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

1、<p><b>　　外文翻譯</b></p><p>　　Reduction of noise of loaded and unloaded misaligned gear drives</p><p>　　Faydor L. Litvina, Daniele Vecchiatoa, Kenji Yukishimaa, Alfonso Fuentesb,

2、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, USAbDepartment of

3、Mechanical Engineering, Polytechnic University of Cartagena, C/Doctor Fleming, s/n, 30202, Cartagena, Murcia, SpaincGear R&D Group, Research and Development Center, Yamaha Motor Co., Ltd., 2500 Shingai, </p>

4、<p><b>　　Abstract</b></p><p>　　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 i

5、s 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

6、 is accompanied with edge contact, and then inside the cycle of meshing, the meshing becomes a mixed one: (i) as surface-</p><p>　　Keywords: Gear drives; Transmission errors; Tooth contact analysis (TCA); Fi

7、nite element analysis; Reduction of noise </p><p>　　1. Introduction</p><p>　　Simulation of meshing of gear drives performed by application of tooth contact analysis (TCA) and test of gear drives

8、 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 transmis

9、sion 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)</p><p>　　The reduction of noise proposed by

10、 the authors is achieved as follows: </p><p>　　(1) The bearing contact of tooth surfaces is localized.</p><p>　　(2) A parabolic function of transmission errors is provided. This allows to absorb

11、 linear functions of transmission errors caused by misalignment [7].</p><p>　　(3) One of the pair of mating surfaces is modified by double-crowning (see Section 2). This allows usually to avoid edge contact

12、(see Section 5).</p><p>　　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 develop

13、ed approach is illustrated with numerical examples (see Section 5). </p><p>　　2. Modification of tooth surfaces</p><p>　　Reduction of noise of a gear drive requires modification of one of the pa

14、ir of contacting surfaces. The surface modification is illustrated for three types of gear drives: helical gears, spiral bevel gears, and worm gear drives. </p><p>　　2.1. Helical gear drives</p><p

15、>　　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]. </p><p>　　Profile crowning allows to lo

16、calize 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 parabo

17、lic 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]).</p><p>　　2.2. Spiral bevel gears</p><p&

18、gt;　　Localization of contact of generated spiral bevel gears is provided by application of two mismatched head-cutters Σp and Σg used for generation of the pinion and the gear, respectively [7]. Two head-cutters Σp and Σ

19、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 Σp and Σg of the head-cutters have only a common sin

20、gle point of tangency, but not a line of tangency. Double-crownin</p><p>　　2.3. Worm gear drives with cylindrical worm</p><p>　　Very often the technology of manufacturing of a worm-gear is based

21、 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 d

22、rive. 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: </p>

23、<p>　　(i) by long-time lapping of the produced gear drive in the box of the drive;</p><p>　　(ii) by running of the gear drive under gradually increased load, up to the maximal load;</p><p&g

24、t;　　(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.</p><p>　　The authors’ approach is based on localization of bearing contact b

25、y application of: (a) an oversized hob, and (b) modification of geometry (see below). </p><p>　　There are various types of geometry of worm gear drives [7], but the preferable one is the drive with Klingelnb

26、erg’s 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). </p>

27、<p>　　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. </p><p>　　3. Types of meshing and ba

28、sic functions of transmission errors</p><p>　　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:

29、 (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 exist

30、ence of edge contact [7]. </p><p>　　The algorithm of TCA for surface-to-surface tangency is based on the following vector equations [7]:</p><p>　　that represent in fixed coordinate system Sf pos

31、ition vectors and surface unit normals . Here, (ui, θi) are the surface parameters and (1, 2) determine the angular positions of surfaces. </p><p>　　The algorithm for surface-to-curve tangency is repres

32、ented in Sf by equations [7]</p><p>　　Here, represents the surface that is in mesh with curve is the tangent to the curve of the edge. </p><p>　　Application of TCA allows to discover both types

33、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]. </p><p>　　By applying double-crowning t

34、o 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 reductio

35、n of noise. (17K) </p><p>　　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 Δ2

36、(1) of transmission errors formed by parabolas. </p><p>　　Application of double-crowning allows to assign ahead that function of transmission errors is a parabolic one, and allows to assign as well the maxim

37、al value of transmission errors as of 6–8″. 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

38、 shows the case wherein due to a large magnitude of error of misalignment, the function of transmission errors is formed by two bra</p><p>　　Fig. 2. Results of TCA of a case of double-crowned helical ge

39、ar drive with a large error Δγ = 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 surf

40、ace; (c) path of contact on gear tooth surface.</p><p>　　4. Transmission errors of a loaded gear drive</p><p>　　The contents of this section cover the procedure of determination of transmission

41、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 f

42、or unloaded and loaded gear drives is represented in Section 5. </p><p>　　4.1. Preliminary considerations</p><p>　　(i) Due to the effect of loading of the gear drive, the maximal transmission er

43、rors are reduced and the contact ratio is increased</p><p>　　(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 e

44、ach configuration of the set of applied configurations.</p><p>　　(iii) Fig. 3 illustrates a configuration that is investigated under the load. TCA allows to determine point M of tangency of tooth surfaces Σ1

45、 and Σ2, 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 apply

46、ing the torque to the gear. The illustrations of Fig. 3(b) and (c) are based on discrete presentations of the contacting surfaces. (25K) </p><p>　　Fig. 3. Illustration of: (a) a single configuration; (

47、b) and (c) discrete presentations of contacting surfaces and surface normals N1 and N2. </p><p>　　(iv) Fig. 4 shows schematically the set of configurations in 2D space. The location of each configuration (be

48、fore the elastic deformation will be applied) is determined by TCA. (25K) </p><p>　　Fig. 4. Illustration of set of models for simulation of meshing of a loaded gear drive. </p><p>　　4.2. A

49、pplication of finite element analysis for determination of function of transmission errors of a loaded gear drive</p><p>　　The described procedure is applicable for any type of a gear drive. The following is

50、 the description of the required steps: </p><p>　　(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 an

51、alytically.</p><p>　　(ii) Related angular positions are determined by (a) applying of TCA for Nf configurations (Nf = 8–16), and (b) observing the relation</p><p>　　(iii) A preprocesso

52、r 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 gear’s axis (Fig. 5). Prescribed torque is ap

53、plied to this surface. (7K) </p><p>　　(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 (

54、ii) the elastic approach .</p><p>　　5. Numerical examples</p><p>　　A helical gear drive with design parameters given in Table 1 is designed. The following conditions of meshing and contact of th

55、e drive are considered: </p><p>　　(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 pin

56、ion. Mismatched rack-cutter profiles yield the so-called profile crowning.</p><p>　　(2) The misalignment of gear drive is caused by an error of the shaft angle, Δγ ≠ 0.</p><p>　　(3) A

57、predesigned parabolic function for absorption of transmission errors caused by Δγ ≠ 0 is provided. （Such a function for a double-crowned pinion tooth surface is obtained by plunging of the generating disk, or b

58、y modified roll of the grinding worm.）</p><p>　　(4) TCA (tooth contact analysis) for unloaded and loaded gear drives are applied for determination of transmission errors caused by Δγ. This enables to investi

59、gate the influence of the load on the magnitude and shape of the function of transmission errors.</p><p>　　(5) Application of a computer program for finite element analysis [3] enables to determine the stres

60、ses of a loaded gear drive.</p><p>　　(6) Formation of bearing contact is investigated.</p><p><b>　　Table 1. </b></p><p>　　Design parameters </p><p>　　(i) Ex

61、ample 1: An aligned gear drive (Δγ = 0) is considered. The gear drive is unloaded. A parabolic function with the maximal value of transmission errors Δ2(1) = 8″ is provided (Fig. 6(a)). The cycle of m

62、eshing is . The bearing contact on the pinion and gear tooth surfaces is oriented almost longitudinally (Fig. 6(b) and (c)). (24K) </p><p>　　Fig. 6. Results of computation for an unloaded gear drive wi

63、thout misalignment: (a) function of transmission errors; (b) and (c) paths of contact on pinion and gear tooth surfaces.</p><p>　　6. Comparison of the power of noise for two functions of transmission errors&

64、lt;/p><p>　　6.1. Conceptual consideration of applied approach</p><p>　　Determination of the power of the signal of noise is based on the assumption that the velocity of oscillation of the generated

65、 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 applicati

66、on of a complex dynamic model of the gear drive. </p><p>　　We emphasize that the proposed approach is applied for the following conditions: </p><p>　　(a) The goal is the determination of differe

67、nce of power of signals, but not the determination of absolute values of signals.</p><p>　　(b) The difference of power of signals is the result mainly of the difference of first derivatives of two smooth fun

68、ctions of transmission errors.</p><p>　　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 yi

69、elds the simulation of the intensity (the power) of the signal defined as</p><p>　　Here ω2(1)′ represents the deviation of the angular velocity of the gear from the average value, and ωrms represents the des

70、ired rms value. The definition of function of transmission errors yields that 2 = m211 + Δ2(1), where m21 is the gear ratio. By differentiation with respect to time, we obtain the angular velocity of

71、the gear as</p><p>　　wherein is assumed as constant. The second term on the right side of Eq. (8) represents the sought-for fluctuation of velocity</p><p>　　The definition above assumes that the

72、 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 gi

73、ven points ((1)i, (Δ2)i) (i = 1, … , n). The given data of points have to be interpolated by continuous functions for application of Eq. (7).) </p><p>　　6.2. Interpolation by a

74、piecewise linear function</p><p>　　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 determi

75、ned as follows:</p><p><b>　　(5K) </b></p><p>　　Fig. 7. Interpolation of function of transmission errors by application of a piecewise linear function. </p><p>　　Dat

76、a points have been chosen as follows: (i) an increment (1)i ? (1)i?1 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

77、). Based on this assumption, the ratio of two magnitudes of power by application of the mentioned functions is represented as</p><p>　　7. Conclusion</p><p>　　The previously presented discussions

78、, computations, and numerical examples enable to draw the following conclusions: </p><p>　　(1) Errors of alignment of a gear drive (if modification of surfaces is not provided enough) may cause a mixed meshi

79、ng: (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).</p><p>　　(2) The investigation of influen

80、ce 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 membe

81、r of the gear drive, usually of the pinion (or the worm, in case of a worm gear drive).</p><p>　　(3) Determination of transmission errors of a loaded gear drive requires application of a general purpose fini

82、te 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 misalignme

83、nt. 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 d</p><p>　　Acknowledgements</p>

84、<p>　　The authors express their deep gratitude to the Gleason Foundation, and the Yamaha Motor Co., Japan, for the financial support of the projects. </p><p>　　References</p><p>　　[1] J. A

85、rgyris, 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. 1057–1095. SummaryPlus | Full Text + Links | P

86、DF (1983 K) </p><p>　　[2] Gleason Works, Understanding Tooth Contact Analysis, Rochester, New York, 1970. </p><p>　　[3] Hibbit, Karlsson & Sirensen, Inc., ABAQUS/Standard User’s Manual, 1800

87、 Main Street, Pawtucket, RI 20860-4847, 1998. </p><p>　　[4] Klingelnberg und Söhne, Ettlingen, Kimos: Zahnkontakt-Analyse für Kegelräder, 1996. </p><p>　　[5] F.L. Litvin et al., H

88、elical and spur gear drive with double crowned pinion tooth surfaces and conjugated gear tooth surfaces, USA Patent 6,205,879, 2001. </p><p>　　[6] F.L. Litvin, A. Fuentes and K. Hayasaka, Design, manufacture

89、, stress analysis, and experimental tests of low-noise high endurance spiral bevel gears, Mech. Mach. Theory 41 (2006), pp. 83–118. SummaryPlus | Full Text + Links | PDF (1234 K) </p><p>　　[7] F.L. Litvin an

90、d A. Fuentes, Gear Geometry and Applied Theory (second ed.), Cambridge University Press, New York (2004). </p><p>　　[8] J.J. Moré, B.S. Garbow, K.E. Hillstrom, User Guide for MINPACK-1, Argonne National

91、 Laboratory Report ANL-80-74, Argonne, Illinois, 1980. </p><p>　　[9] A.D. Pierce, Acoustics. An Introduction to Its Physical Principles and Applications, Acoustical Society of America (1994). </p><

92、;p>　　[10] J.D. Smith, Gears and Their Vibration, Marcel Dekker, New York (1983). </p><p>　　[11] H.J. Stadtfeld, Gleason Bevel Gear Technology—Manufacturing, Inspection and Optimization, Collected Publicat

93、ions, The Gleason Works, Rochester, New York (1995). </p><p>　　[12] O.C. Zienkiewicz and </p><p>　　對降低齒輪傳動裝載和卸載時因誤差引起的噪音的研究</p><p>　　作者 弗萊德 L.萊特芠那, 丹尼爾.文科黑特</p><p>　　摘要

94、 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

95、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。</p><p>　　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</p><p><b>　　1緒論</b></p><p>　　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

96、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)）</p><p>　　為減少噪音而依下列的計劃進行：</p><p>　?。?）牙齒接觸表面被局部化</p><p>　　（2）提供一個傳輸誤差的

97、函數(shù)。這種傳輸錯誤是由未對準的一函數(shù)所引起的[7]。</p><p>　　（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)]</p><p>　　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)]</p><p><b>　　2齒牙表面的修正&l

98、t;/b></p><p>　　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：</p><p>　　螺旋狀的齒輪，螺旋狀的斜齒輪，蝸桿齒輪。</p><p>　　2.1 螺旋狀的齒輪傳動</p><p>　　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]。</p>

99、<p>　　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]）。</p><p>　　2.2 螺旋狀的斜齒輪</p><p>　　應(yīng)用提供兩個有誤差的刀尖Σp 和Σg 而有局部接觸會產(chǎn)生螺旋狀的斜齒輪：Σp和Σg二者

100、是分別用來產(chǎn)生小齒輪和齒輪的[7]。倆個刀尖Σp和Σg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Σp和Σg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]。</p><p>　　2.3 圓柱型蝸桿齒輪傳動</p><p

101、>　　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：</p><p>　　(1)長期在齒輪箱中研磨加工而使齒輪傳動畸形；</p><p>　　(2)齒輪傳動在長期的運轉(zhuǎn)下產(chǎn)生負載，