A MODEL TO PREDICT SUN GEAR RADIAL ORBIT OF A PLANETARY GET SET HAVING MANUFACTURING ERRORS THESIS

Transcription

1 A MODEL TO PREDICT SUN GEAR RADIAL ORBIT OF A PLANETARY GET SET HAVING MANUFACTURING ERRORS THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Anindo Banerjee, B.S. Graduate Program in Mechanical Engineering The Ohio State University 2012 Master's Examination Committee: Dr. Ahmet Kahraman, Advisor Dr. Sandeep Vijayakar

2 Copyright by Anindo Banerjee 2012

3 ABSTRACT In this study, a two-dimensional analytical model is proposed to predict radial motions of the floating sun gear of an n-planet gear set ( n [3,5] ) due to gear run-out errors, carrier eccentricity and planet pin hole position errors. This analytical model is based on an earlier model proposed by Hidaka [1] for a three-planet gear set having only gear run-out errors. The proposed model extends Hidaka s formulations to include four- and fiveplanet gear sets as well as pin hole position errors on the carrier. The model is used to perform parametric studies on the relation between errors (types, amplitudes and relative orientation angles) and the resultant sun gear radial motions. A frequency-domain representation of sun gear motions is introduced to provide means to use any measured sun orbit as a diagnostics tool such that the error content of the gear set can be directly identified. At the end, the proposed analytical model results are compared to sun orbit measurements of Boguski et al. [2] and predictions of a finite element based computational model to demonstrate its accuracy. ii

4 This document is dedicated to my advisor Dr. Ahmet Kahraman, my family, and Lord Sri Narasimha Saraswati. iii

5 ACKNOWLEDGEMENTS I would like to express my heartfelt gratitude to my advisor, Dr. Kahraman, for having faith in me and motivating me at every step to be able to drive this work to completion. His invaluable teachings and suggestions will serve as a guiding beacon to me throughout my life. I would also like to thank my senior, Dr. David Talbot for helping me with a number of technical concepts throughout my program here. My sincerest appreciation goes out to Dr. Sandeep Vijayakar for not only letting me use CALYX but also guiding me all along the way, and accepting to be a part of my thesis defense committee. A special thanks to Mr. Jonny Harianto and Mr. Samuel Shon for their technical and administrative support. To all my friends and colleagues, I thank you greatly for making my time here interesting and fun filled. Finally, I owe everything I am and ever will be, to my parents and my grandmother. It is nothing but their sacrifice and dedication which I see in my success today. iv

6 VITA October 30, Born Rajkot, India January - February, Engineering Intern General Electric Co. Bangalore, India June, B.S., Mechanical Engineering, Visvesvaraya Technological University Bangalore, India July, November, Engineer, General Electric Co. Bangalore, India September 2010 to present...graduate Research Associate, Department of Mechanical Engineering The Ohio State University Columbus, OH FIELD OF STUDY Major Field: Mechanical Engineering v

7 TABLE OF CONTENTS ABSTRACT ii DEDICATION...iii ACKNOWLEDGEMENTS... iv VITA...v LIST OF TABLES... ix LIST OF FIGURES... x NOMENCLATURE... xv CHAPTER 1: INTRODUCTION Background and Motivation Literature Review Scope and Objective Thesis Outline... 8 CHAPTER 2: AN ANALYTICAL MODEL TO DETERMINE SUN GEAR ORBIT Introduction... 9 vi

8 2.2 Kinematic Analysis of Sun Orbit of a Three-Planet Gear Set Two-Dimensional Model Displacement of the Sun Gear due to Run-out Errors Influence of Planet Pin Hole Position Errors Kinematic Analysis of Sun Orbit of Four and Five-Planet Gear Sets Effect of Run-out Errors of Various Members Influence of Planet Pinhole Position Errors Parametric Study on the Effect of Various Errors on Sun Gear Orbit Run-Out Errors of Central Members Run-Out Errors of Planets Planet Pin Hole Position Errors Effect of Planet Mesh Phasing Combined Effect of Manufacturing Errors CHAPTER 3: COMPARISON OF THE ANALYTICAL MODEL TO A COMPUTATIONAL MODEL AND EXPERIMENTS Introduction Computational Study Computational Model: PLANETARY2D Comparison of Results of the Analytical and Computational Models Comparison of the Analytical Model to Experiments Summary vii

9 CHAPTER 4: SUMMARY AND CONCLUSIONS Thesis Summary Main Conclusions Recommendations for Future Work REFERENCES viii

10 LIST OF TABLES Table: Page: Table 2.1 Basic design parameters of the 4-planet gear set used in the study. IP: In Phase, SP: Sequentially Phased, CP: Counter Phased, S: Sun, P: Planet, R: Ring...31 Table 3.1 Planet pin hole position errors of the carrier used in the experiments of Boguski et al [2] simulated in this study. Orientation angles measured as in Figure Table 3.2 Error magnitudes and orientations assumed in the analytical model to match experimental results of Boguski et al [2]...68 ix

11 LIST OF FIGURES Figure: Page: Figure 2.1 Two-dimensional lumped parameter model of a single-stage planetary gear set Figure 2.2 An equivalent static model to calculate sun gear displacement Figure 2.3 Influence of various run-out errors on sun gear motion Figure 2.4 Influence of planet pin-hole position errors for a threeplanet gear set Figure 2.5 Influence of planet pin-hole position errors for a four-planet gear set Figure 2.6 (a) Radial sun orbit of a four-planet gear set with E = 40 µ m and E = 40 µ m, and (b) the Fourier spectrum c s of xt () Figure 2.7 (a) Radial sun orbit of a four-planet gear set with, E = 40 µ m, E = 40 µ m and E = 40 µ m, β = 0 and (b) c s r the corresponding Fourier spectrum of x(t) r x

12 Figure 2.8 (a) Radial sun orbit of a four-planet gear set with E p1 = 40 µ m and (b) the Fourier spectrum of xt ()...37 Figure 2.9 (a) Radial sun orbit of a four-planet system with E p1 = 15μm, β p1 = 15, E p2 = 34μm, β p2 = 312, E p3 = 26μm, β p3 = 225 and E p4 = 22μm, β p4 = 172, and (b) the Fourier spectrum of x(t) Figure 2.10 (a) Sun gear orbit of a four-planet gear set for E = 40μm, s E = 40μm β = 0, E 1 = 40μm, γ 1 = 90 and c c E 2 = 20μm, γ 2 = 90, E 3 = 0μm, E 4 = 20μm γ 4 = 270, and (b) the Fourier spectrum of x(t) Figure 2.11 (a) Sun gear orbit for the case of E = 40μm, E = 40μm s c β = 0 and pin hole position errors E = 40 µ m, 20 µ m, c γ 40 µ m,20 µ m, i = 90, 270, 90, 90, and (b) The i Fourier spectrum of the x displacement Figure 2.12 (a) Sun gear orbit of a sequentially phased four-planet gear set with E = 40 µ m and E = 40 µ m, and (b) the Fourier c s spectrum of x(t) xi

13 Figure 2.13 (a) Sun gear orbit of a counter phased four-planet gear set with E = 40 µ m and E = 40 µ m, and (b) the Fourier c s spectrum of x(t) Figure 2.14 (a) Sun gear orbit of a four-planet system with E = 40 µ m c, E s = 40 µ m, E r = 10 µ m, β r = 0, E = 15 µ m, 34 µ m,26 μm, 22μm, β pi = 15, 312, 225,172, and (b) the pi Fourier spectrum of x(t) Figure 3.1 Sun orbit and X ( ω ) of an in-phase gear set with E c = 30 µm E s = 75 µm and E = 70 µ m, β = 120 using (a1, r a2) the analytical model and (b1, b2) Planetary2D...53 Figure 3.2 Sun orbit and X ( ω ) spectrum of an in-phase gear set with E c = 30 µm, E s = 75 µm, E 1 = 40 µ m, γ 1 = 60, E 2 = 30 µ m, γ 2 = 75, E 3 = 10 µ m, γ 3 = 90 and E 4 = 20 µ m, γ 4 = 340 (Figure 2.5) using (a1,a2) the analytical model and (b1,b2) Planetary2D Figure 3.3 Sun orbit and X ( ω ) spectrum of an in-phase gear set with E p1 = 18 µ m, β p1 = 40, p2 p2 p3 p3 r E = 10 µ m, β = 310, E = 15µ m, β = 120 and xii

14 E = 12µ m, β = 250, (Figure 2.5) using (a1,a2) the p4 p4 analytical model and (b1,b2) Planetary2D... Figure 3.4 Sun orbit and X ( ω ) spectrum of a sequentially-phased gear set (Table 2.1) with E c = 30 µm, E s = 75 µm and E = 70 µ m, β = 120 using (a1,a2) the analytical model r r and (b1) Planetary2D Figure 3.5 Sun orbit and X ( ω ) spectrum of an in-phase gear set with E c = 30 µm, E s = 75 µm, E 1 = 40 µ m, γ 1 = 60, E 2 = 30 µ m, γ 2 = 75, E 3 = 10 µ m, γ 3 = 90, E 4 = 20 µ m, γ 4 = 340, E p1 = 18 µ m, β p1 = 40, E p2 = 10 µ m, β p2 = 310, p3 p3 p4 p4 E = 15µ m, β = 120 and E = 12 µ m, β = 250 (Figure 2.5) using (a1,a2) the analytical model and (b1,b2) Planetary2D Figure 3.6 (a) Measured [2] and (b) predicted sun orbits for an inphase, four-planet gear set Figure 3.7 (a1, a2) Measured [2] and (b1,b2) predicted X ( ω) and Y ( ω ) spectra for an in-phase, four-planet gear set...62 Figure 3.8 (a) Measured [2] and (b) predicted sun orbits for a sequentially-phased, four-planet gear set xiii

15 Figure 3.9 (a1, a2) Measured [2] and (b1,b2) predicted X ( ω) and Y ( ω ) spectra for a sequentially-phased, four-planet gear set Figure 3.10 (a) Measured [2] and (b) predicted sun orbits for a counterphased, four-planet gear set Figure 3.11 (a1, a2) Measured [2] and (b1,b2) predicted X ( ω) and Y ( ω ) spectra for a counter-phased, four-planet gear set...67 xiv

16 NOMENCLATURE Symbol E E p Definition Run out error vector of a central member Run out error vector of a planet gear E i E it E t F H i i j k Planet pinhole position error vector Vector sum of the effect of all pinhole position errors Total effective carrier eccentricity error Force vector inducing mesh stiffness reactions Harmonic orders in the Fourier spectrum Planet index Unit vector along the x axis Unit vector along the y axis Combined mesh stiffness of the sun-planet and ring-planet interfaces k sp Mesh stiffness along the line of action of the sun-planet interface k rp Mesh stiffness along the line of action of the ring-planet interface K K Stiffness transformation matrix element Stiffness transformation matrix n Total number of planets xv

17 n r R xt ( ) X ( ω ) X y( t ) Z α β γ Γ Γ δ Unit vector tangent to the carrier Radius of the sun gear orbit Mesh stiffness reaction force vector Sun gear horizontal displacement Fourier spectrum of the horizontal sun gear displacement Sun gear displacement vector Sun gear vertical displacement Number of gear teeth Pressure angle Initial clocking angle of an eccentricity vector Clocking angle of the planet pinhole position error vector Phase component of sun orbit due to planet eccentricity Phase component of sun orbit due to planet eccentricity Mesh displacement vector θ ( t) Rotation angle of a member ϕ ξ s ω Angle at which a planet branch is located Line of action vector along a sun-planet mesh interface Angular velocity of a member xvi

18 Subscript i s r c p n M V Definition Planet/Pin hole Index Sun gear Ring gear Planet carrier Planet gear Total number of planets Mean mesh displacement due to applied torque Time varying mesh displacement due to run-out errors xvii

19 CHAPTER 1 INTRODUCTION 1.1 Background and Motivation Planetary gear sets have been used commonly in automotive and aerospace industries due to their innate characteristics, which align with the design goals of these sectors. In the automotive industry, they can be found in almost all automatic and CVT transmissions, transfer cases and certain differentials. Likewise, helicopter gearboxes and jet engine geared turbofans are a few examples of their application in the aerospace/aviation industries. Apart from this, they can also be found as speed reducers/increasers in wind turbines, off-highway vehicles, and many other applications ranging from material handling to mining systems. The main reason for their extensive popularity is that planetary gear sets offer an efficient and compact construct. The input torque is split among the various planet branches before being transferred to the output member. This parallel loading arrangement, while permitting the use of smaller components, also ensures a longer 1

20 lifespan since none of the gear meshes are loaded to the extent of equivalent parallel axis gears. Apart from compactness, lower pitch-line velocities associated with smaller components and increased contact ratios often result in reduced noise levels during their operation. With these advantages in hand, certain unique demands of planetary gear sets must be met in order for them to function as intended. One such demand is associated with the distribution of the torque to planet branches equally. In principle, all gear mesh forces cancel each other in a planetary gear set having equally spaced planets. However, this is often not possible due to errors present in the gear set such as run-out errors of gears, eccentricity of the carrier and position errors of pin holes on the carrier. Hence, while in principle there should not be any need for piloting all of its central members radially, it has been shown that allowing certain central members (often the sun gear) to float radially provides the means for the gear set to self-center itself in the presence of such errors. In case of a floating sun gear, the sun gear moves radially along a trajectory defined by the magnitudes and orientations of these errors as well as number of planets and the phasing conditions of planets. The main purpose of this research is to study radial motions of the sun gear under the influence of typical gear and carrier errors. Since a planetary gear set contains various structurally and functionally distinct components behaving together as a unit, knowledge of this behavior will prove useful in identifying the effect various errors have 2

21 on the system-level behavior. It can also be used to estimate the appropriate value of certain design parameters, such as backlash, to be used in the gear set. 1.2 Literature Review In recent gear literature, planetary gear sets have been a main topic for two primary concerns. One concern is their dynamic behavior. Numerous studies were published to characterize the dynamic behavior of planetary gear sets from the dynamic load (reliability) and noise perspectives. As quasi-static behavior is of main concern here, these planetary dynamics studies will not be reviewed here. The second primary concern is the planet-to-planet load sharing. Ideally, each planet branch of an n-planet gear set should carry a 1 n share of the total torque applied to its central input member. Gear run-out errors, eccentricities and pin hole position errors of the planet carrier might create conditions to prevent such ideal (equal) planet load sharing conditions. This problem has been known for almost a century as Lanchester et al [3] was one of the first to report the benefits of a three-planet gear set over a four-planet one (in a qualitative sense) in terms of their load sharing performance. Much later, Seager [4] came up with a simplified analytical model to demonstrate the importance of floating the sun gear on the load sharing behavior of an n-planet ( 3 n 6) gear set. He considered load sharing issues to arise from two sources, namely planet pin hole position errors and gear tooth thickness errors. He concluded that equal load sharing is possible in a three-planet gear set with a floating sun gear irrespective of the error configurations. Working on the same problem, Imwalle [5] further classified planetary gear systems based on the degree of 3

22 static determinacy imposed by the constraints on the various members. According to him, increasing the degrees of freedom of the system (by eliminating radial bearings of some of the members, like the sun gear for instance) aided in terms of equalizing planet loads. In a series of experiments carried out on the Stoeckicht Type 2K-H, single-stage planetary gear set, Hidaka [6, 7] was able to draw major conclusions regarding the static and dynamic planet load sharing behavior. He also developed simplified analytical models for three-planet gear sets on the lines of his experimental results. From these studies he concluded that at low speeds where dynamic phenomena can be neglected, the displacement of the sun gear in a three planet system served to compensate for the effect of various manufacturing and assembly errors to achieve equal load sharing conditions. He also stated that this static planet load sharing behavior remains the same under dynamic conditions as well, with additional dynamic gear mesh forces are introduced regardless of planet load sharing. This second conclusion of Hidaka was confirmed later by Kahraman [8] through his time-varying dynamic model which has shown that the gear mesh forces along each planet branch are a product of a static planet load sharing factor and a dynamic load factor. A novel method of introducing an intermediate ring gear between the planet gears and their shafts to equalize planet load imbalances was studied by Nakahara et al. [9]. They successfully reasoned that since the floating sun gear was not effective in achieving load equalization for a gear set with more than three planets, and reduced natural 4

23 frequencies rendered ring gear deflections ineffective for the same purpose, increasing the compliance of the planets by inserting the intermediate annulus would yield positive results and also increase bearing life. In the quasi-static zone, however, the ring gear flexibility was found to be of some usefulness to equalize planet loads as was concluded by Kahraman and Vijayakar [10] from their deformable body simulations. The problem of quasi-static load sharing behavior of planetary gears was further studied by Bodas and Kahraman [11] who used the same deformable body finite element algorithm as ref. [10] to model an actual four-planet transmission gear set. They characterized the various sources of errors on the basis of their variation in time and on assembly. From their simulations, they inferred that time-invariant, assembly independent errors such as the ones associated with the planet carrier are the most detrimental to the planet load sharing behavior. Subsequently, at least two different analytical models to predict load sharing behavior in the presence of such time-invariant carrier errors were proposed by Ligata et al. [12] and Singh [13]. In the former, Ligata et al. proposed a discrete analytical model in which they adopted a translational representation of the torsional system to establish the initial configuration of the system and its behavior under subsequent deformation due to increasing load. Not only was the analytical model coherent with earlier experimental results [14-16], but was also further validated by both experiments and simulations. Recently, Boguski et al. [2] carried out experiments to study the load sharing behavior of a four-planet gear set by using strain gauges mounted on planet pins as opposed to tooth root strain based measurements used conventionally. 5

24 Literature on sun gear motion in a planetary gear set is sparse apart from studies carried out by Hidaka et al. [1] in the seventies. Recognizing that eccentricities in the various members play a key role in the time dependent sun gear displacement, he proposed a kinematic model to predict the same in which he assumed constant and equal gear mesh stiffnesses with negligible motion of the ring gear. He supplemented this model with induction based measurements of the sun gear displacement on a Stoeckicht gear set [7]. August et al. [19] looked into this problem from the viewpoint of predicting the static and dynamic loads on the various planets. Gear tooth mesh stiffnesses were treated as being variable and the effect of sun bearing stiffness, gear quality and position errors of various components were included in the model. Further, the model also accounted for non conjugate action (pressure angle change) caused by sun gear motion in the transverse plane. Boguski et al., in their studies mentioned previously [2], carried out measurements of the sun gear orbital motion using precision proximity probes positioned at right angles to the sun shaft sleeve. They presented a family of measured sun orbits for four-planet gear sets with apparent consistent qualitative features without fully describing them. 1.3 Scope and Objective While measured sun orbits are available for three-planet [7] and four-planet [2] gear sets, the analytical treatment of sun orbits is limited to the work of Hidaka [1] on three-planet gear sets only. No analytical study of sun orbits on four- and five-planet gear sets has been performed to date while most automotive and aerospace planetary gear 6

25 sets have more than three planets. As such, this study aims at providing an analytical framework to relate sun gear orbital motions of a single-stage planetary gear set to various manufacturing errors in the gear set. For this goal, the following major thesis objectives are identified. Generalize the three-planet load sharing model of Hidaka [1] to predict sun gear orbits of an n-planet gear set ( n [3,5] ) as a function of eccentricities (or run-out errors) of sun, ring, planets and carrier. Further generalize the analytical model by including the effects of planet pin hole position errors on the carrier. Assess the accuracy of the proposed analytical model through direct comparisons to a state-of-the-art finite elements model and experiments of Boguski et al. [2]. Perform a parametric study to demonstrate the link between various errors with their relative orientation and the resultant sun gear orbits. Provide a frequency-domain presentation of sun orbits such that the amplitudes and frequencies of the harmonic content of the spectra can be used to diagnose what errors of what magnitudes present the gear set. The scope of this study is kept limited to the effect of pitch line run-outs in all gears, eccentricity of the carrier and pin hole position errors of the carrier. Time-invariant, assembly dependent errors such as planet tooth thickness errors, indexing and spacing 7

26 errors, planet bearing needle diameter errors and planet bore and/or pin diameter errors have not been included in the study. All the results have been presented for a four-planet system, although similar results can be obtained for three- and five-planet gear sets as well. 1.4 Thesis Outline The first part of Chapter 2 is devoted to explaining the kinematic framework for the three-planet system, which forms the basis for this thesis. This is followed by an extension of this formulation to four- and five-planet gears sets. Chapter 2 concludes with a parametric study intended to establish the relation between various errors and the resultant sun gear orbit. Chapter 3 presents the results of a validation effort where the results of the analytical model developed in Chapter 2 are compared to those from a commercial finite elements model and published experiments. A summary and the main conclusions as well as a list of recommendations for future work are provided in Chapter 4. 8

27 CHAPTER 2 AN ANALYTICAL MODEL TO DETERMINE SUN GEAR ORBIT 2.1 Introduction As mentioned in the previous chapter, every planetary gear set is subject to certain assembly and manufacturing errors. As such, the planets in an n-planet planetary gear set might carry unequal loads as opposed to an ideal 1/n share of the total input torque. However, maintaining a radially floating configuration for central members like the sun and ring gear helps mitigate the effects of such errors [4]. Since the motion of the ring gear as a rigid body is comparatively small [7], the cumulative compensatory displacement of the sun gear over time in response to the planet load imbalance, referred to as the sun gear orbit, can be used to characterize the kind and magnitude of errors present in the gear set. A systematic study, both experimental and theoretical (using a lumped parameter model), by Hidaka et al. [1] for a three-planet system showed the effect of planet load sharing imbalances on the sun gear orbit. 9

28 This chapter focuses on the analytical study of the sun gear orbits of single stage, three, four and five-planet gear sets under quasi-static conditions. Accordingly, it starts with an insight into the kinematic formulation proposed by Hidaka et al. [1] for threeplanet gear sets and goes further into expanding the same for four and five-planet gear sets. Apart from the pitch line run-out errors of the various members, a methodology of incorporating planet pinhole position errors, which may be present in the planet carrier, has also been developed. The proposed formulation is restricted to only one kinematic power flow condition, i.e., the fixed ring gear configuration. 2.2 Kinematic Analysis of Sun Orbit of a Three-Planet Gear Set Two-Dimensional Model Due to the nature of constraints imposed by various gear meshes on the floating sun gear, a three-planet system lends itself to a simple deterministic evaluation of the sun gear displacement to equalize planet loads under static conditions [1]. This fact was used by Hidaka in his theoretical treatment of sun gear orbits. Consider a three-planet gear set as shown in Figure 2.1. Mesh stiffnesses at the sun-planet interfaces along the line of action are represented as k spi and at the ring gear-planet interface as k rpi, where the sub-subscript i represents the planet index. This lumped parameter model employs several assumptions. First, the sun gear is assumed to be the only floating central member, as the ring gear exhibits a negligible amount of rigid body displacement under the same conditions [7]. Second, all gear blanks 10

29 y π α+θ c +ϕ 1 2 k rp2 P 2 π +α+θ c +ϕ 1 2 k sp2 k sp3 S k sp1 P 1 k rp1 x P 3 k rp3 Figure 2.1 Two-dimensional lumped mass model of a single-stage planetary gear set. 11

30 along with planet bearings are considered rigid. In addition, gear mesh stiffnesses are assumed to be constant with a linear force-displacement relation over a mesh cycle, which implies that the sun gear displacement due to mesh stiffness variations is neglected. Also, this is treated as a two-dimensional problem, with the sun gear motion confined to its plane of rotation. The input torque is assumed constant, planet load sharing equal (or almost equal), and friction at the tooth contact interfaces is neglected. Under such conditions, the model of the gear set can be replaced by an equivalent model as shown in Figure 2.2. When the sun gear is assumed to move as a rigid body, with adjacent planets rotating on rigid bearings, the springs k spi and krpi behave as if they are in an in-series connection with an equivalent spring stiffness given by k i kspikrpi =. (2.1) ( k + k ) spi rpi Let the geometric center of the perfect planetary gear set (i.e. without any errors) be the origin of the fixed coordinates. The position of the displaced sun gear is shown as having the position X = [ x y] T. The operating pressure angle α is assumed to be same for the sun-planet and ring-planet meshes and any change in α due to this displacement has been neglected as per ref. [1]. The reaction force in spring i (along its axis) due to the movement of the sun gear by an amount x is R i = ki xcos π α+θ ( ) 2 c t +ϕ i. (2.2a) 12

31 ξ si δ π α+θ C +ϕ 1 2 M y j i α i k 1 π α+θ c +ϕ 2 2 δ Vi () t ( xy, ) x π α+θ C +ϕ 3 2 k 2 k 3 δ M δ M Figure 2.2 An equivalent static model to calculate sun gear displacement. 13

32 Here θ c() t is the rotation angle of the carrier and ϕ i = 23 ( i 1), [1, 3] π i. Since the springs are at arbitrary orientations relative to the x-y coordinate frame, any displacement of the sun gear along any of these axes evokes a reaction force from the springs, which has components along both axes. The reaction force vector of spring i along the coordinate axes caused by any sun gear displacement x along its respective axis is R 2 { kxcos π ( t) 2 } i { kx i cos π c() t i sin π c() t i } = α+θ +ϕ i i c i + α+θ +ϕ α+θ +ϕ 2 2 j (2.2b) where i and j are the unit vectors along the x and y axes, respectively. Similar expressions can be derived for the reaction force in the same springs due to a displacement of the sun gear in the y direction. Summing all force components (spring reactions) acting on the sun gear along the x and y directions, a force-displacement relationship is established as KX= F (2.3a) where = [ x y] T X is the sun gear position vector and K is the 2 2 stiffness matrix K K K = K21 K, (2.3b) 22 14

33 isin [ c( ) i], i= 1 K = k θ t α+ϕ icos [ c() i] sin [ c() i], i= 1 K = K = k θ t α+ϕ θ t α+ϕ icos [ c( ) i]. i= 1 K = k θ t α+ϕ While in operation, the kind of loads that gear teeth in mesh experience can be divided into two parts; an equivalent mean load due to the applied torque and a timevarying load caused by run-out errors, which is a function of the meshing position. Representing a displacement vector at a particular mesh due to the mean load component as δ M and that due to the variable load component by Vi, the total mesh displacement δ (spring compression) is given as the vector sum δ i () t = δ M Vci () t + Vsi () t + Vpi () t + Vri () t δ δ δ δ. (2.4) Here, additional subscripts c, s, p and r following subscript V denote the contribution from the run-out errors of the carrier, sun, planet and ring, respectively. The negative sign is due to the fact that the mesh displacement due to run-out errors is considered to be positive when they are in the direction of δ M mean would give rise to a time varying sun gear motion. and any mesh displacement above the F is the total sun force vector due to δ i () t is expressed as, 15

34 3 ki( i() t ) cos π δ α+θ () 2 c t +ϕ i i= 1 F () t =. (2.4c) 3 ki( i() t ) sin π δ α+θ () 2 c t +ϕ i i= 1 The solution to Eq. (2.3a) represents the general form of displacement of the sun gear as the carrier rotates and is given as ( K F K F ) x( θ c) =, (2.5a) ( K K K ) ( K F K F ) y( θ c) =. (2.5b) ( K K K ) Displacement of the Sun Gear due to Run-out Errors As previously mentioned, sun gear displacement as a result of mesh stiffness variation over a mesh cycle can be neglected due to its small magnitude as compared to the effects due to various run-out errors. In this direction, Figure 2.3 shows the effect each run-out error has on both the sun-planet and ring-planet mesh interfaces. Let E s, E r, E c and E pi denote the pitch line run-out error vectors of the sun gear, ring gear, carrier and planet i, respectively and β s, β r, β c and β pi denote the initial angles of these errors with respect to the x axis as shown in Figure 2.3. At the initial ( t = 0 ) position, [ cos( ) sin( ) ] E = E β i+ β j. (2.6) s s s s 16

35 j y i θ p +β p1+ϕ1 E p1 ξ r1 π +α+θ c +ϕ 1 2 E s E c θ +β s s ξ s1 E r r θ +β c θ +β r c π α+θ c +ϕ 1 2 x Figure 2.3 Influence of various run-out errors on sun gear motion 17

36 The other errors can similarly be resolved into their components along the coordinate axes. The gear mesh displacement along any given line of action due to the various runout errors can be given by their components in the direction of that line of action. It is this mesh displacement that generates the additional tooth loads that cause the floating sun gear to move in response at every time instant. Let ξsi be the unit vector along the line of action of the sun gear and planet i and ξri be the unit vector of the ring gearplanet i line of action as shown in Figure 2.3. Then, the total mesh displacement due to the sun gear and ring gear run-out errors can be expressed as { π } δ = E cos θ () t +β α+θ () t +ϕ Vsi E s ξ si = s s s 2 c i, (2.7) { π } δ = E cos θ () t +β +α+θ () t +ϕ Vri E r ξ ri = r r r 2 c i. (2.8) Here θs () t and θ r () t denote the rotation angles of the sun and ring gears at a particular time instant with the counter-clockwise direction being positive. An important point to note here is that the terms θ () t θ () t and θ () t θ () t inside the above braces s c represent the angular speeds of the sun and ring gears with respect to the carrier and contribute to the nature of the periodicity seen in the sun orbit of a particular system. r c Secondly, E s affects only the sun-planet meshes and E r affects only the ring-planet meshes with the planets at a time and hence can be represented at each sun/ring-planet 18

37 mesh with a phase difference of ϕ i. Meanwhile, E c affects both the sun-planet and ring - planet meshes simultaneously. Proceeding in the same manner as Eq. (2.7) and Eq. (2.8), the mesh displacements due to E c in the ξ si and ξ ri directions are defined, respectively, as, π ( 2 ) π ( ) Ec ξsi = Ec cos βc +α ϕi, Ec ξri = Ec cos βc α ϕ. 2 i (2.9a) With the rigid planet body approximation, the deflections at the sun and ring meshes of a planet add up in torsional sense, so that these equations can be combined into one single component along the eccentricity becomes ξ si direction. Then, the total mesh displacement due to carrier δ = E ξ + E ξ = 2E cos αsin( β ϕ ). (2.9b) Vci c si c ri c c i It is evident from this equation that δ Vci is constant. Finally, the run-out error of planet i can be resolved into the mesh displacements along its interfaces with the sun and the ring gears as { π 2 } π { } E pi ξsi = Epi cos θ p () t +β pi +ϕi α+θ c() t +ϕi, E pi ξri = Epi cos θ p () t +β pi +ϕi +α+θ (). 2 c t +ϕ i (2.10a) Combining these equations gives 19

38 δ Vpi =Epi ξsi + Epi ξ ri = 2Epi cosαsin θp () t θ c() t +βpi. (2.10b) The above equation reflects the direct influence of the run-out error of a planet i on deflection of its own meshes. All the above equations have ξ Si components that represent the variable mesh displacement along lines of action in the reference frame of the carrier. As previously mentioned, the sun gear is floating and hence it offers no reaction force to this and its motion in response is identical to the sum of these mesh displacements, in the opposite direction. In order to obtain the instantaneous displacements of the mesh springs in the fixed coordinate frame due to, for instance, the sun gear eccentricity, δ Vsi has to be substituted into the right side of the generalized force displacement relation, Eq. (2.3a). Assuming equal mesh stiffnesses for simplicity, and realizing that the mean tooth load component δ M sums up to zero over the three planets (due to equal planet load sharing assumption), the force displacement relation becomes 3 3 cos π ( ) 0 δ Vsi α+θ 2 c t +ϕi 2 xt ( ) i= y( t) 3 2 Vsi sin π ( ) 2 c t i i= 1 =. (2.11) δ α+θ +ϕ Evaluating the above expression, the displacement of the sun gear in fixed coordinates due to the sun gear eccentricity can be expressed as 20

39 ( ) ( ) ( t) ( t) xt cos θ s +βs X ( t) = = Es. (2.12) y t sin θ s +βs Similar equations can be derived for the sun gear displacement due to the effect of the carrier and ring eccentricity. Since an eccentricity in the ring gear affects only the ring planet mesh, (and with the ring exhibiting a negligible amount of displacement, ref. [7]) an additional transformation by an angle π 2α has to be carried out to extract its effect along the line of action of the sun gear-planet mesh. The kinematic relation between absolute velocities of the central members of a planetary gear set is given by Willis equation: Z ω + Z ω = ( Z + Z ) ω (2.13a) s s r r s r c where Z s and Z r are the number of teeth of the sun and ring gears, and ω s, ω r and are the rotational speeds of the sun and ring gears and the carrier. For the case of a fixed ring gear ( ω r = 0 ), one obtains ω c Z ω (1 r s = + ) ω c, (2.13b) Zs Z ω (1 r p = ) ω c, (2.13c) Z p where Z p and ωp are the number of teeth and absolute rotational speed of planet gears. 21

40 Substituting Eq. (2.8) and (2.9b) in Eq. (2.3a) and expressing the result in the form of Eq. (2.12), the cumulative sun gear displacement vector due to eccentricity of the central members (with a fixed ring) is obtained as Z cos (1 r + ) ω ct +βs xt () Zs cos ( ω ct +βc) X ( t) = = Es + 2Eccos α yt () Z sin ( ct c) sin (1 r ω +β + ) ω ct +βs Z s cos( π 2 α+βr ) + Er sin( π 2 α+βr ) (2.14) where xt () and yt () represent the respective sun gear displacements in the fixed coordinates. are found as By following a similar procedure, the sun gear orbit due to planet eccentricities 1 n 2 n xt ( ) = cos α Epi cos( ϕi βpi α ) + Epi sin( ϕi βpi α) n i= 1 i= 1 Z cos (1 r + ) ω ct +Γi Z p 1 n 2 n cos α Epi cos( ϕ i +βpi α ) + Epi sin( ϕ i +βpi α ) n i= 1 i= 1 Z cos (1 r ) ω ct +Γi, Z p (2.15a) 22

41 1 n 2 n yt ( ) = cos α Epi cos( ϕi βpi α ) + Epi sin( ϕi βpi α) n i= 1 i= 1 Z sin 1 r + ω ct +Γ i Z p 1 n 2 n cos α Epi cos( ϕ i +βpi α ) + Epi sin( ϕ i +βpi α ) n i= 1 i= 1 Z sin 1 r ω ct +Γ i Z p (2.15b) where n sin( ϕi βpi α) 1 tan i= 1 Γ i =, n cos( ϕi βpi α) i= 1 n sin( ϕ i +βpi α) 1 tan i= 1 Γ i = n cos( ϕ i +βpi α) i= 1 In these equations, n = 3 since the gear set has three planets. 23

42 2.2.3 Influence of Planet Pin Hole Position Errors Pin holes on the carrier on which planet bearing pins (serving as the inner races) are mounted are subject to variations in their nominal position due to errors associated with the manufacturing of them. Such variations are known to cause load sharing issues by moving a planet or a number if planets towards (or away from) the normal meshing positions. Since these errors maintain their value as the carrier rotates, they can be labeled as time-invariant, assembly independent errors. A number of studies in the past have focused on determining the effect of such errors on planet load sharing analytically [12, 14, 18], computationally [11, 12] and experimentally [15]. A total of six parameters fully define the carrier pinhole position errors of a threeplanet system: magnitudes, E 1 to E 3 and position angles, γ 1 to γ 3. These errors can be visualized as affecting the sun orbit by effectively translating the center of the sun gear in a manner similar to E c, which translates the center of the carrier. Since they are timeinvariant in the carrier reference frame, they can be included in the carrier eccentricity term in Eq. (2.14). Further, after resolving the pinhole errors in the radial and tangential directions it can be seen that: Errors in the radial directions do not contribute to variations in the load sharing behavior of the planets, and hence, they do not influence the shape and diameter of the orbit. However, they contribute to how far the sun gear can move radially before mesh locking can be expected. 24

43 Errors in the tangential direction are the ones, which can be thought to be effectively equivalent to the carrier eccentricity with certain influence on the overall form and diameter of the sun gear orbit. In order to obtain the sun gear displacement due to pin hole errors, first, it can be inferred that the component of the pin hole position errors in the tangential direction compresses the mesh (modeled as springs) in that direction as shown in Figure 2.4 for a three planet system. Consequently, the sun gear must displace to achieve force equilibrium. Since it is the sun gear that moves and this displacement remains fixed in the carrier reference frame, the effect manifests itself as a term analogous, but opposite in direction to to be satisfied E c. Assuming equal mesh stiffnesses as before, the following equation has n i= 1 [ ] Ei ni + Eit ni ni = 0 (2.16a) where ni is the direction cosine vector tangent to the carrier at the i th planet. In this case of a three-planet gear set, the resultant errors (Figure 2.4) is given as Eit vector due to pin hole [ ] { } E 1 1 it = E3sin γ3 E2sin γ2 i+ E 3 2sin γ 2 + E3sin γ3 2E1sin γ1 3 j. (2.16b) 25

44 j E 2 γ 2 2 i E c E i E 1 E it β t E t 1 γ 1 3 γ 3 E 3 Figure 2.4 Influence of planet pin-hole position errors for a three-planet gear set. 26

45 It is combined with the carrier eccentricity to obtain the total effective eccentricity of the carrier as Et = Ec E it. (2.16c) Here, the first planet is assumed to be at ϕ 1 = 0 without loss of generality with subsequent planets defined in the counter-clockwise direction. 2.3 Kinematic Analysis of Sun Orbit of Four and Five-Planet Gear Sets Effect of Run-out Errors of Various Members The basic premise of the three-planet formulation, namely the ability of the sun gear to move to a position to distribute the load to each planet equally is not generally valid for planetary gear sets with more than three planets. It was shown both experimentally and theoretically that, in presence of certain errors, some of the planets might end up carrying more load than others [11, 14, 15]. Equation (2.15) that accounted for the effect of planet run-out errors was derived by assuming equal load sharing amongst the planets as described in Section As long as this condition can be met reasonably well in four- and five-planet systems, Eq. (2.15) can be used for four- and five-planet gear sets by setting, n = 4 and n = 5, respectively. 27

46 2.3.2 Influence of Planet Pinhole Position Errors Similar to the three-planet case, equations to account for the effect of planet pinhole position errors can be derived for the case of the four and five planet systems. Referring to Figure 2.5 for a four-planet carrier with pin hole errors of E1 to E 4 at angles γ 1 to γ 4, the resultant pin hole position error vector is defined ( sin sin ) ( sin sin ) Eit = E γ E γ i+ E γ E γ j. (2.17) The same methodology is extended to a five-planet system with pin-hole errors of E 1 to E 5 at angles γ 1 to γ 5 to obtain ( ) ( ) Eit = 0.38 E2sin γ2 E5sin γ E3sin γ3 E4sin γ4 i ( ) ( ) E3sin γ 3+ E4sin γ E2sin γ 2 + E5sin γ5 0.4 E1 j. (2.18) In Eq. (2.17) and (2.18), the first planet is located along the x axis at ϕ 1 = Parametric Study on the Effect of Various Errors on Sun Gear Orbit Every error explored in the preceding sections has a unique effect on the sun gear orbit. This section is devoted to a systematic study of the influence of combinations of the aforementioned errors on the sun gear orbit. It is also valuable from the diagnostics point of view to be able to determine what kind of errors the gear set has through an analysis of the resultant sun gear orbit. 28

47 E 2 γ 2 j 2 i E c E it E 1 γ 3 3 E it β c E t β t 1 γ 1 E 3 4 γ 4 E 4 Figure 2.5 Influence of planet pin-hole position errors for a four-planet gear set. 29

48 Every case of errors in the system is dealt with in the following manner. A particular error or a certain combination of errors of certain magnitudes is considered and the resultant sun gear orbit determined by using the proposed analytical model. Along with this, a Fourier analysis of the orbit is also performed with xt () (horizontal displacement of the sun gear center) as the representative time signal. The corresponding Fourier spectrum X ( ω) is obtained with its frequency axis normalized to represent carrier orders. Examining the harmonic content of X ( ω ), the errors present in the gear set are related to the harmonic amplitudes and frequencies Run-Out Errors of Central Members First, combinations of run-out errors of the sun gear and the carrier are examined. A sample four planet gear set used previously for experiments of Boguski et al [2] as specified in Table 2.1 is chosen as the example gear set for this study. By themselves, E s and E c yield circular orbits with a period corresponding to the speed of the member in which the error is present. A system with run-out errors of Es = Ec = 40 µm oriented in the same direction initially is presented as the first example. A trochoidal orbit with 12 distinct loops is obtained as shown in Figure 2.6(a). This can be described by the kinematics of the gear set chosen for the study. The example gear set has in-phase planet meshes, which means that, at any given point in time, all the sun planet and ring planet meshes are at the same point in their meshing cycle, respectively. 30

49 Table 2.1 Basic design parameters of the 4-planet gear set used in the study. IP: In Phase, SP: Sequentially Phased, CP: Counter Phased, S: Sun, P: Planet, R: Ring IP SP CP Parameter S P R S P R S P R Number of Teeth Normal Module [mm] Pressure Angle [deg] Helix Angle [deg] Center Distance [mm] Active Face Width [mm]

50 According to Ref. [19, 20], this corresponds to the mathematical conditions Z s n Z = integer and r integer n = ( = and 96 = 24 in this case). Now, from Eq. 4 (2.13b), ω s = 17 5 ω, which means that for every 17 absolute rotations of the sun gear, c the carrier completes 5 rotations. Additionally, the speed of the sun gear with respect to 12 the carrier, ω sc / =ωs ω c= ω c, indicating that the sun gear completes 12 rotations 5 relative to the carrier during this time span. Since the ring gear is stationary, these 5 carrier rotations also represent 5 rotations of the ring gear relative to the carrier. In other words, the length of a trochoidal motion segment (all 12 loops) brings the sun gear and ring gear back to their starting positions with respect to the carrier. Hence, each such loop represents one complete rotation of the sun gear with respect to the carrier. The Fourier spectrum X ( ω) of the xt () component of the same example case is shown in Figure 2.6(b). This spectrum shows a harmonic component at the carrier output rotational order ( H c =ωc ω c = 1) at amplitude equal to 75 µm (this corresponds to 2Ec cos α= 2(40) cos(21.88 ) = 75 µm). The second harmonic term appears at the output order of the sun gear ( H s =ωs ω c = ) at an amplitude of 40 µm that is equal to the Es magnitude used in this example. The order of these peaks follows from the formulation presented in Eq. (2.14) where the amplitude of the first two terms corresponds to the amplitude of the peaks in the Fourier spectrum. The first term of X ( ω ) is zero since the orbit is centered at the origin. The radius of the envelop of the 32

51 orbit in Figure 2.6(a) is about 114 µm which corresponds to r = E + 2 E cos( α), since there are no carrier pin hole position errors present. s c An eccentricity in the ring gear is responsible for moving the sun orbit as a whole to a position where its center is at a distance of Er and at an angle of π 2α+βr from the geometric center of the gear set. In the above example, in addition to Es = Ec = 40 µm, E r = 40 µm is applied next. The orbit shown in Figure 2.7(a) is identical to that or Figure 2.6(a) except is center is shifted at an angle of π 2α+β r = 180 2(21.88 ) 0 = by an amount of 40 µm. The corresponding X = E r π α+β r X ( ω) in Figure 2.7(b) reveals a zero order at amplitude of cos( 2 ) = 40 cos( ) = µm Run-Out Errors of Planets In case of a three-planet gear set, the proposed analytical formulation for sun orbit assumes an equal load sharing between the planets. In higher number of planet cases, it is seen that sun orbits due to planet run-outs can be successfully predicted using Eq. (2.15) provided the sun gear and ring gear are both maintained in the floating configuration. Hence assuming such a dual floating configuration, the effect of having a run-out error of one planet of a four-planet gear set ( E p1 = 40 µm and E = E = E = ) is shown in Figure 2.8. p2 p3 p4 0 33

52 150 (a) y(t) [µm] x(t) [µm] 100 (b) 80 H c X(ω) [µm] H s Carrier Order Figure 2.6 (a) Radial sun orbit of a four-planet gear set with E = 40 µ m, and (b) the Fourier spectrum of x(t). s E = 40 µ m and c 34

53 150 (a) y(t) [µm] x(t) [µm] 100 (b) 80 H c X(ω) [µm] X H s Carrier Order Figure 2.7 (a) Radial sun orbit of a four-planet gear set with, E c = 40 µ m, E s = 40 µ m and E r = 40 µ m, β r = 0 and (b) the corresponding Fourier spectrum of x(t). 35

54 The Fourier spectrum shows peaks at orders Hpc / ± Hc= 2.31 and The amplitude of the peaks follows from the same equation. Next consider an arbitrary case with E p1 = 15 µm, E p2 = 34 µm, E p3 = 26 µm and E p4 = 22 µm, clocked initially at angles of 15, 312, 225 and 172, respectively, from the positive x axis. Like the previous case, the system is devoid of any other run-out errors. The resultant sun orbit is shown in Figure 2.9(a) with the corresponding X ( ω) in Figure 2.9(b). In this case, the planets have been numbered in a manner similar to the planet pin holes as shown in Figure 2.5. From this, it can be concluded that peaks at Hpc / ± Hc= Hpc / ± 1 orders are indicative of planet eccentricities present in the system, unless, diametrically opposite planet eccentricities have the same magnitude and orientation, in which case, sun gear does not execute any translation as a rigid body Planet Pin Hole Position Errors As mentioned in Section 2.2.3, planet pin hole position errors fall under the category of errors, which maintain their values in the carrier reference frame. It was also explained that these errors can be visualized to move the sun gear (as opposed to the carrier center which the E c term does) from its geometric center, and hence, can be combined with the carrier eccentricity term. 36

55 80 60 (a) y(t) [µm] x(t) [µm] 100 (b) 80 X(ω) [µm] H pc / H c H pc / + H c Carrier Order Figure 2.8 (a) Radial sun orbit of a four-planet gear set with E p1 = 40 µ m and (b) the Fourier spectrum of xt (). 37

56 80 60 (a) y(t) [µm] x(t) [µm] 100 (b) 80 X(ω) [µm] H pc / H c 20 H pc / + H c Carrier Order Figure 2.9 (a) Radial sun orbit of a four-planet system with E p1 = 15μm, β p1 = 15, E p2 = 34μm, β p2 = 312, E p3 = 26μm, β p3 = 225 and E p4 = 22μm, β p4 = 172, and (b) the Fourier spectrum of x(t). 38

57 For the first gear set, the following pin hole position errors and angles are used: E 1 = 40 µm, E 2 = 20 µm, E 3 = 0 µm, E 4 = 20 µm, γ 1 = 90, γ 2 = 90, γ 3 = 0, γ 4 = 270 E = 40 µm and E = 40 µm. The result for the first case is shown in Figure s c Although X ( ω ) shows peaks at the same Hc and H s orders, the orbit exhibits a much larger radius of about 150 µm as can be predicted from r = Es + 2 E t cos( α) where E t is defined by Eq. (2.16c). For a second configuration with E 1 = 40 µm, E 2 = 20 µm, E 3 = 40 µm, E 4 = 20 µm, γ 1 = 90, γ 2 = 270, γ 3 = 90, γ 1 = 90, E s = 40 µm and E = 40 µm, the orbit can be seen to shrink in size to an approximate radius of about 70 c µm as shown in Figure Similar conclusions can be drawn for three and five planet systems as well., Effect of Planet Mesh Phasing To this point, the example planetary gear set considered had in-phase planet meshes since Z s n Z = integer and r integer n =. In this section, the effect of two other phasing conditions available for a four-planet gear set, namely sequentially-phased and counter-phased, are examined. In a sequentially phased four-planet gear set, all the sunplanet and ring-planet meshes are at a 90 degree phase difference between two adjacent planets. 39