In this paper, the dynamics modeling of the gearbox ... mechanical gearbox power transmission components. It consists .... The finite element software ANSYS. ©.
Begg, C. D.; Byington, C. S.; and Maynard, K. P, Dynamic Simulation of Mechanical Fault Transition, Proceedings of the 54th Meeting of the Society for Machinery Failure Prevention Technology, Virginia Beach, VA, May 1-4, 2000, p. 203-212.
DYNAMIC SIMULATION OF MECHANICAL FAULT TRANSITION Colin D. Begg, Carl S. Byington, and Kenneth P. Maynard The Pennsylvania State University, Applied Research Laboratory Condition-Based Maintenance Department University Park, Pennsylvania 16804 Abstract: Over the past few decades many vibration and signal analysis techniques have been investigated, developed, and used to provide operational response information about mechanical power transmission systems for monitoring and diagnosis of components and their worn/faulted condition. The presence of multiple sources of excitation and forcing in a complex distributed mechanical structure (rotor transmission shafting, gearing, bearings, casing, and foundation) presents problems for algorithms that are designed to present a single feature associated with a single fault mechanism. The consequence of the mechanical structures dynamic interaction is not always clear but an understanding of it is critical for the successful development of effective integral signal processing algorithms and automated reasoning components used in CBM systems. As part of a combined experimental-theoretical analysis effort, ARL is investigating mechanical fault evolution in damaged rotating components. A dynamics model of the system using component fault models was developed for response simulations. Comparison of experimentally measured and simulated results of system vibratory responses allow physical insights into vibratory measurement sensor placement and specification, dynamic system response to a fault, and the development of fault detection signal processing algorithms. In this paper, the dynamics modeling of the gearbox rotor/bearing- foundation system using the Finite Element method is outlined and it’s relevance to diagnostics and prognostics is highlighted. Key Words: Condition-Based Maintenance; diagnostic features; dynamic systems modeling; mechanical fault simulation; model-based diagnosis; prognostics. Condition-Based Maintenance: CBM has been driven by the demand to increase system efficiency through elimination of unnecessary maintenance in a system. Numerous authors have highlighted the cost and safety benefits of CBM.[1][2] This approach to maintenance relies on monitoring the condition of a system in order to detect anomalies and on the ability to diagnose the health of critical components. An ultimate goal is to develop a prognosis or prediction of Remaining Useful Life (RUL) with an associated functional impact assessment so that appropriate maintenance can be scheduled. The maturation of technologies in the areas of: measurement sensors, signal processing theory, digital processing hardware, dynamic system simulation, multi- sensor data analysis, and approximate reasoning are making CBM possible. The fault detection phase involves comparing historical and nominal values for a statistically significant change. During the diagnostics process, specific fault recognition measures (figures of merit) are typically compared to threshold limits. Additional processing may determine a signature pattern in one, or multiple, fault measure(s). Automated reasoning is used to identify the fault type (cracked shaft or gear tooth, bearing spall), location, affected component, and severity. Prognosis builds upon the
diagnostic assessment with a tracked parameter that is related to damage and a future damage state prediction. These diagnostic and prognostic analyses can be based on either extensive statistical experimental data with an associated empirical model of the particular system, or from an estimate made using predictions from a detailed systems dynamics model, or from a combination of both. The current work investigates the development of a dynamics model that is useful in this vein. MDTB and Transitional Failures: The Mechanical Diagnostics Test Bed (MDTB) [1] was built as an experimental research station for the study of fault evolution [2] in mechanical gearbox power transmission components. It consists of a motor, gearbox, and generator on a steel platform. Gearboxes are instrumented with accelerometers, thermocouples, acoustic emission sensors, and oil debris sensors, and tests are run at various load profiles while logging measurement signals for later analysis. The test gearbox is mounted on a pedestal structure and is driven by a 30 HP variable speed AC motor through a torque cell shaft via gear couplings. A torsion load is supplied to the gearbox by a 75 HP AC (absorption) motor connected through an output torque cell, in the same manner as the input. The MDTB is shown in Figure 1.
Figure 1. Penn State ARL Mechanical Diagnostics Test Bed
a. Helical Reduction Gear
b. Overhung Pinion Shafting
Figure 2. Examples of Component Material Detachment and Structural Dislocation
Reduction gears, overhung geared shafting, and roller element bearings are typically the first components to experience damage due to wear from operational loads. As examples, Figure 2 shows, a) a gearbox helical reduction gear with partially missing and damaged teeth, and b) a totally dislocated overhung pinion shaft, both of which came from gearbox accelerated fault-to-failures that were induced by controlled overloading. From dynamics and fracture mechanics it is well known that accelerated crack nucleation and micro-crack formation in components can occur due to start-ups and shutdowns, transient load swings, higher than expected intermittent loads, or defective component materials. More commonly, normal wear causes configuration changes (loose fit of assembled parts, work hardened surfaces, and reduced structural section areas) that contribute to dynamic loading conditions. High cycle dynamic loads cause micro-crack incubation [3] and formation at material grain boundaries in stress concentrated regions (especially between hardened surfaces and softer subsurface material interfaces, and at acute changes in component material geometry). The majority of crack growth evolves in a sub-critical propagation process of crack tip blunting, unstable crack formation, and crack elongation. As super-critical loading in the cracked material region is approached, growth accelerates and material dislocation and detachment results. Sub-critical crack evolution is highly dependent on a component’s material, geometry, loading conditions, and the particulars of the unique component crack growth cycle. During the evolution, the opportunity to take corrective or compensatory actions exists if we are able to detect the existence of the fault, isolate it to the specific component, and assess its severity. Model-Based Methods and Considerations: The development of model-based prognostic capability for CBM requires a proven methodology to create and validate physical models that capture the dynamic response of the system under normal and faulted conditions. For a majority of systems, operational demands induce a slow evolution in material property and/or component configuration changes. The potential thus exists to track the fault through the filter of the system’s behavior via its dynamic (vibratory) response. Validation through comparison and correlation between simulated and experimental responses of the system, with and without a specified fault, could also facilitate model refinement. Figure 3 illustrates some of the components of a model-based diagnosis. System Control
Component Fault Diagnosis
Specific System Dynamics Model Component Wear and Fracture Models
Operational Conditions Monitoring
Component Loads
Damage Initiation and Progression
Figure 3. Elements of Model Based Diagnosis
A consideration that differentiates the modeling of the MDTB from more common rotordynamic systems is the fact that the rotor system contains a pair of meshing gears. One of the most powerful and popular tools for modeling a rotordynamic system has been the finite element method (FEM).[4] Gearbox dynamics problems differentiate themselves from other structural dynamic systems by the branching of transmitted power through a gear mesh that leads to parametric excitation. Some common practices have been established in dynamic modeling of geared rotor power transmission systems with full- face width hub gearing. [5-8] The base rotor hub is treated as a rigid disk with gear tooth contact, body, and root deflections lumped together to represent a dependent function of both pinion and gear rigid rotational motion. The dynamic response between gear pairs can be treated as a transmission error [9] or by defining the dynamic forces using effective gear tooth deflection forces and apparent variable stiffness [10]. The latter more accurately characterizes a system in terms of effective parameters for dynamic system analysis. Methods also exist to improve accuracy in a general FEM representation of an actual system [11-14] used for response predictions, and in the lumped parameter estimates of rotor/bearing [15-17], gear teeth elements [18][19], and shaft coupling misalignment.[20] A procedure has been developed for composite modeling of gearbox systems for response simulation.[21] A full discussion is beyond the scope of this paper, but the references are listed for the interested reader to pursue greater detail. It is recognized that there are many research activities that need to continue before the entire gap is bridged between dynamics modeling of lumped parameter structural (finite element model) and in situ micro- mechanical systems. The authors believe though, that validated, theoretical fault models could glean some practical insights into bridging this gap and provide a path towards the development of reliable dynamic fatigue models. MDTB Dynamic Model: The topology of the MDTB mechanical structure is shown in Figure 4. The rotor system finite element model of the MDTB is made up of five subsystems: 1) drive motor, 2) torque transducer at gearbox input, 3) single reduction helical gearbox, 4) torque transducer at gearbox output, and 5) load motor. The subsystems are linked with 3 gear and 1 chain couplings, which are modeled using lumped mass polar moments of inertia and elastic gear tooth mesh compliance. The system rotor model is comprised of 36 structural finite elements and 38 nodal points. The structural finite elements include: rotational axisymmetric, axial translational, and 2D bending type elements for circular shafts.[22] A translational spring (representing gear mesh tooth stiffness) is incorporated in to a rigid hub/elastic tooth gearbox pinion and gear coupling matrix.[23] The nodal points include: 16 single degrees-of-freedom axisymmetric rotational nodes at rotary torsional element connections of the driveline outside of the gearbox, and 22 six degree-of-freedom nodes along the gearbox shafts. Nodes are placed at discrete steps in shafts, at the axial center of shaft couplings, and at the center of gearbox shaft bearing seats. Only torsionally driven axisymmetric rotations about the system driveline shaft are considered. Shaft axial and bending type displacements of the rotor train are eliminated at the input and output gear couplings due to the effective kinematic joint associated with the gear coupling.
Gear Couplings Drive Motor 6 1
Chain Coupling
Torque/Tach Sensor
Gearbox
9
•
Load Generator
19
•
x x y
•
x x
38
•
30
x
33
20 Gear Mesh
Rigid Hub/Elastic Tooth • - node Coupling - lumped mass bearing seat x - gear contact point
Figure 4. MDTB Topology and Rotordynamic Model with Node Points
Figure 5 shows the FEM torsional model used for free-free boundary condition testing. The finite element software ANSYS was used to generate the model. Gear mesh connectivity was modeled using lumped stiffness elements to depict nominal gear tooth stiffness and nodal constraint equations (ui=Riθi ) representing the translational displacements at the ends of zero free length springs as a function of the angular motion of the gear rotor hub. The full width hubs were taken to be rigid disks.
Drive Motor
Torque Cell
Couplings
Gearbox
Torque Cell
Load Generator
Couplings
Figure 5. Finite Element Model of the MDTB The system complete nominal finite element model will be assembled using all degreesof- freedom of the rotor elements in the input and output gearbox shafts, and lumped parameter characterization of the roller element bearings. Disturbances will be considered as harmonic synchronous shaft speeds and N per revolution (N being an integer) periodic forces depending upon the disturbance anomaly. Fault Models and Diagnostics: Few structural dynamic models of dynamic, in situ, gear tooth fracture appear in the literature. However, variable stiffness tooth profiles have been modified for use in dynamic simulation of a root fracture in a gear tooth. [24] The
damaged tooth’s stiffness profile is lessened by some degree (that is assumed proportional to the damage) per damaged gear mesh contact cycle. The nominal lumped parameter (FEM) system model (inertia-[M], damping-[C], gyroscopic-[G], and stiffness-[K] parameters) will be modified to incorporate system faults for response simulations. Faults will be incorporated into the overall system model through time varying stiffnesses, perturbations in those stiffnesses, and perturbation forces as prescribed by current fault models, see Equation (1).
[M ]&s& + ([C] + [G]) s& + ([K ]− [∆K( t) ]) s = ω2 Re iωt + Sg
(1)
Composite Gear Tooth Stiffness About Damaged Tooth 3.0
k
2.0 Softening due to Tooth Crack 1.0 0.571
0.619
0.667
0.714
0.762
0.810
P
Figure 6. Nominal Representation of Gear Tooth Stiffness Variation Due to Fault Many frequency and time- frequency domain methods have been developed primarily for vibratory response signals.[25][26] Diagnostic methods generally focus on the identification of certain types of faults such as gear tooth fracture [27-31]. For gear tooth faults on the MDTB, some specific signal processing techniques have provided indications of damage well before macroscopic damage of the gear teeth was evident. These include interstitial processing [31], where envelope spectral, kurtosis, and other statistical techniques are applied to casing acceleration data that has been pre-processed by bandpass filtering between higher harmonics of gear mesh frequency. FM4 was developed to detect changes in the vibration pattern resulting from damage on a limited number of gear teeth.[32] A difference signal is constructed by removing from the time-synchronous averaged data, the shaft frequency, its harmonics, and the first order sidebands, as well as the primary meshing frequency, its harmonics, and the first order sidebands. FM4 is then calculated by applying the fourth normalized statistical moment to this difference signal. Figure 6 shows a comparison of FM4 and the interstitial enveloping peak [31], along with the figures of merit M6A and M8A.[33][34] FM4, M6A, and M8A all show early indications of damage. However, they are weak for indicating the damage progression.
1
Interstitial RMS Interstitial Kurtosis
Normalized Figure of Merit
0.8
2:00 No visible damage
M6A M8A FM4
0.6
3:00 One broken tooth, one cracked
0.4
5:00 Two broken teeth
0.2
8:15 am: 8 teeth missing
0 94
96
98
100
102
104
106
108
110
112
114
Time (Hours)
Figure 6. Comparison of Interstitial Envelope Peak, FM4, M6A, and M8A (Run 14) 1.E+00
Log Normalized Figure of Merit
Interstitial Env Peak NA4
2:00 No visible damage
NA4*
1.E-01
3:00 One broken tooth, one cracked 1.E-02
5:00 Two broken teeth
1.E-03
8:15 am: 8 teeth missing
1.E-04 94
96
98
100
102
104 106 Time (Hours)
108
110
112
114
Figure 7. Comparison of Interstitial Kurtosis and RMS with NA4, NA4* (Run 14)
NA4 was developed to detect the onset of damage and to continue to react to this damage as it spreads and increases in magnitude.[33][34] A residual signal is constructed by removing the shaft frequency and harmonics, and the primary meshing frequency and harmonics from the time-synchronous averaged data. NA4 is then determined by dividing the fo urth statistical moment of this residual signal by the current run time averaged variance of the residual signal, raised to the second power. NA4* was developed as an enhanced version of NA4, and was expected to be more robust when progressive damage occurs.[33][34] This added robustness is incorporated into NA4* by normalizing the fourth statistical moment with the residual signal variance for a gearbox in good condition instead of the running variance, which is used for NA4. Figure 7 shows an example of a run in which interstitial envelope spectral peak values at gear output speed give strong indication of gear damage before that damage is visible via borescopic inspection (three of four accelerometers). Note that all three parameters show early damage indication, as well as some correlation with damage progression. Application to Prognostics: Figure 7 is a notional design of the parts of the diagnostic and prognostic process.[35] The dynamic model representing system operation is used to estimate the static and dynamic load of a gear. Damage such as tooth breakage could be evaluated with an FEM-based crack propagation model to predict growth of gear tooth fatigue crack. The crack geometry and tooth stiffness (or stress concentration factor) calculated by the FEM model could be feed back to the virtual gearbox which will, in turn, predict the new vibration and loading. In actual operation, the model-to-actual comparison may be accomplished explicitly using generalized residuals or implicitly using learned association (neural network) methods. Developing this capability will require dedicated research in this area and will likely involve significant effort. Diagnosis
Experiment Gear run-to-fail test
Fault Tracking
(fatigue crack, pitting)
Development of sensor observables
Prognosis Methods
Diagnostic Remaining
Assessment Probabilistic
Virtual System
Fault isolation
Life
Model
Prediction at
and severity
Gear fault
Useful
Gear Failure
Expected Loads
Meshing stiffness (FEM)
Equations of motion
Dynamic (vibration)
Stress/Strain Analysis and Fracture Mechanics Crack Prop
Gear static/dynamic loads
Model (FEM)
Expected Future Loads
Figure 7. Eleme nts of Model-Based Machinery Diagnostics and Prognostics
Summary: An overview of the MDTB research modeling effort and considerations has been presented in the context of improving diagnosability for mechanical systems and machinery prognostics. The parametric system dynamics model for the transitional gearbox (MDTB) test bed was presented with the objective of understanding the transitional data feature analysis and evaluating model-based diagnostic/prognostic methods to track component faults as they evolve. The evaluation is accomplished through the comparison of the system experimentally observed (vibratory) behavior, and the behavior derived from the dynamics model. The dynamic model of the MDTB, based on a finite element development, provides a numerical test bed for studies that may be correlated with experimental data. Developing prognostics will require future work using the system and subsystem models concentrating on multiple fault circumstances as well as advancing capabilities in tracking and prediction. Acknowledgment: The support by the Office of Naval Research through the Multidisciplinary University Research Initiative for Integrated Predictive Diagnostics (Grant Number N00014-95-1-0461) is gratefully acknowledged. REFERENCES: 1. Kozlo wski, J.D., and Byington, C.S., 1996, Mechanical Diagnostics Test Bed for ConditionBased Maintenance, ASNE Intelligent Ships Symposium II, November 25-26, 1996. 2. Byington, C.S., and Kozlowski, J.D, 1997, Transitional Data for Estimation of Gearbox Remaining Useful Life, 51st Meeting of the MFPT, April 1997. 3. Kanninen, M.E., 1985, Advanced Fracture Mechanics , Oxford University Press, New York. 4. LaLanne, M. and Ferraris, G., 1998, Rotordynamics Prediction in Engineering, 2 nd Ed, John Wiley and Sons, Chichester, England. 5. Choy, F.K., et al., 1992, Modal Analysis of Multistage Gear Systems Coupled with Gearbox Vibrations, Journal of Mechanical Design, Vol. 114, pp. 486-497. 6. Kahraman, A., Ozguven, H.N., Houser, D.R., and Zakrajsek, J.J., 1992, Dynamic Analysis of Geared Rotors by Finite Elements, Journal of Mechanical Design, Vol. 114, pp. 507-514. 7. Kahraman, A., 1993, Effect of Axial Vibrations on the Dynamics of a Helical Gear Pair, ASME Journal of Vibration and Acoustics, V. 115, pp. 33-39. 8. Vinayak, H., et al., 1995, Linear Dynamic Analysis of Multi-Mesh Transmissions Containing External, Rigid Gears, Journal of Sound and Vibration, Vol. 185, No. 1, pp.1-32. 9. Mark, W.D., 1989, The Generalized Transmission Error of Parallel-Axis Gears, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 111, pp. 414-423. 10. August, R., and Kasuba, R., 1986, Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System , Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, pp. 348-353. 11. Rouch, K.E., McManis, T.H., Stephenson, R.W., Emerick, M.F., 1992, Modeling of Complex Rotor Systems by Combining Rotor and Substructures Models, Finite Elements in Analysis and Design, Vol. 10, pp. 89-100. 12. Mohiuddin, M.A., et al., 1998, Dynamic Analysis and Reduced Order Modeling of Flexible Rotor-Bearing Systems, Computers and Structures, Vol. 69, pp. 349-359. 13. Avitabile, P.; and O'Callahan, J.C., 1991, Understanding Structural Dynamic Modification and the Effects of Truncation, International Journal of Analytical and Experimental Modal Analysis, Vol. 6, no. 4, pp. 215-235. 14. Donley, M., et al., S.G., 1996, Validation of Finite Element Models for Noise/Vibration/ Harshness Simulations, S V Sound and Vibration, Vol. 30, no. 8, pp. 18-23.
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