Measurement of Vibrations of Gears Supported by Compliant Shafts. Thesis

Transcription

1 Measurement of Vibrations of Gears Supported by Compliant Shafts Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Ma Ru Kang Graduate Program in Mechanical Engineering The Ohio State University 2009 Thesis Committee: Dr. Ahmet Kahraman, Advisor Dr. Dennis Guenther

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3 ABSTRACT In this study, an accelerometer-based measurement method is proposed for measurement of motions of parallel-axis gears in torsional, translational and rotational directions. This method uses a family of tri-axial accelerometers that are mounted on a shaft flange at a given radius. Acceleration signals from the accelerometers are processed based on a novel formulation to quantify the motions of a gear in the torsional, rotational (rocking), transverse (line-of-action and off-line-of-action) and axial directions. With the similar instrumentation implemented on the mating gear, the dynamic transmission error of the gear pair is also measured. This measurement system is applied for example to spur and helical gear pairs held by different shafts representative of different support compliances. The transverse motions of the spur pinion in the line-ofaction direction, as well as line-of-action, rocking and axial motions of helical gears are shown to be significant when they are supported compliantly. At the end, the same experiments are simulated by using a previously developed gear-shaft-bearing dynamics model to identify the shapes of the modes excited and to show the agreement between the predicted and measured forced responses. ii

4 To my wife iii

5 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Ahmet Kahraman, for his invaluable guidance and support throughout my research and graduate studies. Without his efforts, this work would not have been possible. I am also grateful to Dr. Dennis Guenther for serving on my examination committee. I would also like to thank Sam Shon and Mohammad Hotait for their help in the lab with the gear dynamic test rig as well as with LabView. I am also thankful to David Talbot for his help on Complex Shaft Module of LDP. I would also like to acknowledge the financial support of the many sponsors of the Gear and Power Transmission Research Laboratory at The Ohio State University. Finally, I am as ever, especially indebted to my parents, Mr. Dong-un Kang and Mrs. Myungbok Kim, for their love and support throughout my life. I would also like to thank my beautiful wife, Dayeon Lee, for her dedicated support and encouragement to pursue this degree. iv

6 VITA December 6, Born Jejudo, South Korea. February, B.S. Mechanical Engineering Korea National Maritime University 2007-Present... Graduate Research Associate The Ohio State University FIELDS 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 Chapter 1: INTRODUCTION 1.1 Background and Motivation Literature Review Theoretical Studies Experimental Studies Scope and objectives Thesis Outline...9 Chapter 2: A TEST METHODOLOGY FOR MEASURING GEAR VIBRATIONS 2.1 Introduction Test Machine Gear Test Specimens Test Gear Shafts...20 vi

8 2.5 Measurement System Measurement of Dynamic Transmission Error Measurement of the Transverse Pinion Motions Measurement of the Rocking and Axial Motions of the Pinion Instrumentation and Data Analysis...36 Chapter 3: EXPERIMENTAL RESULTS 3.1 Introduction Test Matrix Spur Gear Test Results Steady-state Forced Response Transient Response Helical Gear Test Results Summary...74 Chapter 4: COMPARISION TO PREDICTIONS 4.1 Introduction Model Predictions Comparison between Experiments and Predictions...84 Chapter 5: CONCLUSION 5.1 Overview Conclusions Recommendations for Future Research...99 vii

9 Appendix A: Mablab Post-Processing Scripts A.1 Steady-State A.2 Transient References viii

10 LIST OF TABLES 2.1 Basic design parameters of the test spur gear pair Basic design parameters of the test helical gear pairs Basic dimensions and bending stiffness values of the test gear shafts considered in this study The test matrix implemented in this study...43 ix

11 LIST OF FIGURES 1.1 A schematic of a gear pair showing the base and pitch circles and the line of action Gear dynamic test machine used in this study. The safety guards are removed for demonstration purposes The spur gear pair used in this study The helical gear pair used in this study Three different test gear shafts considered in this study A helical test gear pair with three tri-axial and two uni-axial accelerometers A schematic of the test gear pair showing the accelerometer locations Block Diagram for the data processing scheme for computation of DTE The components of the radial accelerations of the pinion Generation of sin ω t and cos ω t from the once-per-rev tach signal Illustration of the motions of the pinion along the LOA and OLOA directions Block diagram for the digital data acquisition system Instrumentation used for the data acquisition and analysis Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 100 N-m from a speed-up test...45 x

12 3.2 Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 100 N-m from a speed-down test Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 200 N-m from a speed-up test Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 200 N-m from a speed-down test Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 300 N-m from a speed-up test Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 300 N-m from a speed-down test Comparison of the rms DTE values of the spur gear pair with shaft A at 100, 200 and 300 Nm Comparison of the rms DTE values of the spur gear pair with shaft B at 100, 200 and 300 Nm The rms LOA amplitudes of the pinion of the spur gear pair with shafts B at 100 Nm The rms OLOA amplitudes of the pinion of the spur gear pair with shafts B at 100 Nm The rms LOA amplitudes of the pinion of the spur gear pair with shafts B at 200 Nm The rms OLOA amplitudes of the pinion of the spur gear pair with shafts B at 200 Nm The rms LOA amplitudes of the pinion of the spur gear pair with shafts B at 300 Nm The rms OLOA amplitudes of the pinion of the spur gear pair with shafts B at 300 Nm Comparison of the rms LOA vibration amplitudes of the spur gear pair with shaft B at 100, 200 and 300 Nm...63 xi

13 3.16 Waterfall plots of the radial acceleration of the spur gear pair with shaft B at 100 N-m during (a) speed-up, and (b) speed-down sweeps Waterfall plots of the radial acceleration of the spur gear pair with shaft B at 200 N-m during (a) speed-up, and (b) speed-down sweeps Waterfall plots of the radial acceleration of the spur gear pair with shaft B at 300 N-m during (a) speed-up, and (b) speed-down sweeps The rms DTE amplitudes of the helical gear pair with shafts C at 100 N-m The rms LOA amplitudes of the helical gear pair with shafts C at 100 N-m The rms rocking motion amplitudes of the helical gear pair with shafts C at 100 N-m The rms axial motion amplitudes of the helical gear pair with shafts C at 100 N-m Dynamic models of (a) the spur gear pair with shafts B and (b) the helical gear pair with shafts C Predicted ( q LOA) rms amplitudes of the spur gear pair with shafts B at 100, 200 and 300 Nm Predicted ( DTE ) rms amplitudes of the spur gear pair with shafts B at 100, 200 and 300 Nm Shapes of the two modes excited by the spur gear mesh excitations at (a) 2583 Hz, and (b) 3383 Hz Predicted ( DTE ) rms amplitudes of the helical gear pair with shafts C at 100 Nm Predicted ( q LOA) rms amplitudes of the helical gear pair with shafts C at 100, Nm Predicted ( ψ ) rms amplitudes of the helical gear pair with shafts C at 100 Nm Predicted ( z ) rms amplitudes of the helical gear pair with shafts C at 100, Nm...88 xii

14 4.9 Shapes of the three modes excited by the helical gear mesh excitations at (a) 1058Hz, (b) 2620 Hz, and (c) 2915 Hz Comparison of the predicted and measured ( q LOA) rms amplitudes of the spur gear pair with shafts B at 100 Nm Comparison of the predicted and measured ( DTE ) rms amplitudes of the spur gear pair with shafts B at 100 Nm Comparison of the predicted and measured ( DTE ) rms amplitudes of the helical gear pair with shafts C at 100 Nm Comparison of the predicted and measured ( q LOA) rms amplitudes of the helical gear pair with shafts C at 100 Nm Comparison of the predicted and measured ( ψ ) rms amplitudes of the helical gear pair with shafts C at 100 Nm Comparison of the predicted and measured ( ) rms z amplitudes of the helical gear pair with shafts C at 100 Nm...96 xiii

15 CHAPTER 1 INTRODUCTION 1.1 Background and Motivation Investigation of the dynamic behavior of high-speed gear systems is a critical step in their design. The main reasons for this necessity are the structure-borne vibrations and noise, as well as the durability of the gear system. The dynamic forces created at the gear mesh interfaces create dynamic bearing forces that are transmitted to supporting structures such as the gearbox housing to cause structure-borne vibrations and noise. On the other hand, the dynamic gear mesh forces are fluctuations around the static force that the gear pairs are designed to carry. As a result, the maximum gear mesh forces and the number of force cycles are dictated by the dynamic response of the gear system, with potential detrimental effects on the fatigue life of the gears and bearings. The motion transmission error (TE) has been recognized as one of the main excitations leading to gear vibrations. TE is defined as the the difference between the actual position of the output gear and the position it would occupy if the gear drive were 1

16 perfectly conjugate [1]. Referring to Figure 1.1, TE of a gear pair along its line of action is defined as TE( t) = rpθ p ( t) + rg θ g ( t) (1.1) where is r p and r g are the base circle radii and p θ and θ g are rotational displacements of the pinion (p) and gear (g) forming the pair. TE can be classified as a displacement excitation applied at the gear mesh between the contacting gear teeth. It is typically a periodic function with a fundamental frequency equal to the gear mesh (tooth passing) frequency. As any dynamic system, the TE excitation is amplified under dynamic conditions, causing larger motion transmission errors, called dynamic transmission error (DTE), and larger dynamic loads and stresses, which drastically reduce the life cycles of the gears. During the last several decades, significant emphasis has been placed on measurement and prediction of TE under unloaded and loaded conditions within the quasi-static and dynamic operating speed ranges. In reality, however, the dynamic behavior of a geared system cannot be fully represented by TE as gears rotate and translate in the directions other than the torsional displacements θ p and θ g that are accounted for in Eq. (1.1). This is especially true for the gear systems supported by flexible shafts and bearings and gear types such as helical gears that are inherently threedimensional in their motions. Several theoretical studies [2-11] predicted that such gear systems exhibit coupled torsional-rotational-transverse vibration, rather than purely 2

17 pitch circle radii base circle radii line of action θ p r g r p θ g pinion p gear g Figure 1.1: A schematic of a gear pair showing the base and pitch circles and the line of action. 3

18 torsional motions as captured by the definition of TE. There is very little published in the experimental side to confirm these predictions regarding the influence of shaft flexibilities and gear type on the overall dynamic response of a gear pair. This study aims at filling this void. 1.2 Literature Review This literature review will focus mainly on the effect of shaft deflections on the dynamic behavior of gears. Both theoretical and experimental work will be cited as well as the measurement methods developed for this purpose Theoretical Studies As one of the earlier modeling studies on the impact of shafts on gear pair dynamics, Iida et al [2] considered a system having spur gears mounted on a flexible driven shaft while the driving shaft was assumed to be rigid. The geared shaft system was modeled as a 4-degree-of-freedom model to quantify the differences from a purely torsional model in terms of normal modes (torsional verses coupled torsional-flexural). They showed the relation between the natural frequencies of the system and the flexural stiffness of the shaft to conclude that the natural frequencies are altered by the shaft flexibility. Several studies including Neriya et al [3,4], Kahraman et al [5] Litak and Friswell [6], and Rao et al [7] extended the study of Iida et al [2] by considering the flexibility of both driving and driven shafts in the dynamic model. For instance, Neriya [3] used the 4

19 normal mode analysis to calculate the dynamic tooth loads and the frequency response (shaft deflection) in the transverse direction. Later, Neriya et al [4] investigated the effect of coupling between torsional and transverse motions by using the finite element models of the shafts. Similarly, Kahraman et al [5] developed a finite element model of a geared rotor system to investigate the influences of flexible shafts and bearings. This model considered the coupling between the torsional and transverse motions of gears. The forced response in both torsional and transverse displacements as well as gear mesh and bearing forces was predicted as well as mode shapes corresponding to the natural frequencies. Kahraman et al [5] concluded that the shaft and bearing compliances may significantly affect the dynamic behavior of a geared rotor system in both torsional and transverse directions. They also pointed to the fact that a coupled transverse-torsional motion dictates gear pair vibrations when the shafts and bearings are flexible. The above studies all considered a spur gear pair so that torsional and transverse motions (on the transverse plane of the gears) are sufficient to describe the resultant motions. In case of a helical gear pair, rotational and axial motions due to a helix angle should be taken into account. Neriya et al [8] investigated the vibratory behavior of a helical geared system considering the coupling between torsional, flexural, rotational, and axial motions as well as clearance nonlinearity and time varying tooth stiffness, using a matrix exponential method. Likewise, Kahraman [9] investigated the effect of axial vibrations on the dynamic behavior of a helical gear pair. In order to account for the shaft and bearing flexibilities, he developed a linear dynamic model, including the coupling with torsional, transverse, axial, and rocking motion caused by a helix angle in gear mesh. He considered a single stage helical gear system supported by flexible shafts and rolling 5

20 element bearings, and used a ten degree-of-freedom model to simulate it. The eigenvalue solutions and the modal summation method were used to find free and forced response of the helical gear pair. He concluded that the axial and rocking motions are significantly affected by the helix angle and indicated that the analysis of a helical geared system should be considered with a helix angle. Kubur et al [10] developed a dynamic model of a multi-stage, counter-shaft helical gear system by using a finite element method. The study of Kubur et al [10] was focused on not only the gear pair but also the shafts and bearings. They demonstrated that the free and forced vibrations are significantly influenced by shaft dimensions and bearing stiffness values. Also, Choi et al [11] developed a new analytical model on a turboset consisting of turbine, generator, and a double helical gear pair to predict the coupling with torsional, lateral, and axial motions by using a finite element method Experimental Studies While all these studies cited above pointed to the shaft flexibility as a key parameter impacting gear dynamics, they were all in the theoretical side. With the exception of the study by Kubur et al [10] that presented DTE measurements for a gear pair on flexible shafts, there is little experimental proof on the influence of shaft flexibility. Most of the work on measurements of gear motions focused on the gear transmission error. Smith [12] and Remond [13] measured TE under quasi-static conditions (at very low speeds) by using optical encoders mounted on shafts of test gear 6

21 pairs. Tordion et al [14] and Houser at al [15] employed and accelerometer-based measurement method to measure dynamic transmission error at higher speed conditions. Kahraman and Blankenship [16] investigated the nonlinear dynamic behavior of a geared system in the form of dynamic transmission error using the same accelerometer-based measurement method. They later [17, 18] investigated the influence of involute contact ratio and tip relief on the torsional vibrations for a spur gear, again using the same measurement approach. The above studies focused on torsional motions of the gears while there are very few experimental studies regarding the influence of shaft flexibilities and gear type on the dynamic response of a gear pair. In one such study, Mitchell and Mellen [19] first showed experimentally that the dynamic coupling between torsional and transverse vibrations may increase significantly when the shafts have high compliances. In order to observe this phenomenon, vertical motions of shafts were measured by using proximity probes. They obtained the critical speeds in vertical direction, and found that the speeds were observed around torsional critical speeds measured previously. Umezawa et al [20] investigated experimentally the vibration of a helical gear system having various gear ratios. They implemented sets of accelerometers to measure motions of gears in transverse direction as well to suggest that the shafts and bearings should be taken into consideration for precise assessment of the geared system dynamics. 7

22 1.3 Scope and Objectives Review of the literature above reveals that there is a need for an experimental study of all motions of gears under dynamic conditions. This is partly because the measurement methods are not well established for rocking and axial motions as well as transverse motions. This study focuses on development and demonstration of such a measurement system. Building on the data collection and analysis methods developed earlier by Heskamp [21], the main objective of this study is to devise methods to measure various components of gear dynamic motions including (i) DTE through measurement of the motions in the torsional directions, (ii) transverse motions in the transverse plane of the gears along the line-of-action (LOA) and off-line-of-action (OLOA) directions (iii) the axial motions in the direction of gear rotation axes, and (iv) rocking motions that are in the directions perpendicular to the transverse plane. The specific objectives of this thesis are followed as: Develop an accelerometer-based measurement method to investigate the impact of shaft deflections on the dynamic motions of spur and helical gear pairs. Apply this measurement method to a number of spur and helical gear pairs supported by a number of shafts of varying diameters and length to quantify the impact of shaft flexibilities and gear type on the dynamic motions of the gears. 8

23 Quantify the influence of gear profile modifications and operating conditions (speed and torque) on the gear motion within the resonance and off-resonance regions. Investigate the contributions of the nonlinear effects due to gear backlash to the resultant motions of the gears supported by various shafts. Employ a previously developed multi-mesh gear-shaft system model [22] to correlate and describe the measured behavior. As the focus will be on the dynamic behavior, the low-speed, quasi-static behavior will not be investigated in this study. The measurement systems will be devised for accurate measurements under high-speed dynamic conditions, rather than low-speed conditions. In addition, the types of gears will be kept limited to parallel-axis gears of spur and helical type while the other types of gears will not be included in this study. 1.4 Thesis Outline Chapter 2 introduces the test machine set-up, the data acquisition system, and the data analysis methods for measuring the steady-state and transient responses. Several variations of test spur and helical test gears used in this study will be introduced. The gear shafts of different flexural flexibility (diameter and length) will also be described in detail. A new measurement method by using tangentially-mounted, tri-axial accelerometers will be proposed. The formulations to extract the transverse torsional axial and rotational components of the gear motions from the tri-axial accelerometer signals will be presented. 9

24 Chapter 3 proposes test matrices designed to quantify the impact of the shaft flexibility on the gear motions. This chapter also presents the results from the experiments performed in accordance with these test matrices to exhibit the changes in dynamic behavior with gear type and shaft flexibilities. In Chapter 4, some of the experimental set-ups are simulated by using an existing gear dynamics model [22]. The experimental results are compared to the results of these simulations to identify the natural modes excited by the gear mesh excitations. Finally, major conclusions from this study and a set of recommendations for future work are presented in Chapter 5. 10

25 CHAPTER 2 A TEST METHODOLOGY FOR MEASURING GEAR VIBRATIONS 2.1 Introduction This chapter first describes the test machine used in this study. The details of the spur and helical test gear specimens employed in this study will be given next, including the basic gear design parameters and the tooth profile modifications. In line with the objectives listed in the previous chapter, several gear shafts having different lengths and diameters will be considered here. These shaft variations will be introduced next. With the test machine and test gear pair-shaft arrangements defined, an accelerometer-based measurement system designed to measure angular (torsional and rotational), translational (in the line-off-action and off-line-of-action directions), and axial motions of the mating gears under dynamic conditions will be proposed. Formulations to extract these motions from various accelerometer measurements will be presented and the data collection and analysis system designed to implement these formulations will be described. 11

26 2.2 Test Machine In this study, the test machine shown in Figure 2.1 was employed to precisely measure vibrations for a gear pair under specific torque and speed values. This machine was developed originally by Kahraman and Blankenship for collecting benchmark spur and helical gear transmission error data [23]. Further details of the test machine can be found in this reference. The test machine has a four-square layout that allows a power circulation with a closed mechanical loop formed by two gear pairs (test and reaction gear pair) of the same speed ratio and flexible shafts connecting the respective gears of these pairs. A split coupling mounted on one of the connecting shafts is used to apply a torque value to the loop. For this purpose, one flange of the coupling is held stationary and a user-defined torque is applied to the other flange via a torque arm and calibrated weights. The bolts of the split coupling are tightened before the torque arm is removed and the fixed flange of the coupling is released. This traps a constant torque value within the loop causing both gear meshes to be loaded equally. With this, a small DC motor is sufficient to provide the required amounts of torque to the loop externally to overcome power losses caused by the gear meshes and the bearings. While this four-square arrangement has been used commonly in gear durability testing, its use for dynamics measurements presents a primary issue that must be addressed. Here, the goal is to measure the dynamic behavior of the test gear pair. However, any vibrations created at the reaction gear pair mesh are likely to influence the dynamic behavior of the test gear pair. For this reason, the reaction and test gear pairs 12

27 13 Figure 2.1: Gear dynamic test machine used in this study. The safety guards are removed for demonstration purposes.

28 must be isolated dynamically. Three different measures were employed in the test machine of Figure 2.1 to decouple the test and reaction gear pairs: The shafts connecting the test and reaction gearboxes were made sufficiently long to have stiffnesses that are much softer than the gear mesh stiffnesses and provide sufficient compliance between the gearboxes. A pair of flywheels of large inertias was mounted in the connecting shafts to act as mechanical filters. One of these flywheels also acts as the split coupling whose function was described above. A pair of elastomeric couplings was placed between the shafts of the test gear pair and the connecting shafts to further increase the compliances between the test and reaction gear pairs. In addition, as a precautionary measure, the tooth counts of the test and reaction gears were designed to be different (50 tooth gears for test side and 63 tooth gears for the reaction side) such that any influence of the reaction gear pair on the vibration spectra from the test side can be identified and removed. In this configuration, a speed range of 0 to 4500 rpm and a torque range of 0 to 500 Nm were obtained. The operating torque value within this range was applied manually while the operating speed value was controlled tightly by a speed controller. 14

29 2.3 Gear Test Specimens Both spur and helical test gears were used for the experiment in this study. The spur gear pair is shown in Figure 2.2. This gear pair has perfect involute profiles (no tooth modifications) and has an involute contact ratio of The design parameters of this spur gear pair are listed in Table 2.1. Here, it is noted that the gear pair has a unity (1:1) ratio, a module of 3.0 mm and a pressure angle of 20 degrees. The gear pair operates at a fixed center distance of 150 mm. The second gear pair was helical type as shown in Figure 2.3. This gear pair and the spur gear pair of Figure 2.2 have the same transverse geometry. With a helix angle of degrees, a normal pressure angle of degrees and normal module of mm was obtained with these helical gears. Table 2.2 lists all of the relevant design parameters of this gear pair. This gear pair also employed no tooth modifications. The total contact ratio of this gear pair was 2.75 (an involute contact ratio of 1.75 and a face contact ratio of 1.0). While they were comparable, the spur and helical pairs are expected to exhibit significantly different dynamic responses. First of all, given its increased contact ratio, the helical gear pair should exhibit lower torsional vibration amplitudes while having larger axial and rotational motions due to its helical shape. As the mesh stiffness fluctuations are reduced for the helical gears, the resultant dynamic response is likely to be linear [10] while this spur gear pair was reported to have strongly nonlinear behavior [16-18, 23]. Use of these two gear pairs allows both linear and nonlinear responses to be investigated in this study. 15

30 Figure 2.2: The spur gear pair used in this study 16

31 Table 2.1: Basic design parameters of the test spur gear pair. Design Parameter Pinion Gear Number of teeth Module (mm) 3.00 Pressure angle (deg) Center Distance (mm) Base diameter (mm) Major diameter (mm) Minor diameter (mm) Circular tooth thickness (mm)

32 Figure 2.3: The helical gear pair used in this study 18

33 Table 2.2: Basic design parameters of the test helical gear pairs Design Parameter Pinion Gear Number of teeth Normal Module (mm) Normal Pressure angle (deg) Helix angle (deg) Center Distance (mm) Base diameter (mm) Major diameter (mm) Minor diameter (mm) Circular tooth thickness (mm)

34 2.4 Test Gear Shafts Three different test gear shafts were used in this study. These shafts will be named as A, B and C. Figure 2.4 shows the shapes of these shafts to indicate that they vary in length L (distance between the two spherical roller bearings supporting the test gear shaft at both sides of the gear) and diameter d between the bearings. In this configuration, the gear is nearly at distance L 2 from a bearing. The test machine shown in Figure 2.1 has two bearing pedestals supporting the shafts of the test gear pair. The pedestal to the left is movable to different discrete axial locations to accommodate shafts of different length. Shafts A and C in Figure 2.4 have the same length ( LA = LC = 255 mm or 10 inches), but they differ in terms of their diameters as d A = 39.9 mm and d = 20.1 mm. Meanwhile, shaft B has a diameter of d = 39.9 mm (the same as shaft C A), but its length is longer ( L B = 382 mm or 15 inches). The test gear shafts shown in Figure 2.1 are B type. B As listed in Table 2.3, these three shafts result in three different shaft bending stiffness values. Shaft C has the lowest bending stiffest of all with a stiffness value of 6 4.8(10) N/m. In comparisons to shaft C, shaft A and shaft B are 15.4 and 4.6 times stiffer, respectively, as evident from the values listed in Table 2.3. This provides a sufficient range of bending stiffness (more than one order of magnitude) for studying the effect of shaft bending stiffness on gear motions. It is noted here that all three sets of shafts were made of the same material. All test gear shafts were supported by spherical roller bearings that are designed to rotate 20

35 Shaft A: bearing d A gear bearing L A Shaft B: d B L B Shaft C: d C L C Figure 2.4: Three different test gear shafts considered in this study. 21

36 Table 2.3: Basic dimensions and bending stiffness values of the test gear shafts considered in this study Shaft A Shaft B Shaft C Length (mm) Diameter (mm) Bending Stiffness (N/m) (10) 2.2(10) 4.8(10) 22

37 freely in the shaft bending direction, encouraging more shaft deflections than other types of rolling element bearings. Shafts A and B will be used for the spur gear tests while shafts C will be used for helical gears. In an earlier study by Heskamp [21], shafts A were used to measure dynamic TE (DTE) with the reasonable assumption that transverse motions of the gears were negligible since the shafts are very rigid. This study will compare DTE values corresponding to shafts A and B with spur gears. In order to help increase the amplitudes of the helical gear motions, they will be tested with shafts C which are the most compliant of the shafts considered. 2.5 Measurement System In order to measure various components of vibratory gear motions under dynamic conditions, an accelerometer-based measurement system was devised. Use of uni-axial accelerometers mounted tangentially to measure torsional vibrations of each gear and DTE are not a new concept as it was employed in several earlier studies [16-18, 21, 24]. The use of tri-axial accelerometers that allow computation of motions in other directions through post-processing of the acceleration signals is relatively new, as it will be described in this section. The measurement system was designed to perform two different functions: (i) Measure torsional vibrations θ p and θ g of both gears so that DTE can be determined according to Eq. (1.1). 23

38 (ii) Measure the vibratory motions of a gear (say pinion) in transverse (line-of-action (LOA) and off-line-of-action (OLOA)) directions, axial direction as well as rocking motions in the directions of shaft bending. The flange of the shaft of the pinion was machined such that three tri-axial accelerometers (PCB Piezotronics, Model: 354C10, sensitivity: nearly 10 mv/g, frequency range: 0 to 8 khz) can be mounted tangentially at 0, 90 and 180-degree angles as shown in Figure 2.5. In this arrangement, each sensor i ( i = 1 3) measures tangential ait ( t ), radial air ( t) and axial aia ( t ) accelerations. These components of the accelerations are illustrated schematically in Figure 2.6. In addition, two uni-axial accelerometers (PCB Piezotronics, Model: 353B18, sensitivity: nearly 10 mv/g, frequency range: 0 to 10 khz) are mounted 180 degrees apart from each other on the flange of the gear shaft next to the gear as shown in Figure 2.5. These uni-axial accelerometers, denoted as accelerometers 4 and 5 in Figure 2.6, measure tangential accelerations a4 T ( t ) and a 5 T ( t ). With these 11 acceleration signals ( a1 T ( t ), a1 R ( t ), a1 A ( t ), a2 T ( t ), a2 R ( t ), a2 A ( t ), a3 T ( t ), a3 R ( t ), a3 A ( t ), a4 T ( t ) and a5 T ( t ) ) in hand, the DTE of gear pair and the motions of the pinion will be determined using the formulation proposed in the next three sections. 24

39 Uni-axial accelerometer Tri-axial accelerometer Figure 2.5: A helical test gear pair with three tri-axial and two uni-axial accelerometers. 25

40 y g a 4T y p a3r a3t ωt a 3A x g z g a 5T Gear z p a 2T a 2R a 2A ρ a 1T ωt a 1R a 1A x p Pinion Figure 2.6: A schematic of the test gear pair showing the accelerometer locations 26

41 2.5.1 Measurement of Dynamic Transmission Error Dynamic Transmission Error is defined as the difference between the actual position of the output gear and the position it would occupy if the gear pair is perfectly conjugate, and it has the same mathematical definition as Eq. (1.1). In order to measure DTE, two tangential accelerometers for each gear mounted diametrically opposite of each other are sufficient. According to Figure 2.6, two tangential acceleration signals of interest, a1 T ( t ) and a 3 T ( t ), measured by sensors 1 and 3 on the pinion shafts contain gravity and the rotational acceleration of the gear as a1 T ( t ) = ρθ p ( t ) + g sin( ωt ), (2.1a) a3 T ( t ) = ρθ p ( t ) g sin( ωt ) (2.1b) where g is the gravitational acceleration, ω t is the angle between the horizontal reference position and the angle of the 1 st sensor, and ρ is the radius at which the accelerometer are mounted. Adding a1 T ( t ) and a 3 T ( t ), the gravity terms can be cancelled out such that the angular acceleration of the pinion can be found as 1 θ p ( t) = [ a1t ( t) + a3t ( t) ] 2ρ. (2.2) Similarly one writes a4 T ( t ) and a5 T ( t ) in terms of their components as a4 T ( t ) = ρθ g ( t ) + g sin ωt, (2.3a) 27

42 a5 T ( t ) = ρθ g ( t ) g sin ωt (2.3b) such that their sum yields the angular acceleration of the gear 1 θ g ( t) = [ a4t ( t) + a5t ( t) ] 2ρ. (2.4) According to the procedure defined in Figure 2.7, each signal is multiplied by the respective base radii defined in Figure 1.1, added to each other and integrated twice with respect to time to obtain the corresponding DTE( t ) along the line of action as DTE( t) = r pθ p ( t) + r gθg ( t) dt dt. (2.5) Measurement of the Transverse Pinion Motions With the same 11 measured acceleration components in hand, the transverse vibrations of the pinion can be computed in both the LOA and OLOA directions. This method can also be applied to the gear, provided it is instrumented with tri-axial accelerometers in the same manner as the pinion shown in Figures 2.5 and 2.6. For obtaining the transverse vibrations of the pinion, two radial acceleration signals, a2 R ( t ) and a 3 R( t ), that are oriented radially from the rotational centerline of the pinion shaft are required. Here, a1 R ( t ) and a2 R ( t ) could also have been used for the same purpose. These accelerations are written in terms of their components as 28

43 Pinion Accelerometer 1 data + Accelerometer 3 data Gear + DTE Accelerometer 4 data + Accelerometer 5 data Figure 2.7: Block Diagram for the data processing scheme for computation of DTE 29

44 a ( t) = x( t)sin ωt y( t)cos ω t + g cos ωt, (2.6a) 2 R a ( t) = x( t)cos ω t + y( t)sin ωt g sin ωt (2.6b) 3 R where x( t) and y( t) are the horizontal and vertical accelerations of the pinion, and the angle ω t is the rotational position angle of the first sensor from the horizontal axis, as shown in Figure 2.8. In order to determine the acceleration in the horizontal x direction, multiply Eq. (2.6a) and (2.6b) by sin ωt and cos ω t, respectively, to obtain 2 a2 R( t)sin ω t = x( t)sin ωt y( t)cos ωt sin ω t + g cos ωt sin ωt, (2.7a) 2 a3 R( t)cos ω t = x( t)cos ω t + y( t)sin ωt cos ωt g sin ωt cos ωt. (2.7b) Further, these two equations are added and solved for x( t) to obtain x( t) = a2r ( t)sin ωt a3r ( t)cos ωt. (2.8) Likewise, first multiplying Eq. (2.6a) and (2.6b) by cos ω t and sin ω t, respectively, one obtains 2 2 a2 R( t)cos ω t = x( t)sin ωt cos ωt y( t)cos ω t + g cos ωt, (2.9a) 2 2 a3 R( t)sin ω t = x( t)cos ωt sin ω t + y( t)sin ωt g sin ωt. (2.9b) 30

45 y p ẏ a 3R ωt Accel. 3 ωt ẋ ωt g ωt x p ẏ ρ ωt Accel. 1 Accel. 2 a 2R ωt g ωt ẋ Figure 2.8: The components of the radial accelerations of the pinion 31

46 Subtracting Eq. (2.9a) from (2.9b), the acceleration y( t) in the vertical direction can be obtained by y( t) = g a2r( t)cos ω t + a3r ( t)sin ωt. (2.10) In order for Eq. (2.8) and (2.10) can be used for finding x( t) and y( t), one must determine the instantaneous angular position ω t of the sensors. This was done by generating a once-per-revolution tach signal using a magnetic pick-up. A small metallic object was glued on one of the flywheels to produce an impulse at every instant when the first sensor on the pinion is at its horizontal position. With this tach signal, the sin ωt and cos ω t terms can be constructed as shown in Figure 2.9. The Matlab program written to do this task is given in Appendix A. With x( t) and y( t) determined, the accelerations of the pinion along the LOA and OLOA can be determined from the involute geometry parameters. Defining these displacements as q LOA and q OLOA in Figure 2.10, accelerations of the pinion in these directions are given as q LOA ( t) = x( t)sin φ+ y( t)cos φ, (2.11a) q OLOA ( t) = x( t)cos φ y( t)sin φ, (2.11b) where φ is the transverse pressure angle of the gears. Finally, these equations can be integrated twice to obtain the vibrations of the pinion along the LOA and OLOA 32

47 2 tach( t) sin ωt cosωt Figure 2.9: Generation of sin ω t and cos ω t from the once-per-rev tach signal 33

48 q LOA y LOA φ ri φ rj x Pinion Base Circle Gear Pitch Circle q OLOA Figure 2.10: Illustration of the motions of the pinion along the LOA and OLOA directions 34

49 directions as qloa( t) = qloa( t) dtdt, (2.12a) qoloa ( t) = qoloa ( t) dtdt. (2.12b) Measurement of the Rocking and Axial Motions of the Pinion The axial components of the accelerations from the first and third tri-axial accelerometers can be used to determine the rotational motions of the pinion about the x and y axes, as shown in Figure 2.6. These motions will be called here the rocking motions. Starting with the axial acceleration signals a1 A ( t ) and a 3 A( t ), the rotational acceleration of the pinion about the rotating axis connecting the points at which sensors 1 and 3 located is given as 1 ψ ( t) = [ a1 A( t) a3a( t) ]. (2.13) 2ρ This acceleration can be decomposed into two angular acceleration components ψ x( t ) and ψ y ( t ) about the x and y axes as ψ x( t ) = ψ ( t )cos ωt, (2.14a) ψ y ( t ) = ψ ( t )sin ωt. (2.14b) 35

50 As the other accelerations, these also be must integrated twice to obtain ψ x( t ) and ψ ( ) y t ψ x( t) = ψx( t) dtdt, (2.15a) ψ y( t) = ψ y( t) dtdt, (2.15a) The axial acceleration signals a1 A ( t ) and a3 A ( t ) might have components due to both rocking and axial motions of the pinion. As the rocking motions would result in the acceleration components that are opposite of each other, the average of a1 A ( t ) and a3 A ( t ) simply represents the axial acceleration of the pinion, i.e. 1 z( t) = [ a1 A( t) + a3a( t) ] (2.16) 2 A double integration of Eq. (2.16) with respect to time gives z( t) = z( t) dtdt. (2.17) 2.6 Instrumentation and Data Analysis In order to implement the accelerometer-based measurement procedure outlined above, a set of instrumentation was put together. The flowchart shown in Figure 2.11 describes the instrumentation as well as the speed control done through the computer. 36

51 Slip Rings (Michigan Scientific, Model: SR10M #2320) Test Rig Slip Ring Signal Conditioner (PCB Piezotronics, Slip Model: 483M92) Ring A/D Converter (National Instruments Model: PXI ) Tach Signal Speed Controller Figure 2.11: Block diagram for the digital data acquisition system. PXI Chassis (National Instruments Model: PXI -1042) PC (LabVIEW, VSI Rotate) 37

52 The data from the rotating accelerometers are transmitted to the fixed frame through the end-of-shaft type slip rings (Michigan Scientific, Model: SR10M). Signals from the output of the slip rings are fed into a multi-channel signal conditioner (PCB Piezotronics ICP Model 483M92) to condition and amplify the data.. Then, the signals are transmitted to an Analog/Digital converter (National Instruments, Model: PXI-4472) that digitizes the analog signals at a user defined sampling rate defined in Labview application. As the Nyquist frequency must be at least twice the maximum frequency of the accelerometers, the sampling frequency was greater than 20 khz. Next, the digital data was fed into a general-purpose chassis for PXI (National Instruments, Model: PXI- 1042) and then sent to the PC for analyzing and processing data through various Labview and Matlab applications. The Labview program used here is based on the program developed by Haskamp [21]. Here, the Labview program was used primarily for the data acquisition purposes while the bulk of the data analysis was done using Matlab. Figure 2.12 shows a picture of the data acquisition system. In this study, the focus was kept on the steady-state behavior of the test gear pairs. As such, constant speed data were collected within a gear rotational speed range of 500 to 4,200 rpm with an increment of 50 rpm. The speed control was automated such that the speed would be increased to a certain value, the machine would be operated at that particular speed for a while such that speed smear limits are met, the steady-state data would be collected, and speed would be increased to the next increment to repeat this process. At the end all the data was written a very large file that was decimated by the Matlab program off-line into individual steady-state segments. Once this was done, each 38

53 39 Signal Conditioner Figure 2.12: Instrumentation used for the data acquisition and analysis. NI PXI4472 NI PXI1042

54 data segment was processed to computed DTE, q LOA, q OLOA, ψ ( t) and z( t) according to the formulations presented in the previous section. Double integration of the acceleration signals were done in using a pseudo-integration scheme, focusing only on the first three gear mesh harmonics. In order to view the data in the frequency domain, Fast Fourier Transforms (FFT) were taken at the end. This provided the amplitudes of the harmonics of each vibratory motion as well as the root-mean-square value that was defined by using the first three gear mesh harmonics as 3 2 Arms = An, (2.18) n where A n is the magnitude at the n-th mesh harmonic. Finally, the gear mesh RMS values were plotted with respect to the rotational gear speed. 40

55 CHAPTER 3 EXPERIMENTAL RESULTS 3.1 Introduction In this chapter, test results from the spur and gear pairs with various shafts will be presented. First the test matrix used for these tests will be presented. Next spur gear test results will be presented with shafts A and B in two different formats: (i) steady-state response amplitudes within a speed range of 500 to 4200 rpm, and (ii) waterfall plots representing transient up and down speed sweeps. Due to the nonlinear behavior observed with the spur gears, the steady-state response will be presented for both speedup and speed down conditions individually such that the softening type behavior observed at the resonance regimes can be captured completely. The spur gear data will include DTE, q LOA and q OLOA motions since no tangible axial (z) and rocking ( ψ ) motions should be expected from a spur gear with zero helix angle mounted in the midspan between the bearings. Influence of the torque transmitted on the forced response will be also demonstrated. 41

56 In another individual section, the helical gear results will be presented with the most compliant shafts (shafts C). These results will include not only DTE, q LOA and q OLOA motions, but also the axial and rocking motions. No displacements data will be presented at speeds below 500 rpm since the double integration of the acceleration signals elevates the noise floor significantly at such low speed ranges. 3.2 Test Matrix The test matrix used in this study is shown in Table 3.1. This test matrix was designed to demonstrate the viability of the proposed measurement scheme in measuring transverse (LOA and OLOA), axial and rocking motions of the gears, in addition to the DTE of the pair. In the process, a demonstration of the impact of shaft flexibility on such motions will also be attempted. According to Table 3.1, the first group of tests is with spur gears and shafts A. As these shafts are the stiffest of the three shaft types considered in this study, these test will form the baseline for nearly purely tosional conditions. Three torque values of 100, 200 and 300 Nm were considered in these tests within a speed range of 500 to 4200 rpm. The same tests were repeated with the same spur gear pair, but now with shafts B. Comparisons of tests from these two groups should demonstrate any shaft effects The third group of tests listed in Table 3.1 was performed with the helical gear pair of Figure 2.4 that was mounted on shafts C. The speed range was kept the same for 42

57 Table 3.1: The test matrix implemented in this study Test Gear Shaft Torque Speed Group type type [Nm] [rpm] 1 Spur Shaft A 100, 200, Spur Shaft B 100, 200, Helical Shaft C

58 these tests (500 to 4200 rpm) while the torque level was kept limited to 100 Nm since shaft C was not designed to endure higher torque values. An automatic transmission fluid will be used in all tests. High-pressure jets of lubricant will be delivered to the gear mesh and the spherical roller bearings at room temperature. With no preload applied to the bearings, this test arrangement was reported to correspond to a torsional damping ratio of nearly 0.02 [16-18, 24]. A gear pair operating at such levels of damping can be considered as lightly damped. In case of spur gears, such lightly damped conditions encourage tooth separations and parametric resonances [16] as some of these will be demonstrated in the next section 3.3 Spur Gear Test Results Steady-state Forced Response Figures 3.1 and 3.2 show the rms DTE amplitudes of the spur gear pair at a torque value of 100 Nm. In Figure 3.1, the DTE response of spur gear pair with shafts A and B are compared for a steady-state forced response test under the speed-up conditions. Meanwhile, Figure 3.2 represents the same comparison under the speed-down conditions. Several observations can be made from these figures in terms of the characteristics of the forced responses: The speed-up and speed-down DTE amplitudes are nearly identical for most of the speed increments where the dynamic response is linear, i.e. there is no tooth separation. These speed ranges dominate the entire data set except in the vicinity 44

59 16 Shaft A Shaft B 12 ( DTE ) rms [ µ m] Speed [rpm] Figure 3.1: Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 100 N-m from a speed-up test. 45

60 16 Shaft A Shaft B 12 ( DTE ) rms [ µ m] Speed [rpm] Figure 3.2: Comparison of the rms values of DTE of the spur gear pair with shaft A and shaft B at 100 N-m from a speed-down test. 46

61 of the resonance at about 3000 rpm (at a gear mesh frequency of (3000)/60*50=2500 Hz for this system with 50-tooth gears). A double stable motion region is observed between the jump-up speed of about 2,700 rpm (2250 Hz mesh frequency) in Figure 3.1 and the jump-down speed of about 2,400 rpm (2000 Hz) in Figure 3.2. This is a direct result of tooth separations taking place in certain portion of the gear mesh cycle, effectively reducing the gear mesh stiffness to exhibit a softening type of nonlinear behavior [16, 24]. The response curves for the stiffer shafts A exhibit resonance peaks at 3000, 1500 and 1000 rpm (2500, 1250 and 830 Hz). These all related to the same natural frequency (primary torsional natural mode frequency) with the peak at 3000 rpm corresponding to the primary resonance (excited by the first or fundamental harmonic of the transmission error excitation) and the other two corresponding to super-harmonic resonances (excited by the second and third harmonics of the transmission error excitation). No additional resonance activity is evident in data for shaft A. Curves for the more compliant shafts B differ from those of shaft A in two basic ways. First, the rms amplitudes are increased with shafts B, especially in the speed ranges away from the resonance peaks. Secondly, while the peaks associated with the same mode at 3000, 1500 and 1000 rpm are preserved with slight changes in frequency, some additional peaks are observed at speeds 4000, 2000 and 1330 rpm. This suggests that another natural mode is present with the 47