1、 STD.AGMA 94FTM7-ENGL 1994 b87575 0004b14 205 = 94FTM9 1 Analytical and Experimental Vibration Analvsis of a 4 I Damaged Gear by: E Choy, M. Braun, and V. Polyshchuk University of Akron; J. Zakrajsek,and D. Townsend, Lewis Research Center, NASA; and R. Handschuh, US Army Research Center TECHNICAL PA
2、PER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGMA SLIFTMS-ENGL 1794 0687575 0004635 141 W Analytical and Experimental Vibration Analysis of a Damaged Gear F. Choy, M. Braun, and V. Polyshchuk, University of Akron; J. Zakrajsek, and D. Townse
3、nd Lewis Research Center, NASA; and R. Handschuh, US Army Research Center The statements and opinions contained herein are those of the author and should not be construedas an officiai action or opinion of the American Gear Manufacturers Association. ABSTRACT: A comprehensive analytical procedure wa
4、s developed for predicting faults in gear transmission systems under normal operating conditions. A gear tooth fault model is developed to simulate the effects of pitting and wear on the vibration signal under normal operating conditions. The model uses changes in the gear mesh stiffness to simulate
5、 the effects of gear tooth faults. The overaii dynamics of the gear transmission system is evaluated by coupling the dynamics of each individual gear-rotor system through gear mesh forces generatedbetween each gear-rotor system and the bearing forces generated between the rotor and the gearbox struc
6、ture. The predicted results were compared with experimental results obtained from a spiral bevel gear fatigue test rig at NASA Lewis Research Center. The Wigner-Vile Distribution (WVD) was used to give a comprehensive comparison of the predicted and experimental results. The WVD method applied to th
7、e experimental results were also compared toother fault detection techniques to verify the WVDs ability to detect the pitting damage, and to determine its relative perfomce. Overail results show good correlation between the experimental vibration data of the damaged test gear and the predicted vibra
8、tion from the model with simulated gear tooth pitting damage. Results also verified that the WVD method can successfully detect and locate gear tooth wear and pitting damage. Copyright O 1994 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 223 14 October, 19
9、94 ISBN 1-55589444-8 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STDSAGMA 74FTM7-ENGL A974 W 0b87575 0004bLb O88 W ANALYTICAL AND EXPERIMENTAL VIBRATION ANALYSIS OF A DAMAGED GEAR F. K. Choy, Professor M. J. Braun, Professor V. Polyshchuk, Researc
10、h Associate The University of Akron, Akron, Ohio 44325 J. J. Zakrajsek, Aerospace Engineer D. P. Townsend, Senior Research Engineer NASA Lewis Research Center, Cleveland, Ohio 44135 R. F. Handschuh, US Army Research Laboratory, Cleveland, Ohio 44135 - I. INTRODUCTION In the last two decades, the use
11、 of gear transmissions in both defense and commercial applications has substantially increased. With the demand for higher power and performance, premature failures in transmissions often result in financial losses, and sometimes even lead to catastrophic consequences. In the aerospace industry, one
12、 of the major concerns is with gear fatigue failures in rotorcraft transmission systems. Large vibrations in gear transmission systems usually result in excessive gear tooth wear and possible tooth crack formation which, in turn, leads to premature gear failure. Thus, it is important to understand t
13、he dynamics of a transmission system over a variety of fault conditions, as well as under nominal conditions. With this, methods can be explored to detect and assess the magnitude of the gear damage present. Due to limitations in the number and types of experiments that can be performed, the only pr
14、actical means of obtaining this type of data is through analytical simulations. The major objective of the research reported herein is to develop and verify a model to predict the vibration of a transmission system with the effects of gear surface pitting and wear. To simulate the vibration of the t
15、ransmission system, the equations of motion were established individually for each rotor-gear-bearing system. The effects of tooth wear or surface pitting are simulated by changes in the magnitude and phase of the mesh stiffness. These localized changes in the gear mesh are incorporated into each ge
16、ar-rotor model for dynamic simulationl-31. The dynamics of each gear-rotor system are coupled with each other through the gear mesh interacting forces and the bearing support forces. The global vibrations of the system are evaluated by solving the transient dynamics of each rotor system simultaneous
17、ly with the vibration of the casing. In order to minimize the computational effort, the number of degrees- of-freedom of the system are reduced by using a modal synthesis procedurei,2. To verify the analytical model with experimental data, the dynamics of a single spiral bevel pinion with various de
18、grees of gear tooth damage was simulated. Results from the model were compared to experimental results using a joint time-frequency analysis method. This approach was chosen because of the large amount of information represented in the joint time-frequency results which can not be represented separa
19、tely in either the time domain or the frequency domain. The joint time-frequency analysis will provide an instantaneous frequency spectrum of the system at every instant of the revolution of the pinion while a Fourier Transform can only provide the average vibration spectrum of the signal obtained d
20、uring one complete revolution. In other words, the time-changing spectral density from the joint time-frequency spectra will provide information concerning the frequency distribution concentrated at, that instant around the excited instantaneous frequency which cannot be obtained in a regular vibrat
21、ion frequency spectrum. The joint time- frequency analysis approach applies the Wigner-Ville Distribution (WVD) 4-6 on the time vibration signai of the system. Some success has been achieved in applying the WVD to gear transmission systems 7,8 to recognize faults at various locations of the gear. Ot
22、her fault detection techniques, including frequency domain analysis9,10 and several time discriminant methods such as: FM411, NA4*, and NB4*12-141 are also used to compare with and verify with the WVD approach. Based on results of this study, some conclusions are made on the ability of the developed
23、 model to simulate gear tooth surface damage, and the ability of the WVD method to detect damage sufficient to verify with the model. - II. ANALYTICAL PROCEDURE The dynamics of the ith individual gear-shaft system can be evaluated through the equations of motion for the vibrations of a individual ro
24、tor-bearing-gear system as shown in Figure 11,2, given in matrix form, as COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Serviceswhere Mj and KSI are respectively the mass and shaft stiffness matrices of the rotor, W; is the general displacement vector of the
25、 ith rotor in its local coordinate system, and, Fbi(t), Fgi(t), and F,(t) are respectively the force vectors acting on the ith rotor system due to bearing forces, gear mesh interactions, and mass-imbalances. In this model, the dynamics between the gearbox and the rotor are coupled through the bearin
26、g forces, which are evaluated by the relative motion between the rotor and the gearbox. The interactions between each individual rotor are coupled through the gear forces generated by the relative motion of the two mating gears at the mesh point. The equations of motion of the gearbox with p rotor s
27、ystems can be expressed as P i=l fMCI WC -I- KC wC) = fTciI Fbi(t)I (2) where Tci represents the coordinate transformation between the ith rotor and the gearbox. The bearing forces Fbi(t) for the ith rotor can be evaluated as (3) where chi and K . are res ectively the damping and stiffness of the ba
28、ring, ITic is the coordinate transformation matrices for the gearbox with respect to the ith rotor, and Wci is the casing displacements at the rotor locations. The gear forces generated from the gear mesh interaction(l,2 can be written as (4) where Fn(t) is the vector containing the gear forces and
29、moments resulting from the relative rotation between the two mating gears and Fti(t) is the vector containing gear forces and moments due to the translational motion between the two gearsl5,16. In order to calculate the transient/steady state dynamics of the system, all the coupled rotor and casing
30、equations of motion have to be solved simultaneously. To minimize the computational effort, the modal transformation 1,2 procedure was applied to reduce the the undamped mode shapes of the rotor system and the undamped mode shapes of the gearbox, the rotor and the gearbox displacements were transfor
31、med into the modal coordinates to reduce the number of degrees of freedom in the system to minimize the computational effort. The modal equations of motion for the rotor systems and the gearbox vibration were solved simultaneously for the modal accelerations. A numericai integration scheme was used
32、to integrate the accelerations to obtain velocities and displacements at each time step for transient calculations. ULEXPERIMENTALSTUDY degrees of free d om of the global equations of motion. Using Using the spiral bevel gear fatigue test rig illustrated in Figure 2, the resulting fatigue damage on
33、the pinion is illustrated in Figures 3 and 4. The primary purpose of this rig is to study the effects of gear tooth design, gear materials, and lubrication types on the fatigue strength of aircraft quality gears 17. Because spiral bevel gears are used extensively in helicopter transmissions to trans
34、fer power between nonparallel intersecting shafts, the use of this fatigue rig for diagnostic studies is practical. Vibration data from an accelerometer mounted on the pinion shaft bearing housing was captured using a personal computer with an analog to digital conversion board and antialiasing filt
35、er. The 12 tooth test pinion, and the 36 tooth gear have: 0.5141 in diametrai pitch, 35 degree spiral angle, 1 in. face width, 90 degree shaft angle, and 22.5 degree pressure angle. The pinion transmits 720 hp at nominal speed of 14,400 rpm. The test rig was stopped several times for gear damage ins
36、pection. The test was ended at 17.79 operational hours when a. broken portion of a tooth was found during one of the shutdowns. Three major methodologies: A) the joint time- frequency approach, B) the frequency domain approach, and C) the time domain techniques, were used in this study. The followin
37、g is a description of the three methodologies: (A) Joint Time-Frequency Technique To examine the vibration signal in a joint time- frequency domain, the Wigner-Ville method 4,5 was used in this study. While the Fast Fourier Transform (FFT) technique can provide the spectral contents of the time sign
38、al, it cannot distinguish time phase change during a complete cycle of operation. In other words, it assumes that the time signals are repeatable for each time data acquisition window without considering the effects of any magnitude and phase changes during the sampling period. The Wigner-Ville dist
39、ribution will provide an interactive relationship between time and frequency during the period of the time data window. The comprehensive representation of the vibration signal using the WVD method is the primary reason that it was used to compare the predicted and experimental vibration results. Th
40、e WVD (Wigner-Ville Distribution), in a discrete .form, can be written as: WX (nT, f) = 2T Ozguven, H.N.; Houser, D.R.; and Zakrajsek, J.J.;“Dynamic Analysis of Geared Rotors by Finite Element“, NASA TM-102349, AVSCOM-TM-89-C- 006, 1990. 4. Boashash, B. and Black, P.J. “An efficient Real Time Implem
41、entation of the WignerViile Distribution“ IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. ASSP35, NO. 11, November 1987. 5. Claasen, T.A.C.M., and Mecklenbniker, W.F.G , “The Wigner Distribution A tool for TimeFrequency Signal Analysis,“ Part I Philip J. Res. 35, 1980. 6. Shin, YS. and
42、 Jeon, J.J. “Pseudo WignerVille TimeFrequency Distribution and its Application to Machinery Condition Monitoring“, J. of Shock and Vibration, Vol. 1, Issue 1, 1993/1994, pp. 65. 7. Forrester, B.D., “Analysis of Gear Vibration in the TimeFrequency Domain,“ Proc. of the 44th Meeting of the Mechanical
43、Failure Prevenention Group. Feb. 1990. 8. Mcfadden, P.D. and Wang, W.J. “TimeFrequency Domain Analysis of Vibration Signal for Machinery Diagnostics (II) the Weighted WignerVille Distribution“, University of Oxford, Report No. OUEL 1891, 1991. 9. Randall, R.B., “A new Method of Modeling Gear Faults,
44、“ Journal of Mechanical Design, Apr. 1982, Vol. 1041259. 10. Taylor, J.I., “Fault Diagnosis of Gears Using Spectrum Analysis,“ Second International Conference on Vibrations in Rotating Machines, Cambridge, Sept. 24, 1980, I. Mech. E., London, pp. 163168. 11. Stewart, R.M., “Some Useful Data Analysis
45、 Techniques for Gearbox Diagnostics,“ Institute of Sound and Vibration Research, Paper MHM/R/10/77, 1977. 12. Zakrajsek, J.J., Towsend, D.P. and Decker, H.J. “An Analysis of Gear Fault Detection Methods as Applied to Pitting Fatigue Failure Data“, NASA TM105950, presented at the 47th Mechanical Fail
46、ure Prevention Group Meeting, Virginia Beach, Virginia, April 1315, 1993. 13. Zakrajsek, J.J., Handschuh, R.F., and Decker, H.J. “Application of Fault Detection Techniques to Spiral Bevel Gear Fatique Data“, NASA TM106467, presented in the 48th Mechanical Failures Prevention Group Meeting, Wakefield
47、, Massachusetts, April 1921, 1994. 14. Decker, H.J., Handschuh, R.F., and Zakrajsek, J.J., “An Enhancement to the NAU Gear Vibration Diagnostic Parameter,“ NASA TM106553, presented at the 18th Annual Meeting of the Vibration Institute, Hershey, PA, June 2023, 1994. 15. Choy, F.K.; Townsend, D.P.; an
48、d Oswald, F.B., “Dynamic Analysis of Multimesh-Gear Helicopter Transmissions, NASA TP-2789, 1988. 16. Boyd, L.S., and Pike, J.A., “Epicycle Gear Dynamics“, AIAA Journal, vo1.27, No.5, May 1989. 17. Handschuh, R., “Effect of Lubricant Jet Location on Spiral Bevel Gear Operating Temperatures,“ NASA TM
49、105656, AUSCOM TR91C033, presented at the 6th International Power Transmission and Gearing Conference, ASME Sept. 1316, 1992, Phoenix, AZ. ft Beating forces Bearing forces Gear forces Figure 1.-Schematic of a rotor-gear bearing system. Figure 2.-Spiral bevel gear test rig at NASA Lewis Research center. COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesFigure 3.-Picturea of the damaged pinion teeth. (a) 5.5 hours. (b) 6.55 hours. (c) 8.55 hours. (d) 10.03 hours Figure 4.-Pictures of the
