外文翻譯--對降低齒輪傳動裝載和卸載時因誤差引起的噪音的研究

外文翻譯 Reduction of noise of loaded and unloaded misaligned gear drives Faydor L. Litvina, Daniele Vecchiatoa, Kenji Yukishimaa, Alfonso Fuentesb, Ignacio Gonzalez-Perezb and Kenichi Hayasakac aGear Research Center, Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor St., Chicago, IL 60607-7022, USA bDepartment of Mechanical Engineering, Polytechnic University of Cartagena, C/Doctor Fleming, s/n, 30202, Cartagena, Murcia, Spain cGear R&D Group, Research and Development Center, Yamaha Motor Co., Ltd., 2500 Shingai, Iwata, Shizuoka 438-8501, Japan Received 22 February 2005； revised 6 May 2005； accepted 17 May 2005. Available online 25 January 2006. Abstract Transmission errors are considered as the main source of vibration and noise of gear drives. The impact of two main functions of transmission errors on noise is investigated: (i) a linear one, caused by errors of alignment, and (ii) a predesigned parabolic function of transmission errors, applied for reduction of noise. It is shown that a linear function of transmission errors is accompanied with edge contact, and then inside the cycle of meshing, the meshing becomes a mixed one: (i) as surface-to-surface tangency, and (ii) surface-to-curve meshing when edge contact starts. Application of a predesigned parabolic function of transmission errors enables to absorb the linear functions of transmission errors caused by errors of alignment, reduce noise, and avoid edge contact. The influence of the load on the function of transmission errors is investigated. Elastic deformations of teeth enable to reduce the maximal transmission errors in loaded gear drives. Computerized simulation of meshing and contact is developed for loaded and unloaded gear drives. Numerical examples for illustration of the developed theory are provided. Keywords: Gear drives ； Transmission errors； Tooth contact analysis (TCA)； Finite element analysis； Reduction of noise 1. Introduction Simulation of meshing of gear drives performed by application of tooth contact analysis (TCA) and test of gear drives have confirmed that transmission errors are the main source of vibrations of the gear box and such vibrations cause the noise of gear drive 1, 2, 4, 5, 6, 7, 10 and 11. The shape of functions of transmission errors depends on the type of errors of alignment and on the way of modification of gear tooth surfaces performed for improvement of the drive (see Section 2). The reduction of noise proposed by the authors is achieved as follows: (1) The bearing contact of tooth surfaces is localized. (2) A parabolic function of transmission errors is provided. This allows to absorb linear functions of transmission errors caused by misalignment 7. (3) One of the pair of mating surfaces is modified by double-crowning (see Section 2). This allows usually to avoid edge contact (see Section 5). The authors have compared the results of application of TCA for loaded and unloaded gear drives. It is shown that transmission errors of a loaded gear drive are reduced. The developed approach is illustrated with numerical examples (see Section 5). 2. Modification of tooth surfaces Reduction of noise of a gear drive requires modification of one of the pair of contacting surfaces. The surface modification is illustrated for three types of gear drives: helical gears, spiral bevel gears, and worm gear drives. 2.1. Helical gear drives Profile crowning of helical gears may be illustrated considering that the mating surfaces are generated by two rack-cutters with mismatched profiles 5 and 7. Profile crowning allows to localize the bearing contact. Double-crowning in comparison with profile crowning allows to: (i) avoid edge contact (caused by errors of crossing angle and different helix angles of mating gears), and (ii) provide a parabolic function of transmission errors. Double-crowning is performed by plunging of the disk that generates the pinion (see details in Chapter 15 of Ref. 7). 2.2. Spiral bevel gears Localization of contact of generated spiral bevel gears is provided by application of two mismatched head-cutters p and g used for generation of the pinion and the gear, respectively 7. Two head-cutters p and g have a common line C of generating tooth surfaces (in the case when profile crowning is provided). In the case of double-crowning, the mismatched generating surfaces p and g of the head-cutters have only a common single point of tangency, but not a line of tangency. Double-crowning of a generated gear may be achieved by tilting of one of the pair of generating head-cutters, or by proper installment of one of the head-cutters. It is very popular for the modern technology that during the generation of one of the mating gears, usually of the pinion, modified roll is provided 7. 2.3. Worm gear drives with cylindrical worm Very often the technology of manufacturing of a worm-gear is based on the following approach. The generation of the worm-gear is performed by a hob that is identical to the worm of the gear drive. The applied machine-tool settings simulate the meshing of the worm and worm-gear of the drive. However, manufacture with observation of these conditions causes an unfavorable bearing contact, and high level of transmission errors. Minimization of such disadvantages may be achieved by various ways: (i) by long-time lapping of the produced gear drive in the box of the drive； (ii) by running of the gear drive under gradually increased load, up to the maximal load； (iii) by shaving of the worm-gear in the box of the drive by using a shaver with minimized deviations of the worm-member, etc. The authors approach is based on localization of bearing contact by application of: (a) an oversized hob, and (b) modification of geometry (see below). There are various types of geometry of worm gear drives 7, but the preferable one is the drive with Klingelnbergs type of worm. Such a worm is generated by a disk with profiles of a circular cone 7. The relative motion of the worm with respect to the generating disk is a screw one (in the process of generation). Very often localization of bearing contact in a worm gear drive is achieved by application of a hob that is oversized in comparison with the worm of the drive. 3. Types of meshing and basic functions of transmission errors It is assumed that the tooth surfaces are at any instant in point tangency due to the localization of contact. Henceforth, we will consider two types of meshing: (i) surface-to-surface, and (ii) surface-to-curve. Surface-to-surface tangency is provided by the observation of equality of position vectors and surface unit normals 7. Surface-to-curve meshing is the result of existence of edge contact 7. The algorithm of TCA for surface-to-surface tangency is based on the following vector equations 7: (1) (2) that represent in fixed coordinate system Sf position vectors and surface unit normals . Here, (ui, i) are the surface parameters and ( 1, 2) determine the angular positions of surfaces. The algorithm for surface-to-curve tangency is represented in Sf by equations 7 (3) (4) Here, represents the surface that is in mesh with curve is the tangent to the curve of the edge. Application of TCA allows to discover both types of meshing, surface-to-surface and surface-to-curve. Computerized simulation of meshing is an iterative process based on numerical solution of nonlinear equations 8. By applying double-crowning to one of the mating surfaces, it becomes possible to: (i) avoid edge contact, and (ii) obtain a predesigned parabolic function 7 (Fig. 1). Application of a predesigned parabolic function is the precondition of reduction of noise. (17K) Fig. 1. Illustration of: (a) transmission functions 1 of a misaligned gear drive and linear function 2 of an ideal gear drive without misalignment； (b) periodic functions 2( 1) of transmission errors formed by parabolas. Application of double-crowning allows to assign ahead that function of transmission errors is a parabolic one, and allows to assign as well the maximal value of transmission errors as of 68. The expected magnitude of the predesign parabolic function of transmission errors and the magnitude of the parabolic plunge of the generating tool have to be correlated. Fig. 2 shows the case wherein due to a large magnitude of error of misalignment, the function of transmission errors is formed by two branches: of surface-to-surface contact and of surface-to-curve contact. (31K) Fig. 2. Results of TCA of a case of double-crowned helical gear drive with a large error = 10: (a) function of transmission errors wherein corresponds to surface-to-surface tangency and correspond to surface-to-curve tangency； (b) path of contact on pinion tooth surface； (c) path of contact on gear tooth surface. 4. Transmission errors of a loaded gear drive The contents of this section cover the procedure of determination of transmission errors of a loaded gear drive by application of a general purpose FEM computer program 3. Transmission errors of an unloaded gear drive are directly determined by application of TCA. Comparison of transmission errors for unloaded and loaded gear drives is represented in Section 5. 4.1. Preliminary considerations (i) Due to the effect of loading of the gear drive, the maximal transmission errors are reduced and the contact ratio is increased (ii) The authors approach allows to reduce the time of preparation of the model by the automatic generation of the finite element model 1 for each configuration of the set of applied configurations. (iii) Fig. 3 illustrates a configuration that is investigated under the load. TCA allows to determine point M of tangency of tooth surfaces 1 and 2, before the load will be applied (Fig. 3(a), where N2 and N1 are the surface normals (Fig. 3(b) and (c). The elastic deformations of tooth surfaces of the pinion and the gear are obtained as the result of applying the torque to the gear. The illustrations of Fig. 3(b) and (c) are based on discrete presentations of the contacting surfaces. (25K) Fig. 3. Illustration of: (a) a single configuration； (b) and (c) discrete presentations of contacting surfaces and surface normals N1 and N2. (iv) Fig. 4 shows schematically the set of configurations in 2D space. The location of each configuration (before the elastic deformation will be applied) is determined by TCA. (25K) Fig. 4. Illustration of set of models for simulation of meshing of a loaded gear drive. 4.2. Application of finite element analysis for determination of function of transmission errors of a loaded gear drive The described procedure is applicable for any type of a gear drive. The following is the description of the required steps: (i) The machine-tool settings applied for generation are known ahead, and then the pinion and gear tooth surfaces (including the fillet) may be determined analytically. (ii) Related angular positions are determined by (a) applying of TCA for Nf configurations (Nf = 816), and (b) observing the relation (5) (iii) A preprocessor is applied for generation of Nf models with the conditions: (a) the pinion is fully constrained to position , and (b) the gear has a rigid surface that can rotate about the gears axis (Fig. 5). Prescribed torque is applied to this surface. (7K) (vi) The total function of transmission errors for a loaded gear drive is obtained considering: (i) the error caused due to the mismatched of generating surfaces, and (ii) the elastic approach . (6) 5. Numerical examples A helical gear drive with design parameters given in Table 1 is designed. The following conditions of meshing and contact of the drive are considered: (1) The gear and pinion rack-cutters are provided with a straight-line and parabolic profiles as cross-section profiles, respectively, for generation of the gear and the pinion. Mismatched rack-cutter profiles yield the so-called profile crowning. (2) The misalignment of gear drive is caused by an error of the shaft angle, 0. (3) A predesigned parabolic function for absorption of transmission errors caused by 0 is provided. （ Such a function for a double-crowned pinion tooth surface is obtained by plunging of the generating disk, or by modified roll of the grinding worm.） (4) TCA (tooth contact analysis) for unloaded and loaded gear drives are applied for determination of transmission errors caused by . This enables to investigate the influence of the load on the magnitude and shape of the function of transmission errors. (5) Application of a computer program for finite element analysis 3 enables to determine the stresses of a loaded gear drive. (6) Formation of bearing contact is investigated. Table 1. Design parameters Number of teeth of the pinion, N1 21 Number of teeth of the gear, N2 77 Normal module, mn 5.08 mm Normal pressure angle, n 25 Hand of helix of the pinion Left-hand Helix angle, 30 Face width, b 70 mm Parabolic coefficient of pinion rack-cutter, aca 0.002 mm1 Radius of the worm pitch cylinder, rwa 98 mm Parabolic coefficient of pinion modified roll, amrb 0.00008 rad/mm2 Applied torque to the pinionc 250 N m (i) Example 1: An aligned gear drive ( = 0) is considered. The gear drive is unloaded. A parabolic function with the maximal value of transmission errors 2(1) = 8 is provided (Fig. 6(a). The cycle of meshing is . The bearing contact on the pinion and gear tooth surfaces is oriented almost longitudinally (Fig. 6(b) and (c). (24K) Fig. 6. Results of computation for an unloaded gear drive without misalignment: (a) function of transmission errors； (b) and (c) paths of contact on pinion and gear tooth surfaces. 6. Comparison of the power of noise for two functions of transmission errors 6.1. Conceptual consideration of applied approach Determination of the power of the signal of noise is based on the assumption that the velocity of oscillation of the generated acoustic waves is proportional to the fluctuation of the instantaneous value of the velocity of the gears. This assumption (even if not accurate in general) is good as the first guess, since it allows to avoid application of a complex dynamic model of the gear drive. We emphasize that the proposed approach is applied for the following conditions: (a) The goal is the determination of difference of power of signals, but not the determination of absolute values of signals. (b) The difference of power of signals is the result mainly of the difference of first derivatives of two smooth functions of transmission errors. The proposed approach is based on the comparison of the root mean square of the signals (in rms) caused by two functions of transmission errors 9. Such comparison yields the simulation of the intensity (the power) of the signal defined as (7) Here 2( 1) represents the deviation of the angular velocity of the gear from the average value, and rms represents the desired rms value. The definition of function of transmission errors yields that 2 = m21 1 + 2( 1), where m21 is the gear ratio. By differentiation with respect to time, we obtain the angular velocity of the gear as (8) wherein is assumed as constant. The second term on the right side of Eq. (8) represents the sought-for fluctuation of velocity (9) The definition above assumes that the function of transmission errors (FTE) is a continuous and differentiable one. In the case of computation of a loaded gear drive simulated by FEM (finite element method), this function is defined by a finite number of given points ( 1)i, ( 2)i) (i = 1, , n). The given data of points have to be interpolated by continuous functions for application of Eq. (7).) 6.2. Interpolation by a piecewise linear function In this case (Fig. 7), two successive data points are connected by a straight line. The derivative (velocity) between point i and i 1 is constant and is determined as follows: (10) (5K) Fig. 7. Interpolation of function of transmission errors by application of a piecewise linear function. Data points have been chosen as follows: (i) an increment ( 1)i ( 1)i1 is considered as constant for each interval i, and (ii) as the same for the two functions (FTE) represented in Examples 2 and 3 (in Section 5). Based on this assumption, the ratio of two magnitudes of power by application of the mentioned functions is represented as (11) 7. Conclusion The previously presented discussions, computations, and numerical examples enable to draw the following conclusions: (1) Errors of alignment of a gear drive (if modification of surfaces is not provided enough) may cause a mixed meshing: (i) surface-to-surface and (ii) edge contact (as surface-to-curve). Edge contact may be usually avoided by application of a predesigned parabolic function (PPF). (2) The investigation of influence of a parabolic function of transmission errors shows that application of PPF enables to reduce the noise and vibration of the gear drive. Application of PPF requires modification of generation of at least of one member of the gear drive, usually of the pinion (or the worm, in case of a worm gear drive). (3) Determination of transmission errors of a loaded gear drive requires application of a general purpose finite element computer program. A loaded gear drive is accompanied with elastic deformation of teeth, the increase of the contact ratio, and as a result, the decrease of transmission errors of the drive caused by misalignment. The time for preparation of the models is substantially reduced due to application of the authors approach of automatic generation of finite element models 1 for determination of transmission errors of a loaded gear drive. Acknowledgements The authors express their deep gratitude to the Gleason Foundation, and the Yamaha Motor Co., Japan, for the financial support of the projects. References 1 J. Argyris, A. Fuentes and F.L. Litvin, Computerized integrated approach for design and stress analysis of spiral bevel gears, Comput. Methods Appl. Mech. Engrg. 191 (2002), pp. 10571095. 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Zienkiewicz and 對降低齒輪傳動裝載和卸載時因誤差引起的噪音的研究 作者 弗萊德 L.萊特芠那 , 丹尼爾 .文科黑特 摘要 齒輪傳動時產(chǎn)生震動和噪音的主要原因是傳輸誤差。有關(guān)影響噪音傳輸誤差的兩個主要函數(shù)已被查明：（ 1）一個是線性的對應(yīng)誤差；（ 2）一個是初步設(shè)計使用傳輸誤差以減少噪音而引起的。它顯示 了傳輸誤差的線性關(guān)系，在一個周期內(nèi)形成了混合的循環(huán)嚙合：（ 1）如點對點接觸；（ 2）當從表面以曲線形式移動到起始點時就產(chǎn)生嚙合。使用初步設(shè)計傳輸誤差能夠減少因為線性對應(yīng)函數(shù)而引起的傳輸誤差，減少噪音和避免移動接觸。引起傳輸誤差的負載函數(shù)已被研究。齒牙的損壞能夠使在裝載的齒輪傳動中減少最大的傳輸誤差。用計算機處理的模擬齒輪嚙合，且齒輪傳動裝載和卸貨技術(shù)已發(fā)展相當水平。 關(guān)鍵字 齒輪傳動；傳輸誤差；齒牙嚙合分析（ YCA）；限定的元素分析；噪音的減少 1 緒論 模擬的齒輪傳動嚙合執(zhí)行應(yīng)用齒牙接觸分析（ TCA）和測試齒 輪傳動已被證實傳輸誤差的主要原因是齒輪箱的震動，這樣的震動引起齒輪傳動的噪音 1，2， 3， 4， 5， 6， 7， 10和 11。傳輸誤差函數(shù)的類型依賴對應(yīng)錯誤的類型且齒輪齒牙表面為了進一步的傳動在進行改善。（見第二節(jié)） 為減少噪音而依下列的計劃進行： （ 1）牙齒接觸表面被局部化 （ 2）提供一個傳輸誤差的函數(shù)。這種傳輸錯誤是由未對準的一函數(shù)所引起的 7。 （ 3）對雙層表面之一進行最高倍數(shù)的修正。 見第 2 節(jié) 這通常是避免表面摩擦。 見第 5 節(jié) 已經(jīng)對裝載和卸載齒輪傳動應(yīng)用 TCA 進行了比 較，它顯示裝載的齒輪傳動的傳輸誤差較少。其發(fā)展的方式與數(shù)字進行一起舉例。 見第 5 節(jié) 2 齒牙表面的修正 減少齒輪傳動的噪音需要修正接觸的雙表面之一。要修正齒輪傳動接觸表面有三種類型： 螺旋狀的齒輪，螺旋狀的斜齒輪，蝸桿齒輪。 2.1 螺旋狀的齒輪傳動 螺旋狀的齒輪最高剖面可能相交而表面產(chǎn)生兩個齒條刀形成錯誤的輪廓 5和 7。 完美輪廓允許接觸方向的局部化。最完美的輪廓比較是允許的：（ 1）避免邊緣接觸（交叉角和不同形狀角的相交齒輪）（ 2）提供一個傳輸誤差的拋物線函數(shù)。雙倍完美的執(zhí)行突進的圓盤而產(chǎn)生小齒輪（ 見 REF 的第 15 章資料。 7）。 2.2 螺旋狀的斜齒輪 應(yīng)用提供兩個有誤差的刀尖 p 和 g 而有局部接觸會產(chǎn)生 螺旋狀的斜齒輪： p 和 g 二者是分別用來產(chǎn)生小齒輪和齒輪的 7。倆個刀尖 p 和 g再齒呀的表面產(chǎn)生一個共同線 C。（當提供外層輪廓的情況下）再加倍的情況下產(chǎn)生配合誤差表面 p 和 g 刀尖 只有接觸的通常單一點，但不是一條接觸的線。加倍可能產(chǎn)生齒輪而形成有斜齒的刀尖，或者是刀尖特有的部分。她是近代科技生產(chǎn)的齒輪當中教授歡迎的齒輪之一，通常小齒輪都被改良為滾動的 7。 2.3 圓柱型蝸桿齒輪傳動 通 常蝸輪制造工藝是以下列的方式為基礎(chǔ)。蝸輪的生產(chǎn)和蝸桿齒輪傳動一樣都是由一個滾刀運行的。應(yīng)用的機床設(shè)置模擬蝸桿和蝸輪嚙合而形成齒輪傳動。然而，觀察發(fā)現(xiàn)在這些條件下的制造引起不宜的軸接觸，和高度傳動誤差。為把這些誤差減少到最低限度可用以下不同的方法完成： (1)長期在齒輪箱中研磨加工而使齒輪傳動畸形； (2)齒輪傳動在長期的運轉(zhuǎn)下產(chǎn)生負載，近而達到最大負載； (3)蝸輪在蝸輪箱中被刨且傳動裝置利用刨削蝸桿部分背離減少到最小化，等等。 制造者的方法是應(yīng)用接觸局限為基礎(chǔ)的：（ a）一個特大號的滾齒刀，和（ b）幾何學的 修正。（見下面）。 有蝸輪傳動幾何學的各種不同類型 7，但是一個較好的是有 Klingelnberg類型的蝸桿。 這種蝸桿是由圓盤輪廓和錐形圓作成的 7。有關(guān)蝸桿傳動要考慮圓盤的一個螺紋的產(chǎn)生（在生產(chǎn)的方法中）時常，再蝸輪傳動局限接觸中以達成應(yīng)用滾刀且是比較特大號的蝸輪傳動。 3 嚙合的類型和傳動誤差的基本函數(shù) 它假定齒牙表面任何點相切是正當?shù)木窒薅ㄎ?。此后，我們考慮兩種嚙合：（ 1）面與面，（ 2）面與曲線。面與面相切是平等觀察表面的位置向量和表面單位提供 7。面與曲線嚙合是曲線邊緣實在接觸的結(jié)果 7。 面與面相切的 TCA 運算法則是以下列的矢量為基礎(chǔ)的方程 7： 在固定的同等系統(tǒng) Sf位置矢量 和表面常態(tài) 中表現(xiàn)。這里， (ui, i)是表面的參數(shù)而且 ( 1, 2)決定表面的角位置。 面與曲線的運算法則是用 Sf方程來表現(xiàn)的 7： 在這里 描述表面的嚙合曲線 是邊緣曲線的切線。 TCA 的允許應(yīng)用而發(fā)現(xiàn)兩種嚙合的類型，面與面和面與曲線。計算機處理的嚙合模擬是以一個反復(fù)的程序為基礎(chǔ)的非線性方程的數(shù)字解決方案 8 應(yīng)用最高的相交表面之一，它可能變成：（ 1）避免邊緣接觸，（ 2）獲得一個初步設(shè)計的拋物線函數(shù) 7（圖 1）。初步設(shè)計的拋物線函數(shù)功能的應(yīng)用是減少噪音的先決條件。 圖 1例證：（ a）齒輪驅(qū)動的一個不成直線的傳動函數(shù) 1和沒有欠對準的理想的線性函數(shù) 2；（ b）周期函數(shù)拋物線形成的傳動誤差 2( 1)。 應(yīng)用最高的允許向前分配傳動的誤差函數(shù)的是一個拋物線，而且允許分配同樣最大 誤差值的 6-8 。初步設(shè)計預(yù)期大小的傳動誤差拋物線函數(shù)和投入大量生產(chǎn)的工具是有關(guān)聯(lián)的。圖 2表示在何處由于欠對準的誤差的大小，傳動誤差函數(shù)形成兩個支流： 面對面接觸和 面與曲線接觸。 圖 2一個螺旋狀齒輪的最大 TCA 誤差結(jié)果 = 10: （ a）傳動誤差函數(shù)在何處 符合面與面相切和何處 符合面與曲線相切；（ b）在小齒輪齒面上相切的路徑：（ c）在齒輪表面的接觸路徑。 4.裝載齒輪傳動的傳動誤差 這一部分內(nèi)容覆蓋了一般用途 FEM 電腦程序應(yīng)用裝載齒輪驅(qū)動的傳動誤差果斷程序 3。 TCA 決定直接應(yīng)用卸載齒輪驅(qū)動的傳動誤差。描述 比較裝載和