Center for Rotating Machinery Dynamics and Control

Pesch, A.H. and Sawicki, J.T., 2017, “Active Magnetic Bearing Online Levitation Recovery through μ-Synthesis Robust Control,” to be published in Special Issue “Active Magnetic Bearing Actuators” in Actuators, MDPI AG.

Abstract A rotor supported on active magnetic bearings (AMBs) is levitated inside an air gap by electromagnets controlled in feedback. In the event of momentary loss of levitation due to an acute exogenous disturbance or external fault, reestablishing levitation may be prevented by unbalanced forces, contact forces, and the rotor’s dynamics. A novel robust control strategy is proposed for ensuring levitation recovery. The proposed strategy utilizes model-based μ-synthesis to find the requisite AMB control law with unique provisions to account for the contact forces and to prevent control effort saturation at the large deflections that occur during levitation failure. The proposed strategy is demonstrated experimentally with an AMB test rig. First, rotor drop tests are performed to tune a simple touchdown-bearing model. That model is then used to identify a performance weight, which bounds the contact forces during controller synthesis. Then, levitation recovery trials are conducted at 1000 and 2000 RPM, in which current to the AMB coils is momentarily stopped, representing an external fault. The motor is allowed to drive the rotor on the touchdown bearings until coil current is restored. For both cases, the proposed control strategy shows a marked improvement in relevitation transients.

Kulesza, Z. and Sawicki, J.T., 2016, “Parametrically Induced Damping in a Cracked Rotor,” ASME Journal of Engineering for Gas Turbines and Power, 139(1), 012505, 8 pages.

Abstract A transverse shaft crack in a rotor is usually modeled as a local change in shaft stiffness. This local stiffness change is not constant and varies as a result of a so-called breathing mechanism, explained with periodical opening and closing of crack faces under the load of external forces applied to the rotor. The rotor with a periodically varied stiffness can be modeled as a parametrically excited linear system. In the presence of a parametric excitation, the vibrations of the system can be amplified or damped at specific excitation frequencies. Usually, the frequencies at which the vibrations are amplified are important, since they can affect stability of the system. However, the increased damping at specific frequencies is a significant feature of a parametrically excited system that can have some potentially useful applications. One of such applications can be an early detection of a shaft crack. This paper presents results of numerical analysis of the influence of Rayleigh’s damping and gyroscopic effects on the increase in damping in a parametrically excited rotor with a cracked shaft. It is shown that the increase in damping in a parametrically excited system is rather a rare phenomenon that can be observed only at properly selected values of the excitation frequency and Rayleigh’s damping. Furthermore, gyroscopic effects influence the exact values of antiresonance frequencies at which the phenomenon appears.

Kulesza, Z. and Sawicki, J.T., 2015, “Damping by Parametric Excitation in a Set of Reduced-Order Cracked Rotor Systems,” Journal of Sound and Vibration, 354, pp. 167-179.

Abstract A common tool utilized for the stability analysis of parametrically excited linear systems, such as rotors with cracked shafts, is Floquet׳s method. The disadvantage is a long calculation time needed to evaluate the monodromy matrix and instability zones. An efficient alternative is the generalized Bolotin׳s method, where the instability zones are evaluated quickly, yet the matrices that must be calculated are of large dimensions.

In the present paper, the stability analysis is conducted with both Floquet׳s method and the generalized Bolotin׳s method. However, the order of the model is reduced to two modes only and stability analyses are performed for the second-order systems obtained with various combinations of the reducing modes. Then, the results of such analyses are collected in an overall stability map. The stability map obtained in this way closely reconstructs the stability map calculated with the full-order model of the rotor, yet the calculation time needed to generate the collected map as well as the dimension of the problem are considerably reduced.

The approach is demonstrated with a mathematical model of the machine with the breathing crack modeled using the rigid finite element method. The rotor is not rotating, yet the stiffness of the shaft is varied periodically to simulate the parametric excitation.

An interesting indication of the developing shaft crack observed in the generated stability maps is the presence of anti-resonant zones, where the rotor vibration amplitudes quickly decay. It is anticipated that this phenomenon of increased damping at specific excitation frequencies may have potential application for shaft crack detection.

Smirnov, A., Pesch, A.H., Wroblewski, A.C., Pyrhönen, O. and Sawicki, J.T., 2015, “Feedback Compensation of Tool Deflection in a High Speed AMB Machining Spindle,” accepted for publication in JSME Mechanical Engineering Journal.

Abstract

Sawicki, J.T. and Kulesza, Z., 2015, “Stability of a Cracked Rotor Subjected to Parametric Excitation,” ASME Journal of Engineering for Gas Turbines and Power, 137(5), 052508, 8 pages.

Abstract It is well known that parametric vibrations may appear during the rotation of a rotor with a cracked shaft. The vibrations occur due to periodic stiffness changes being the result of the crack breathing. A parametrically excited system may exhibit parametric resonances and antiresonances affecting the stability of the system. In most cases the destabilizing effect due to parametric resonances is studied. Antiresonant cases seem to be uninteresting. However, the antiresonances have a unique property of introducing additional artificial damping to the system, thus improving its stability and reducing the vibration amplitude. Apart from different control applications, this stabilizing effect may be interesting for its probable ability to indicate the shaft crack. The possible application of the additional damping introduced by parametric excitation for the shaft crack detection is analyzed in the present paper. The approach is demonstrated with a mathematical model of a rotor with a cracked shaft. The stability analysis of the rotor is performed analytically by employing the averaging method. Stability boundaries for different frequencies of the parametric excitation and for different crack depths are derived. The results of this analysis are checked numerically by means of the Floquet’s theory. Next, possible applications of the parametric excitation for the shaft crack detection are validated numerically.

Smirnov, A., Pesch, A.H., Pyrhönen, O., and Sawicki, J.T., 2015, “High-Precision Cutting Tool Tracking with a Magnetic Bearing Spindle,” ASME Journal of Dynamic Systems, Measurement and Control, 137(5), 051017, 8 pages.

Abstract A method is presented for tool tracking in active magnetic bearing (AMB) spindle applications. The method uses control of the AMB air gap to achieve the desired tool position. The reference tracking problem is transformed from the tool coordinates into the AMB control axes by bearing deflection optimization. Therefore, tool tracking can be achieved by an off-the-shelf AMB controller. The method is demonstrated on a high-speed AMB boring spindle with a proportional integral derivative (PID) control. The hypothetical part geometries are traced in the range of 30μm. Static external loading is applied to the tool to confirm disturbance rejection. Finally, a numerical simulation is performed to verify the ability to control the tool during high-speed machining.

Madden, R. and Sawicki, J.T., 2015, “Model Validation for Identification of Damage Dynamics,” Paper GT2014-27341, ASME Journal of Engineering for Gas Turbines and Power, 137(6), 062506, 7 pages.

Abstract Robust control techniques have allowed engineers to create more descriptive models by including uncertainty in the form of additive noise and plant perturbations. As a result, the complete model set is robust to any discrepancies between the mathematical model and actual system. Experimental unfalsification of the model set leads to the guarantee that the model and uncertainties are able to recreate all experimental data points. In this work, such a robust control relevant model validation technique is applied to structural health monitoring in order to (1) detect the presence of damage and (2) identify the damage dynamics when used in conjunction with model-based identification. Additionally, the robust control relevant model validation technique allows for a novel quality measure of the identified damage dynamics. Feasibility of the method is demonstrated experimentally on a rotordynamic crack detection test rig with the detection and identification of a change in structure. Further insight is gained from application of the method to seeded damage on a rotor levitated on active magnetic bearings (AMBs) in the form of a local reduction in stiffness.

Wroblewski, A.C., Pesch, A.H., and Sawicki, J.T., 2014, “Structural Change Quantification in Rotor Systems Based on Resonance and Antiresonance Frequencies Extracted from Frequency Response Functions,” ASME Journal of Engineering for Gas Turbines and Power, 136(2), 022506, 6 pages.

Abstract A structural change quantification methodology is proposed in which the magnitude and location of a structural alteration is identified experimentally in a rotor system. The resonance and antiresonance frequencies are captured from multiple frequency response functions and are compared with baseline data to extract frequency shifts due to these features. The resulting expression contains sufficient information to identify the dynamic characteristics of the rotor in both the frequency and spatial domains. A finite element model with carefully selected tunable parameters is iteratively adjusted using a numerical optimization algorithm to determine the source of the structural change. The methodology is experimentally demonstrated on a test rig with a laterally damaged rotor and the frequency response functions are acquired through utilization of magnetic actuators positioned near the ball bearings.

Litak, G., and Sawicki, J.T., 2013, “Regular and chaotic vibrations in the rub impact model of a Jeffcott rotor with a fractional restore force,” The European Physical Journal – Applied Physics, 64(3), 31303, 5 pages.

Abstract We study the solutions and bifurcations of the Jeffcott rotor with a rubbing effect. The model of horizontal rotor possesses such nonlinear effects as inertia, dry friction, and contact loss between the rotor and stator. By the exceeding of the rotor-stator radius clearance, the rotor can penetrate into the limiting rubbers with a fractional power in the restore force. The system response is analyzed by a bifurcation diagram. The specific cases are additionally clarified by means standard methods and quantified by the test 0-1 which is sensitive to chaotic behavior.

Sawicki, J.T., 2013, “Control Driven Advances in Smart Rotating Machinery,” Solid State Phenomena, Special Issue: Mechatronic Systems and Materials, 198, pp. 457-467.

Abstract In the last twenty years, there has been significant progress in the design of rotating machinery equipped with smart components and embedded functions. This was accompanied by developments in actuators, sensors and power electronics technologies, advances in data acquisition and signal processing, as well as developments in control theory. This paper will provide a brief overview of the current state of art and it will illustrate examples of the applications of smart technologies applied to rotating machines. Several technologies, either most recently developed or under development will be presented, which involve active control and are relevant to smart solutions applied to rotating machinery.

Wroblewski, A.C., Pesch, A.H., and Sawicki, J.T., 2013, “Structural Change Quantification in Rotor Systems Based on Resonance and Antiresonance Frequencies Extracted from Frequency Response Functions,” Journal of Engineering for Gas Turbines and Power.

Abstract A structural change quantification methodology is proposed in which the magnitude and location of a structural alteration is identified experimentally in a rotor system. The resonance and antiresonance frequencies are captured from multiple frequency response functions and are compared with baseline data to extract frequency shifts due to these features. The resulting expression contains sufficient information to identify the dynamic characteristics of the rotor in both the frequency and spatial domains. A finite element model with carefully selected tunable parameters is iteratively adjusted using a numerical optimization algorithm to determine the source of the structural change. The methodology is experimentally demonstrated on a test rig with a laterally damaged rotor and the frequency response functions are acquired through utilization of magnetic actuators positioned near the ball bearings.

Sawicki, J. T., 2013, “Control Driven Advances in Smart Rotating Machinery,” Solid State Phenomena, 198, pp. 457-466.

Abstract In the last twenty years, there has been significant progress in the design of rotating machinery equipped with smart components and embedded functions. This was accompanied by developments in actuators, sensors and power electronics technologies, advances in data acquisition and signal processing, as well as developments in control theory. This paper will provide a brief overview of the current state of art and it will illustrate examples of the applications of smart technologies applied to rotating machines. Several technologies, either most recently developed or under development will be presented, which involve active control and are relevant to smart solutions applied to rotating machinery.

Kulesza, Z., and Sawicki, J.T., 2012, “New Finite Element Modeling Approach of a Propagating Shaft Crack,” Journal of Applied Mechanics, 80(2), 021025, 17 pages.

Abstract Transverse shaft cracks are one of the most dangerous malfunctions of the rotating machines, including turbo- and hydrogenerators, high-speed machine tool spindles, etc. The undetected crack may grow slowly and not disturb normal machine operation. However, if it extends to a critical depth, the immediate shaft fracture may completely damage the machine, resulting in a catastrophic accident. Therefore, in-depth knowledge of the crack propagation process is essential to ensure reliable and safe operation of rotating machinery. The article introduces a new model of the propagating shaft crack. The approach is based on the rigid finite element (RFE) method, which has previously proven its effectiveness in the dynamical analysis of numerous complicated machines and structures. The crack is modeled using several dozen spring-damping elements (SDEs), connecting the faces of the cracked section of the shaft. By controlling the exact behavior of individual SDEs, not only the breathing mechanism, but also the crack propagation process can be simply introduced. In order to accomplish this, the stress intensity factors (SIFs) along the crack edge are calculated using the novel approach based on the modified virtual crack closure technique (VCCT). Based on the SIF values, the crack propagation rate is calculated from the Paris law. If the number of load cycles is greater than the constantly updated threshold number, then the crack edge is shifted by a small increment. This way, starting from the first initially cracked SDE, the crack is extended little by little, continuously changing its shape. The approach is illustrated with numerical results, demonstrating the changes in the rotor vibration response and in the crack shape and also explaining some issues about the breathing mechanism due to the propagating shaft crack. The increasing amplitude of the 2X harmonic component is recognized as an evident propagating crack signature. The numerical results correspond well with the data reported in the literature. The RFE model of the rotor is validated by comparing the vibration responses obtained experimentally and numerically. A good agreement between these data confirms the correctness and accuracy of the proposed model. The suggested approach may be utilized for a more reliable dynamic analysis of the rotating shafts, having the potential to experience propagating transverse cracks.

Kulesza, Z., Sawicki, J. T., and Gyekenyesi, A. L., 2012, “Robust Fault Detection Filter using Linear Matrix Inequalities Approach,” Journal of Vibration and Control, 19(9), pp. 1421-1440.

Abstract Detecting cracks in rotating shafts is a challenging problem when using vibration-based diagnostics. This is due to the fact that a localized crack has a minimal influence on the global vibration response of the system. To increase sensitivity and reliability, the vibration response needs to be coupled with additional sources of information such as a mathematical model of the machine. Modern control theory techniques offer system-level mathematical models for both control and diagnostics. Focusing on the latter, a new and promising approach involves the use of unknown input observers. Such observers can be designed to employ robust fault detection filters (RFDFs) for isolating fault signatures while reducing the influence of real-world disturbances and noise. For the present study, a modified design procedure coupled with robust fault detection is utilized for shaft crack detection. The filter is designed using the linear matrix inequalities (LMI) technique. The LMI approach is applied to obtain the solution of the mixed H-/H∞ optimization problem, which arises during the synthesis of the RFDF. By reformulating the LMI conditions, the proposed RFDF design procedure is simplified and thus requires less iteration steps to find the optimal solution. A new feature of the present approach involves the application of the rigid finite element method for the formulation of the mathematical model of the rotor and the shaft crack. The numerical and experimental results confirm the advantages of the designed robust fault detection filter and its ability to detect shaft cracks. The filter is minimally sensitive to measurement noise while allowing for the identification of shallow cracks (2% or 5% deep). The cracks are manifested through the observance of very subtle vibration response changes. The results also confirm the effectiveness and accuracy of the rigid finite element modeling concerning the cracked rotor.

Wroblewski, A. C., Sawicki, J. T., and Pesch, A. H., 2012, “Rotor Model Updating and Validation for an Active Magnetic Bearing Based High-Speed Machining Spindle,” Journal of Engineering for Gas Turbines and Power, 134(12), pp. 122509.

Abstract This paper presents an experimentally driven model updating approach to address the dynamic inaccuracy of the nominal finite element (FE) rotor model of a machining spindle supported on active magnetic bearings. Modeling error is minimized through the application of a numerical optimization algorithm to adjust appropriately selected FE model parameters. Minimizing the error of both resonance and antiresonance frequencies simultaneously accounts for rotor natural frequencies as well as for their mode shapes. Antiresonance frequencies, which are shown to heavily influence the model’s dynamic properties, are commonly disregarded in structural modeling. Evaluation of the updated rotor model is performed through comparison of transfer functions measured at the cutting tool plane, which are independent of the experimental transfer function data used in model updating procedures. Final model validation is carried out with successful implementation of robust controller, which substantiates the effectiveness of the model updating methodology for model correction.

Kulesza, Z., and Sawicki, J. T., L., 2012, “Controlled Deflection Approach for Rotor Crack Detection,” Journal of Engineering for Gas Turbines and Power, 134(9).

Abstract A transverse shaft crack is a serious malfunction that can occur due to cyclic loading, creep, stress corrosion, and other mechanisms to which rotating machines are subjected. Though studied for many years, the problems of early crack detection and warning are still in the limelight of many researchers. This is due to the fact that the crack has subtle influence on the dynamic response of the machine and still there are no widely accepted, reliable methods of its early detection. This paper presents a new approach to these problems. The method utilizes the coupling mechanism between the bending and torsional vibrations of the cracked, nonrotating shaft. By applying an external lateral force of constant amplitude, a small shaft deflection is induced. Simultaneously, a harmonic torque is applied to the shaft inducing its torsional vibrations. By changing the angular position of the lateral force application, the position of the deflection also changes opening or closing of the crack. This changes the way the bending and torsional vibrations are being coupled. By studying the coupled lateral vibration response for each angular position of the lateral force one can assess the possible presence of the crack. The approach is demonstrated with a numerical model of a rotor. The model is based on the rigid finite element method (RFE), which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. The RFE method is extended and adopted for the modeling of the cracked shafts. An original concept of crack modeling utilizing the RFE method is presented. The crack is modeled as a set of spring-damping elements (SDEs) of variable stiffness connecting two sections of the shaft. By calculating the axial deformations of the SDEs, the opening/closing mechanism of the crack is introduced. The results of numerical analysis demonstrate the potential of the suggested approach for effective shaft crack detection.

Madden, R. J., and Sawicki, J. T., 2012, “Rotor Model Validation for an Active Magnetic Bearing Machining Spindle using Mu-Synthesis Approach,” Journal of Engineering for Gas Turbines and Power, 134(9), pp. 092501-1 – 092501-6.

Abstract Model-based identification and l-synthesis are employed for model updating of the rotor for a high-speed machining spindle supported on active magnetic bearings. The experimentally validated model is compared with a nominal engineering model to identify the unmodeled dynamics. The extracted missing dynamics from the nominal rotor model provides engineering insight into an effective model correction strategy. The corrected rotor model is validated by successful implementation of a number of l-synthesized controllers, providing robust and stable levitation of the spindle over its entire operating speed range.

Sawicki, J. T., and Kulesza, Z., 2012, “A New Rotor Crack Detection Method Based on Auxiliary State Observer,” Advances in Vibration Engineering, 11(2), pp. 131-141.

Abstract A new model-based approach for rotor crack detection and crack location is presented. The finite element model of the rotor-bearing system accounts for the breathing mechanism of the transverse crack. First, the model of the rotor system is enhanced by adding an auxiliary one-degree-of-freedom oscillator. Then, the observer is designed and the estimates of its two auxiliary state variables are proposed as crack indicators. Identification of the crack location along the shaft is determined by designing a set of observers, which yield the values of these indicators for different possible crack locations along the shaft. The proposed method is validated numerically and the results demonstrate its capability for an on-line crack detection and location.

Kulesza, Z., and Sawicki, J.T., 2012, “Rigid Finite Element Model of a Cracked Rotor,” Journal of Sound and Vibration, 331(18), pp. 4145-4169.

Abstract The article introduces a new mathematical model for the cracked rotating shaft. The model is based on the rigid finite element (RFE) method, which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. In this article, the RFE method is extended and adopted for the modeling of rotating machines. An original concept of crack modeling utilizing the RFE method is developed. The crack is presented as a set of spring–damping elements of variable stiffness connecting two sections of the shaft. An alternative approach for approximating the breathing mechanism of the crack is introduced. The approach is simple and allows one to intuitively and systematically prepare and analyze the model of a cracked rotor.
The proposed method is illustrated with numerical and experimental results. The experiments conducted for the uncracked free–free rotor as well as the numerical results obtained with other software confirm the accuracy of the RFE model. The numerical analysis conducted for a set of cracked rotors has shown that, depending on the eccentricity and its angular location, the breathing behavior of the crack may take different forms. In spite of this, the frequency spectra for different cracks are almost identical.
Due to its simplicity and numerous advantages, the proposed approach may be useful for rotor crack detection, especially if methods utilizing the mathematical model of the rotor are applied.

Sawicki, J. T., and Madden R., 2011, “Identification of Missing Dynamics in Rotor System using Robust Control Theory Approach,” Vibration Problems ICOVP 2011, 330, pp. 581-587.

Abstract Analytical models only approximate the true dynamics of analyzed rotating machines, due to the presence of components that are inherently difficult to model. Such models of rotating machines are driven by the best engineering knowledge and experience, and very often are updated based on experimental results. The problem of un-modeled or missing dynamics can be exacerbated in the presence of rotor structural damage such as a transverse crack on a shaft. This paper will present an effective approach for model updating using advanced tools developed in the robust control theory, specifically μ-synthesis. The methodology will be applied to the identification of minute changes in the dynamics of the rotor due to the presence of a transverse crack on a shaft. Experimental data collected from the cracked rotor rig will be utilized to validate the developed approach.

Sawicki, J. T., Friswell, M. I., Kulesza, Z., Wroblewski, A., and Lekki, J. D., 2011, “Detecting Cracked Rotors using Auxiliary Harmonic Excitation,” Journal of Sound and Vibration, 330, pp. 1365-1381.

Abstract Cracked rotors are not only important from a practical and economic viewpoint, they also exhibit interesting dynamics. This paper investigates the modelling and analysis of machines with breathing cracks, which open and close due to the self-weight of the rotor, producing a parametric excitation. After reviewing the modelling of cracked rotors, the paper analyses the use of auxiliary excitation of the shaft, often implemented using active magnetic bearings to detect cracks. Applying a sinusoidal excitation generates response frequencies that are combinations of the rotor spin speed and excitation frequency. Previously this system was analyzed using multiple scales analysis; this paper suggests an alternative approach based on the harmonic balance method, and validates this approach using simulated and experimental results. Consideration is also given to some issues to enable this approach to become a robust condition monitoring technique for cracked shafts.

Sawicki, J. T., Storozhev, D. L., and Lekki, J. D., 2011, “Exploration of NDE Properties of AMB Supported Rotors for Structural Damage Detection,” Journal of Engineering for Gas Turbines and Power, 133(10), pp. 102501-102509.

Abstract Recent advancements in actuator technology, power electronics, sensors, and signal processing have created a rapid development of smart machine technologies for rotating machinery. Ranging from machine condition monitoring and diagnostics to full active control of machine behavior, the integration of electrical and computer systems has produced significant advances in machine performance and reliability. Magnetic bearings are a typical mechatronics product. The hardware is composed of mechanical components combined with electronic elements such as sensors and power amplifiers and an information processing part, usually in the form of a microprocessor. In addition, an increasingly important part is software, which specifies the coordination of bearing forces to sensed rotor motion and consequently dictates the dynamic properties of the complete system. The inherent ability for sensing, information processing, and actuation gives the magnetic bearing the potential to become a key element in smart and intelligent machines.

Kulesza, Z., and Sawicki, J.T., 2010, “Auxiliary State Variables for Rotor Crack Detection,” Journal of Vibration and Control, 17(6), pp. 857-872.

Abstract In the present study, a new model-based method for rotor crack detection and crack location is proposed. The finite-element model of the rotor-bearing system accounts for the breathing mechanism of the crack. The model of the rotor system is augmented with an auxiliary single-degree-of-freedom oscillator. The observer is designed and the estimates of its two auxiliary state variables are proposed as crack indicators. The crack location along the shaft is determined by designing a set of observers, which calculate the values of these indicators for different possible crack locations along the shaft. The proposed method is validated numerically and the results prove its capability to detect and locate the crack. Further study will include experimental and numerical investigations to make the approach more robust.

Gyekenyesi, A., Sawicki, J. T., and Haase, W. C., 2010, “Modeling Disk Cracks in Rotors by Utilizing Speed Dependent Eccentricity,” Journal of Materials Engineering and Performance, 19(2), pp. 207-212.

Abstract This paper discusses the feasibility of vibration-based structural health monitoring for detecting disk cracks in rotor systems. The approach of interest assumes that a crack located on a rotating disk causes a minute change in the system’s center of mass due to the centrifugal force induced opening of the crack. The center of mass shift is expected to reveal itself in the vibration vector (i.e., whirl response; plotted as amplitude and phase versus speed) gathered during a spin-up and/or spin-down test. Here, analysis is accomplished by modeling a Jeffcott rotor that is characterized by analytical, numerical, and experimental data. The model, which has speed dependent eccentricity, is employed in order to better understand the sensitivity of the approach. For the experimental set-up emulated here (i.e., a single disk located mid-span on a flexible shaft), it appears that a rather sizable flaw in the form of a through-thickness notch could be detected by monitoring the damage-induced shift in center of mass. Although, identifying actual disk cracks in complex “real world” environments, where noncritical crack lengths are small and excessive mechanical and/or electrical noise are present, would prove to be rather challenging. Further research is needed in this regard.

Litak, G. and Sawicki, J.T., 2009, “Crack Identification by Multifractal Analysis of a Dynamic Rotor Response,” ZAMM – Journal of Applied Mathematics and Mechanics, 89(7), pp. 587-592.

Abstract Multifractal analysis has been used to diagnose cracked and healthy rotors. Is has been shown that the complexity and regularity criteria of the dynamical systems defined by the multiple scaling of the time series can indicate the damages of the rotating shaft. Relation to the standard power spectrum technique has been also discussed.

Sawicki, J. T., Sen, A. K., and Litak, G., 2009, “Multiresolution Wavelet Analysis of the Dynamics of a Cracked Rotor,” International Journal of Rotating Machinery, 2009, 8 pages.

Abstract We examine the dynamics of a healthy rotor and a rotor with a transverse crack, which opens and closes due to its self weight. Using discrete wavelet transform, we perform a multiresolution analysis of the measured vibration signal from each of these rotors. In particular, the measured vibration signal is decomposed into eight frequency bands, and the rms amplitude values of the healthy and cracked rotors are compared in the three lowest-frequency bands. The results indicate that the rms vibration amplitudes for the cracked rotor are larger than those of the healthy rotor in each of these three frequency bands. In the case of externally applied harmonic force excitation to the rotor, the rms values of the vibration amplitude of the cracked rotor are also found to be larger than those of a healthy rotor in the three lowest-frequency bands. Furthermore, the difference in the rms values between the healthy and cracked rotors in each of the three lowest-frequency bands is more pronounced in the presence of external excitation than that with no excitation. The obtained results suggest that the present multiresolution approach can be used effectively to detect the presence of a crack in a rotor.

Litak, G., and Sawicki, J.T., 2009, “Intermittent Behaviour of a Cracked Rotor in the Resonance Region,” Chaos Solutions and Fractals, 42(3), pp. 1495-1501.

Abstract Vibrations of the Jeffcott rotor are modeled by a three degree of freedom system including coupling between lateral and torsional modes. The crack in a rotating shaft of the rotor is introduced via time dependent stiffness with off-diagonal couplings. Applying the external torque to the system allows to observe the effect of crack “breathing” and gain insight into the system. It is manifested in the complex dynamic behavior of the rotor in the region of internal resonance, showing a quasi-periodic motion or even non-periodic behavior. In the present paper report, we show the system response to the external torque excitation using nonlinear analysis tools such as bifurcation diagram, phase portraits, Poincaré maps and wavelet power spectrum. In the region of resonance, we study intermittent motions based on laminar phases interrupted by a series nonlinear beats.

Litak, G., Sawicki, J.T., and Kasperek, R., 2009, “Cracked Rotor Detection by Recurrence Plots,” Nondestructive Testing and Evaluation, 24(4), pp. 347-351.

Abstract Recurrence plots (RPs) analysis has been used to distinguish cracked and healthy rotor responses. It has been shown that the recurrence criteria of the dynamical systems defined by the RPs can indicate the damages of the rotating shaft using relatively short time series.

Sawicki, J. T., 2009, “Rotor Crack Detection using Active Magnetic Bearings,” Solid State Phenomena, Special Issue: Mechanical Systems and Materials, 144, pp. 9-15.

Abstract Well-established procedures exist to monitor and diagnose fairly severe problems with rotating machinery but little progress has been made in developing techniques to detect subtle changes in machine condition for both improved diagnostics, and to develop prognostic procedures for determining remaining service life. Of all machine faults, crack initiated problems present probably the most significant safety and loss hazard in modern turbomachinery, including aircraft engines and power generation units. Different approaches are used to model, detect, and localize crack-induced damage in rotating structures. This paper presents novel application of active magnetic bearings (AMBs) for on-line rotor crack detection. AMB-actuators provide convenient means to apply a broad spectrum of known dynamic forces and monitor responses in a rotor-bearing system, which facilitates more sensitive and precise diagnostics. The paper presents theoretical modeling and description of the experimental facility for proof-of-concept testing.

Sawicki, J. T., Johansson, S. A., Rumbarger, J. H., and Sharpless, R. B., 2008, “Fatigue Life Prediction for Large-Diameter Elastically Constrained Ball Bearings,” Journal of Engineering for Gas Turbines and Power, 130(2). pp. 022506-022513.

Abstract The application of large-diameter bearing rings and the thereof inherited low stiffness make them susceptible to local distortions caused by their surrounding structures, which are often under heavy loads. The standard accepted design criteria for these bearings are based on the estimation of the internal load distribution of the bearing, under the assumption of rigid circular and flat supporting structures, that keep the bearing inner and outer races in circular, flat, i.e., not deformed shapes. However, in the presence of structural distortions, the element load distribution can be severely altered and cannot be predicted via the standard design criteria. Therefore, the application of large-diameter ball and roller bearing rings as the critical components in rotating machines becomes more of a design task than making a catalog selection. The analytical and finite element approach for fatigue life prediction of such a bearing application is presented. The undertaken approach and the results are illustrated based on the analysis and fatigue life simulation of the computed tomography scanner’s main rotor bearing. It has been demonstrated that flexibility of the rings can significantly reduce the fatigue life of the ball bearing.

Pesch, A. H., and Sawicki, J. T., 2012, “Application of Robust Control to Chatter Attenuation for a High-Speed Machining Spindle on Active Magnetic Bearings,” The 13th International Symposium on Magnetic Bearings (ISMB13), Washington, DC, USA.

Abstract The purpose of this work is to demonstrate a robust control strategy to avoid machining chatter in a high-speed machining spindle levitated on active magnetic bearings. Chatter avoidance is an important aspect in machining because chatter limits the material removal rate and inhibits production. A method is implemented which uses μ-synthesis AMB control with a cutting force model in the controller design process. Such an approach causes the controller to stabilize the naturally unstable cutting process, avoiding chatter and resulting in smoother surface finish, longer tool life and increased material removal rate. Stability lobe diagrams are calculated for the new control strategy showing increased critical feed rate and confirmed through numerical simulations. The developed method is implemented experimentally on the high-speed AMB machining spindle. Impulse hammer testing is performed to measure the controlled spindle’s transfer function and evaluate chatter free machining suitability.

Sawicki, J. T., and Madden, R., 2011, “Identification of Missing Dynamics in Rotor Systems using Robust Control Theory Approach,” International Conference on Vibration Problems, Prague, Czech Republic.

Abstract Analytical models only approximate the true dynamics of analyzed rotating machines, due to the presence of components that are inherently difficult to model. Such models of rotating machines are driven by the best engineering knowledge and experience, and very often are updated based on experimental results. The problem of un-modeled or missing dynamics can be exacerbated in the presence of rotor structural damage such as a transverse crack on a shaft. This paper will present an effective approach for model updating using advanced tools developed in the robust control theory, specifically μ-synthesis. The methodology will be applied to the identification of minute changes in the dynamics of the rotor due to the presence of a transverse crack on a shaft. Experimental data collected from the cracked rotor rig will be utilized to validate the developed approach.

Wroblewski, A. C., Sawicki, J. T., and Pesch, A. H., 2011, “High-Speed AMB Machining Spindle Model Updating and Validation,” SPIE Smart Structures/NDE, San Diego, California, USA.

Abstract High-Speed Machining (HSM) spindles equipped with Active Magnetic Bearings (AMBs) have been envisioned to be capable of automated self-identification and self-optimization in efforts to accurately calculate parameters for stable high-speed machining operation. With this in mind, this work presents rotor model development accompanied by automated model-updating methodology followed by updated model validation. The model updating methodology is developed to address the dynamic inaccuracies of the nominal open-loop plant model when compared with experimental open-loop transfer function data obtained by the built in AMB sensors. The nominal open-loop model is altered by utilizing an unconstrained optimization algorithm to adjust only parameters that are a result of engineering assumptions and simplifications, in this case Young’s modulus of selected finite elements. Minimizing the error of both resonance and anti-resonance frequencies simultaneously (between model and experimental data) takes into account rotor natural frequencies and mode shape information. To verify the predictive ability of the updated rotor model, its performance is assessed at the tool location which is independent of the experimental transfer function data used in model updating procedures. Verification of the updated model is carried out with complementary temporal and spatial response comparisons substantiating that the updating methodology is effective for derivation of open-loop models for predictive use.

Barbaraci, G., Pesch, A. H., and Sawicki, J. T., 2010, “Experimental Investigations of Minimum Power Consumption Optimal Control for Variable Speed AMB Rotor,” ASME International Mechanical Engineering Congress and Exposition, Vancouver, Canada.

Abstract The purpose of this paper is to present a method for development of the optimal speed-dependent control matrix for a rotor supported on active magnetic bearings (AMBs) with the provision of minimum control power consumption over the operating speed range. The speed dependency of the optimal control matrix is the result of the dynamics of rotating machines. Most of published works on optimal control use a stationary optimal control matrix derived for the non-rotating system and thus neglecting the effect of gyroscopic phenomena. This paper employs the minimum energy consumption condition to derive the speed varying optimal control for rotating AMB rotor system. In the presented approach the control matrix is characterized by a second order polynomial matrix with the angular speed as a variable. This leads to a more compact and lower computational burden for controller implementation. Calculations are performed for a 4-axis AMB rotor test rig. Testing with rotor speed ramps is performed and experimental values for power consumption are presented. These results are compared to results with speed invariant optimal control and PID control.

Pesch, A. H., and Sawicki, J. T., 2010, “Uncertainty Range Estimation for μ-Synthesis Control of AMB Spindle,” 10th International Conference on Motion and Vibration Control (MOVIC2010), Tokyo, Japan.

Abstract Active magnetic bearings (AMBs), being inherently nonlinear, are often modeled with linear approximations about the operating point. μ-synthesis control which offers robust stability in the presence of uncertain parameters also provides a method for overcoming the limitations that arise from this linear approximation. In the past, experimental measurements have been taken at the zero deflection point to determine the uncertainty range in the current stiffness and position stiffness parameters. This paper proposes an analytical method for deriving, from the non linear force equation, the nominal (minimum) range of uncertainty for the position stiffness and current stiffness in the μ-synthesis control of AMBs. This is of particular interest in applications where zero deflection is not expected at the bearing location. Stiffness range calculations for a high speed magnetic bearing machining spindle are also presented as a case study.

Sawicki, J. T., Friswell, M. I., Pesch, A. H., and Wroblewski, A., 2008, “Condition Monitoring of Rotor using Active Magnetic Actuator,” Proceeding of ASME Turbo Expo 2008, Berlin, Germany.

Abstract It has been widely recognized that the changes in the dynamic response of a rotor could be utilized for general fault detection and monitoring. Current methods rely on the monitoring of synchronous response of the machine during its transient or normal operation. Very little progress has been made in developing robust techniques to detect subtle changes in machine condition caused by rotor cracks. It has been demonstrated that the crack-induced changes in the rotor dynamic behavior produce unique vibration signatures. When the harmonic excitation force is applied to the cracked rotor system, nonlinear resonances occur due to the nonlinear parametric excitation characteristics of the crack. These resonances are the result of the coexistence of a parametric excitation term and different frequencies present in the system, namely critical speed, the synchronous frequency, and excitation frequency from the externally applied perturbation signals. This paper presents the application of this approach on an experimental test rig. The simulation and experimental study for the given rig configuration, along with the application of active magnetic bearings as a force actuator, are presented.

Sawicki, J.T. and Kulesza, Z., 2015, “Damping in a Parametrically Excited Cracked Rotor,” Mechanisms and Machine Science, 21, pp. 335-345, Springer.

Abstract Conducted analyses of rotors with cracked shafts show that the stability of such systems deteriorates with an increasing crack depth. Instability areas near parametric resonances enhance as a result of increasing periodic stiffness changes due to a breathing mechanism of a developing shaft crack. However, some recent studies on the dynamics of linear structures with periodically altered stiffness present an interesting phenomenon of an increase in damping. It has been demonstrated that under certain conditions a parametrically excited mechanical structure can increase its stability. When the structure falls into the parametric anti-resonant area, its vibration amplitudes quickly decay. For a long time these anti-resonant zones seemed to be not interesting, yet they can introduce additional artificial damping into the system, improving its stability and leading to further studies of their possible applications. The present paper analyzes the possibility of the appearance of such a phenomenon (the increase in damping of the parametrically excited system) in a rotor with a cracked shaft. The approach is demonstrated with a mathematical model of the machine. The breathing crack is modeled using the rigid finite element method that has previously proven its robustness and efficiency in similar applications. The stability analysis is conducted numerically by the Floquet’s technique. The conditions required for the appearance of parametric anti-resonances for different crack depths are provided. Finally, a possible application of the additional damping introduced by parametric excitation for rotor crack detection is analyzed.

Madden, R.J., Pesch, A.H. and Sawicki, J.T., 2015, “Model Validation for Damage Identification and Determination of Local Damage Dynamics,” Mechanisms and Machine Science, 21, pp. 769-778, Springer.

Abstract Robust control techniques have allowed engineers to create model sets which are robust to deviations from the actual system through the use of model uncertainty in the form of both additive noise and plant perturbations. To ensure the quality of the uncertain model, experimental unfalsification is employed to guarantee that the model set is able to recreate all experimental data points. In previous work, such a robust control relevant model validation technique was employed to identify the presence of damage on a rotordynamic test rig. Additionally, model-based identification was employed to model the overall change in dynamics due to the damage, as well as to provide a novel quality measure for the identified damage dynamics. In the present work, the technique is extended to include a rotordynamic model. This advancement allows for locating the damage source and to determine the local change in dynamics at the damage location. The method is demonstrated experimentally on a rotordynamic test rig through the identification of a wire EDM cut in the shaft and determination of the local change in dynamics, including axial position along the shaft.

Pesch, A.H., Smirnov, A. Pyrhönen, O., and Sawicki, J.T., 2015, “Magnetic Bearing Spindle Tool Tracking through μ-Synthesis Robust Control,” IEEE Transactions on Mechatronics, 20(3), pp. 1448-1457.

Abstract A method is presented for tooltip tracking in active magnetic bearing (AMB) spindle applications. The proposed tool tracking approach uses control of the AMB air gap to achieve the desired tool position. A μ-synthesis-based controller is designed for the AMBs with the goal of robustly minimizing the difference between the tool reference and estimated tool position. In such a way, the model-based control approach concurrently addresses the tracking problem and the inability to directly measure real-time tool position in the presence of machining disturbances. To ensure the tractability of the control problem, a model of the desired tracking dynamics is included in the plant. The method is demonstrated on a high-speed AMB boring spindle. To confirm the tool tracking capability, characteristic part geometries are traced including stepped, tapered, and convex profiles. Tool tracking is demonstrated for the rotating AMB spindle in the range of 90 μm. Also, static and dynamic external loading is applied to the spindle tool location to confirm the disturbance rejection ability of the closed-loop system.