42}}) = {F_{n46}}\cos {\alpha _{46}}\sin \beta + {F_{n56}}\cos {\alpha _{56}}\sin \beta \\[-2pt] {J_6}{{\ddot \theta }_6} = - {T_6} + {r_6}({F_{n46}}\cos {\alpha _{46}} + {F_{n56}}\cos {\alpha _{56}})\cos \beta \end{gathered} \right\} $$ (21) $$ \left. \begin{gathered} {m_{b4j}}{{\ddot x}_{b4j}} + {C_b}_{4jx}{{\dot x}_{b4j}} + {C_{s4x}}({{\dot x}_{b4j}} - {{\dot x}_6}) + \\[-2pt] \qquad {K_{s4x}}({x_{b4j}} - {x_6}) = {F_{b4jy}} \\[-2pt] {m_{b4j}}{{\ddot y}_{b4j}} + {C_b}_{41y}{{\dot y}_{b4j}} + {C_{s4y}}({{\dot y}_{b4j}} - {{\dot y}_6}) + \\[-2pt] \qquad {K_{s4y}}({y_{b4j}} - {y_6}) = {F_{b4jy}} \\[-2pt] {m_{b4j}}{{\ddot z}_{b4j}} + {C_b}_{41z}{{\dot z}_{b4j}} + {C_{s4z}}({{\dot z}_{b4j}} - {{\dot z}_6}) + \\[-2pt] \qquad {K_{s4x}}({z_{b4j}} - {z_6}) = {F_{b4jz}} \\ \qquad (j = 1,2) \\ \end{gathered} \right\} $$ (22) 为避免上述微分方程组中部分参数的数量级相差过大导致求解速度降低甚至求解失败, 因此引入无量纲参数τ = ωnt和半值齿侧间隙bm12对上述方程组进行量纲一处理, 其中$ {\omega _n} = \sqrt {{K_{a12}}/{m_{e12}}} $, $ {m_{e12}} = {J_1}{J_2}/({J_1}r_2^2 + {J_2}r_1^2) $. 由于篇幅限制, 对于无量纲后得到的振动微分方程组不再展示. 此外, 本文涉及的研究对象是一个具有时变参数的非线性系统, 使用解析法求解上述微分方程组是相对困难的, 因此在求解过程中采用龙格库塔数值方法进行计算.
3. 非正交偏置面齿轮传动系统非线性行为
3.1 激励频率对系统非线性动力学行为的影响
此部分研究激励频率Ω变化对非正交偏置面齿轮系统非线性动力学行为的影响. 当Ω = 1.9时, 分流级、汇流级齿轮副啮合线位移的时域响应、相图以及Poincaré映射分别如图4和图5所示.
由图4可知, 当Ω = 1.9时, 位移xn12和xn13具有相似的运动状态, 此时分流级啮合线位移表现为倍周期运动状态, 结合其相图以及Poincaré映射可知, 此时分流级处于多周期状态; 同样地, 从时域图、相图以及庞加莱截面可知, 位移xn46和xn56的运动状态也是类似的, 并且汇流级两条啮合线位移均处于周期2运动状态.
利用快速傅里叶变换(FFT), 得到啮合线位移xn12, xn13, xn46和xn56的频域响应如图6所示, 其中ωT为无量纲频率, A为无量纲波动幅值. 由图6可知, 各啮合线位移的频率成分复杂, 且随激励频率的增大, 其波动幅值不断变化.
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图 6 系统随激励频率Ω增大的频域响应(续)Figure 6. The frequency domain response of the system with the increase of excitation frequency Ω (continued)以激励频率Ω为分岔参数, 绘制啮合线位移xn12, xn13, xn46和xn56随激励频率Ω增大的李雅普诺夫指数和分岔图分别如图7和图8所示. 从图7和图8可知, 分流级啮合线位移xn12和xn13的状态变化规律相似, 当Ω∈(1.5, 1.75)时, 分流级处于单周期运动状态, 在Ω = 1.75处发生分岔, 当Ω∈(1.75, 2)时, 处于多周期运动状态, 对应李雅普诺夫指数小于0; 当激励频率继续增大, 即Ω∈(2, 2.82)时, 分流级处于混沌状态, 对应李雅普诺夫指数大于0; 结合图7和图8可知, 而当Ω∈(2.82, 2.9)时, 分流级跳出混沌. 而对于汇流级啮合线位移xn46和xn56, 当Ω∈(1.5, 1.71)时, 处于单周期运动状态; 当Ω = 1.71时, 运动状态产生突变, 当Ω∈(1.71, 1.98)时, 汇流级处于多周期运动状态; 于Ω = 1.98处突变为概周期运动状态.
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图 8 系统随激励频率Ω增大的分岔图(续)Figure 8. The bifurcation diagram of the system with the increase of excitation frequency Ω (continued)3.2 啮合阻尼比对系统非线性动力学行为的影响
该部分研究啮合阻尼比变化对系统非线性动力学行为的影响. 当ζm = 0.05时, 系统各级齿轮副之间啮合线位移xn12, xn13, xn46和xn56的时域响应以及Poincaré映射如图9和图10所示. 此时系统各部分的时域图中存在多个波峰, 且运动显然无周期性, 而此时Poincaré映射同样表现出大量混乱、无序的吸引子共存的形式, 表明当ζm = 0.05时, 此系统各部分均处于混沌状态.
同样地, 使用FFT得到不同啮合阻尼比条件下, 系统频域响应如图11所示. 从图中可以看出, 其频率成分主要是啮合频率及其倍频, 当啮合阻尼比较小时, 系统各部分均存在多种频率信号, 随啮合阻尼比的增大, 分流级的低频信号逐渐消失, 其频域信号组成变得简单, 但汇流级仍然有较强的低频信号.
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图 11 系统随啮合阻尼比ζm增大的频域响应Figure 11. The frequency domain response of the system with the increase of meshing damping ratio ζm以啮合阻尼比为分岔参数, 绘制系统随啮合阻尼比增大的李雅普诺夫指数、以及分岔图分别如图12和图13所示. 随啮合阻尼比增大, 分流级位移xn12与xn13的状态演化可视为逆倍化周期分岔过程. 啮合阻尼比较小时, 分流级处于混沌状态; 由分岔图可知, 当ζm增大到0.072时, 开始出现由混沌向多周期运动转变的趋势; 随后于ζm = 0.093处突变为周期2T运动; 当啮合阻尼比继续增大, 分流级于ζm = 0.124处转化为单周期运动, 随后一直保持. 如图13所示, 对于分流级的两条啮合线, 其跳出混沌的临界点分别在ζm = 0.083以及ζm = 0.089 (对应李雅普诺夫指数小于0).
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图 13 系统随激励频率Ω增大的分岔图Figure 13. The bifurcation diagram of the system with the increase of excitation frequency Ω与分流级啮合线位移的变化规律有所不同, 随啮合阻尼比ζm增大的过程中, 汇流级啮合线位移的运动状态不总是变得更加简单. 由图12和图13可知, 当ζm较小时, 汇流级在较大范围内处于混沌状态, 但在混沌运动中也可能会出现小范围的窗口使其跳出混沌运动(例如ζm在0.102 ~ 0.108区间内, 此时汇流级两条啮合线的李雅普诺夫指数小于零); 随后当啮合阻尼比继续增大到0.15时, 汇流级跳出混沌, 此后李雅普诺夫指数小于零, 系统保持稳定的单周期运动.
4. 结论
(1)航空非正交偏置面齿轮分汇流传动系统考虑轴承滚动体变形、齿侧间隙和综合传递误差等因素时表现出丰富的非线性动力学行为, 结合时域图、频域图、相图、Poincaré映射以及分岔理论对其非线性动力学行为的演化进行了研究.
(2)系统随激励频率的变化表现出混沌和分岔等非线性行为. 对于分流级, 当激励频率在(1.76, 2)区间内发生分岔现象, 在(2, 2.82)区间内处于混沌状态, 在(1.5, 1.76)∪(2.82, 2.9)区间内保持单周期运动状态; 对于汇流级, 当激励频率在(1.5, 1.71)区间内保持单周期运动状态, 在(1.71, 1.98)区间内发生分岔现象, 在(1.98, 2.9)区间内表现为概周期运动状态; 选取合理的激励频率能获得相对稳定的状态, 为面齿轮分汇流传动系统的故障预防、故障诊断提供了理论基础.
(3)系统随啮合阻尼比的变化也表现出混沌和分岔等非线性行为. 啮合阻尼比的增大总是使得分流级运动状态变得更加简单, 其状态变化可视为逆倍化周期分岔过程; 与分流级不同, 虽然当啮合阻尼比增大到较大值将使汇流级部分跳出混沌, 但是当啮合阻尼比较小时, 也可能出现特定区间ζm∈(0.102, 0.108), 使其达到概周期运动状态; 为面齿轮分汇流传动系统的设计和润滑条件的选择提供了参考依据.
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图 6 系统随激励频率Ω增大的频域响应(续)
Figure 6. The frequency domain response of the system with the increase of excitation frequency Ω (continued)
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图 8 系统随激励频率Ω增大的分岔图(续)
Figure 8. The bifurcation diagram of the system with the increase of excitation frequency Ω (continued)
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图 11 系统随啮合阻尼比ζm增大的频域响应
Figure 11. The frequency domain response of the system with the increase of meshing damping ratio ζm
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