FAULT IDENTIFICATION IN DRIVETRAIN COMPONENTS USING VIBRATION SIGNATURE ANALYSIS. A Dissertation. Presented to

Transcription

1 FAULT IDENTIFICATION IN DRIVETRAIN COMPONENTS USING VIBRATION SIGNATURE ANALYSIS A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Pirapat Arunyanart May 215

2 FAULT IDENTIFICATION IN DRIVETRAIN COMPONENTS USING VIBRATION SIGNATURE ANALYSIS Pirapat Arunyanart Dissertation Approved: Advisor Dr. Fred Choy Committee Member Dr. Xiaosheng Gao Committee Member Dr. Gregory Morscher Committee Member Dr. Shengyong Wang Accepted: Department Chair Dr. Sergio Felicelli Dean of the College Dr. George K. Haritos Interim Dean of the Graduate School Dr. Rex Ramsier Date Committee Member Dr. Dale Mugler Committee Member Dr. Gerald Young ii

3 ABSTRACT The vibration problems associated with bearing, shaft, and gear have drawn attention of much engineering work. The advancements in preventive maintenance of rotor gear transmission systems are currently being sought for the improvement to study a fault diagnosis in the machine health monitoring system. Previously, the analytical procedure is widely employed by researchers and designers of large rotating machinery to study the dynamic behavior of rotating systems. Presently, it has been developed to include different types of fault and used together with some online vibration monitoring methods. These online monitoring methods do not require to shutdown machinery and can be used as an in-flight diagnostic. However, very little work has been carried out on the fault detection under the combination effects of bearing, shaft, and gear in a transmission system. In this dissertation, both numerical simulations and experimental investigations were performed to identify different damage scenarios involving different combinations of bearing damage, residual shaft bow, and gear tooth damage. The comprehensive numerical models were developed to study the transient response of rotating machinery systems including the effects of individually defected components. In order to increase the calculating efficiency and reduce the computational effort, a modal analysis method iii

4 was applied to the equations of motion and solved the overall dynamics of the system. The nonlinear bearing force is considered as the result of the contacts between the ball and the raceways using Hertzian contact deformation theory and this model was extended considering the effect of localized defect on inner race. In the case of residual shaft bow, the bow effect was introduced as the forcing function in the equations of motion. In the gear transmission system model, the gear-mesh forces result from a nonlinear periodic gear meshing stiffness. The change in gear surface profile due to wear is modelled as the shift in amplitude of the meshing stiffness. The experimental results used in this study were obtained from the high-speed ball bearing test rig. During these tests, vibration signatures due to each damaged effect: namely bearing defect, residual shaft bow, and gear damage/wear were acquired for identification and validation. The vibration signatures obtained from both numerical simulations and experiments were examined in the time domain, the frequency domain, and the joint time-frequency domain. The diagnostic technique was suitably chosen to perform signal analysis for detecting and identifying each damaged characteristic. The analytical model of gear-rotor-bearing systems was finally extended to investigate the damaged effects under combination scenarios of defects in bearing, shaft, and gear. The numerical results were also validated by those acquired from the test rig experiment. The signature analysis techniques were carefully applied on both numerical and experimental results to examine and characterize its root cause. iv

5 DEDICATION This is to my parents v

6 ACKNOWLEDGEMENTS I would like to thank my parents, Parkpian and Dr. Orasri, and my sister, Dr. Suparuedee, for their continuous support and encouragement over the years because this journey would not have been possible without them. I respectfully express my appreciation in memory to my late academic advisor, Dr. Fred K. Choy who passed on after a brief illness. He was an important and steady guide in my professional development. Always ready with advice, and willing to work with me to final solutions for difficult issues. This dissertation would not have been possible without his patient and kind guidance. I also thank my dissertation advisor and my committee members, Dr. Xiaosheng Gao, Dr. Gregory Morscher, Dr. Shengyong Wang, Dr. Dale Mugler, Dr. Gerald Young, for their time and valuable comments. I am grateful to my colleagues, Jie Wen, Chia-Hsuan Shen, and Heng-Hsueh Chen, for their help and support. vi

7 TABLE OF CONTENTS Page LIST OF TABLES... x LIST OF FIGURES... xi CHAPTER I. INTRODUCTION Overview Literature Overview Research Objectives... 5 II. BACKGROUND OF MODAL ANALYSIS OF ROTOR-BEARING SYSTEMS AND VIBRATION SIGNATURE ANALYSIS Overview Dynamic Response of the Rotor-Bearing System Signal Analysis Methods Time Domain Analysis Frequency Domain Analysis Continuous Wavelet Transform Summary III. ANALYTICAL SIMULATION OF THE BEARING DAMAGE WITH EXPERIMENTAL CORRELATION Objective Dynamic of ball bearing vii

8 3.2.1 Curvature and relative curvature Surface deformations Load-deformation relationships Restoring Force Inner race localized defect Discussion of Results Numerical Simulation Results Experimental Results Conclusions IV. DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM UNDER RESIDUAL SHAFT BOW EFFECT Objective Dynamic Response of bowed shaft Discussion of Results Numerical Simulation Results Experimental Results Conclusions V. THE EFFECT OF SPUR GEAR TOOTH IMPERFECTION ON THE DYNAMIC CHARACTERISTICS OF GEAR TRANSMISSION SYSTEM Objective Gear Meshing Stiffness Axial Compressive, Bending, and Shear Energies Hertzian Energy The total gear mesh stiffness Dynamics of the Gear-Shaft System viii

9 5.4 Discussion of Results Numerical Simulation Results Experimental Results Conclusions VI. IDENTIFICATION OF DAMAGE IN GEAR-ROTOR-BEARING ELEMENT IN A TRANSMISSION SYSTEM Objective Method of Approach Numerical Dynamics of Gear-rotor-bearing System Experimental Descriptions Component Damaged Condition Discussion of Numerical Results Bearing Component Analysis Shaft Component Analysis Gear Component Analysis Discussion of Experimental Results Bearing Component Analysis Shaft Component Analysis Gear Component Analysis Conclusions... 2 VII. CONCLUSIONS REFERENCES ix

10 LIST OF TABLES Table Page 3.1 Dimensionless Contact Parameters Summary of Component Fault Identification Parameters x

11 LIST OF FIGURES Figure Page 2.1 Example of CWT of a sine wave with a discontinuity Radial ball bearing showing diametral clearance Geometry of contacting bodies The additional deflection of ball due to defect on inner race Test Rig Model of Rotor-Bearing System Schematic of Rotor-Bearing System for Numerical Simulation Frequency Response Function of System Measured at Left Rotor Frequency Response Function of System Measured at Right Rotor Numerically Simulated Vibration of Undamaged Bearing in x-direction Numerically Simulated Vibration of Undamaged Bearing in y-direction Numerically Simulated Vibration of Damaged Bearing in x-direction Numerically Simulated Vibration of Damaged Bearing in y-direction Frequency Spectrum of Undamaged Bearing in x-direction Frequency Spectrum of Undamaged Bearing in y-direction Frequency Spectrum of Damaged Bearing in x-direction Frequency Spectrum of Damaged Bearing in y-direction Frequency Spectrum of Absolute Vibration for Undamaged Bearing Frequency Spectrum of Absolute Vibration for Damaged Bearing xi

12 3.18 Photograph of the High-speed Ball Bearing Test Rig Photograph of Fafnir Bearing Photograph of Inner Race Damage of Bearing# Photograph of Inner Race Damage of Bearing# Photograph of Inner Race Damage of Bearing# Inner Race Damaged Profile of Bearing# Inner Race Damaged Profile of Bearing# Inner Race Damaged Profile of Bearing# Experimental Vibration Signature of Bearing# in x-direction Experimental Vibration Signature of Bearing# in y-direction Experimental Vibration Signature of Bearing#1 in x-direction Experimental Vibration Signature of Bearing#1 in y-direction Experimental Vibration Signature of Bearing#2 in x-direction Experimental Vibration Signature of Bearing#2 in y-direction Experimental Vibration Signature of Bearing#3 in x-direction Experimental Vibration Signature of Bearing#3 in y-direction Frequency Spectrum of Bearing# in x-direction Frequency Spectrum of Bearing# in y-direction Frequency Spectrum of Bearing#1 in x-direction Frequency Spectrum of Bearing#1 in y-direction Frequency Spectrum of Bearing#2 in x-direction Frequency Spectrum of Bearing#2 in y-direction xii

13 3.4 Frequency Spectrum of Bearing#3 in x-direction Frequency Spectrum of Bearing#3 in y-direction Frequency Spectrum of Absolute Vibration for Bearing# Frequency Spectrum of Absolute Vibration for Bearing# Frequency Spectrum of Absolute Vibration for Bearing# Frequency Spectrum of Absolute Vibration for Bearing# The Frequency Spectrum Amplitude with the Experimental Bearings Schematic of Rotor-Bearing System with centrally bent shaft for Numerical Simulation Numerically Simulated Displacement of Straight Shaft in x-direction Numerically Simulated Displacement of Straight Shaft in y-direction Numerically Simulated Displacement of Residual Bow Shaft in x-direction Numerically Simulated Displacement of Residual Bow Shaft in y-direction Frequency Spectrum of Straight Shaft in x-direction Frequency Spectrum of Straight Shaft in y-direction Frequency Spectrum of Residual Bow Shaft in x-direction Frequency Spectrum of Residual Bow Shaft in y-direction Photograph of Straight Shaft Photograph of Residual Bow Shaft Photograph of the High-speed Ball Bearing Test Rig with sets of sensors Experimental Displacement Signature of Straight shaft in x-direction Experimental Displacement Signature of Straight shaft in y-direction Experimental Displacement Signature of Residual Bow shaft in x-direction xiii

14 4.16 Experimental Displacement Signature of Residual Bow shaft in y-direction Frequency Spectrum of Straight Shaft in x-direction Frequency Spectrum of Straight Shaft in y-direction Frequency Spectrum of Residual Bow Shaft in x-direction Frequency Spectrum of Residual Bow Shaft in y-direction Elastic force on a tooth of gear Schematic of Gear-Shaft System for Numerical Simulation Schematic of the Change in Gear Tooth Surface Profile Numerically Simulated Change in Gear Mesh Stiffness Due to Wear in Gear Tooth Profile CWT of Vibration Signal with No Damaged Gear Tooth CWT of Vibration Signal with Single Damaged Gear Tooth Photograph of Gear Test Rig Photograph of Gear with No Damaged Tooth Photograph of Gear with Single Damaged Tooth CWT of Vibration Signal with No Damaged Gear Tooth CWT of Vibration Signal with Single Damaged Gear Tooth Photograph of Gear Test Rig Numerically Simulated Vibration of GG_GS_GB in x-direction Numerically Simulated Vibration of GG_GS_GB in y-direction Numerically Simulated Vibration of GG_BS_BB in x-direction Numerically Simulated Vibration of GG_BS_BB in y-direction Numerically Simulated Vibration of BG_GS_BB in x-direction xiv

15 6.7 Numerically Simulated Vibration of BG_GS_BB in y-direction Numerically Simulated Vibration of BG_BS_GB in x-direction Numerically Simulated Vibration of BG_BS_GB in y-direction Numerically Simulated Vibration of BG_BS_BB in x-direction Numerically Simulated Vibration of BG_BS_BB in y-direction Frequency Spectrum of Absolute Vibration of GG_GS_GB Frequency Spectrum of Absolute Vibration of GG_BS_BB Frequency Spectrum of Absolute Vibration of BG_GS_BB Frequency Spectrum of Absolute Vibration of BG_BS_GB Frequency Spectrum of Absolute Vibration of BG_BS_BB Numerically Simulated Displacement of GG_GS_GB in x-direction Numerically Simulated Displacement of GG_GS_GB in y-direction Numerically Simulated Displacement of GG_BS_BB in x-direction Numerically Simulated Displacement of GG_BS_BB in y-direction Numerically Simulated Displacement of BG_GS_BB in x-direction Numerically Simulated Displacement of BG_GS_BB in y-direction Numerically Simulated Displacement of BG_BS_GB in x-direction Numerically Simulated Displacement of BG_BS_GB in y-direction Numerically Simulated Displacement of BG_BS_BB in x-direction Numerically Simulated Displacement of BG_BS_BB in y-direction Frequency Spectrum of Displacement of GG_GS_GB in x-direction Frequency Spectrum of Displacement of GG_GS_GB in y-direction xv

16 6.29 Frequency Spectrum of Displacement of GG_BS_BB in x-direction Frequency Spectrum of Displacement of GG_BS_BB in y-direction Frequency Spectrum of Displacement of BG_GS_BB in x-direction Frequency Spectrum of Displacement of BG_GS_BB in y-direction Frequency Spectrum of Displacement of BG_BS_GB in x-direction Frequency Spectrum of Displacement of BG_BS_GB in y-direction Frequency Spectrum of Displacement of BG_BS_BB in x-direction Frequency Spectrum of Displacement of BG_BS_BB in y-direction CWT of Absolute Vibration Signal of GG_GS_GB CWT of Absolute Vibration Signal of GG_BS_BB CWT of Absolute Vibration Signal of BG_GS_BB CWT of Absolute Vibration Signal of BG_BS_GB CWT of Absolute Vibration Signal of BG_BS_BB Normalized Amplitude Ratio Parameters for CWT Comparison of Normalized Amplitude Ratio Parameters for CWT Experimental Vibration Signature of GG_GS_Bearing# in x-direction Experimental Vibration Signature of GG_GS_Bearing# in y-direction Experimental Vibration Signature of GG_BS_Bearing#3 in x-direction Experimental Vibration Signature of GG_BS_Bearing#3 in y-direction Experimental Vibration Signature of BG_GS_Bearing#3 in x-direction Experimental Vibration Signature of BG_GS_Bearing#3 in y-direction Experimental Vibration Signature of BG_BS_Bearing# in x-direction xvi

17 6.51 Experimental Vibration Signature of BG_BS_Bearing# in y-direction Experimental Vibration Signature of BG_BS_Bearing#3 in x-direction Experimental Vibration Signature of BG_BS_Bearing#3 in y-direction Frequency Spectrum of Absolute Vibration of GG_GS_Bearing# Frequency Spectrum of Absolute Vibration of GG_BS_Bearing# Frequency Spectrum of Absolute Vibration of BG_GS_Bearing# Frequency Spectrum of Absolute Vibration of BG_BS_Bearing# Frequency Spectrum of Absolute Vibration of BG_BS_Bearing# Experimental Displacement Signature of GG_GS_Bearing# in y-direction Experimental Displacement Signature of GG_BS_Bearing#3 in y-direction Experimental Displacement Signature of BG_GS_Bearing#3 in y-direction Experimental Displacement Signature of BG_BS_Bearing# in y-direction Experimental Displacement Signature of BG_BS_Bearing#3 in y-direction Frequency Spectrum of Displacement of GG_GS_Bearing# in y-direction Frequency Spectrum of Displacement of GG_BS_Bearing#3 in y-direction Frequency Spectrum of Displacement of BG_GS_Bearing#3 in y-direction Frequency Spectrum of Displacement of BG_BS_Bearing# in y-direction Frequency Spectrum of Displacement of BG_BS_Bearing#3 in y-direction CWT of Absolute Vibration Signal of GG_GS_Bearing# CWT of Absolute Vibration Signal of GG_BS_Bearing# CWT of Absolute Vibration Signal of BG_GS_Bearing# CWT of Absolute Vibration Signal of BG_BS_Bearing# xvii

18 6.73 CWT of Absolute Vibration Signal of BG_BS_Bearing# Normalized Amplitude Ratio Parameter for CWT Comparison of Normalized Amplitude Ratio Parameters for CWT xviii

19 CHAPTER I INTRODUCTION 1.1 Overview Gears-rotor-bearing system is the most commonly used in rotating machinery such as transmission systems in automobile, aircraft gas turbine, and heavy machinery. During the last three decades, the numbers of usage in gear and bearing systems have substantially increased in both defense and commercial applications. Accounting with the increase in higher operating speed, larger carried load, and lighter weight, the premature failures due to excessive wear and material fatigue occur in components of transmission systems. Such premature failures in the transmission system are always subjected to losses in time and financial or even sometimes they may lead to catastrophic consequences. The fault detection has an important role associated with high speed rotating machinery. This could be a benefit if a fault in components is detected at its early stage so a corrective action can be taken promptly. The preventive maintenance can also be made in advance for replacement in damaged part. Thus, in order to assure a safety and reliability in operation of a gear transmission system, it is necessary to understand the dynamics of the system under various possible failure conditions. 1

20 Presently, the prevention and management of premature failures in equipment has become a vital part of the maintenance program. Many researches have done for finding a reliable monitoring strategy in gear transmission systems. A variety of fault detection procedures have been developed. Mostly impractical procedure could be visual inspection because it is not easily to visualize faults in micro scale unless costly, specialized equipment is used. However, it is quite impossible to examine gear transmissions during operation. Practically, visual inspection is used mainly after machine failure has been experienced. One of the most promising procedures for detecting incipient faults in gear-rotorbearing system is the vibration analysis. The advantage of vibration signature analysis is no requirement of machinery shutdown and it can be carried out online by a computerbased machine health monitoring. However, current on-board condition monitoring systems for gear-rotor and bearing systems often fail to provide sufficient time between warning and failure otherwise safety procedures can be implemented. At times, a small fault in transmission system can quickly develop into a dangerous failure mode without any notable signs. Additionally, inaccurate interpretation of operational conditions may result in false alarms and unnecessary repairs and downtime. In the case of effects due to combined damage in bearing, shaft, and gears, vibration signatures usually cannot be identified immediately without special treatment. Therefore, all these needs motivate for the development and research in detecting and identifying faults in ball bearing, shaft, and gear before inducing system failure. 2

21 1.2 Literature Overview Traditionally, the research in rotating machinery attempted to predict the performance of the rotor dynamic system during the operation. An amount of work [1-8] has been reported in machine life prediction based on statistical models where the fatigue life analysis was included in their models. Their prediction is based on statistical approach developed by Lundberg and Palmgren [4, 5] while the reliability models are based on classical fatigue theory and the Weibull failure distribution. However the condition of machine components and their corresponding under various operating environment were not considered. Generally, rolling element bearing systems generate vibration even if they are in perfect condition. The sources of excitation are due to the external disturbance, the internal excitation, and the varying structure compliance [9, 1]. The varying compliance vibration is generated by the relative motions between ball bearing and inner or outer race because the number of rolling elements and their position in the load zone changes with bearing rotation [11]. However, the presence of a defect causes a significant increase in the vibration level so it is important to understand the behavior of vibration signatures under different types of defect. Most of researchers studying on the detection and diagnosis of localized defects in bearing have done experimental works while others developed models for detecting defects and compared their results with experimental results. McFadden and Smith [12, 13] proposed a model for the high-frequency vibration produced by a single 3

22 defect or multiple defects on the inner race of a rolling element bearing under radial load and the model performance was confirmed experimentally by NASA researchers [14]. One of the common faults in large rotating machinery is the residual shaft bow. The existence of residual bow in shaft may result from various effects such as, gravity sag, thermal distortion, uneven shrink fits, and mechanical bow due to previous large unbalance. In the case of gravitational sag, it mostly occurs in the large horizontal turbines or compressors which are allowed to rest for long period of times. This may affect the shaft temporarily. The turbomachinery operating in particularly high temperature such as gas/steam turbines and water pumps in nuclear reactors is possible to developed temporary bow condition because the temperature distribution along the shaft temperature is uneven. However, uneven shrink fits of impellers or spacers on a shaft may produce a permanent mechanical bow due to the rubbing of a shaft on a seal. The research in the dynamics of the gear transmission system has generated a wealth concerning the gear vibrations. Boyd and Pike [15, 16], and Choy [17] used the work done by Cornell [18, 19] to study the dynamics of multimesh gear transmission systems. Mark [2] applied the transfer matrix method to study gear system dynamics. In case of gearbox vibration analysis, some work using finite element analysis has been reported by Lim [21]. The study of gearbox vibration was developed by considering the coupling between the gear system and the casing [22-24]. The global dynamics of a gear transmission system was studied in [23] using the modal analysis. The need to increase the reliability of the gear transmissions encouraged extensively the experimental studies of gear vibration. Some works [25-28] have 4

23 performed the experimental study of vibration in gear system. The experimental and analytical studies of the vibration due to faulty gear have been presented in several literatures [26, 27, 29]. The study of vibration due to fault component in gear-bearing system is somewhat limited [3-33]. Additionally, none of work has been concerned the combined effect of bearing fault, shaft damage, and gear tooth damage in a rotor transmission system. 1.3 Research Objectives In order to develop an on-line health monitoring to detect the combination effects due to defect in bearing, shaft and gear in the gear-rotor transmission system, it is necessary to understand the fundamental characteristics of fault condition generated by each component. The work in this research presents the development of dynamic simulation of transmission systems under the effects of localized defect in bearing element, shaft residual bow, and imperfection in gear tooth profile and the validation of numerical results by those of experiment. Also, the vibration signatures of each case obtained from both numerical simulation and experiment are analyzed by using either frequency method or Wavelet Transform and then the characteristic results of each case are used to classify root course under combination effects. 5

24 The accomplishments of this study can be summarized as follows: Develop the numerical simulation procedure for rotor-bearing transmission systems with the effect of localized defect in bearing element and validate the results with those obtained from experiment. Develop the fault investigation methods suitably for bearing defect scenario. Develop the numerical simulation procedure for rotor-bearing transmission systems with the effect of residual shaft bow and validate the results with those obtained from experiment. Develop the fault investigation methods suitably for residual shaft bow scenario. Develop the numerical simulation procedure for gear-rotor-bearing transmission systems with the effect of gear tooth geometry imperfection and validate the results with those obtained from experiment. Develop the fault investigation methods suitably for gear tooth defect scenario. Develop the numerical simulation procedure for gear-rotor-bearing transmission systems with the combination effects of localized defect in bearing element, shaft residual bow, and gear tooth geometry imperfection. Identify the fault characteristic in dynamic signature obtained from combination scenarios in defected components of bearing, shaft, and gear. 6

25 CHAPTER II BACKGROUND OF MODAL ANALYSIS OF ROTOR-BEARING SYSTEMS AND VIBRATION SIGNATURE ANALYSIS 2.1 Overview In the design of rotating machinery and other high speed turbo machinery, the machinery is often operating through several rotor critical speeds in order to reach its full power operating level. Determination of the motion response of the rotor system is important for a design point of view so the equations of motion were developed and continuously improved for obtaining solutions close to actual system response. Jeffcott [34] had intensively investigated the dynamics of simple rotors using twodegree-of-freedom model including damping effects. He further explained the meaning of critical speed and introduced the new terminology of whirl instability. Newkirk [35] made an intensive investigation on stability of rotor-bearing systems and concluded that the rotor dynamic behavior could not be attributed to a critical speed resonance and the reduction of unbalance had no effect upon rotor whirl amplitudes. Holzer [36] introduced the way to find natural frequencies of torsional systems. This method was later adapted by Myklestad [37, 38] to calculate the natural frequencies of airplane wings coupled in bending and torsion. At about the same time Prohl [39] showed how this method could be 7

26 applied to rotor-bearing systems. It became one of the most powerful tools for solving problem in rotor dynamics. Later, Myklestad and Prohl had made a slight extension of this work to include other effects like shear deformation and gyroscopic moments. This matrix transfer method of Myklestad and Prohl has been fully extended into the analysis of critical speeds [4, 41], stability [42, 43], and forced response [44] of complex turbo rotors. Other methods such as the finite difference and finite element [45-47] approach were also used in study of rotor dynamics. The rotor system is simulated by using the combination of large numbers of small elastic shaft elements where rotor weights and inertia moment effects are lumped at the mass stations, interconnected between two node points of the shaft elements. By using the advanced high-speed computers, transient response motion of rotor-bearing systems were investigated through real time numerical integration of rotor acceleration. Shen [48] presented a formulation for flexible rotor analysis using the influence coefficient approach. Kirk [49] further discussed the transient motion of multi-mass rotor systems with effects of nonlinear bearing support. In his conclusion, the linear stability analysis of the system can be verified using transient response results. However, the use of transient response motion to predict nonlinear stability of the system was not pointed out. Hitching [47] outlined transient approach using finite element approach and also discussed the effects of random excitation functions and the application of Taylor series and curve fitting to the transient solution. In the case for complex rotor system with a large number of mass stations, the use of modal method seems efficiently providing an alternative approach for solving large numbers of equations of motion with less time consuming because the modal method 8

27 greatly reduces the number of degree of freedom of the system. This approach has been proven successful in computing transient response of rotor-bearing systems [5-57]. Childs [52, 53] performed a complete investigation in using undamped modes calculated from the averaged vertical and horizontal support stiffness of the system. Choy [54-56] developed a modal analysis for rotor systems including the effects of gyroscopic, unbalance, disk skew, nonlinear bearings, and rotor acceleration. The primary goal of the vibration signature analysis for machine health monitoring is to aid fault detection and identification. Various signal processing techniques have been developed and applied for fault detection and diagnosis in rotating machinery. The signal processing methods for machine health monitoring may be classified into time domain analysis, frequency domain analysis and joint time-frequency domain analysis. The time domain methods analyze the amplitude and phase information of the vibration time signal to detect the fault of gear-rotor-bearing system [57-61]. One of the traditional techniques is FM developed by Stewart [62], where he used this fault detection parameter to investigate the changes in a gearbox vibration signal due to gear damage. Some statistical measurements and comparisons of the energy/amplitude of different components of the vibration signal are used in some later developed traditional fault detection techniques. The vibration signal is often averaged over a large number of cycles, and by removing some components at desired frequencies the residual signal is obtained. The statistical parameters are then calculated. Some of the techniques include 9

28 Root mean Squared [63], Energy ratio [64], Crest Factor, Kurtosis [65], FM [62], FM4 [62], NA4 [66-69], NB4 [68, 7], M6 [71], M8A [71], etc. Spectral analysis is the classical technique to diagnose bearing, gear, and shaft. The spectrum of a damaged component is compared to one from the healthy condition, so some defect could be detected [57, 72-75]. The Fourier Transform is the most fundamental frequency domain technique where the Fast Fourier Transform (FFT) is widely used computational technique for converting time signal to frequency domain due to its efficiency and less calculation time [72-74]. Others use the difference of power spectral density of the signal to identify the damage due to fault of gear and bearing [75]. The Wigner-Ville Distribution (WVD), wavelet transform (WT), and short time Fourier transform (STFT) are the examples of joint time-frequency analysis [76, 77]. The major difference among these transforms is their resolution properties along time and frequency scales. The joint time-frequency analysis provides information about the vibration energy changes in the system where presenting as an interactive relationship between time and frequency during the period of the time data window. Therefore, the damage could be revealed at much earlier stage. The STFT had been considered by McFadden [78] and suggested that it can potentially be a useful tool for fault detection and localization. The property of WVD was investigated by McFadden [79] and he successfully applied a technique to determine the change in vibration pattern when damage occurs. Further improve of WVD by reducing the cross-term effect [8, 81] was found in McFadden s researches. The early works on the application of WVD in gear diagnostics demonstrated the relation between the gear fault and the distribution pattern 1

29 [79, 8, 82]. Recently it was further established the correlation between a WVD pattern and the severity of the gear fault [27, 29, 83]. Another useful joint time-frequency domain method in vibration analysis is the wavelet method [7, 8, 84, 85]. Because of its superior potential time-frequency technique, the use of wavelet analysis for fault analysis and detection in gear box has been gaining researchers attention [86-89]. The Wavelet Transform has achieved a high frequency resolution to signify the presence of damage in the gearbox vibration signal. The application of wavelet analysis was recently developed to diagnose a fault in gear [9] and combination between gear and bearing [33]. 2.2 Dynamic Response of the Rotor-Bearing System The general equations of motion for multimass shaft-rotor system can be written in the matrix form as [M]{W } + [C]{W } + [K]{W} = {F(t)} (2.1) where {W} is the generalized displacement vector due to translational and rotational displacement and it can be composed into displacement vectors in x direction {U} and y direction {V} as follows: {W} = { {U} {V} } (2.2) 11

30 and the general mass-inertia matrix can be expressed as [M] = [ [M] x [M] y ] (2.3) For the generalized damping matrix, it can be decomposed into the gyroscopic damping and bearing damping submatrices as follows [C] = [ [C] xx [C] xy ] [C] yx [C] yy G + [ [C] xx [C] xy [C] yx [C] yy ] B (2.4) The stiffness matrix may also be decomposed into three submatrices including the symmetric shaft stiffness matrix, the linear bearing stiffness matrix, and the skew symmetric acceleration matrix as follows [K] = [ [k] xx ] + [ [K] xx [K] xy ] + [ [K] xx [K] xy ] [k] yy [K] S yx [k] yy [K] B yx [k] yy A (2.5) The generalized force vector on the right hand side of the equations is the external time dependent force and moment forcing function vector and can be expressed as: F(t) = { {F(t)} x {F(t)} y } (2.6) 12

31 Thus, the equation of motion in (2.1) can be separate into x direction as [M] x {U } + [C] xx {U } + [C] xy {V } + [K] xx {U} + [K] xy {V} = {F(t)} x (2.7) Similarly into y direction as [M] y {V } + [C] yy {V } + [C] yx {U } + [K] yy {V} + [K] yx {U} = {F(t)} y (2.8) where [C] xx and [C] yy are the overall damping matrix in x and y direction respectively; [C] xy and [C] yx are cross-coupling damping matrix. Similarly, [K] xx and [K] yy are the overall stiffness matrix in x and y direction respectively; [K] xy and [K] yx are crosscoupling stiffness matrix. The overall external force vector acting to the system represents as {F(t)} x and {F(t)} y for x and y direction respectively. The transient and steady state dynamic response of the rotor-bearing system is obtained by solving the coupled equations of motion simultaneously. In order to minimize the computational effort, the modal transformation procedure is applied to the equations of motion. This also reduces the degrees of freedom of the global equations of motion. According to the principles of modal expansion, the rotor translational and rotational response amplitudes can be represented by using a set of undamped orthonomal modes of the system. That is 13

32 N {U(t)} = A k (t){φ k } k=1 N {V(t)} = B k (t){φ k } k=1 (2.9) (2.1) where {A k (t)}, {B k (t)} are the modal transient time functions. Using the modal expansions from equation (2.9) and (2.1), the dynamics of motion equation (2.7) and (2.8) respectively can be written as [M] x [Φ]{A } + [C] xx [Φ]{A } + [C] xy [Φ]{B } + [K] xx [Φ]{A} + [K] xy [Φ]{B} = {F(t)} x (2.11) and [M] y [Φ]{B } + [C] yy [Φ]{B } + [C] yx [Φ]{A } + [K] yy [Φ]{B} + [K] yx [Φ]{A} = {F(t)} y (2.12) Premultiplying equation (2.11) and (2.12) by [Φ] T and using the orthogonality conditions, the equation of motion can be written in the compact modal coordinates as {A } + [C x ]{A } + [D x ]{B } + [G x ]{A} + [H x ]{B} = {xf} (2.13) 14

33 and {B } + [C y ]{B } + [D y ]{A } + [G y ]{B} + [H y ]{A} = {yf} (2.14) Combining Equation (2.13) with (2.14) and rearranging the terms, we have { A } = { xf B yf } [[C x] [D x ] [D y ] [C y ] ] {A } [ [G x] [H x ] B [H y ] [G y ] ] {A B } (2.15) Thus, the modal acceleration terms can be calculated from the above equation with initial conditions for the modal displacement given by the following relations { A() T B() } = [Φ x Φ ] [ M x y M ] [ U() y V() ] (2.16) Similarly, the modal velocity is given by { A () T B () } = [Φ x Φ ] [ M x y M ] [ U () y V () ] (2.17) With the initial modal acceleration and velocity terms calculated as above, the modal transient motion can be computed by numerical integration in time. The rotor 15

34 forced response motion can be obtained by back substitution of the modal coefficients into the following equation: x(t) θ(t) {W(t)} = y(t) { ψ(t)} = { [Φ x]{a(t)} [Φ y ]{B(t)} } (2.18) 2.3 Signal Analysis Methods In machine health monitoring, the vibration signal processing methods can be classified into time domain analysis, frequency domain analysis and joint time-frequency domain analysis Time Domain Analysis The time domain methods try to analyze the amplitude and phase information of the vibration time signal to detect the fault of gear-rotor-bearing system. The methods include Signal Averaging Technique, Demodulation Methods, Figures of Merit (FM, FM4, NA4, NA4 *, NB4, NB4 * ), Crest Factor, Energy Ratio and Sideband Level Factor, etc. They are using the difference of vibration amplitude and phase due to the damage of gear-rotor-bearing system to detect the fault of gear and bearing. 16

35 2.3.2 Frequency Domain Analysis The Frequency domain methods include Fast Fourier Transform (FFT), Hilbert Transform Method, Power Cepstrum Analysis, and etc. They are using the difference of power spectral density of the signal due to the fault of bearing, shaft and/or gear to identify the damage of elements. The Fourier Transform is one of the most basic and widely used signal transforms. The Fourier Transform decomposes the original time signal into different frequencies and amplitudes. This is often expressed in the form of Frequency Spectrum or Frequency Representation of the time domain signal. The common form of Fourier Transform can be defined as F(ω) = f(t)e iωt dt (2.19) where F(ω) is the Fourier domain signal and f(t) is the time domain signal. The coefficient, F(ω), is essentially obtained by measuring the level of similarity between the complex sinusoid of frequency, ω, and the time domain signal, which represents the significance of the particular frequency component within the signal. 17

36 2.3.3 Continuous Wavelet Transform The Continuous Wavelet Transform (CWT) is one of the favored joint timefrequency analysis techniques among Short Time Fourier Transform (STFT), Wigner- Ville Distribution (WVD), Choi-Williams Distribution (CWD). This method converts a time series into its time-frequency representation (TFR), providing an interactive relationship between time and frequency during the period of time data window. The equation for the continuous wavelet transform (CWT) is: C(a, b) = 1 a + t b f(t)ψ ( a ) dt (2.2) where f(t) is the time signal input function, ψ * (t) is the complex conjugate of the mother wavelet function, ψ(t), and 1 a is a normalization factor. The mother wavelet is scaled and translated by a scaling parameter, a, corresponding to frequency and translating parameter, b, corresponding to time. Thus, a set of coefficients, C(a,b), is a twodimensional dependent variable function of both scaling and translating parameters. The mother wavelet is an oscillating waveform with finite duration and energy. There are numbers of basic functions being used as mother wavelet in wavelet family, such as Haar, Daubechies, Biorthogonal, Morlet, and Mexican Hat. The basic algorithm of CWT begins with choosing a mother wavelet and comparing it with a section at the beginning of original signal. It is important to choose the appropriate mother wavelet because the characteristics of CWT result are determined by the mother wavelet. Basically, the 18

37 mother wavelet is chosen closely correlated to the original time domain signal. Then, the wavelet is shifted to the right next windows and either stretched or compressed the wavelet for comparison. This process occurs repeatedly until it covers the whole original signal. Finally, the set of coefficients is obtained and each of these coefficients represents the diversity of a scale at different sections of the original signal. Figure 2.1 shows an example of CWT due to a discontinuity in a sine wave. One can notice that the discontinuity is able to be observed in the time-frequency map but not in the frequency spectrum. This is an important property of wavelet transform that it is able to provide an instantaneous characteristic of the original signal. This advantage is useful for examining a vibration signal on non-stationary equipment, such as the signal from a damaged rotating component. 19

38 Time (s) FFT Frequency (Hz) Figure 2.1 Example of CWT of a sine wave with a discontinuity 2

39 2.4 Summary In this chapter, we introduced two fundamental topics of research, the use of modal analysis method for numerical simulation of rotor-bearing systems and the use of signal processing techniques. For the simulation model, the modal method transforms the equations of motion into modal coordinates to reduce the degrees-of-freedom of the system. The advantage of this method is useful for analyzing the transmission system with geometric complex. For the vibration signal analysis, the frequency domain analysis and joint time-frequency domain analysis based on the Continuous Wavelet Transform (CWT) are the two common domain analysis techniques used in machinery health condition monitoring. 21

40 CHAPTER III ANALYTICAL SIMULATION OF THE BEARING DAMAGE WITH EXPERIMENTAL CORRELATION 3.1 Objective With the increasing requirements for long life and safe operation in mechanical systems, accurate bearing fault identification and failure prognostication systems which are cable of on-line bearing health monitoring as well as machine life prediction are being developed. Signature analysis of machine vibration/acoustic signals is one of the advanced fault identification procedures used in high-speed rotor-bearing systems. The traditional approaches including the time signal and frequency spectrum analysis had gained considerable success. For preventing unexpected failure in rolling element bearing systems, different methods are used for predicting the dynamic performance of ball bearing with defect. Basically radially loaded rolling element bearings generate vibration even if they are in perfect condition. The sources of excitation are due to the external disturbance, the internal excitations, and the varying structure compliance [9, 1]. In general, the varying compliance vibration is generated by the relative motions between ball bearing and inner or outer race because the number of rolling elements and their position in the load zone 22

41 changes with bearing rotation [11]. However, the presence of a defect causes a significant increase in the vibration level so it is important to understand the behavior of vibration signatures under different types of defect. The defects in bearing may be classified as distributed and localized defects. Distributed defects include surface roughness, waviness, misaligned races and off-size rolling elements [91] which are usually caused by manufacturing error, improper installation or abrasive wear [92]. Where localized defects including cracks, pits, and spalls are caused by fatigue in material surface, a fatigue crack begins below the surface and propagates towards the surface until a piece of material breaks away [93]. When a defect in one surface of a rolling element bearing is intact with another surface, it generates an impulse which may excite resonances in the bearing and in the machine Most of the researchers studying on the detection and diagnosis of localized defects in bearing have done experimental work while others developed models for detecting defects and compared their results with experimental results. The early work was performed by McFadden and Smith [12, 13]. They proposed a model for the highfrequency vibration produced by a single defect or multiple defects on the inner race of a rolling element bearing under radial load and the model performance was confirmed experimentally by NASA researchers [14]. Tandon and Choudhury [94] proposed an analytical model for predicting vibration frequency and amplitude of frequency components due to defect in bearing elements including local defects on the rolling elements, inner or outer ring under the axial and radial loads. Sopanen and Mikola [95, 96] presented a dynamic model for a deep-groove ball bearing including localized and 23

42 distributed defects, effect of internal radial clearance and unbalance excitation of the system. Braun and Datner [97] investigated a bearing containing new forming local defects and they concluded that every defect gave a vibration signal forming at a certain period. Franco et al. [98] investigated the vibrations caused by multiple physical defects on the bearing. If the ball bearing element with defect was a ring, rolling element will pass over the place with defect at regular intervals as concluded by Yhland and Johansson [99]. They showed that frequency of this event can be calculated using the base bearing geometry and rotating speed of ball bearing. The objective in this chapter was to develop a numerical model to study the effect of local defect in bearing element on variation in vibration signatures. A single defect lying on inner race surface of bearing is considered. Also, the experiments have been performed for the validation of a proposed dynamic model. The vibration results from both simulation and experiment were presented in both time and frequency domains. In frequency domain, the Fast Fourier Transformation (FFT) method was adopted for investigating the vibration analysis. 3.2 Dynamic of ball bearing The ball bearing can be illustrated in its simple form as in Figure 3.1. From this Figure, one can easily see that the bearing pitch diameter is approximately equal to the mean of the bore and O.D. or 24

43 d m 1 (bore + O. D. ) (3.1) 2 More precisely, however, the bearing pitch diameter is the mean of the inner and outer ring raceway contact diameters. Therefore, d m = 1 2 (d i + d o ) (3.2) Generally, ball bearings and other radial rolling bearings such as cylindrical roller bearings are designed with clearance. From Figure3.1, the diametral clearance is as follows: P d = d o d i 2D (3.3) Curvature and relative curvature Considering two bodies of revolution having different radii of curvature, the contact between the bodies through a pair of principal planes may contact each other at a single point under the condition of no applied load. This condition is called point contact. Figure 3.2 demonstrates this condition. 25

44 In Figure 3.2, the upper body is denoted by I and the lower body by II. The principal planes are denoted by 1 and 2. Therefore, the radius of curvature of body I in plane 2 is denoted by r I2. Since r denotes radius of curvature, curvature can be defined as ρ = 1 r (3.4) Although the radius of curvature always has positive sign, curvature may be positive or negative, convex surfaces being positive and concave surfaces being negative. 26

45 Figure 3.1 Radial ball bearing showing diametral clearance 27

46 Figure 3.2 Geometry of contacting bodies 28

47 The following definitions are used for describing the contact between two mating surfaces of revolution. Curvature sum: ρ = 1 r I1 + 1 r I2 + 1 r II1 + 1 r II2 (3.5) Curvature difference: F(ρ) = (ρ I1 ρ I2 ) + (ρ II1 ρ II2 ) ρ (3.6) The sign convention for convex and concave surfaces is used in Equation (3.5) and (3.6). Moreover, it must be careful to check that F(ρ) is positive Surface deformations In order to evaluate elastic deformation between two contacting bodies, the classical solution established by Hertz is used. The relative approach of remote points between two contacting deforming bodies is given by δ = δ [ 3Q 2 ρ ((1 ν I E I 2 ) + (1 ν II 2 2 ) 3 ρ )] E II 2 (3.7) 29

48 where Q is normal force load, ν is Poisson s ratio, and E is the modulus of elasticity of each body. If the contacting bodies are made of steel, Equation (3.7) becomes δ = δ Q ρ1 (3.8) The value of the dimensionless quantities δ * as function of F(ρ) is given in Table 3.1 [1] 3

49 Table 3.1 Dimensionless Contact Parameters F(ρ) δ *

50 3.2.3 Load-deformation relationships The load-deformation relationship for contacting between rolling elements and raceways is developed. For a given ball-raceway contact (point loading), Equation (3.8) can be inverted and expressed as Q = K p δ 3/2 (3.9) For steel-ball raceway contact 1 K p = ρ (δ ) 3 2 (3.1) Considering a rolling element under the load between inner and outer raceways, the total normal approach is the sum of the approaches between rolling element and each raceway. Hence δ n = δ i + δ o (3.11) Therefore, the effective elastic modulus K t for the bearing system is obtained as 1 K t = [ (1 K i ) (1 K o ) 2 3 ] 3 2 (3.12) 32

51 3.2.4 Restoring Force The overall contact deformation for the j th rolling element, δ j, is a function of inner race displacement relative to the outer race in the x- and y-direction (x, y), angular position of the j th rolling element (θ j ), and internal radial clearance (P d ). This is given by δ j = xcosθ j + ysinθ j P d (3.13) The angular position of the j th rolling element is time dependent function and can be calculated by θ j = 2π(j 1) N b + ω c t + θ (3.14) Where N b is the number of rolling elements, ω c is the angular velocity of cage, and θ is the initial angular position of the first rolling element. The total restoring force is the sum of the restoring forces from each rolling element. Thus, the total restoring forces component in the x- and y-direction can be calculated by N b F x = K λ j δ j 1.5 cosθ j j=1 N b F y = K λ j δ j 1.5 sinθ j j=1 33 (3.15)

52 Since compression occurs only in the loading zone, then only positive deformation values δ j are considered. Thus, for convenient in calculation the loading zone parameter λ j is introduced for the j th rolling element as λ j = { 1 if δ j > if δ j (3.16) Inner race localized defect The single point defect on inner race surface is modeled as circumferential step function. The location of this defect does not remain stationary because the inner race rotates at the shaft speed ω s. When the angular position of the j th rolling element coincides with defect angular α, the contact deformation δ j varies due to the additional deflection Δ s as shown in Figure 3.3. Thus, the total contact deformation of j th rolling element can be expressed as δ j = xcosθ j + ysinθ j P d s (3.17) Note that an additional deflection Δ s equals to zero when there is no rolling element approaching in defect region. 34

53 Figure 3.3 The additional deflection of ball due to defect on inner race 35

54 3.3 Discussion of Results In order to investigate and understand the dynamic characteristics of the rotorbearing system, the numerical model was developed to simulate the dynamic vibration response. The results obtained from numerical model are used for comparison to those of experiment and also suitable vibration signature analysis method are selected to examine the defect for both numerical and experimental data Numerical Simulation Results The model consists of two identical rotors of.625 inches long and 6 inches in diameter mounted on a rigid shaft of 21 inches long and.625 inches in diameter and supported by two bearings shown in Figure 3.4. A schematic model of the rotor-bearing system is given in Figure 3.5. First, the frequency response function (FRF) of the system was performed [11, 12] to study the dynamic characteristics of the experimental test rig. FRF is a fundamental measurement that isolates the inherent dynamic properties of a mechanical structure. In modal testing, FRF measurements are usually made under controlled conditions, where the test structure is artificially excited. The most popular testing method used today is obtained by Impact testing. Then, the input force is measured from the load cell attached to the head of impact hammer and the output response is measured by an accelerometer at a fixed point on the structure. Thus, the FRF results describe the relationship as frequency function between input and output [72, 36

55 13]. The test was performed on the test rig using an impact hammer and measuring the response due to the excitation on the left and right rotors. The FRF results of left and right rotors are given in Figures 3.6 and 3.7 respectively. It is observed that the major frequency responses from both left and right rotors are similar which is excited at 117 and 367 Hz. Therefore, these results were served as the first and second critical speeds of the rotor-bearing system in numerical model. 37

56 Figure 3.4 Test Rig Model of Rotor-Bearing System 38

57 Figure 3.5 Schematic of Rotor-Bearing System for Numerical Simulation 39

58 Y(f) Y(f) 15 Frequency response function Frequency (Hz) Figure 3.6 Frequency Response Function of System Measured at Left Rotor 15 Frequency response function Frequency (Hz) Figure 3.7 Frequency Response Function of System Measured at Right Rotor 4

59 The overall dynamic transient response of the system was calculated for an operating speed at 4 Hz (24 rpm). The bearing ball carrier speed was calculated as to be 2.62 times slower than the operating speed. Thus, the ball pass frequency for inner race can be determined by multiplying the relative frequency of operating speed to cage speed by the number of balls, resulting BPFI of 272 Hz. In order to study the dynamic effect due to a bearing defect, it is assumed that the material on an inner race surface of right bearing has been removed. A step function, serving as a damaged profile, was incorporated into a bearing model. The vibration signatures resulted from the numerical simulation of rotor-bearing system are shown in Figures 3.8 and 3.9 for healthy bearing and faulty bearing in Figures 3.1 and These results are extracted only at the location of right bearing in both x- and y-directions from the overall shaft stations. For a comparison, the results in x- direction shown Figures 3.8 and 3.1 are used. It is noticed that the signal amplitudes are increased when the defect is introduced in a bearing. This effect is also induced to the vibrations in y-direction shown in Figures 3.9 and However, this can t indicate that amplitude excitation comes from inner race defect in bearing. By using Fast Fourier Transform method (FFT), the vibration results are transformed to frequency spectra domain. Figures 3.12 and 3.13 depict the frequency spectra of healthy bearing in x- and y-direction respectively. Similarly, the frequency spectra of faulty bearing in x- and y-direction show in Figures 3.14 and For undamaged bearing, the results in both directions are occupied by two major frequencies at 4 and 117 Hz which are corresponding to operating and natural frequencies 41

60 respectively. When damaged model was used, it is observed the additional two major components increasing significantly at 232 and 312 Hz compared with undamaged case, while other components are increased indistinctly. These two frequencies are corresponding to sideband frequencies of ball pass frequency for inner race and shaft rotational frequency (BPFI ± SF). For further investigation, the frequency spectra of absolute vibrations are used. Figures 3.16 and 3.17 depict the frequency spectra for healthy bearing case and faulty bearing case respectively. For defect-free case, the frequency spectrum contains dominant peaks at 78, 156, and 234 Hz. These frequencies are corresponding to sideband of natural frequency and shaft rotational frequency and its harmonics. (ω n -SF). However, the result of defected case in Figure 3.17 shows that amplitude spectra are increased over all frequencies but the dominant spectrum occurs at 272 Hz. This frequency is in good agreement with the ball pass frequency for inner race (BPFI). 42

61 Acceleration (In/s 2 ) Acceleration (In/s 2 ) 1 Vibration of undamaged bearing in x-direction Time (s) Figure 3.8 Numerically Simulated Vibration of Undamaged Bearing in x-direction 1 Vibration of undamaged bearing in y-direction Time (s) Figure 3.9 Numerically Simulated Vibration of Undamaged Bearing in y-direction 43

62 Acceleration (In/s 2 ) Acceleration (In/s 2 ) 1 Vibration of damaged bearing in x-direction Time (s) Figure 3.1 Numerically Simulated Vibration of Damaged Bearing in x-direction 1 Vibration of damaged bearing in y-direction Time (s) Figure 3.11 Numerically Simulated Vibration of Damaged Bearing in y-direction 44

63 Amplitude Amplitude 3 Frequency spectrum of undamaged bearing in x-direction Frequency (Hz) Figure 3.12 Frequency Spectrum of Undamaged Bearing in x-direction 3 Frequency spectrum of undamaged bearing in y-direction Frequency (Hz) Figure 3.13 Frequency Spectrum of Undamaged Bearing in y-direction 45

64 Amplitude Amplitude 3 Frequency spectrum of damaged bearing in x-direction Frequency (Hz) Figure 3.14 Frequency Spectrum of Damaged Bearing in x-direction 3 Frequency spectrum of damaged bearing in y-direction Frequency (Hz) Figure 3.15 Frequency Spectrum of Damaged Bearing in y-direction 46

65 Amplitude 6 Frequency spectrum of undamaged bearing Frequency (Hz) Figure 3.16 Frequency Spectrum of Absolute Vibration for Undamaged Bearing 47

66 Amplitude 6 Frequency spectrum of damaged bearing Frequency (Hz) Figure 3.17 Frequency Spectrum of Absolute Vibration for Damaged Bearing 48

67 3.3.2 Experimental Results The study of damage identification for a ball bearing was carried out by using a high-speed test rig as shown in Figure The taper ball bearing named Fafnir is used in the experiment shown in Figure The bearing consists of 11-ball rolling element with the following characteristics: Bearing ID is 1.7 inches Bearing OD is inches Bearing length is.563 inches Ball diameter is.312 inches Pitch diameter is inches The test rig was driven by a variable speed electric motor through a flexible coupling on the left hand side of the shaft. During the vibration tests, only the right hand side bearing is replaced while the left hand side bearing remain unchanged. Four sets of Fafnir bearing are used: a healthy bearing and three bearings with three different damage profiles on the inner race surface. The photos of bearings with three different inner race damages are shown in Figures 3.2 to 3.22 and the 3-dimensional damage profiles are shown in Figures 3.23 to One can notice the dimension from the damage profiles of bearing#1 is around.3 inch wide and.15 inch deep, bearing#2 is around.4 inch wide and.25 inch deep, and bearing#3 is around.5 inch wide and.4 inch deep. The motor operating speed was set at 4 Hz (24 rpm) and vibration data were acquired through a set of accelerometers mounted on bearing housings. The vibration 49

68 data were sampling at 1, Hz which is approximately 25 samples per revolution. The vibration data of each case was acquired for 1 seconds and stored in computer memory for fault identification signature analysis. In order to perform an overall vibration signature analysis, the vibration data were collected from the accelerometers in both x- and y-direction with different 4 cases: bearing#, #1, #2, #3. The vibration data acquired through accelerometers mounted on right bearing housing of each case show in Figures 3.26 to It is noticed that the time vibration signatures obtained from experimental test rig are much noisier. However, the zoomed in views show that the time vibration data are also sinusoidal. From these figures, it is noticed that there are increases in the vibration signals for both x- and y- direction due to the damages in bearings#1 to #3, but they can t be differentiated by the damage levels. The vibration data were transformed to frequency spectra domain. The frequency results of 4 cases are shown in Figures 3.34 to For all cases, it is observed that the dominant frequencies in x-direction are the operating speed and its sidebands while two times of operating speed is the major frequency in y-direction. Compared with undamaged case, only the increases in the frequency spectra are noticeable for damaged cases. Although, the change in amplitude spectra verify a presence of the damage, it still can t identify which bearing is more severe. Further investigation showed that using the frequency spectra of the absolute vibration signals provides a better quantification of damage in bearings. Figures 3.42 to 3.45 show the frequency spectra of the absolute vibration signals from each bearing. From these figures, the amplitude frequency at 272 Hz, the relative ball pass frequency on the inner race, 5

69 increases accordingly with the damaged volumes on the inner race surfaces. The amplitudes of frequency spectrum at relative ball pass frequency on the inner race with different damaged volume of bearing inner race surface are shown in Figure From this figure, one can notice that the more serious inner race damage, the higher spectral amplitude. 51

70 Figure 3.18 Photograph of the High-speed Ball Bearing Test Rig 52

71 Figure 3.19 Photograph of Fafnir Bearing 53

72 Figure 3.2 Photograph of Inner Race Damage of Bearing#1 54

73 Figure 3.21 Photograph of Inner Race Damage of Bearing#2 55

74 Figure 3.22 Photograph of Inner Race Damage of Bearing#3 56

75 Depth (In) Damaged profile bearing#1 x Width (In) Length (In).15.2 Figure 3.23 Inner Race Damaged Profile of Bearing#1 57

76 Depth (In) Damaged profile bearing#2 x Width (In) Length (In).15.2 Figure 3.24 Inner Race Damaged Profile of Bearing#2 58

77 Depth (In) Damaged profile bearing#3 x Width (In) Length (In).15.2 Figure 3.25 Inner Race Damaged Profile of Bearing#3 59

78 Figure 3.26 Experimental Vibration Signature of Bearing# in x-direction Figure 3.27 Experimental Vibration Signature of Bearing# in y-direction 6

79 Figure 3.28 Experimental Vibration Signature of Bearing#1 in x-direction Figure 3.29 Experimental Vibration Signature of Bearing#1 in y-direction 61

80 Figure 3.3 Experimental Vibration Signature of Bearing#2 in x-direction Figure 3.31 Experimental Vibration Signature of Bearing#2 in y-direction 62

81 Figure 3.32 Experimental Vibration Signature of Bearing#3 in x-direction Figure 3.33 Experimental Vibration Signature of Bearing#3 in y-direction 63

82 Amplitude Amplitude 3 Frequency spectrum of bearing# in x-direction Frequency (Hz) Figure 3.34 Frequency Spectrum of Bearing# in x-direction 3 Frequency spectrum of bearing# in y-direction Frequency (Hz) Figure 3.35 Frequency Spectrum of Bearing# in y-direction 64

83 Amplitude Amplitude 3 Frequency spectrum of bearing#1 in x-direction Frequency (Hz) Figure 3.36 Frequency Spectrum of Bearing#1 in x-direction 3 Frequency spectrum of bearing#1 in y-direction Frequency (Hz) Figure 3.37 Frequency Spectrum of Bearing#1 in y-direction 65

84 Amplitude Amplitude 3 Frequency spectrum of bearing#2 in x-direction Frequency (Hz) Figure 3.38 Frequency Spectrum of Bearing#2 in x-direction 3 Frequency spectrum of bearing#2 in y-direction Frequency (Hz) Figure 3.39 Frequency Spectrum of Bearing#2 in y-direction 66

85 Amplitude Amplitude 3 Frequency spectrum of bearing#3 in x-direction Frequency (Hz) Figure 3.4 Frequency Spectrum of Bearing#3 in x-direction 3 Frequency spectrum of bearing#3 in y-direction Frequency (Hz) Figure 3.41 Frequency Spectrum of Bearing#3 in y-direction 67

86 Amplitude 6 Frequency spectrum of bearing# Frequency (Hz) Figure 3.42 Frequency Spectrum of Absolute Vibration for Bearing# 68

87 Amplitude 6 Frequency spectrum of bearing# Frequency (Hz) Figure 3.43 Frequency Spectrum of Absolute Vibration for Bearing#1 69

88 Amplitude 6 Frequency spectrum of bearing# Frequency (Hz) Figure 3.44 Frequency Spectrum of Absolute Vibration for Bearing#2 7

89 Amplitude 6 Frequency spectrum of bearing# Frequency (Hz) Figure 3.45 Frequency Spectrum of Absolute Vibration for Bearing#3 71

90 Frequency Spectrum Amplitude Bearing# Bearing#1 Bearing#2 Bearing#3 Figure 3.46 The Frequency Spectrum Amplitude with the Experimental Bearings 72

91 3.4 Conclusions The results of this chapter demonstrate the feasibility of the developed rotor-bearing system codes to simulate the vibration signatures of the healthy and damaged bearing. The numerical results are then compared with data obtained from the experiments and both of analytical and experimental results are in agreement with each other. However, the numerical results provided a much cleaner/easier identification of the damage in bearing. The vibration signature analysis method was used to investigate the effects of the damaged bearing in vibration results obtained from both simulation and experiment. The figures were used to present results in this chapter including the vibration time signal and FFT. Based on results of this study, some specific conclusions can be stated as follow: 1. The numerical model was developed to simulate the dynamics of rotor-bearing system with effects of damaged bearing. The damage on inner race surface of bearing can be represented as step function which can be incorporated into bearing restoring force. 2. The numerically simulated model can provide an accurate dynamic representation of the results obtained from the experimental rotor-bearing test rig without the complication due to various operational environments. 3. The use of time signature data can provide information of abnormal condition due to the overall increase of vibration amplitude without any specific indication of type of component failure. 73

92 4. The use of frequency spectrum analysis for individual orientation can provide some indication of component failure with the existence of large side band components. It can provide some information on the severity of damage. 5. The use of frequency spectrum analysis of absolute orientation provides a good identification as well as the quantification of the damage in bearing component. 74

93 CHAPTER IV DYNAMIC ANALYSIS OF ROTOR-BEARING SYSTEM UNDER RESIDUAL SHAFT BOW EFFECT 4.1 Objective One of the common faults in large rotating machinery is the residual shaft bow. The existence of residual bow in shaft may result from various effects such as, gravity sag, thermal distortion, uneven shrink fits, and mechanical bow due to previous large unbalance. In the case of gravitational sag, it mostly occurs in the large horizontal turbines or compressors which are allowed to rest for long period of times. This may affect the shaft temporarily. The turbomachinery operating in particularly high temperature such as gas/steam turbines and water pumps in nuclear reactors is possible to developed temporary bow condition because the temperature distribution along the shaft temperature is uneven. However, uneven shrink fits of impellers or spacers on a shaft may produce a permanent mechanical bow due to the rubbing of a shaft on a seal.thermal bow effect induced by shaft rubbing was first reported by Newkirk [14]. Dimarogonas [15] investigated further the shaft residual bow and its influence on rotor system stability. Nicholas et al. [16] investigated the effect of the residual shaft bow on unbalance response of a single mass flexible rotor. Salamone and Gunter [17] extended 75

94 the work of Kikuchi [18] to analyze multi-mass rotors in fluid film bearings with both shaft warp and disk skew. The dynamic matrix transfer equations for the unbalance response of a multi-mass rotor with shaft warp were formulated [17, 18]. Flack et al. [19] compared the theoretical and experimental synchronous unbalance response of a bowed Jeffcott rotor using different five types of fluid film bearings. Salamone et al. [11] analyzed the effects of a bowed shaft and skewed impeller wheels on the dynamic response and balancing of a double overhung compressor operating near the third critical speed. Flack et al. [111] studied the unbalance response of a Jeffcott rotor with shaft bow and/or runout theoretically and experimentally. The objective of this chapter was to develop a numerical model that can simulate the effect of residual shaft bow on vibration signatures. The numerical results are validated by the experimental data acquired from bow shaft test rig. Both numerical and experimental results were examined in time domain, frequency domain 4.2 Dynamic Response of bowed shaft The effects of mechanical shaft bow can be easier to understand by using a simple rotor-bearing system. The shaft bow effect can be put into the right-hand side of dynamic equation of (2.1) and considered as part of the driving force of the system as follows [54]: [M]{W } + [C]{W } + [K]{W} = [K] S {W} d + {F(t)} (4.1) 76

95 where {W} d is the vector of mechanical bow shaft. expanded into By using the homogeneous property, the general dynamic equation of (4.1) can be [M]{W } + [[C] G + [C] B ]{W } + [[K] S + [K] B + [K] A ]{W} = [K] S {W} d + {F(t)} (4.2) The damping matrix can be decomposed into the gyroscopic damping matrix [C] G and bearing damping matrix [C] B as follows [C xx ] [C xθ ] [C xy ] [C xψ ] [ωi p ] [C θx ] [C θθ ] [C θy ] [C θψ ] [C] = + [C yx ] [C yθ ] [C yy ] [C yψ ] [ [ ωi p ] ] G [[C ψx ] [C ψθ ] [C ψy ] [C ψψ ]] B (4.3) The general stiffness matrix may also be decomposed into three submatrices representing the symmetric shaft stiffness matrix [K] S, the linear bearing stiffness matrix [K] B, and the skew symmetric acceleration matrix [K] A as follows 77

96 [k xx ] [k xθ ] [k [K] = [ θx ] [k θθ ] ] [k xx ] [k xθ ] [k θx ] [k θθ ] S [K xx ] [K xθ ] [K xy ] [K xψ ] [K θx ] [K θθ ] [K θy ] [K θψ ] + [K yx ] [K yθ ] [K yy ] [K yψ ] [[K ψx ] [K ψθ ] [K ψy ] [K ψψ ]] B [ ω + 2 I p] [ [ ω 2 I p] ]A (4.4) The generalized displacements and rotational vector can be written in modal coordinate in both x and y directions as {U(t)} = [ [Φ e] ] {A(t)} (4.5) [Φ β ] {V(t)} = [ [Φ e] ] {B(t)} (4.6) [Φ β ] where [Φ e ] is a set of translational orthonormal mode and [Φ β ] is a set of rotational orthonormal mode. Incorporating the total effects due to unbalance, disc skew, shaft bow, and rotor acceleration, the set of modal equation in the x-z plane can be written as follows 78

97 [ [M] [I t ] ] [[Φ e] [Φ β ] ] {A } + [ [C xx] [C xθ ] [C θx ] [C θθ ] ] [[Φ e] [Φ β ] ] {A } + [ [k xx + K b ] [k xθ ] [k θx ] [k θθ ] ] [[Φ e] [Φ β ] ] {A} [ [k xx] [k xθ ] [k θx ] [k θθ ] ] [x d θ ] + [ [K xx K b ] [K xθ ] d [K θx ] [K θθ ] ] [[Φ e] [Φ β ] ] {A} + [[C xy] [C xψ ] [C θy ] [C θψ ] ] [[Φ e] [Φ β ] ] {B } + [ [ωi p ] ] [[Φ e] [Φ β ] ] {B } + [ [K xy] [K xψ ] [K θy ] [K θψ ] ] [[Φ e] [Φ β ] ] {B} + [ ω 2 I p ] [ [Φ e] [Φ β ] ] {B} M u e u [ω 2 cos(ωt + α) + ω sin(ωt + α)] = [ τ(i p I t )[ω 2 cos(ωt + β) + ω sin(ωt + β)] ] (4.7) According to coupled modal analysis, the orthonormal modes based on the average bearing stiffness [K b ] give better approximation of the system dynamics. The bow of the shaft can be transform into modal coordinates as follows: { (x d) (θ d ) } = [[Φ e] [Φ β ] ] {R x} { (y d) (ψ d ) } = [[Φ e] [Φ β ] ] {R y} (4.8) and [ [k xx] [k xθ ] [k θx ] [k θθ ] ] [x d θ ] = [ [k xx + K b ] [k xθ ] d [k θx ] [k θθ ] ] [[Φ e] [Φ β ] ] {R x} [ [K b] ] [[Φ e] [Φ β ] ] {R x} (4.9) 79

98 and {R x } can be represented by [ [Φ T e] [Φ β ] ] [ [M] T [I t ] ] {(x d) (θ d ) } = [[Φ e] [Φ β ] ] [ [M] [I t ] ] [[Φ e] [Φ β ] ] {R x} = {R x } (4.1) and therefore {R x } = [ [Φ e] [Φ β ] ] T [ [M] [I t ] ] {(x d) (θ d ) } = {[A e] T [M]{x d } + [A β ] T [I t ]{θ d }} (4.11) Substituting this relationship into Equation (4.7) and premultiplying the equation by [ [Φ e] [Φ β ] ] T gives the following matrix equation. 8

99 {A } + {[Φ e ] T [C xx ][Φ e ] + [Φ e ] T [C xθ ][Φ β ] + [Φ β ] T [C θx ][Φ e ] + [Φ β ] T [C θθ ][Φ β ]} {A } + [ω i 2 ]{A} + {[Φ e ] T [K xx K B ][Φ e ] + [Φ e ] T [K xθ ][Φ β ] + [Φ β ] T [K θx ][Φ e ] + [Φ β ] T [K θθ ][Φ β ]} {A} + {[Φ e ] T [C xy ][Φ e ] + [Φ e ] T [C xψ ][Φ β ] + [Φ β ] T [C θy ][Φ e ] + [Φ β ] T [C θψ ][Φ β ]} {B } + {[Φ β ] T [ωi p ][Φ β ]} {B } + {[Φ e ] T [K xy ][Φ e ] + [Φ e ] T [K xψ ][Φ β ] + [Φ β ] T [K θy ][Φ e ] + [Φ β ] T [K θψ ][Φ β ]} {B} + {[Φ β ] T [ ω 2 I p] [Φ β ]} {B} = [Φ e ] T {F x } + [Φ β ] T {M x } + {[ω i 2 ] [Φ e ] T [K b ][Φ e ]} {[Φ e ] T [M]{x d } + [Φ β ] T [I t ][θ d ]} (4.12) The equations can be written in the compact form as {A } + [C x ]{A } + [Λ]{A} + [K x ]{A} + [D x ]{B } + [CM x ]{B } + [E x ]{B} + [EA x ]{B} = {xf} + {xb} (4.13) 81

100 Similarly, the equation for the y-z plane can be written as {B } + [C y ]{B } + [Λ]{B} + [K y ]{B} + [D y ]{A } + [CM y ]{A } + [E y ]{A} + [EA y ]{A} = {yf} + {yb} (4.14) Combining Equation (4.13) with (4.14) and rearranging the terms, we have { A B (xf) + (xb) } = { (yf) + (yb) } [ [C x ] [D x + CM x ] ] { A } [D y + CM y ] [C y ] B [ [Λ + K x] [E x + EA x ] [E y + EA y ] [Λ + K y ] ] {A B } (4.15) Thus, the modal acceleration terms can be calculated from the above equation with initial conditions for the modal displacement given by the following relations { A() T B() } = [Φ x Φ ] [ [M] x ] [ U() y [M] y V() ] (4.16) Similarly, the modal velocity is given by { A () B () } = [Φ x Φ y ] T [ [M] x [M] y ] [ U () V () ] (4.17) 82

101 With the initial modal acceleration and velocity terms calculated as above, the modal transient motion can be computed by numerical integration in time. The rotor forced response motion can be obtained by back substitution of the modal coefficients into the following equation. x(t) [ [Φ e] θ(t) [Φ β ] ] {A(t)} {W(t)} = = y(t) { ψ(t)} [ [Φ e] { [Φ β ] ] {B(t)} } (4.18) 4.3 Discussion of Results In order to validate and understand the dynamic characteristics of the rotorbearing system under the effect of residual bow shaft, the numerical simulations with straight shaft and residual bow shaft are performed. Under similar operating condition, the experiment is carried out using a test rig and the experimental results are used for validation. Additionally, the vibration signature analysis is used for damage identification Numerical Simulation Results The numerical model developed in previous chapter is used as reference. The major model has kept the same where it is composed of two identical rotors of.625 inches long and 6 inches in diameter mounted on a shaft of 21 inches long and

102 inches in diameter and supported by two bearings. For the case of bow shaft, it is assumed that the maximum bow occurred at the center of the shaft with the height of.8 inches which is in agreement with experimental test rig. Figure 4.1 depicts the schematic model of rotor-bearing system with a centrally bent shaft. The overall dynamic response of the system is obtained by solving the equations of motion (equation 4.14). The system was assumed to be operated at slow speed as 2 Hz (12 rpm). 84

103 Figure 4.1 Schematic of Rotor-Bearing System with centrally bent shaft for Numerical Simulation 85

104 The numerical simulation data of straight shaft and bow shaft are plotted in Figures 4.2 to 4.5. These figures present displacements of shaft at the middle span with and without residual bow effect in both x- and y-direction, respectively. As comparing displacements in x-direction of straight shaft in Figure 4.2 with bow shaft in Figure 4.4, one can notice that the signal amplitude is substantially increased in the bow case. Similarly, it can be observed in y-direction displacements shown in Figures 4.3 and 4.5. Consequently, the displacement data are transformed to frequency spectra domain for investigating the source of excitation. The frequency spectra of straight shaft in x- and y-direction are shown in Figures 4.6 and 4.7 respectively. In the same manner, the residual bow shaft results are shown in Figures 4.8 and 4.9. For spectra results of straight shaft in Figures 4.6 and 4.7, it is observed that the major frequency dominates at 2 Hz with small amplitude, which is corresponding to operating speed. However, amplitudes of operating speed are greatly increased when the bow shaft profile is used in simulation model. This aspect can be recognized in the results from both directions as shown in Figures 4.8 and

105 Displacement (In) Displacement (In).5 Displacement of straight shaft in x-direction Time (s) Figure 4.2 Numerically Simulated Displacement of Straight Shaft in x-direction.5 Displacement of straight shaft in y-direction Time (s) Figure 4.3 Numerically Simulated Displacement of Straight Shaft in y-direction 87

106 Displacement (In) Displacement (In).5 Displacement of residual bow shaft in x-direction Time (s) Figure 4.4 Numerically Simulated Displacement of Residual Bow Shaft in x-direction.5 Displacement of residual bow shaft in y-direction Time (s) Figure 4.5 Numerically Simulated Displacement of Residual Bow Shaft in y-direction 88

107 Amplitude Amplitude.8 Frequency spectrum of straight shaft in x-direction Frequency (Hz) Figure 4.6 Frequency Spectrum of Straight Shaft in x-direction.8 Frequency spectrum of straight shaft in y-direction Frequency (Hz) Figure 4.7 Frequency Spectrum of Straight Shaft in y-direction 89

108 Amplitude Amplitude.8 Frequency spectrum of residual bow shaft in x-direction Frequency (Hz) Figure 4.8 Frequency Spectrum of Residual Bow Shaft in x-direction.8 Frequency spectrum of residual bow shaft in y-direction Frequency (Hz) Figure 4.9 Frequency Spectrum of Residual Bow Shaft in y-direction 9

109 4.3.2 Experimental Results In order to provide better understanding and verification of the numerical simulation due to the effect of the bent shaft, the tests were carried out by using the rotorbearing test rig. The shaft test rig is driven by a variable speed electric motor on the left hand side. The driving torque is transmitted from motor to test rig by a flexible coupling. Two sets of shafts were used in the experiments. One is the straight shaft taken as a reference and the other is the bent shaft as shown in Figures 4.1 and 4.11 respectively. For the bent shaft, the maximum measured eccentricity is about 8 mils located at the central of the shaft. The motor operating speed was set at 2 Hz (12 rpm). The vibration data were collected through two sets of 5mm non-contacting proximitor sensors located approximately at mid span between two rotor disks and two sets of accelerometers mounted on two bearing housings in both horizontal and vertical directions as shown in Figure The vibration data of each case were acquired through a multi-channel analog-to-digital converter and stored in computer memory for fault identification signature analysis. 91

110 Figure 4.1 Photograph of Straight Shaft 92

111 Figure 4.11 Photograph of Residual Bow Shaft 93

112 Figure 4.12 Photograph of the High-speed Ball Bearing Test Rig with sets of sensors 94

113 The experiments were carried out under the similar operating condition and the vibration data were collected for both cases, straight and bent shafts. In this scenario, the data acquired from non-contacting sensors are useful for vibration signature analysis. The result of a straight shaft is shown in Figures 4.13 and These figures display horizontal and vertical movements of a straight shaft at the mid span respectively. It is observed that these time data signatures are similar to wave functions. Likewise, Figures 4.15 and 4.16 depict the displacement signatures of the shaft due to residual bow effect measured in both horizontal and vertical planes. In this case, the structures of displacement signatures still remain as wave form where it is also noticed that the displacement signature in horizontal plane has a 9 degree phase lead with respect to the one in vertical plane. Moreover, the amplitudes of signature displacements have considerably increased in both horizontal and vertical planes when the tests were performed on bent shaft. To draw additional information from displacement signatures, the data are transformed to frequency spectra domain. Figures 4.17 and 4.18 show the frequency spectra resulted from the signature data in Figures 4.13 and 4.14 respectively for the case of defect-free shaft. From these frequency results, it is observed that there are small amplitudes occur at 1.9, 3.8, 5.8, and 7.5 Hz. These frequencies are recognized as the operating speed and its harmonics. Correspondingly, the results of the bent shaft are shown in Figures 4.19 and 4.2. According to overall frequency results, it is easy to distinguish the results in x-direction between straight shaft in Figure 4.17 and bent shaft in Figure 4.19 where the peak at operating speed of the bent shaft are significantly 95

114 increased. For results in y-direction, a similar trend can be observed where the frequency spectrum at the operating speed increases substantially in Figure

115 Displacement (In) Displacement (In).5 Displacement of straight shaft in x-direction Time (s) Figure 4.13 Experimental Displacement Signature of Straight shaft in x-direction.5 Displacement of straight shaft in y-direction Time (s) Figure 4.14 Experimental Displacement Signature of Straight shaft in y-direction 97

116 Displacement (In) Displacement (In).5 Displacement of residual bow shaft in x-direction Time (s) Figure 4.15 Experimental Displacement Signature of Residual Bow shaft in x-direction.5 Displacement of residual bow shaft in y-direction Time (s) Figure 4.16 Experimental Displacement Signature of Residual Bow shaft in y-direction 98

117 Amplitude Amplitude.8 Frequency spectrum of straight shaft in x-direction Frequency (Hz) Figure 4.17 Frequency Spectrum of Straight Shaft in x-direction.8 Frequency spectrum of straight shaft in y-direction Frequency (Hz) Figure 4.18 Frequency Spectrum of Straight Shaft in y-direction 99

118 Amplitude Amplitude.8 Frequency spectrum of residual bow shaft in x-direction Frequency (Hz) Figure 4.19 Frequency Spectrum of Residual Bow Shaft in x-direction.8 Frequency spectrum of residual bow shaft in y-direction Frequency (Hz) Figure 4.2 Frequency Spectrum of Residual Bow Shaft in y-direction 1

119 4.4 Conclusions The numerical model of rotor-bearing system was developed to study the vibration signatures using both the straight and residual bow shafts. The shaft response data were calculated around the mid-span of the shaft where the maximum bow amplitude is located. The numerical results were then validated by those obtained from experiment where both of analytical and experimental results are in agreement. All time signature data were transformed into frequency spectrum using FFT to identify the effect due to the residual bow shaft. Some specific conclusions can be summarized as follow: 1. The numerical model was developed to simulate the dynamics of rotor-bearing system with effect of residual bow shaft where the residual bow effect is incorporated into the global model as driving force function. 2. The time signature data of displacement response around mid-span of the shaft can provide information of abnormal operation due to the increase of displacement amplitude with some degrees of component failure. 3. The use of frequency spectrum analysis provides a good indication of the component defect where the amplitude are substantially increased and dominated at the component operating speed. 11

120 CHAPTER V THE EFFECT OF SPUR GEAR TOOTH IMPERFECTION ON THE DYNAMIC CHARACTERISTICS OF GEAR TRANSMISSION SYSTEM 5.1 Objective The diagnostic in transmission has increasingly become an important research area of the rotorcraft community as transmission fault related accidents and fleet grounding continue to plague helicopters at an increasing rate. A serious investigation in rotorcraft accidents showed that 28 percent were from engine and 22 percent were from transmission components [112] due to fatigue failures. One of the major fatigue failures in rotor craft transmission system was resulted from the excessive gear tooth wear. Over decades, many researchers have focused on developing the gear model and dynamic analysis as apparent from literature review conducted by Ozguven and Houser [113]. The classic model in evaluating gear mesh stiffness was proposed by Weber [114, 115]. He used strain energy to obtain analytical expressions for the tooth deflections based on an integration of the actual shape of the tooth. This work was extended by Richardson [116, 117] with experimental verification of the dynamic tooth loads caused by the tooth deflections. Later, Cornell [18, 19] extended Weber s work by including the foundation effect and adapted to digital computation. Chakraborty and Hunashikatti [118] 12

121 evaluated the combined mesh stiffness of a spur gear pair. However, the foundation effect in the gear tooth was not included. Nagaya [119,12] used taper Timshenko beams to investigate the effects of moving speeds of dynamic loads on the deflections of gear teeth. Savage and Caldwell [123] extended the deflection model of a tooth [121, 122] by including rim bending effects at the base of the tooth. Yang and Lin [124] extended the previous work [125] by including the bending potential energy and axial compressive energy and used the potential energy principal for calculating the effective stiffness of meshing gear pairs as a function of the rotation angle of the gear and Boyd and Pike [16, 126] developed computer program to predict the dynamic gear tooth load response for single-stage epicyclic gearing systems. Wang and Howard [127] employed a commercial finite element program for analyzing the torsional stiffness of a pair of involute spur gears in mesh. The gear tooth profile has a great influence on the dynamic loading, vibration and noise of the gear system. Tavakoli and Houser [128] were the first authors to investigate the optimization of tooth profile to minimize the static transmission error. The effect of gear profile modifications on static transmission error and dynamic loading of spur gears was implemented by using an analytical computer simulation program [ ]. Cai and Hayashi [132] developed a vibration model to optimize the modification in tooth profile for a pair of spur gears for reducing its rotational vibration to zero. The dynamic gear tooth contact has been modeled for analyzing the influence of shape deviations and mounting errors on gear dynamics [133, 134]. Kahraman and Blankenship [135] proved the influence of profile modifications on the dynamic response with experiments. 13

122 The spur gear dynamic response in the parallel gear shaft system was studied by Lin [136, 137]. David and Park [138] examined the vibration problem peculiar to the gear mesh and other problems excited by the gear mesh in gear coupled rotor systems. Choy et al. [22-25, 139] developed a numerical model to simulate the dynamic gear transmission system using modal analysis method and used experimental results to validate. An analytical model was developed including the effects of surface pitting and wear of the gear tooth and transient simulation results of transmission system was evaluated with experimental results using joint time-frequency method of WVD [29, 14]. Another useful joint time-frequency domain method in vibration analysis is the wavelet method [7, 8, 84, 85]. Because of its superior potential time-frequency technique, the use of wavelet analysis for fault analysis and detection in gear box was gaining researchers attention [86-89]. The Wavelet Transform has achieved a high frequency resolution to signify the presence of damage in the gearbox vibration signal. The application of wavelet analysis was recently developed to diagnose a fault in gear [9]. The major objective of this chapter was to develop a model to simulate and analyze the vibration signatures in presence of the localized gear tooth surface wear/damage. The numerical responses were computed for both healthy and damaged cases and then simulations results are compared with those of the experimental with a gear set under controllable damage. Both numerical simulation and experimental results were examined in time domain, frequency domain, and a joint time-frequency domain using Continuous Wavelet Transform (CWT). 14

123 5.2 Gear Meshing Stiffness The potential energy method is used to analytically derive effective gear meshing stiffness. The total potential energy stored in gear meshing system can be decomposed into four components: axial compressive energy, bending energy, Hertzian energy and shear energy [141]. The elastic deflection of tooth along the direction of tooth load is contributed by axial deformation, bending deformation, and shear deformation. However, the local deformation due to the elastic compression of two elastic bodies occurs in the vicinity of the contact point Axial Compressive, Bending, and Shear Energies The gear tooth is modelled as nonuniform contilever beam. It is assumed that there is no deflection at the base of the beam (at the root of the tooth). The potential energies stored in beam due to axial compressive, bending, and shear energies can be calculated as U a = F2 2k a (5.1) U b = F2 2k b (5.2) U s = F2 2k s (5.3) 15

124 Where k a, k b, and k s represent the effective axial, bending, and shear stiffness in the same direction with the acting force F, respectively. According to the properties of involute profile, the direction of the transmitted load during gear meshing is always along the line of action. This applied load can be decomposed into two perpendicular components, F a and F b as shown in Figure 5.1. F a and F b are the force components acting on tooth profile which are parallel and perpendicular to central line of the tooth, respectively and can be expressed as F a = Fsinθ 1 (5.4) F b = Fcosθ 1 (5.5) The F a component introduces both axial compressive and bending effects while F b component causes bending and shear. 16

125 Figure 5.1 Elastic force on a tooth of gear 17

126 The axial compressive energies stored in a tooth can be expressed as l U a = F a 2 dx (5.6) 2EA x where E is the Young modulus and A x is the area of the section Substitute F a into equation (5.6), and then substitute U a into equation (5.1). The stiffness due to the axial compressive energy yields 1 = sinθ 1 k a EA x l 2 dx (5.7) Correspondingly, the bending potential energy can be obtained as l U b = M x 2 dx 2EI x l = [F b(l x) F a h] 2 dx (5.8) 2EI x where I x represents the area moment of inertia of the section at the distance x from the tooth root. The bending stiffness can be obtained by substituting equation (5.8) into equation (5.2) 18

127 l 1 = [(l x)cosθ 1 hsinθ 1 ] 2 dx k b EI x (5.9) In similar way, the shear potential energy can be expressed as l U s = 1.2F b 2 dx (5.1) 2GA x where G is the shear modulus. Substitute equation (5.1) into equation (5.3), the shear stiffness is as l 1 = 1.2cosθ 1 k s GA x 2 dx (5.11) Hertzian Energy The two mating teeth are considered as two isotropic elastic bodies. According to Hertzain law, the elastic compression of two isotropic elastic bodies can be approximated by two parabolic in the vicinity of the contact. The Hertzian potential energy can be expressed as: U h = F2 2k h (5.12) 19

128 where F represents elastic force between tooth contact and K h represents stiffness of Hertzian contact. The stiffness of Hertzian contact of two meshing teeth is practically a constant along the entire line of action independent to both the position of contact and the depth of interpenetration and can be calculated as: k h = πel 4(1 ν 2 ) (5.13) where E, L, ν represents Young s modulus, tooth width, and Poisson s ratio, respectively The total gear mesh stiffness Finally, the total potential energy for a single pair meshing teeth is obtained by summing axial compressive, bending, shear, and Hertzian energies and can be express as U t = F2 2k t = U h + U a1 + U b1 + U s1 + U a2 + U b2 + U s2 (5.14) where k t represents the total effective meshing stiffness for a gear pair. It is noticeable that all potential energies are function of the stiffness. 11

129 Therefore, the total effective meshing stiffness for a gear pair can be calculated as 1 k t = (5.15) k h k a1 k b1 k s1 k a2 k b2 k s2 5.3 Dynamics of the Gear-Shaft System The equations of motion are developed for simulating the dynamic response of gear-shaft system as shown in Figure 5.2. The equations can be expressed in matrix form as [M]{U } + [C]{U } + [K]{U} = {F b (t)} + {F g (t)} + {F u (t)} (5.16) [M], [C], and [K] are the mass, damping, and stiffness matrices of the system respectively, where {U} is the general displacement vector of the system. {F b (t)}, {F g (t)}, and {F u (t)} are the force vectors acting on the rotor system due to bearing forces, gear mesh interaction, and mass-imbalances, respectively. The gear forces generated by the gear mesh interaction can be written as {F g (t)} = k t {U} (5.17) 111

130 where the vector contains gear forces and moments due to the translational motion of the gear. The transient and steady state dynamics of the system is obtained by solving the coupled equations of motion simultaneously 5.4 Discussion of Results The numerical model of gear-rotor system was developed based on the experimental test rig. The vibration data were obtained from both numerical model and experimental test rig considering non damage and damage cases where the wear surface damage was introduced to one of the gear teeth in defected gear case. Then, the joint time-frequency technique using CWT is applied to vibration signals for identifying and verifying the gear damage in transmission system Numerical Simulation Results The gear model is incorporated in to the model developed in Chapter III. Hence, the entire model is composed of a gear and two identical rotors mounted on a rigid shaft supported by two bearings. In this numerical model, gear has 28 teeth and is located at the left end of the shaft. Therefore, the total reaction force due to the gear mesh is applied only at gear station in the transmission system. The gear-rotor model can be illustrated by 112

131 schematic model in Figure 5.2. The overall dynamic of the system is achieved by solving the global equations of motion (Equation 5.16). In order to examine the sensitivity of the system vibration signal due to the gear tooth surface damage, the gear mesh stiffness of ideal perfect and wear teeth are used in the numerical model. The gear mesh stiffness is simulated by assuming all the gear teeth are in ideal perfect conditions except for damage/wear case. A thin layer material at the leading surface of the driving gear tooth has been removed to resemble the wear profile. Figure 5.3 depicts the change in gear tooth profile due to surface wear. The deviation in gear surface profile due to material removal has affected the change in gear mesh stiffness as shown in Figure 5.4. In this study, the shaft rotating speed was simulated at 1 Hz (6 rpm). Thus, the gear meshing frequency can be calculated by multiplying shaft rotating speed with the number of teeth, obtaining 28 Hz. 113

132 Figure 5.2 Schematic of Gear-Shaft System for Numerical Simulation 114

133 Figure 5.3 Schematic of the Change in Gear Tooth Surface Profile 115

134 Mesh stiffness (lbs/in) 2.4 x 15 no damage damage due to surface wear 1 14 Distance along line of action in term of angle (degree) Figure 5.4 Numerically Simulated Change in Gear Mesh Stiffness Due to Wear in Gear Tooth Profile 116

135 The vibration signatures generated by the numerical results of the gear transmission system excited by two sets of gear show in Figures 5.5 and 5.6. These figure results are obtained using the CWT analysis with simulated time domain signal. For each case, the time domain signal is plotted on the left hand side from to 36 representing one shaft revolution. The time frequency representation is displayed on the right hand side to the time domain signal plot with the color scale representing the magnitude of the transform coefficients of CWT. The normalized frequency spectrum calculated using the FFT is given at the bottom where the shaft rotating speed and the gear meshing frequency can be observed at 9.8 and 283 Hz respectively. In the case with healthy gear, vibration components dominated by gear meshing frequency were found in the time domain signals as well as in the CWT shown in Figure 5.5, where no significant pattern of damage can be found. In the case with damaged gear as shown in Figure 5.6, an oscillation in vibration occurring from to 36 can be observed from the time domain signal. This fluctuation also induces an overall amplitude increase in vibration signal where it can be noticed at the same location by the darker color region in the CWT. Moreover, this darkened area in CWT, more energy intensity area, occurs at same location where an amplitude increase occurs in frequency spectrum which is correlated with shaft rotating speed. Based on the fact that a pair of gear teeth is engaged one time per shaft revolution for 1:1 gear contact ratio, it was initially concluded that this increase can be resulted from the damage of a gear tooth. If one observes closely darkened area in the time-frequency representation of the gear damage case, it can be found that two zero energy spots occur in the darkened area during one shaft revolution. 117

136 These two locations are found in time domain signal where vibration signal intercepts the horizontal axis or has zero amplitude. It was found that the second intercepted point is located at around 282 from the reference point of where this is associated with the damage of gear tooth angle on the shaft location. Thus, the intercepted point where the sign of vibration signal changes from negative to positive values can be used as an indicator for the damage location. 118

137 36 CWT of good gear FFT Hz Figure 5.5 CWT of Vibration Signal with No Damaged Gear Tooth 119

138 36 CWT of damaged gear FFT Hz Figure 5.6 CWT of Vibration Signal with Single Damaged Gear Tooth 12

139 5.4.2 Experimental Results In order to study the gear damage effect on the vibration signatures of the system, the healthy and gear damage experiments were performed on the test rig shown in Figure 5.7. Under the similar operating conditions, the shaft test rig was driven by a variable speed electric motor on the left hand side through a set of gears where the shaft of the test rig was properly aligned with the shaft of the motor in its vertical plane. The motor operating speed was set at 1 Hz (6 rpm). The identical spur gears with 28 teeth were used as driving and driven gears. Therefore, the gear tooth passing frequency is equal to 28 Hz (168 rpm). The vibration data were acquired through a set of accelerometers mounted on bearing housings in both horizontal and vertical directions at 1, Hz sampling rate. The shaft speed was also measured by the optical encoder. Approximately 98 revolutions were acquired each time and stored in computer for further vibration signature analysis. The preset gear damage was produced by removing a material layer from the gear tooth surface. The gear damage was applied to the leading side of the driving gear tooth while the driven gear remains intact. Figure 5.8 shows the gear with no damage and Figure 5.9 shows the gear with single tooth damage. Under same operating condition in numerical approach, the gear meshing frequency is obtained as 28 Hz. 121

140 Figure 5.7 Photograph of Gear Test Rig 122

141 Figure 5.8 Photograph of Gear with No Damaged Tooth 123

142 Figure 5.9 Photograph of Gear with Single Damaged Tooth 124

143 The vibration signatures obtained from filtered experimental data were analyzed using the CWT where the Morlet wavelet was chosen for CWT analysis shown in Figures 5.1 and The results were arranged in the same format to that of the numerical analysis. The time signal representation is plotted on the left hand side from to 36 representing one shaft revolution from the point of triggering. The joint time-frequency representation is displayed on the right hand side to the time signal representation plot. The color scale depicts the magnitude of the coefficients of the CWT. Also, the normalized frequency spectrum calculated using FFT is given below the CWT plot. In healthy gear case as shown in Figure 5.1, there was no noticeable pattern displaying in the CWT display. In damaged gear case, a similar trend as numerical result can be observed where the vibration signature oscillates through a shaft rotating cycle in time domain signal. This fluctuation also makes the vibration amplitudes higher where it can be noticed as a darkened portion at the low frequency and a horizontal wide frequency band around the high frequency on CWT display. In the frequency representation, it was observed some moderate increase in energy levels at the gear meshing frequency component while a very substantial increase can be detected at the shaft rotating component. The gear damage location, the intercepted point with the sign change from negative to positive values, can be found at around 284 from the reference point of in the time domain signal. 125

144 36 CWT of good gear FFT Hz Figure 5.1 CWT of Vibration Signal with No Damaged Gear Tooth 126

145 36 CWT of damaged gear FFT Hz Figure 5.11 CWT of Vibration Signal with Single Damaged Gear Tooth 127

146 5.5 Conclusions A numerical procedure was developed to simulate the dynamics of gear transmission system with the effect of gear tooth damage due to profile deviation. The numerical results are validated with data acquired from the experimental test rig. The study examined the vibration signal from both simulation and experiment using joint timefrequency method through Continuous Wavelet Transform (CWT). Based on results of this study, some specific conclusions can be summarized as follows: 1. The numerical model was developed to simulate the dynamic of gear-rotorbearing system with effect of gear tooth damage. Gear tooth damage due to wear can be simulated by amplitude change in the gear mesh stiffness model and then the force due to gear mesh can be easily incorporated into the global transmission system for dynamic predictions. 2. The use of joint time-frequency method using CWT is very useful to examine and characterize the fault in the vibration signal of the gear system. It can provide diagnostic information on the damage, such as the location and the severity of the damage on gear. 3. The numerically simulated model can provide an accurate dynamic representation of the results from the experimental gear test rig without the complications due to the various operational environments. 128

147 CHAPTER VI IDENTIFICATION OF DAMAGE IN GEAR-ROTOR-BEARING ELEMENT IN A TRANSMISSION SYSTEM 6.1 Objective The objective of this chapter was to develop numerical model to simulate vibration signatures included combined localized damage components among bearing, shaft and gear and to analyze and classify the vibration results. The numerical results were also compared with those of experimental. Results from both numerical simulation and experimental test rig were examined by the vibration analysis techniques in time domain, frequency domain, and a joint time-frequency domain using Continuous Wavelet Transform (CWT). 6.2 Method of Approach In order to study overall vibration signature analysis of the system due to combined defect in bearing, shaft, and gear, five different damage scenarios were performed for both numerical simulation and experiment: 129

148 (a) The undamaged gear, shaft, and bearing (GG_GS_GB) (b) The undamaged gear with damaged shaft and bearing (GG_BS_BB) (c) The damaged gear and bearing with undamaged shaft (BG_GS_BB) (d) The damaged gear and shaft with undamaged bearing (BG_BS_GB) (e) The damaged gear, shaft, and bearing (BG_BS_BB) Numerical Dynamics of Gear-rotor-bearing System The dynamic response of the gear-rotor-bearing system can be evaluated through the equations of motion where it can be expressed in matrix form as [M]{U } + [C]{U } + [K]{U} = {F b (t)} + {F d (t)} + {F g (t)} (6.1) [M], [C], and [K] are the mass, damping, and stiffness matrices of the system respectively, where {U} is the general displacement vector of the system. The transient vectors, {F b (t)}, {F d (t)}, and {F g (t)}, are the force vectors acting on the rotor system due to the bearing forces, shaft bow effects, and gear mesh interactions, respectively. The bearing forces {F b (t)} can be evaluated as N b {F b (t)} = K λ j δ j 1.5 j=1 (6.2) 13

149 where K is the bearing effective elastic modulus, λ j is the loading zone parameter, δ j is the contact deformation of the j th rolling element, and N b is the number of rolling elements in bearing. The shaft bow effect {F d (t)} can be expressed as {F d (t)} = [K] s {U} d (6.3) where [K] s is the shaft stiffness matrix and {U} d is the vector of mechanical bow shaft. The gear forces {F g (t)} can be written as {F g (t)} = [K] t {U} (6.4) where [K] t is the total effective meshing stiffness for a gear pair. The transient and steady state response of the system is obtained by solving the set of equations of motion simultaneously Experimental Descriptions In order to perform the investigation of the effects due to combined damage in bearing, shaft, and gear on the vibration signatures of the system, vibration data were acquired using gear-rotor-bearing test rig shown in Figure 6.1. The test rig was composed of two identical rotors mounted on the shaft. Fafnir bearings with 11-ball rolling elements 131

150 with an OD of inches, an ID of 1.7 inches, a length of.563 inches, and ball diameter of.312 inches were used for supporting the test rig systems. The shaft test rig was driven by a variable speed electric motor on the left hand side through a set of gears. The test rig shaft was properly aligned with the output shaft from motor in the vertical plane. The motor was set for operating speed of 1 Hz (6 rpm). The driving and driven gears are identical spur gear with 28 teeth. The reflective was affixed onto the shaft and the rotating speed was measured by optical encoders. Accelerometers were mounted on bearing housings to acquire vibration data in horizontal and vertical plane and 5 mm noncontacting proximitor sensors was mounted at mid span between two rotor disks to measure shaft displacement in vertical plane. The experimental data were acquired and stored through computer-based high-speed analog-to-digital data acquisition system at sampling rate of 1, Hz Component Damaged Condition The damage types considered for each component were inner race surface damage in bearing, mechanical bow in shaft, and surface tooth wear in gear. In order to provide damage in the experiment, the surface damage was induced on the inner race surface of the rolling element bearing for approximately.5 inch wide and.4 inch deep (bearing#3). The bearing damage was applied only on the right hand side bearing while on the left always kept intact. Mechanical bent force was applied to the shaft to induce bow profile with the maximum 8 mils eccentric located at the shaft central. The preset 132

151 gear damage was produced by removing material from the gear tooth surface on the leading side of driving gear while the rest of the gears remain intact. In similar manner, the damage conditions in experimental components were also applied to the numerical model. During the test, the shaft was running at 1 Hz (6 rpm). The relative ball pass on inner race surface was at 68 Hz, resulted from multiplication of number of rolling elements in bearing and relative speed of shaft operating speed to ball carrier speed. The gear tooth passing frequency for 28 teeth gear was obtained as 28 Hz from multiplying shaft operating speed with number of teeth. 6.3 Discussion of Numerical Results The numerical models were established for simulating dynamic response of all five scenarios (Case (a) to (e)) with similar operating conditions. The vibration signatures were measured at the right bearing station and the shaft displacement data were drawn at the mid span. In order to clarify type of damage, the vibration analysis technique was suitably chosen to examine simulating response data under combined damage effects. The results will be first analyzed the effect due to bearing damage, the later section discuss the results under the effect of shaft damage, and last section discuss the effect caused by gear damage. The vibration results in x- and y-direction of all cases are shown in Figures 6.2 to From these figures, it is noticed that vibration amplitude increased when there is 133

152 presence of damage. Although a substantial increase in vibration amplitude is detected, there is no definite hint to identify damage in the system Bearing Component Analysis In order to investigate the damage on inner race bearing, the vibration signals were transformed to frequency spectra domain by using Fast Fourier Transform method (FFT). Figures 6.12 to 6.16 depict the frequency spectra of absolute vibration signatures from each case. From overall frequency spectrum results, the major frequency components occur at 2 Hz which are corresponding to the 2 nd harmonic of operating speed. The other dominant components shown at around 28 Hz are due to gear tooth passing frequency. In Case (b), (c), and (e), it was observed that the additional components emerged at frequency of 68 Hz shown in Figures 6.13, 6.14, and The appearances at this frequency were the results of impact force where each of rolling elements passed over surface damaged area. This frequency is also the same as relative frequency ball rolling element pass on inner race surface. However, frequency components of 68 Hz are invisible in Case (a) and (d) where bearings were assumed to be healthy. 134

153 6.3.2 Shaft Component Analysis For identifying the vibration due to bow shaft effect, the displacement signals at the mid-span of shaft were used. The displacements in both x- and y-direction are shown in Figure 6.17 to 6.26 for Case (a) to (e), respectively. From these time signal plots, sinewaves with small amplitude can be observed in Case (a) and (c) while displacement amplitudes increased distinctively in Case (b), (d), and (e) where they were included residual bow shaft effects. Furthermore, the frequency spectra domain was used for verifying the excitation source of the amplitude increase in time domain. Figure 6.27 to 6.36 depict the frequency spectra of x- and y-direction displacement from Case (a) to (e). From overview, a major frequency response at 1 Hz can be observed in both directions of every case where this excitation is corresponding to operating speed. For the systems with straight shaft shown in Figures 6.27 and 6.28 for Case (a) and in Figures 6.31 and 6.32 for Case (c), the amplitudes at 1 Hz are practically small. These amplitudes substantially increased under the bow shaft effect shown in Figures 6.29 and 6.3 for Case (b), in Figures 6.33 and 6.34 for Case (d), and in Figures 6.35 and 6.36 for Case (e). Thus, the amplitude increment at 1 Hz provides definite identification of damage of shaft in the system. 135

154 6.3.3 Gear Component Analysis In this section, the vibration signatures were examined by using the CWT where the Morlet wavelet was chosen for CWT analysis. The results of all cases are shown in Figures 6.37 to For results of each case, the time domain signal is plotted on the left hand side from to 36 representing one shaft revolution. The time-frequency representation is displayed on the right hand side to the time domain signal plot with the color scale representing the magnitude of the transform coefficients of CWT. The normalized frequency spectrum calculated using the FFT is given at the bottom where the shaft rotating speed and the gear meshing frequency can be observed at 1 and 28 Hz respectively. In Case (a), as shown in Figure 6.37, time vibration signature plotted on the left side are even wave functions resulted from a major component of gear tooth pass frequency at 28 Hz. There was no remarkable pattern observing in the CWT display. In Case (b), as shown in Figure 6.38, the vibration pattern has substantially changed where the amplitude has increased. The vibration signature oscillated during a shaft rotating revolution where the x-intercepts point was around 18. The increase in time signal amplitude can be observed as the darker region at the lower frequency on the CWT display where it coincides with amplitude increase at 1 Hz in frequency domain. In Case (c), three peaks observed in the time domain signal was also noticeable as darker color interrupting by two zero energy spots in CWT shown in Figure Also, a horizontal wide frequency band can be seen at higher frequency which is correlated to gear meshing 136

155 frequency. The darkened areas in CWT were endorsed by amplitude increases of two component spectra in frequency domain. In Case (d) and (e), as shown in Figures 6.4 and 6.41, both time domain signals behaved similar to result from undamaged gear with damaged shaft and bearing in Case (b). Also, both feature from Case (b) at the low frequency and wide band from Case (c) at high frequency can be seen on CWT displays. However, pattern of gear damage as in Case (c) cannot be seen on CWT display. With closely examination, both shaft rotating frequency and gear tooth passing frequency in Case (d) and (e) were shown as more dominant compared to other three cases where energy levels in Case (d) were slightly higher than in Case (e). Based on above observations, it can be primarily concluded that the effect due to damage in shaft has more influence than the one in gear. In order to illustrate the damaged effect between bow shaft and defect gear in Case (d) and (e), Figure 6.42 presents the numerical comparisons of both shaft operating and gear tooth pass frequency components extracted from average frequency of CWT for each case normalized with respect to no damage scenario of Case (a). Note that Figure 6.42 provides a better representation of the amplification and relationships of the vibration energy levels between both shaft operating and gear tooth passing frequency components due to the shaft and gear tooth damages. It is noticed that under combined damage effect of shaft and gear a very substantial increase in energy levels can be detected at the shaft operating frequency which is related to both shaft and gear vibrations, while moderate increase in energy levels can be observed at gear tooth pass frequency which related to gear vibrations. Furthermore, the numerical results of shaft 137

156 operating frequency components were extracted from different damaged cases namely: single damage in shaft, single damage in gear, combined damage in shaft and gear in both healthy and damaged bearing. In similar manner by normalizing results of each damaged case with respect to Case (a) of no damage, the comparisons of the results of shaft operating frequency were obtained as shown in Figure 6.43 In the cases of healthy bearing, an increase in vibration energy level due to single defect in shaft is higher than the one due to single defect in gear while a very substantial increase can be observed under the combined defect in both shaft and gear (Case (d)). A similar trend can also be noticed for the cases of bearing damage where the higher energy level increases was found in shaft damaged respected to gear damage while the combined damage in both shaft and gear has a very large increase of energy level. 138

157 Figure 6.1 Photograph of Gear Test Rig 139