第 1 頁 共 17 頁英文：Dynamic tooth loads of planetary gear sets having toothprofile wearC. Yuksel a, A. Kahraman b,*a The University of Toledo, Toledo, OH 43606, USAb Department of Mechanical Engineering, The Ohio State University, Room 255, 650 AckermanRoad,Columbus, OH 43202, USAReceived 25 June 2003; received in revised form 12 January 2004; accepted 10 February 2004 AbstractA computational model of a planetary gear set is employed to study the influence of surface wear on thedynamic behavior of a typical planetary gear set. The overall computational scheme combines a wear modelthat defines geometric description of contacting gear tooth surfaces having wear and a deformable-bodydynamic model of a planetary gear set. The wear model employ s a quasi-static gear contact model tocompute contact pressures and Archa rd’s wear model to determine the wear depth distributions. The wornsurfaces are input into the dynamic model to quantify the impact of wear on gear tooth and mesh dynamicforces. The results on a planetary gear set having a fixed planet carrier indicates that the dynamic behavioris nonlinear due to tooth separations in its resonance regions. The results for worn gear surfaces indicatethat surface wear has a signicant inuence in off-resonance speed ranges while its influence diminishes nearresonance peaks primarily due to tooth separations.ó 2004 Elsevier Ltd. All rights reserved. Keywords: Planetary gear sets; Gear dynamics; Gear wear 1. Introduction Planetary gear sets, also known as Epicyclic gear drives, are commonly used in a large number ofautomotive, aerospace and industrial applications. They posses numerous 第 2 頁 共 17 頁advantages over parallel-axis gear trains including compactness of design, availability of multiple speed reduction ratios, and less demanding bearing requirements. Most common examples of planetary gear sets can be foundin automatic transmissions, gas turbines, jet engines, and helicopter drive trains. A typical simple planetary gear set consists of a sun gear, a ring gear and a number of identical planet gears meshing both with the sun and ring gears. A common carrier holds the planets in place.Dynamic analysis of planetary gears is essential for eliminating noise and vibration problems of the products they are used in. The dynamic forces at the sun-planet and ring-planet meshes are the main sources of such problems. Although planetary gear sets have generally more favorable noise and vibration characteristics compared to parallel-axis gear systems, planetary gear set noise still remains to be a major problem. The dynamic gear mesh loads that are much larger than the static loads are transmitted to the supporting structures, in most cases, increasing gear noise. Larger dynamic loads also shorten the fatigue life of the components of the planetary gear set including gears and bearings. Surface wear is considered one of the major failure modes in gear systems. In case of planetary gear sets, experimental data has shown that especially the sun gear meshes might experience significant surface wear when run under typical operating conditions. While wear is a function of a large number of parameters, sliding distance and contact pressure were shown to be most significant parameters influencing gear wear. Wear of tooth profiles results in a unique surface geometry that alters the gear mesh excitations in the form of kinematic motion errors, enhancing the dynamic effects. Modeling of planetary gear set dynamics received significant attention for the last 30 years. A number of studies proposed lumped-parameter models to predict free and forced vibration characteristics of planetary gear sets. These models assumed rigid gear wheels, connected to each other by springs representing the flexibility of the meshing teeth. In these studies, nonlinear effects due to gear backlash and time-varying parameters due to gear mesh stiffness fluctuations were neglected. The corresponding Eigen value solution of the linear equations of motion resulted in natural modes. Modal summation technique was typically used to find the forced response due to external gear mesh displacement excitations defined to 第 3 頁 共 17 頁represent motion transmission errors. These lumped-parameter models vary in degrees of freedom included, from purely torsional models to two or three-dimensional transverse-torsional models. While these models served well in describing the dynamic behavior of planetary gear sets qualitatively, they lacked certain critical features. First, the gear mesh models were quite simplistic with a critical assumption that complex gear mesh contact interaction can be represented by a simple model formed by a linear spring and a damper. These models demand that the values of the gear mesh stiffness and damping, as well as the kinematic motion transmission error excitation, mustbe known in advance. It is also assumed that these parameter values determined quasi-statically remain unchanged under dynamic conditions. In addition, gear rim deflections and Hertzian contact deformations are also neglected. Another group of recent models used more sophisticated finite element-based gear contact mechanics models. These computational models address all of the shortcomings of the lumped-parameter models since the gear mesh conditions are modeled as individual nonlinear contact problems. The need for externally defined gear mesh parameters is eliminated with these models. In addition, rim deflection and spline support conditionsare modeled accurately . These models are also capable of including the influence of the tooth profile variations in the form of intentional profile modifications, manufacturing errors or wear on the dynamic behavior of the system. The study of wear of gear contact is becoming one of the emerging areas in gear technology. A number of recent gear wear modeling effortsform a solid foundation for more accurate, larger system analyses. All of these models use Archard’s wear model in conjunction with a gear contact model and relative sliding calculations. These studies focused on prediction of wear of either spur or helical gear pairs in aparallel-axis configuration. The tooth contact pressures were computed in these models using either simplified Hertzian contact or boundary element formulations under quasi-static conditions. Sliding distance calculations were carried out kinematically by using the involute geometry and Archard’s wear model was used with an empirical wear coefficient to compute the surface wear depth distribution. A number of studies investigated the influence of wear on gear dynamics response . Among them, Kuang and Lin simulated the tooth profile wear process by the 第 4 頁 共 17 頁model proposed by Ref. and predicted the variations of the dynamic loads and the corresponding frequency spectra as a function of wear for a single spur gear pair. Wojnarowski and Onishchenko performed analytical and experimental investigations of the influence of the tooth deformation and wear on spur gear dynamics. They stated that the change in the profiles of the teeth due to wear must be taken into account when dealing with the durability of the gear transmissions as well. These previous models considered surface wear effects for only a single spur gear pair, avoiding multi-mesh gear systems such as the planetary gear sets. They focused on only external gears and used lumped-parameter dynamic models excluding nonlinear and time-varying effects. 1.1 Objectives and scope As none of the previous studies on planetary gear set dynamics took into account the effect of wear, this study is intended to describe to the influence of gear tooth surface wear on dynamic behavior of planetary gears. A deformable-body dynamic model similar to the one proposed earlier will be used to investigate the influence of tooth surface wear on the dynamic behavior of planetary gear sets. The main objective here is to quantify the influence of surface wear on dynamic behavior of planetary gear sets. A planetary gear set formed by spur gears will be considered. A wear prediction model will be proposed to predict the gear surface wear distribution under quasi-static conditions. Different amounts of wear depths will then be introduced in the dynamic model to quantify the differences in dynamic behavior from the baseline behavior representing a gear set having no wear. Several complex dynamic phenomena exhibited by the planetary gear set including nonlinear behavior such as jump discontinuities and tooth separations will be demonstrated. The influence of surface wear on such behavior will also be described.2 Computational model This study relies on two previously developed models for investigation of the effect of surface wear on the dynamics of planetary systems. First, a wear model developed by Bajpai et al. will be used to determine the tooth surface wear profiles after different wear cycles. This model uses quasi-static finite elements-based gear contact model for prediction of the gear contact pressures and employs Archard’s wear model to predict wear of contacting tooth 第 5 頁 共 17 頁surfaces. Predicted tooth surface wear will then be applied to a deformable-body planetary gear dynamic model similar to the one proposed by Kahraman et al. to quantify the impact of gear surface wear on the dynamic behavior of planetary gear sets. 2.1 Wear prediction model For the local point on one of the contact gear surfaces, the induction wear equation can be expressed: ??s0kPdhWhere k is the experimentally determined wear coefficient, h is the cumulative wear depth, P is the contact pressure, and s is the contact pressure between the sliding distance at the point of interest. Where all the parameters except the contact pressure and the sliding distance are calculated by k. These include many materials, heat treatment, surface roughness and lubrication related parameters. Although the wear model can be improved by explicitly describing additional parameters in the formula. It was proved that the work was not good enough for the purpose of the project.2.2 Deformable-body dynamics model Worn surface profiles predicted by the wear model are used in a dynamic model to quantify the changes in dynamic behavior. A commercial gear contact mechanics software 第 6 頁 共 17 頁package is used to develop the dynamic model of the planetary gear set. The model uses finite element (FE) method to compute relative deformations and stresses for points away from the contact zones and semi analytical techniques for the points within the contact zones, is employed . The semi analytical FE approach does not require a highly refined mesh at the contacting tooth surfaces, reducing the computational effort while conventional FE models require a refined mesh at gear tooth region, limiting the model to static analysis only. Therefore, the model used here allows a more accurate and comprehensive study of planetary gear dynamics than the conventional FE models . The gears have complex shapes that an be best modeled by the FE method. The tooth surfaces are modeled by a large number of coordinate nodes, representing the involute shape and surface modifications making it possible to incorporate worn profiles .The width of the contact zone in typical gear applications is two orders of magnitude smaller than the dimensions of the gear teeth themselves, requiring a very fine mesh inside the contact zone.The location of the contact zone changes as the gears enter and exit the mesh. When conventional .FE models are used, besides having an extremely refined mesh, re-meshing is necessary for every contact position. The model used here avoids this problem since deformations near or at the contact zone predicted by using a semi-analytical formulation are matched with the deformations away from the contact predicted by using FE method. The model attaches a reference frame to each individual component and the finite element computations are done for each individual component separately. The mesh stiffness and mesh contact forces, comprising the dynamic excitation for the system, are evaluated internally at each time step . Contact conditions are handled as essentially linear inequality constraints whose convergence is ensured by a revised Simplex solver. A contact analysis determines the contact stresses and deformations of the gears at each time step. The elastic deformations of the gears are much smaller and must be superposed on the rigid body motions. By choosing a gear coordinate frame that follows the rigid body motion, the FE displacement vector xfi for gear i can be represented by a linear system of differential equations. 第 7 頁 共 17 頁The equations for each gear are assembled into the entire planetary gear system to obtain the overall matrix equation of motion. For the solution of the above equation, the contact mechanics model employs a timediscretization scheme based on Newmark method as used successfully in previous studies . 3 Results and discussion An example spur-type planetary system representative of typical gear sets in automatic transmission systems is considered here. Design parameters of the example system are listed in Table 1 and dynamic model is shown in Fig. 2. The sun gear is the input, the internal gear is the output, and the carrier is held stationary. A constant torque of 25 Nm/mm face width (FW) is applied to the input member. The system has four equally spaced planets that are not allowed to float radially. In order to avoid added complexity of ring gear bending modes , the outside diameter of the internal gear is chosen as rigid throughout this study while radial planet bearing flexibilities are included. Dynamic analysis of the model shown in Fig. 2 took a significant computational time. The simulation must be carried out for a reasonably long period to surpass the transient region. For each analysis, first a speed ramp up was simulated for a complete input revolution to pass through the transients, followed by a more refined analysis at the desired speed to cover two complete input revolutions. The steady state response is extracted from the last stage of the analysis. Whenthe input speed is increased by a small increment, as it is the case in an actual speed sweep, the last point of the steady state motion from the previous speed increment was considered as the initial condition followed by a rapid ramp-up and a refined steady state simulation. 第 8 頁 共 17 頁Dynamic analyses were performed within an input speed range up to Xin ? 15; 000 rpm, with a speed increment between 50 and 250 rpm. In each analysis, individual tooth loads at the sun and ring gear meshes of the planet gears were considered the output parameters. The time increment is adjusted at each analysis such that there are nearly 120 data points per tooth mesh cycle that was found to be a sufficient resolution to capture high frequency dynamic effects on tooth loads. Total gear mesh force time histories were obtained by adding all tooth forces at a given mesh and the corresponding frequency spectrum was obtained by using a fast Fourier transform (FFT) routine.In predicting the wear depth, a wear coefficient value of k ? 1018 m2/N was used in this analysis. This value was determined experimentally by Bajpai et al. using a similar automatic transmission final drive planetary gear set formed by case carburized shaved external gears and shaped internal gears. Bajpai, et.al. also pointed out that the wear at thering-planet meshes is simply negligible compared to those measured at the sun-planet meshes. In the power flow configuration considered, a sun gear tooth that mates with four planets will experience four wear cycles per input revolution, while a ring tooth goes through only 4eZs=ZrT ? 4e34=70T ? 1:94 wear cycles for the example system. Therefore, given this kinematic condition and previous experimental observations , wear at the ring-planet mesh was neglected in this study all together for the sake of simplicity.第 9 頁 共 17 頁4 Conclusions In this study, a computational model of a planetary gear set was employed to study thei nfluence of surface wear in the dynamic behavior of a typical automotive automatic transmission planetary gear set. The overall computational scheme combines a gear wear prediction model that gives geometric description of contacting tooth surfaces having wear and a deformable-body dynamic model of a planetary gear set. The wear model employs a quasi-static gear contact mechanics model to compute contact pressures and Archard’s wear model to determine the wear depth distributions. The worn surfaces were input into the dynamic model to quantify the impact of wear on gear tooth and mesh dynamic forces. It was shown that a planetary gear set is inherently nonlinear, and exhibits softening type nonlinear behavior near its resonance peaks, characterized by sudden jumps of dynamic gear mesh force amplitudes. A sun gear experiences the largest amount of wear, compared to other gears in the system as the maximum wear locations are in the dedendum of the sun gear. It is also observed that the tooth surface wear influences the fundamental harmonic of the gear mesh forces the most. While this influence is evident in both resonance and off-resonance regions of the forced response, the impact of wear is limited in the resonance regions dictated by higher harmonics. It is also concluded that wear has a negligible influence on the nonlinear behavior as nearly the same type of tooth separations were observed with or without surface wear. 第 10 頁 共 17 頁翻譯： 行星齒輪裝置動態(tài)載荷的齒廓磨損C. Yuksel, A. Kahrama托萊多大學(xué), 托萊多, OH 43606, 美國 機械工程系,俄亥俄州立大學(xué), 255 號, 650 艾克曼路, 哥倫布, OH 43202, 美國 2003 年 6 月 25 號初稿 2004年 1 月 14 號修訂稿2004 年 2 月 10 號發(fā)表 摘要以一個行星齒輪裝置的計算機模型為研究對象,研究表面磨損對典型的行星齒輪裝置動態(tài)行為的影響。整體的計算機模型的定義包括齒面接觸磨損的幾何描述和可變動載荷系統(tǒng)的行星齒輪組。模型采用準(zhǔn)靜態(tài)過程的磨損齒輪接觸模型接觸壓力的計算和Archard 磨損模型來確定磨損深度分布。表面的磨損程度作為動態(tài)模型的輸入量,用來定量研究了影響齒的磨損和嚙合動態(tài)的力量。結(jié)果表明對于有一個固定行星架的行星齒輪組，動態(tài)行為的非線性是由于輪齒在共振區(qū)域的分散導(dǎo)致的。結(jié)果表明齒面的疲第 11 頁 共 17 頁勞損壞在非共振速度區(qū)表面的缺陷起主要作用，在非共振區(qū)時輪齒的分散性起決定性作用。 關(guān)鍵詞：行星齒輪；齒輪動力學(xué)；齒輪磨損 1 簡介行星齒輪組，也被稱為行星齒輪傳動裝置，通常大量的用在汽車，航空航天和工業(yè)領(lǐng)域。他們擁有了平行軸齒輪系眾多優(yōu)點，包括緊湊的設(shè)計，多種減速比可用性和不高的承載要求。行星齒輪組最常見的例子可以發(fā)現(xiàn)在自動變速器，燃?xì)鉁u輪機，噴氣發(fā)動機，直升機傳動系統(tǒng)。一個典型的簡單行星齒輪組由一個太陽輪齒圈和幾個相同的行星齒輪都與太陽輪和環(huán)形齒輪嚙合。一個共同的載體固定位置的行星。行星齒輪的動態(tài)分析，為消除他們在產(chǎn)品應(yīng)用中的噪音和振動問題有很重要的作用。在太陽輪和行星輪的動力特性是這些問題的主要來源。雖然行星齒輪組相比平行軸齒輪系統(tǒng)一般都更有小的噪音和振動特性，但是行星齒輪組噪音仍然是一個主要問題。動態(tài)齒輪嚙合載荷遠(yuǎn)大于靜載荷傳遞到支撐結(jié)構(gòu)，在大多數(shù)情況下增加齒輪噪聲。較大的動載荷也能縮短包括齒輪和軸承在內(nèi)的行星齒輪組件的疲勞壽命。 表面磨損是齒輪系統(tǒng)失效的主要形式之一。在行星齒輪組的情況下，實驗數(shù)據(jù)顯示，在典型的操作條件下運行尤其是太陽齒輪嚙合表面磨損可能會遇到很大時。雖然磨損是一個大量的參數(shù)，滑動距離和接觸壓力的作用被證明是最重要的影響齒輪磨損參數(shù)。齒廓磨損的結(jié)果在一個獨特的表面幾何形狀，改變了運動中的運動誤差的形成齒輪嚙合激勵，增強的動態(tài)效果。 行星齒輪組動態(tài)建模收到近 30 年來極大關(guān)注。很多建議集總參數(shù)模型來預(yù)測行星齒輪組自由與強迫振動特性的研究。這些模型假定剛性齒輪，連接通過彈簧代表靈活的互相嚙合齒。在這些研究中，由于齒輪間隙和時變參數(shù)，由于齒輪嚙合剛度的波動非線性效應(yīng)這些研究被忽視了。相應(yīng)的本征線性方程組的運動造成的自然價值的解決方案模式。模型求和技術(shù)。通常用于發(fā)現(xiàn)因外部強迫響應(yīng)位移定義為齒輪嚙合傳動誤差激勵代表議案。這些集總參數(shù)模型各不相同的自由度。包括從純粹的扭轉(zhuǎn)模型，二維或三維橫扭模型。雖然這些模型曾在描述行星齒輪組的動態(tài)行為以及定性，他們?nèi)狈δ承╆P(guān)鍵功能。首先，齒輪嚙合模型進行了相當(dāng)關(guān)鍵的假設(shè)，復(fù)雜的齒輪嚙合接觸互動可以由一個線性彈簧和阻尼器組成一個簡單的模型表示簡單化。這些模型要求的齒輪嚙合剛度和阻尼值，以及運動運動傳遞誤差激勵，必須在事先知道。另據(jù)估計，第 12 頁 共 17 頁這些參數(shù)值確定準(zhǔn)靜態(tài)動態(tài)條件下保持不變。此外，齒輪邊緣變形撓度和赫茲接觸也被忽視。另一些最新的模型采用更先進的有限元素的齒輪接觸力學(xué)模型。這些計算模式解決了自齒輪嚙合條件的集總參數(shù)模型的缺點都作為單獨的非線性接問題的藍(lán)本。對于外部定義齒輪嚙合參數(shù)需要消除這些模型。此外邊緣變形和樣條支撐條件是準(zhǔn)確的參照。這些模型也是包括在故意修改，形成了專齒形變化的影響，制造錯誤或磨損在行星齒輪系統(tǒng)的動態(tài)行為的能力。 對齒輪接觸磨損的研究正日益成為齒輪技術(shù)的新領(lǐng)域之一。最近的一些齒數(shù)磨損建模工的形式進行更加準(zhǔn)確的，更大的系統(tǒng)中分析了堅實的基礎(chǔ)。這些機型全部使用Archard 的磨損模型在與齒輪接觸的計算模型和相對滑動結(jié)合。這些研究集中于任直磨損預(yù)報或在一個平行軸的螺旋齒輪副配置。對齒面接觸的壓力均在使用上述兩種模型計算，簡化赫茲接觸或邊界元在準(zhǔn)靜態(tài)條件的公式?；瑒泳嚯x的計算進行了運動學(xué)上使用了漸開線的幾何形狀和 Archard 的磨損模型進行了實證磨損系數(shù)來計算表面磨損深度分布使用。許多研究調(diào)查了磨損對齒輪動力學(xué)響應(yīng)的影響。其中，光，林模擬了由 Ref 提出的齒形磨損過程。并預(yù)測的動態(tài)負(fù)載變化和頻率變化范圍的作為單個齒輪對磨損的依據(jù)。Wojnarowski 和 Onishchenko 做了輪齒變形對齒輪動態(tài)磨損的分析和實驗研究。他們表明，當(dāng)討論齒輪傳動系統(tǒng)的耐久性時，輪齒的配置條件對磨損的影響必須考慮進去。這些以前的模型只考慮了一對齒輪嚙合時的表面磨損，沒有談及類似行星齒輪這種多個嚙合面的情況。他們只關(guān)注于外嚙合齒輪并且是在集中參數(shù)下的動態(tài)模型，而不考慮非線性和時變性的影響因素。 1.1 目的和內(nèi)容 鑒于之前沒有研究齒輪齒面磨損對于行星動態(tài)因素的影響，本研究的目的在于描述行星齒面磨損對于齒輪動態(tài)因素的影響。先前提出一個相似的可變形體動態(tài)模型將用于后續(xù)的研究中。這里主要的目的是量化這個指標(biāo)。由正齒輪組成的行星齒輪組是研究對象。一個計算模型用來預(yù)測在準(zhǔn)靜態(tài)條件下齒面磨損的分布狀態(tài)。磨損深度將作為動態(tài)模型的一個量化指標(biāo)來比較不同動態(tài)行為與無磨損動態(tài)行為的差異。行星齒輪組的幾個復(fù)雜的非線性動態(tài)特性包括齒輪不連續(xù)的跳動性和輪齒的分散性等,表面磨損對這些方面的影響也將進行討論。 2 計算模型 這個研究基于先前開發(fā)的兩個用來研究表面磨損對于動態(tài)特性和行星齒輪組的影第 13 頁 共 17 頁響的模型。首先，由巴杰帕伊