Elsevier

Journal of Sound and Vibration
声音与振动》杂志

Volume 475, 9 June 2020, 115269
第 475 卷,2020 年 6 月 9 日,115269
Journal of Sound and Vibration

Nonlinear analysis of a geared rotor system supported by fluid film journal bearings
由流体薄膜轴颈轴承支撑的齿轮转子系统的非线性分析

https://doi.org/10.1016/j.jsv.2020.115269Get rights and content 获取权利和内容

Abstract 摘要

This paper presents a novel approach for modeling and analyzing a geared rotor-bearing system including nonlinear forces in the gear set and the supporting fluid film journal bearings. The rotordynamics system model has five degrees of freedom that define the transverse displacements of the shaft-gear centerlines and the relative displacement of the gear tooth contact point. The journal bearing nonlinear forces are obtained via a solution of Reynolds equation for lubricant film pressure utilizing the finite element method. Co-existing, steady-state, autonomous and non-autonomous responses are obtained in an accurate and computationally efficient manner utilizing the multiple shooting and continuation algorithms. This yields the full manifolds of the multiple bifurcation system. Chaos is identified with maximum Lyapunov exponents, frequency spectra, Poincaré attractors, etc. The results reveal a dependence of the gear set contact conditions and system nonlinear response characteristics, i.e. jump, co-existing responses, subharmonic resonances and chaos on the choice of journal bearing parameters. The results also show that Hopf bifurcations, which occur along with oil whirl in a journal bearing system, can be attenuated by increasing the gear torque.
本文提出了一种新方法,用于模拟和分析齿轮转子轴承系统,包括齿轮组和支撑流体薄膜轴颈轴承中的非线性力。转子动力学系统模型有五个自由度,分别定义了轴-齿轮中心线的横向位移和齿轮齿接触点的相对位移。轴颈轴承的非线性力是通过利用有限元法求解润滑油膜压力的雷诺方程得到的。利用多重拍摄和延续算法,以精确且计算效率高的方式获得共存、稳态、自主和非自主响应。这产生了多重分岔系统的完整流形。混沌可通过最大 Lyapunov 指数、频率谱、Poincaré 吸引子等进行识别。结果表明,齿轮组接触条件和系统非线性响应特征,即跳跃、共存响应、亚谐共振和混沌,取决于轴颈轴承参数的选择。结果还表明,在轴颈轴承系统中伴随油旋发生的霍普夫分岔可以通过增加齿轮扭矩来减弱。

Keywords 关键词

Nonlinear gear dynamics
Multiple shooting method
Jump phenomena
Multiple co-existing response
Chaos
Journal bearing

非线性齿轮动力学多重射击法跳跃现象多重共存响应混沌期刊轴承

1. Introduction 1.导言

Gearing system speeds and operating torques continue to increase in high-performance machinery. This amplifies the effects of nonlinearities in the gears including tooth backlash and time-varying mesh stiffness. Backlash describes the intentional clearance provided between mating teeth to prevent binding and to include a thin lubricant film between the teeth for heat removal and reduced wear. Backlash causes intermittent loss of contact between the teeth creating a nonlinear force and torque. The mesh stiffness varies periodically with time due to the variation of the number of tooth pairs in contact, and the variation of the point of contact along with the tooth profiles. The time-varying stiffness of the meshing teeth may lead to parametric resonances, which are principal sources of internal excitations and vibrations in gear transmission systems. The backlash forces and time-varying stiffness interact yielding a complex nonlinear, parametrically excited system with both torsional and lateral vibration. Accurate and computationally efficient gear dynamic models, including nonlinear forces and parametric excitations, are required for the effective design of gear sets and the machinery in which they form a critical component.
在高性能机械中,齿轮系统的速度和工作扭矩不断提高。这放大了齿轮中的非线性效应,包括齿隙和随时间变化的啮合刚度。齿隙指的是啮合齿之间故意留出的间隙,以防止咬合并在齿间形成一层薄薄的润滑油膜,从而达到散热和减少磨损的目的。齿隙会导致齿间间歇性失去接触,从而产生非线性力和扭矩。由于接触齿对数量的变化以及接触点随齿廓的变化,啮合刚度随时间周期性变化。啮合齿的时变刚度可能导致参数共振,而参数共振是齿轮传动系统内部激励和振动的主要来源。反向间隙力和时变刚度相互作用,产生了一个复杂的非线性参数激励系统,并伴有扭转和横向振动。要有效设计齿轮组及其关键部件机械,就必须建立精确且计算效率高的齿轮动态模型,包括非线性力和参数激励。

Significant prior research has been performed on the nonlinear dynamic response of geared systems. Kahraman and Singh [1] analyzed the effect of backlash on a single-degree-of-freedom gear model employing both analytical and numerical simulations. They validated their model by comparison with experimental results and found that the nonlinear characteristics caused chaotic and subsynchronous resonance responses. Kahraman and Singh [2] examined interactions between gear backlash nonlinearity and the bearing clearances and identified chaotic and subharmonic responses. Kahraman and Singh [3] included time-varying stiffness and clearance nonlinearity in their numerical model of geared systems and identified strong coupling effects between these characteristics. Blankenship and Kahraman [4] presented an experimental – analytical correlation study of a geared system including backlash nonlinearity and parametric excitation. Their predictions of co-existing solutions with the harmonic balance method were confirmed experimentally. Kahraman and Blankenship [5] observed subharmonic resonances in a geared system experiment, which were demonstrated to be strongly dependent on damping ratio and stiffness variation of the gear mesh. Kahraman and Blankenship [6] experimentally observed chaotic vibration, jump phenomena, and subharmonic response due to parametric and backlash excitations. Ranghothama and Narayanan [7] employed an incremental harmonic balance method, arc-length continuation, Floquet theory, and Lyapunov exponents to examine the bifurcation characteristics of a three-degree-of-freedom geared rotor-bearing model. Theodossiades and Natsiavas [8] introduced a new analytical method for a gear system with time-varying stiffness and backlash using perturbations techniques. Al-shyyab and Kahraman [9,10] investigated the nonlinear response of a multi-mesh gear system using a multi-term harmonic balance method. The effects of gear parameters on the nonlinear behavior were studied for both period-one and sub-harmonic motions. Liu et al. [11] analyzed the effect of gear mesh damping and backlash amplitude on the states of gear meshing and nonlinear behaviors of a gear pair. Yang et al. [12,13] predicted the nonlinear vibration of a gear system subjected to multi-frequency excitations utilizing a multiple time scales method. They confirmed the interaction between different harmonic excitations and the complex nonlinear behaviors caused by the multi-frequency excitations. Yang et al. [14] performed parametric studies to investigate the influence of the contact ratio, spacing error, transmitted load and mesh damping of a gear using a fifth-order Runge-Kutta method. Wang et al. [15] analyzed the effect of modulation internal excitation on the gear system and verified the accuracy of the prediction by comparing its results with the experimental measurements.
之前已经对齿轮系统的非线性动态响应进行了大量研究。Kahraman 和 Singh [1] 通过分析和数值模拟,分析了反向间隙对单自由度齿轮模型的影响。他们通过与实验结果的对比验证了自己的模型,并发现非线性特性会导致混乱和次同步共振响应。Kahraman 和 Singh [2] 研究了齿轮反向间隙非线性与轴承间隙之间的相互作用,确定了混乱和次谐波响应。Kahraman 和 Singh [3] 在他们的齿轮系统数值模型中加入了时变刚度和间隙非线性,并确定了这些特性之间的强耦合效应。Blankenship 和 Kahraman [4] 对包括反向间隙非线性和参数激励的齿轮系统进行了实验-分析关联研究。他们用谐波平衡法预测的共存解决方案在实验中得到了证实。Kahraman 和 Blankenship [5] 在齿轮系统实验中观察到了次谐波共振,并证明其与齿轮啮合的阻尼比和刚度变化密切相关。Kahraman 和 Blankenship [6] 通过实验观察到了参数和反向间隙激励引起的混乱振动、跳跃现象和次谐波响应。Ranghothama 和 Narayanan [7] 采用增量谐波平衡法、弧长延续、Floquet 理论和 Lyapunov 指数研究了三自由度齿轮转子轴承模型的分岔特性。Theodossiades 和 Natsiavas [8] 利用扰动技术为具有时变刚度和反向间隙的齿轮系统引入了一种新的分析方法。Al-shyyab 和 Kahraman [9,10] 采用多期谐波平衡法研究了多啮合齿轮系统的非线性响应。研究了齿轮参数对周期一运动和次谐波运动的非线性行为的影响。Liu 等人[11] 分析了齿轮啮合阻尼和齿隙振幅对齿轮啮合状态和齿轮副非线性行为的影响。Yang 等人[12,13]利用多时间尺度方法预测了齿轮系统在多频激励下的非线性振动。他们证实了不同谐波激励之间的相互作用以及多频激励引起的复杂非线性行为。Yang 等人[14]采用五阶 Runge-Kutta 方法对齿轮的接触比、间距误差、传递载荷和啮合阻尼的影响进行了参数研究。Wang 等人[15 [15] 分析了调制内激振对齿轮系统的影响,并将其结果与实验测量结果进行比较,验证了预测的准确性。

Nonlinear vibration in different types of gears has been investigated. Motahar et al. [16] performed a numerical, nonlinear dynamics study of a bevel gear system. Tip and root modifications were introduced to study their influence on gear vibration. Yang and Lim [17] developed a hypoid gear model considering time varying mesh stiffness, backlash nonlinearity and time-varying bearing stiffness. They showed that the backlash nonlinearity could suppress parametric instability induced by the time-varying bearing stiffness, under certain operating conditions. Wang and Lim [18] studied the effect of gear mesh stiffness asymmetry for the drive and coast sides of the hypoid gear system, and confirmed that the mesh stiffness at the drive side has more significant effect on the nonlinear dynamics. Ambarisha and Parker [19] investigated nonlinear dynamics of a planetary gear system. They applied the profile of the time-varying mesh stiffness obtained from a finite element analysis to improve accuracy. Zhao and Ji [20] performed numerical simulations of a wind turbine gearbox having two planetary gear trains. Complex nonlinear responses of the gearbox were shown to result from a time-varying mesh stiffness, backlash nonlinearity and static transmission error. Xinghui et al. [21] analyzed parametric resonance of a planetary gear subjected to speed fluctuations. The gear model considers time-varying mesh stiffness, and the instability boundaries for the fundamental and combinations resonances were derived based on a perturbation analysis.
对不同类型齿轮的非线性振动进行了研究。Motahar 等人[16] 对锥齿轮系统进行了非线性动力学数值研究。他们引入了齿尖和齿根修正,研究它们对齿轮振动的影响。Yang 和 Lim [17] 建立了一个准双曲面齿轮模型,考虑了时变啮合刚度、反隙非线性和时变轴承刚度。他们的研究表明,在一定的工作条件下,反齿隙非线性可以抑制时变轴承刚度引起的参数不稳定性。Wang 和 Lim [18] 研究了准双曲面齿轮系统驱动侧和沿岸侧齿轮啮合刚度不对称的影响,证实驱动侧的啮合刚度对非线性动力学的影响更大。Ambarisha 和 Parker [19] 研究了行星齿轮系统的非线性动力学。他们应用从有限元分析中获得的时变啮合刚度轮廓来提高精度。Zhao 和 Ji [20] 对具有两个行星齿轮系的风力涡轮机齿轮箱进行了数值模拟。结果表明,齿轮箱的复杂非线性响应是由时变啮合刚度、反向间隙非线性和静态传动误差造成的。Xinghui 等人[21] 分析了速度波动下行星齿轮的参数共振。该齿轮模型考虑了时变啮合刚度,并根据扰动分析得出了基本共振和组合共振的不稳定边界。

Some researchers have explored approaches to suppress vibrations induced by gear nonlinearities. Cheon's [22] simulation study investigated the effect of a one-way clutch to reduce the dynamic transmission error of a geared system. Cheon [23] employed a phasing approach to reduce time-varying mesh stiffness and the resulting vibration, especially at the fundamental resonance.
一些研究人员探索了抑制齿轮非线性引起的振动的方法。Cheon [22] 的模拟研究调查了单向离合器对减少齿轮系统动态传输误差的影响。Cheon[23]采用了一种相位方法来减少时变啮合刚度和由此产生的振动,尤其是在基本共振时。

Stochastic methods have been applied to study the effects of uncertainty in gear parameters. Bonori and Pellicano [24] utilized a stochastic model to analyze the effect of manufacturing error on nonlinear gear dynamics and showed that this could induce chaotic vibrations in the gear system. Wei et al. [25] included modeling uncertainties of a gear system, such as mesh stiffness and damping, and determined the resulting response levels using an interval harmonic balance method.
随机方法已被用于研究齿轮参数不确定性的影响。Bonori 和 Pellicano [24] 利用随机模型分析了制造误差对非线性齿轮动力学的影响,结果表明这会引起齿轮系统的混沌振动。Wei 等人[25] 将啮合刚度和阻尼等齿轮系统的建模不确定性包括在内,并使用区间谐波平衡法确定了由此产生的响应水平。

Various analytical and modeling methods have been applied to gear dynamics simulation. Kim et al. [26] investigated the effect of smoothing functions on clearance nonlinearity of an oscillator and showed how the adjustment of a regulating factor associated with the smoothing functions yielded more reliable predictions. Farshidianfar and Saghafi [27] applied a Melnikov type analysis to investigate homoclinic bifurcations and chaotic responses in a geared system. Gou et al. [28] employed a cell mapping theory to analyze the multi-parameter coupling characteristics of gear parameters. Li et al. [29] used an incremental harmonic balance method to analyze gear systems with internal and external periodic excitations.
各种分析和建模方法已被应用于齿轮动力学模拟。Kim 等人[26]研究了平滑函数对振荡器间隙非线性的影响,并展示了调整与平滑函数相关的调节因子如何产生更可靠的预测结果。Farshidianfar 和 Saghafi [27] 应用梅尔尼科夫分析法研究了齿轮系统中的同轴分岔和混沌响应。Gou 等人[28] 采用单元映射理论分析了齿轮参数的多参数耦合特性。Li 等人[29] 采用增量谐波平衡法分析了具有内部和外部周期性激励的齿轮系统。

Hydrodynamic journal bearings are widely employed in geared systems with high speed and load requirements due to their relatively high stiffness and damping. Theodossiades and Natsiavas [30] investigated the effect of gear and journal bearing parameters on bifurcation, chaos and oil whirl. They represented the journal bearing force with a finite-length impedance method. Baguet and Jacquenot [31] developed a finite element shaft model to study the interactions between a helical gear and a finite-length bearing and showed that a linearized bearing coefficient model does not provide accurate predictions of gear vibrations, especially at high speed and load conditions. Fargère and Velex [32] investigated the effects of the bearing oil inlet location and thermal response on the gear system dynamics. These effects change the journal static equilibrium position, which in turn alters the dynamic response of the system. Liu et al. [33] studied the interactions between tooth wedging effect and journal bearing clearance using the approximate short journal bearing theory. Simulation results showed that varying the operating speed or applied torque may cause the occurrence of oil whirl response of the rotordynamic systems. The effect of tooth wedging on the vibration level of the geared-rotor system is also presented.
流体动力轴颈轴承因其相对较高的刚度和阻尼,被广泛应用于具有高速和高负载要求的齿轮系统中。Theodossiades 和 Natsiavas [30] 研究了齿轮和轴颈轴承参数对分叉、混沌和油旋的影响。他们用有限长度阻抗法表示了轴颈轴承力。Baguet 和 Jacquenot [31] 建立了一个有限元轴模型来研究斜齿轮和有限长度轴承之间的相互作用,结果表明线性化轴承系数模型不能准确预测齿轮振动,尤其是在高速和负载条件下。Fargère 和 Velex [32] 研究了轴承进油位置和热反应对齿轮系统动力学的影响。这些影响改变了轴颈的静态平衡位置,进而改变了系统的动态响应。Liu 等人[33]利用近似短轴颈轴承理论研究了齿楔效应和轴颈轴承间隙之间的相互作用。仿真结果表明,改变运行速度或施加扭矩可能会导致旋转动力系统出现油旋响应。此外,还介绍了齿楔对齿轮-转子系统振动水平的影响。

Kim and Palazzolo [34,35] employed shooting with deflation to study the nonlinear response of a Jeffcott rotor supported by floating ring bearings. The effects of changing parameters such as bearing length-to-diameter (L/D) ratio and including the thermal effect of the lubricant were presented. Kim and Palazzolo [36] studied the bifurcation of a heavily loaded rotor with five-pad tilting pad bearings. A shooting/arc-length continuation approach was utilized to obtain quasi-periodic and chaotic motions, the latter being confirmed by maximum Lyapunov exponents.
Kim 和 Palazzolo [34,35]采用放气射击法研究了由浮动环轴承支撑的 Jeffcott 转子的非线性响应。研究介绍了改变轴承长径比 ( L/D ) 等参数以及润滑剂热效应的影响。Kim 和 Palazzolo [36] 研究了装有五片倾斜垫轴承的重载转子的分叉问题。利用射击/弧长延续方法获得了准周期和混沌运动,后者由最大 Lyapunov 指数证实。

Prior models for coupled gearset-bearing vibration generally utilized lower fidelity or steady-state bearing models, and presented results in less rigorous nonlinear dynamics formats. This may have been motivated by the high computational expense of employing higher fidelity bearing models and presenting results in advanced nonlinear dynamics formats. Bearing forces were typically represented with linear spring and damping constants, or were obtained using short bearing theory, with highly simplified oil film cavitation models. The simplified approaches may lead to significant prediction error especially for steady-state responses with orbits that are relatively large (>15%) with respect to the bearing clearance. The present approach provides a highly accurate, finite element-based solution of the finite-length, Reynold's equation accounting for cavitation at each time step in the numerical integration. Additionally, results are presented in advanced nonlinear dynamics formats including bifurcation diagrams, maximum Lyapunov exponent plots, and Poincaré attractor plots. Computation time is held within practical limits utilizing multiple shooting and continuation algorithms, and with the use of embedded C++ components in the MATLAB code, and parallel processing.
之前的齿轮组-轴承耦合振动模型一般采用较低保真度或稳态轴承模型,并以不太严格的非线性动力学格式呈现结果。这可能是由于采用保真度较高的轴承模型并以高级非线性动力学格式呈现结果的计算成本较高。轴承力通常使用线性弹簧和阻尼常数表示,或使用短轴承理论和高度简化的油膜气蚀模型获得。简化方法可能会导致显著的预测误差,尤其是对于相对于轴承游隙而言轨道相对较大(>15%)的稳态响应。本方法提供了基于有限元的高精度有限长度雷诺方程解决方案,在数值积分的每个时间步长上都考虑了气蚀。此外,计算结果以先进的非线性动力学形式呈现,包括分岔图、最大 Lyapunov 指数图和 Poincaré 吸引子图。利用多重射击和延续算法,以及在 MATLAB 代码中使用嵌入式 C++ 组件和并行处理,将计算时间控制在实际范围内。

The highlight and original contribution of the work is to provide a computationally efficient, high fidelity and rigorously presented modeling approach for the dynamics of the five-degree-of-freedom dual shaft-gear pair system supported on fluid film bearings. This approach involves finite-length bearing models, advanced multiple shooting and continuation methods, gear flexibility and transmission error effects, bifurcation and Poincaré attractor diagrams, and maximum Lyapunov exponents for identifying chaotic behavior. Finally, this approach is applied to parametric studies with varying journal bearing and gear mesh stiffness parameters.
这项工作的亮点和原创性贡献在于为流体薄膜轴承支撑的五自由度双轴齿轮对系统的动力学提供了一种计算效率高、保真度高且严谨的建模方法。该方法涉及有限长度轴承模型、先进的多重射击和延续方法、齿轮柔性和传动误差效应、分岔和泊恩卡莱吸引子图,以及用于识别混沌行为的最大 Lyapunov 指数。最后,该方法被应用于不同轴颈轴承和齿轮啮合刚度参数的参数研究。

2. Modeling of a geared rotor system supported by fluid film journal bearings
2.由流体薄膜轴颈轴承支撑的齿轮转子系统建模

2.1. Five-degree-of-freedom gear-bearing-rotor model
2.1.五自由度齿轮-轴承-转子模型

Fig. 1 shows a centered gear pair attached to parallel rotors that are each supported by fluid film journal bearings. The model is composed of two rigid rotors having mass elements mi, radii Ri and polar moments of inertia Ji. The subscript i denotes the driving (i=1) and driven (i=2) geared rotors.
图 1 显示了连接到平行转子上的对中齿轮,每个转子都由流体薄膜轴颈轴承支撑。该模型由两个刚性转子组成,其质量元素为 mi ,半径为 Ri ,极惯性矩为 Ji 。下标 i 表示驱动转子 ( i=1 ) 和从动转子 ( i=2 ) 。

Fig. 1
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Fig. 1. Gear set supported by hydrodynamic journal bearings.
图 1.由流体动力轴颈轴承支撑的齿轮组。

An external torque T1 is applied to the driving gear. A nonlinear mesh coupling consisting of tooth backlash and time-varying stiffness is modeled to transmit torque between driving and driven gears. The motion coordinates for the model include (θ1, θ2, x1, x2, y1, y2) as shown in Fig. 2.
外部扭矩 T1 作用于驱动齿轮。由齿隙和时变刚度组成的非线性啮合耦合被建模用于在驱动齿轮和从动齿轮之间传递扭矩。模型的运动坐标包括 ( θ1 , θ2 , x1 , x2 , y1 , y2 ) ,如图 2 所示。

Fig. 2
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Fig. 2. Spur gear pair model including hydrodynamic journal bearings.
图 2.包括流体动力轴颈轴承的正齿轮对模型。

The dynamic transmission error (DTE), δ(t) is given by(1)δ(t)=R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t)where er(t) represents the static transmission error. The analytical description of the time-varying mesh stiffness and the static transmission error can be expressed in the form of Fourier series as [3](2)km(t)=k0+i=1sik0cos(iωgtφi)er(t)=e0+j=1pje0cos(jωgtψj)where k0 is a mean mesh stiffness, e0 is a mean static transmission error, and si and pj are the amplitude of the Fourier series components. The phase angles of the Fourier series are represented by φi and ψj, respectively.
δ(t) 的动态传输误差 (DTE) 由 (1)δ(t)=R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t) 得出,其中 er(t) 代表静态传输误差。对时变网格刚度和静态传输误差的分析描述可以用傅里叶级数的形式表示 [3] (2)km(t)=k0+i=1sik0cos(iωgtφi)er(t)=e0+j=1pje0cos(jωgtψj) ,其中 k0 是平均网格刚度, e0 是平均静态传输误差, sipj 是傅里叶级数分量的振幅。傅里叶级数的相位角分别用 φiψj 表示。

The term ωg is the gear mesh frequency represented by(3)ωg=Niωiwhere Ni is the number of gear teeth, ωi is rotor operating frequency and i=1,2. For this study ω1=ω2=ω is used, which follows the convention in the related literature [33]. The pressure angle α is assumed to remain constant during operation. Plain journal bearings support both rigid shafts, and their nonlinear fluid film force models are explained in section 2.2.
ωg 是齿轮啮合频率,由 (3)ωg=Niωi 表示,其中 Ni 是齿轮齿数, ωi 是转子工作频率, i=1,2 。本研究采用 ω1=ω2=ω ,这与相关文献[33]中的惯例一致。压力角 α 假设在运行过程中保持恒定。滑动轴承支撑两个刚性轴,其非线性流体膜力模型将在 2.2 节中解释。

As noted in Ref. [26], a tooth backlash model defined with a piecewise linear function in the governing nonlinear differential equations may result in convergence difficulties when employing a Newton-Raphson method. Therefore, the following smoothening function presented in the same reference is also used in the present study.(4)ρ(t)=12{(δ(t)b)[1+tanh(σ(δ(t)-b))]}+12{(δ(t)+b)[1+tanh(σ(δ(t)+b))]}where ρ(t) represents relative gear mesh displacement considering backlash, b is the half-length of the tooth backlash amplitude (b=b02) and σ is a modulating factor which affects the accuracy of the backlash representation and convergence [26]. The value σ=100 is selected for this study.
正如参考文献[26]所指出的,在使用牛顿-拉夫逊方法时,如果在非线性微分方程中使用片断线性函数来定义齿隙模型,可能会导致收敛困难。[26] 所述,在非线性微分方程中使用片断线性函数定义的齿隙模型在使用牛顿-拉夫逊方法时可能会导致收敛困难。因此,本研究也采用了同一参考文献中的以下平滑函数。 (4)ρ(t)=12{(δ(t)b)[1+tanh(σ(δ(t)-b))]}+12{(δ(t)+b)[1+tanh(σ(δ(t)+b))]} ,其中 ρ(t) 表示考虑了齿隙的齿轮啮合相对位移, b 是齿隙振幅的半长 ( b=b02 ), σ 是影响齿隙表示精度和收敛性的调节因子 [26]。本研究选择的值为 σ=100

The coupling force between the driving and driven gear mesh is given by(5)Fm0=km(t)δ(t)+cmδ˙(t)where cm represents mesh damping, and it is assumed to be constant in this study. δ(t) represents the dynamic transmission error in Eq. (1), and δ˙(t) is its derivative.
驱动和从动齿轮啮合之间的耦合力由 (5)Fm0=km(t)δ(t)+cmδ˙(t) 给出,其中 cm 代表啮合阻尼,在本研究中假定为常数。 δ(t) 代表公式 (1) 中的动态传输误差, δ˙(t) 是其导数。

The equations of motion for the six-degree-of-freedom gear-bearing rotor system are(6)J1θ¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))R1+cmδ˙(t)R1=T1J2θ¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))R2cmδ˙(t)R2=T2m1x¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))sin(α)+cmδ˙(t)sin(α)=Fb1xm1y¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))cos(α)+cmδ˙(t)cos(α)=Fb1ym1gm2x¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))sin(α)cmδ˙(t)sin(α)=Fb2xm2y¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))cos(α)cmδ˙(t)cos(α)=Fb2ym2gBy replacing the term km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er)+cmδ˙(t) with Fm0, the equations become(7)J1θ¨1+R1Fm0=T1J2θ¨2R2Fm0=T2m1x¨1+Fm0sin(α)=Fb1xm1y¨1+Fm0cos(α)=Fb1ym1gm2x¨2Fm0sin(α)=Fb2xm2y¨2Fm0cos(α)=Fb2ym2gwhere Fbix and Fbiy represents the ith bearing forces in the x and y directions, and m1g and m2g terms represent gravity forces. By multiplying each of equations with R1 and R2, the first two become(8)J1R1θ¨1+R12Fm0=R1T1J2R1θ¨2R22Fm0=R1T2
六自由度齿轮轴承转子系统的运动方程为 (6)J1θ¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))R1+cmδ˙(t)R1=T1J2θ¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))R2cmδ˙(t)R2=T2m1x¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))sin(α)+cmδ˙(t)sin(α)=Fb1xm1y¨1+km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))cos(α)+cmδ˙(t)cos(α)=Fb1ym1gm2x¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))sin(α)cmδ˙(t)sin(α)=Fb2xm2y¨2km(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er(t))cos(α)cmδ˙(t)cos(α)=Fb2ym2gkm(t)(R1θ1R2θ2+x1sin(α)x2sin(α)+y1cos(α)y2cos(α)er)+cmδ˙(t) 项替换为 Fm0 后,方程变为 (7)J1θ¨1+R1Fm0=T1J2θ¨2R2Fm0=T2m1x¨1+Fm0sin(α)=Fb1xm1y¨1+Fm0cos(α)=Fb1ym1gm2x¨2Fm0sin(α)=Fb2xm2y¨2Fm0cos(α)=Fb2ym2g ,其中 FbixFbiy 代表 xy 方向上的 i th 轴承力, m1gm2g 项代表重力。将每个方程与 R1R2 相乘,前两个方程变为 (8)J1R1θ¨1+R12Fm0=R1T1J2R1θ¨2R22Fm0=R1T2

Divide the two equation with J1 and J2 respectively, and then subtracting the second equation from the first one, to obtain(9)R1θ¨1R1θ¨2+(R12J1+R22J2)Fm0=R1J1T1+R2J2T2Substituting p=R1θ1R2θ2 and manipulating the equation yields(10)p¨+(J2R12+J1R22J1J2)Fm0=R1J1T1+R2J2T2Dividing through by (J2R12+J1R22J1J2) yields(11)(J1J2J2R12+J1R22)p¨+Fm0=(J1J2J2R12+J1R22)(R1J1T1+R2J2T2)Substitute Je for (J1J2J2R12+J1R22) to obtain(12)Jep¨+Fm0=Tewhere Te=Je(R1J1T1+R2J2T2) is an equivalent input torque term.
将两个等式分别与 J1J2 相除,然后将第二个等式减去第一个等式,得出 (9)R1θ¨1R1θ¨2+(R12J1+R22J2)Fm0=R1J1T1+R2J2T2p=R1θ1R2θ2 代入并运算等式,得出 (10)p¨+(J2R12+J1R22J1J2)Fm0=R1J1T1+R2J2T2(J2R12+J1R22J1J2) 除以 (11)(J1J2J2R12+J1R22)p¨+Fm0=(J1J2J2R12+J1R22)(R1J1T1+R2J2T2)Je 代入 (J1J2J2R12+J1R22) ,得出 (12)Jep¨+Fm0=Te Te=Je(R1J1T1+R2J2T2) ,其中 是等效输入扭矩项。

The term δ(t) was inserted into Eq. (4) to include the backlash nonlinearity effect. Then, from Eqs. (4), (5), the gear meshing force including the backlash nonlinearity effect becomes(13)Fm=kmρ(t)+cmδ˙(t)
在公式 (4) 中插入了 δ(t) 一词,以包含反向间隙非线性效应。然后,根据公式 (4)、(5),包括反向间隙非线性效应的齿轮啮合力变为 (13)Fm=kmρ(t)+cmδ˙(t)

Finally, the equations including the backlash nonlinearity, time-varying mesh stiffness and the static transmission error become(14)Jep¨+Fm=Tem1x¨1+Fmsin(α)=Fb1xm1y¨1+Fmcos(α)=Fb1ym1gm2x¨2Fmsin(α)=Fb2xm2y¨2Fmcos(α)=Fb2ym2gwhere Je is the equivalent inertia of two gears, i.e. (J1J2J2R12+J1R22). The torsional natural frequency ωn of the system is defined as k0Je.
最后,包括齿隙非线性、时变啮合刚度和静态传动误差的方程变为 (14)Jep¨+Fm=Tem1x¨1+Fmsin(α)=Fb1xm1y¨1+Fmcos(α)=Fb1ym1gm2x¨2Fmsin(α)=Fb2xm2y¨2Fmcos(α)=Fb2ym2g ,其中 J e 是两个齿轮的等效惯性,即 (J1J2J2R12+J1R22) 。系统的扭转固有频率 ωn 定义为 k0Je

For validation purposes, the simulation result are compared with experimental measurements [4] in Fig. 3. The experiment was conducted using relatively stiff ball bearings so the x and y journal motions are assumed fixed in the simulation. The gear parameters for backlash, Fourier coefficients of time varying mesh stiffness and amplitude of static transmission error from Ref. [4] are employed in the simulation. The root mean square (RMS) value of the dynamic transmission error is plotted with respect to operating speed, showing good agreement between prediction and test results.
出于验证目的,模拟结果与实验测量结果[4]进行了比较,如图 3 所示。实验是使用相对较硬的球轴承进行的,因此在模拟中假定 xy 轴颈运动是固定的。参考文献[4]中的反向间隙、啮合刚度时变傅里叶系数和静态传输误差振幅等齿轮参数被用于仿真。[4] 中的反向间隙傅里叶系数和静态传输误差振幅。动态传动误差的均方根值随运行速度而变化,显示出预测与测试结果之间的良好一致性。

Fig. 3
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Fig. 3. Comparison of dynamic transmission error with experimental measurements in Ref. [4] (Ref. [37]).
图 3.动态传输误差与参考文献 [4] 中实验测量结果的比较(参考文献 [37] )。参考文献 [4](参考文献 [37])。

Table 1 provides a second validation case through comparison of the five-degree-of-freedom gear-bearing dynamic model's predicted natural frequencies with those provided in Ref. [15]. Since natural frequencies are characteristics of a linear model, the backlash and time varying stiffness were omitted and the bearing forces were represented by the stiffness and damping provided in the reference. The correlation is shown in the table and confirms excellent agreement.
表 1 提供了第二个验证案例,将五自由度齿轮轴承动态模型的预测自然频率与参考文献[15]中提供的自然频率进行了比较。[15].由于自然频率是线性模型的特征,因此省略了反向间隙和时变刚度,轴承力由参考文献中提供的刚度和阻尼表示。表中显示了相关性,并证实两者非常吻合。

Table 1. Comparison of calculated natural frequencies with [15].
表 1.计算的自然频率与 [15] 的比较。

Empty CellCalculated natural frequencies in Ref. [15]Natural frequencies based on current model
1st mode 第一模式0 Hz 0 赫兹0 Hz 0 赫兹
2nd mode 第二模式1149 Hz 1149 赫兹1149.3 Hz 1149.3 赫兹
3rd mode 第三模式1293 Hz 1293 赫兹1293.6 Hz 1293.6 赫兹
4th mode 第四模式1604 Hz 1604 赫兹1604.3 Hz 1604.3 赫兹
5th mode 第 5 模式1799 Hz 1799 赫兹1799.1 Hz 1799.1 赫兹
6th mode 第六模式5043 Hz 5043 赫兹5043.7 Hz 5043.7 赫兹

2.2. Finite element model of plain journal bearing
2.2.滑动轴承的有限元模型

The Reynolds equation [34] for an incompressible lubricant combines the fluid continuity and momentum equations into a partial differential equation for film pressure, and is given by(15)θ(h312μpθ)+z(h312μpz)=RJωJ2hθ+htwhere ωJ is the rotating speed of the journal, and RJ and μ represent the radius of the journal and the viscosity of the lubricant, respectively. The centers of the bearing and the journal are OB and OJ in Fig. 4, respectively.
不可压缩润滑油的雷诺方程 [34] 将流体连续性方程和动量方程合并为油膜压力偏微分方程,其式为 (15)θ(h312μpθ)+z(h312μpz)=RJωJ2hθ+ht ,其中 ωJ 为轴颈的旋转速度, RJμ 分别代表轴颈半径和润滑油粘度。轴承和轴颈的中心分别为图 4 中的 OBOJ

Fig. 4
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Fig. 4. Axial mid-plane section of a journal bearing.
图 4.轴颈轴承的轴向中平面剖面图。

The displacements of the journal center relative to the bearing center in the x and y directions are xJ and yJ, respectively, and p is the pressure in the lubricant film. Expressions for fluid film thickness h and its derivative h(θ)t at θ are given by(16)h(θ)=CBxJcosθyJsinθh(θ)t=x˙Jcosθy˙Jsinθwhere CB represents the bearing radial clearance.
轴颈中心相对于轴承中心在 xy 方向上的位移分别为 xJyJp 为润滑油膜中的压力。液膜厚度 h 及其在 θ 处的导数 h(θ)t 的表达式为 (16)h(θ)=CBxJcosθyJsinθh(θ)t=x˙Jcosθy˙Jsinθ ,其中 CB 代表轴承径向游隙。

The mathematical model assumes rigid shafts and rigid attachments between the bearings and ground. Therefore, the journal motions xJ and yJ are identical to their respective gear centerline motions. Thus x1, y1 are identical to xJ1 and yJ1 , and x2, y2 are identical to xJ2 and yJ2.
数学模型假设轴是刚性的,轴承和地面之间的连接也是刚性的。因此, xJyJ 的轴颈运动与其各自的齿轮中心线运动相同。因此, x1 , y1xJ1yJ1 相同, x2 , y2xJ2yJ2 相同。

The finite element mesh of a fluid film is illustrated in Fig. 5. The coordinate θ corresponds to the circumferential direction of the film and the direction of rotation is from the left (θB) to the right (θE). The axial coordinate is represented with z and only a half-length (L2) of the film is modeled due to its symmetry. The pressure on the bottom (z=0) side of the mesh are set to ambient pressure