THE complex interaction between rotordynamics and

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1 274 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005 Rotordynamic Modeling Using Bond Graphs: Modeling the Jeffcott Rotor Joaquín Campos, Mark Crawford, and Raul Longoria Abstract Visualizing the complex and counterintuitive phenomenology associated with rotordynamics has been problematic since Rankine first scientifically studied it in the early 1900s. To address this, a simple model capable of allowing visualization of the parameters and problematic aspects associated with an imbalanced rotor (Jeffcott rotor) will be proposed using bond graphs, which provide a structured and unified method for modeling a large class of nonlinear, multienergetic systems. The purpose of this paper is to provide a high-fidelity, experimentally proven bond graph model of the Jeffcott rotor to aid in the design and analysis of high-speed rotational machinery often associated with pulsesd energy systems. The intent of the model is the ability to use it as a modular and foundational piece in more complex rotordynamic models. Index Terms Bond graph, Jeffcot rotor, pulsed power, rotordynamic models, rotordynamics. I. INTRODUCTION THE complex interaction between rotordynamics and electromagnetics has been of considerable interest as high-speed rotating electrical machines have become the logical choice for mobile pulsed power applications. Due to the inherent size and operating speeds of the proposed machines, great care must be put into the design and analysis of both the mechanical and electrical aspects of the machine. In this paper, a small portion of the overall analysis problem the basic rotordynamics modeling will be addressed. The intent is to produce an analytical model that is simple, robust, and can be used as a modular and foundational piece to the more complex overall system models. II. PROBLEM STATEMENT A. Dilemma and Approach Visualizing the complex and counterintuitive phenomenology associated with rotordynamics has been prob- Manuscript received December 19, The research reported in this work was performed in connection with Contract DAAD17-01-D-0001 with the U.S. Army Research Laboratory. The views and conclusions contained in this document are those of the authors and should not be interpreted as presenting the official policies or position, either expressed or implied, of the U.S. Army Research Laboratory or the U.S. Government unless so designated by other authorized documents. Citation of manufacturers or trade names does not constitute an official endorsement or approval of the use thereof. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. J. Campos and M. Crawford are with the Institute for Advanced Technology, The University of Texas at Austin, Austin, TX USA ( joaquin_campos@iat.utexas.edu; mark_crawford@iat.utexas.edu). R. Longoria is with the Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX USA ( r.longoria@mail.utexas.edu). Digital Object Identifier /TMAG Fig. 1. Schematic of Jeffcott rotor with flexible shaft. lematic since Rankine first scientifically studied it in the early 1900s [1]. In order to address this, a simple model capable of allowing visualization of the parameters and problematic aspects associated with a simple imbalanced rotor system (Jeffcott rotor) will be proposed using a formulation technique known as bond graphs. This technique inherently allows the modeler to see causality in physical systems, which for the current application greatly aids in understanding system phenomena. Specifically, the method described in this paper shows how a Lagrangian formulation can be readily integrated with the bond graph approach. The analytical formulation followed in a Lagrangian approach is especially useful in this application where the modular Jeffcott (Lagrangian) model can be integrated with a bond graph model representing an electromechanical drive train. B. Physical System and Kinematics The system used as a case study is the classical Jeffcott Rotor developed by Jeffcott et al. to study the whirl phenomena of imbalanced rotors [1] [3]. The Jeffcott rotor consists of a rigidly supported isotropic shaft (free to rotate along its bearing axis) with finite elasticity, on which a geometrically symmetrical but inertially asymmetrical rotor is rigidly fastened (see Fig. 1). Using a Cartesian coordinate system with an Eulerian angle description, the following diagram is constructed (Fig. 2), where the position vector and spin angle initially define the system s center of mass with respect to the inertial reference frame. Breaking the position vector down into its components yields Differentiating the position vector with respect to time yields the velocity vector (1) (2) /$ IEEE

2 CAMPOS et al.: ROTORDYNAMIC MODELING USING BOND GRAPHS 275 Fig. 2. End view of Jeffcot rotor with flexible shaft. Fig. 4. Orthogonal spring and dashpot physical model of Jeffcot rotor. Fig. 3. Free body diagram of the shaft-rotor system. but since the magnitude of the eccentricity with time the equation becomes does not change Differentiating the position vector once more yields the acceleration vector for the center of mass of the Jeffcott rotor. Now that the acceleration vector has been attained, the free body diagram of the system can be constructed (Fig. 3). In this figure, is the deflection of the rotor shaft, is the shaft input torque, is the stiffness of the rotor shaft, is the rotational damping, and is the internal damping of the shaft. III. PHENOMENOLOGY The principal phenomena to address with the model are those associated with whirl. There are two types of whirl, the first being the most common form known as synchronous whirl, which produces a forcing function frequency that is the same as the drive or spin frequency. The second form of whirl is asynchronous whirl. This phenomenon is not as common, but is the most dangerous, since vibrations due to it are exceedingly violent [4] and are usually caused by system instabilities at very high speeds (a term that is relative to the properties of the system, i.e., shaft natural frequency). Synchronous whirl is a very well characterized phenomenon and has been the subject of numerous papers and journal articles. As such, there is a great deal of information on the subject matter [5]. The primary focus of study for synchronous whirl is the critical speed, which is the speed at which the amplitude of vibration increases dramatically, a phenomenon often incorrectly associated with the natural frequency of the rotor-shaft system. Asynchronous whirl, as its name implies, is a vibrational phenomenon that is not synchronous with the spin frequency of (3) (4) the rotor. These vibrations usually occur at multiples of the natural frequency of the shaft-rotor system and are typically at supercritical speed (speeds that are usually twice the natural frequency) [6], [7]. This phenomenon and its subphenomena are investigated analytically, and in a limited form, experimentally. Due to the high speeds and the size of the rotor-shaft system that was constructed, experimental testing is kept under supercritical speed. Thus, the test setup reported in this paper will not experience any of the phenomena associated with asynchronous whirl. The analytical results will also be compared to those in literature to measure the level of fidelity of the proposed model. IV. PHYSICAL MODEL The first, most common, and simplest method for the physical modeling of this system is to construct it by supporting the imbalanced flywheel at its geometric center by two orthogonal springs [2], [4] with the same stiffness, each in parallel with a dashpot, as in Fig. 4 [4]. There are two coordinate sets for this method, one that is fixed to the flywheel (rotating with it), and the second that is the inertial frame. If an initial point of view is taken from the inertial frame, the flywheel, when rotating, imparts a sinusoidal forced vibration on the two orthogonal springs. This method best describes the case where the supports are much more compliant than the shaft on which the flywheel rotates, which is not the case in this study. The second and less common method of modeling this system uses an inertial cylindrical coordinate system as shown in Fig. 5, in which the compliance is a single spring dashpot system that rotates with the body [2], [8]. The loading on the structural member modeled in this system is constant, as opposed to the above case where the loading is sinusoidal. This is an important difference that greatly affects the outcome of the models where system damping is concerned. This method is less popular because of the mathematical complexity introduced by the cylindrical coordinates, but it has its conceptual advantage as being more realistic to an actual system, where the supports are much more rigid than the shaft, as in the current application. In creating the model for this study, elements from both simplified models will be used to construct a more appropriate one. In the current case, it is known that the supporting structure is much more rigid than the shaft, so the second model is a more viable candidate for application. In order to simplify the analytical portion of this study, the spring force

3 276 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005 that uses Lagrange s equation to construct the state equations. Recall the Lagrangian (6) Fig. 5. Single spring and dashpot physical model of Jeffcot rotor. is decomposed into its and components to avoid the need for cylindrical coordinates. This in effect produces two springs of equivalent stiffness as in the first model above. It is important at this point to emphasize that, even though the system looks like the first model, it is still conceptually equivalent to the second model. The reason for this is to emphasize that the internal damping in the system is present in the shaft only if the shaft is being cyclically loaded and/or has some. These effects will be seen primarily during acceleration and deceleration of the rotor and during whirl, where instabilities are caused by internal damping. The internal damping affects the critical speed and the maximum amplitude reached at critical speed. The rotational or viscous damping is a form of braking of the flywheel, decreasing the angular velocity and in turn decreasing the amplitude of apparent vibration. where is the Lagrangian, is the kinetic coenergy, and is the potential energy. The Hamiltonian formulation starts with the identification of the kinetic and potential energies and forms the equations of state from the derivative of these two energy functions [9] in the form of Lagrange s equation. Since the Jeffcott rotor only has one inertia the rotor the formulation will be mainly the description of its rate of change of motion. The general form of the Hamiltonian is Using the procedure as shown by Vance [10], the following state equations are derived: (7) V. FORMULATION Two formulation techniques will be used to construct the system s state equations. This will be done to compare and contrast a commonly used technique known as Hamilton s method to the proposed technique of bond graphing. From the free-body diagram (Fig. 2), the following equations of motion are derived: direction: direction: direction: This set of equations of motion will be the starting point from which the comparative methods of modeling will be constructed. Each equation of motion will be verified using each of the formulations methods proposed. These equations of motion should be reproduced by the integration of the state equations, to be derived by the methods described above. So they will be used as checks on the formulation techniques. A. Hamiltonian Formulation The most common analytical technique used to develop the system s state equations in rotordynamics is known as Hamilton s method. This technique is an energy formulation (5) This formulation is particularly tedious for the problem studied. Albeit a common method for rotordynamics, it would not be this author s preferred method for obtaining the system s state equations. B. Bond Graph Formulation The bond graph has an advantage. Besides being a formulation technique, it acts as a conceptual aid enabling the user to view the flow of power and track its effects a tool that is very useful for the subject of rotordynamics, where there are so many counterintuitive phenomena [4]. There does not appear to be a great deal of work in open literature on the subject of modeling rotordynamics using bond graphs other than the notes by Beaman and Paynter [11], text by Karnopp et al. [12], and one reference for explicit bond graph modeling of rotordynamics, from Hubbard [13]. The bond graph technique that will be used in this paper will be the Lagrangian bond graph technique as laid out by Beaman and Paynter [14]. The basic structure of the Lagrangian bond graph is as a multiport capacitive element; although it represents storage of both kinetic and potential energies, all the energy is (8)

4 CAMPOS et al.: ROTORDYNAMIC MODELING USING BOND GRAPHS 277 Fig. 6. Basic Langrangian bond graph of Jeffcot rotor. Fig. 8. Final bond graph model of Jeffcot rotor. TABLE I MATERIAL PROPERTIES FOR THE SHAFT AND THE ROTOR Fig. 7. Initial condition input bond graph. representatively stored in the capacitive element ( ). The formulation is similar to Lagrange s method, only the equations of motion are expressed in the form of first-order ordinary differential equations (ODEs). The bond graph structure for the system under study will have the general structure of the bond graph in Fig. 6. The transformation matrix,, in this case, is an identity matrix, and the inputs to the Lagrangian are the generalized moments,, and. The generalized conservative efforts,,, and are the causal outputs for which the Lagrangian returns the flows,, and. The nonconservative inputs to the system are,, and. In order to account for the losses in the system (nonconservative forces) and to account for the transformation of the input torque (effort) into three different flows in the three directions,, and spin the structure shown in Fig. 7 is needed. Putting the entire bond graph together gives Fig. 8. Using the same notation as Beaman and Paynter [14], the general form of the Lagrange equation is where is the generalized flow variable, is the generalized velocity, is the Lagrangian, is the kinetic coenergy function, and is the potential energy function. The procedure used for the Lagrangian bond graph generally followed that presented by Beaman and Paynter [14], which gives the following state equations: (9) (10) VI. EXPERIMENTAL SETUP The experimental setup for this study was constructed as an instrumented Jeffcott-like rotor. Although great care was put into the design and the construction of the test device, it is difficult to produce an ideal Jeffcott rotor, and any test setup will have inconsistencies and imperfections in the shaft-rotor system as well as the supporting structure. The test setup is a simple shaft-rotor system suspended on rigid bearings and whose rotor accommodates eccentric placement of mass in order to precisely control the position of the center of mass of the rotor, so that an imbalance can be induced. Table I shows the material properties for the shaft and the rotor. The rotor is constructed of 6061-T6 aluminum, using 19-mm plate stock, and has a diameter of 300 mm. Controlled eccentricity of the center of mass is accomplished by means of three rows of 1 /4-20 tapped holes spaced 20 mm apart and symmetrically located 180 on either side of the center of the rotor. The tapped holes provide a consistent mechanism for introducing an eccentric mass by means of either 1-in-long 1 /4-20 steel hex screws having an average mass of 6.03 g each, or 0.6-in-long 1 /4-20 steel hex screws, having an average mass of 4.96 g each. The rotor shaft is 303 stainless steel with a 0.50-in diameter. The shaft length is in, of which only in is used as the supporting length of the rotor. The rotor was driven by a Kollmorgen servo motor that was limited to 5000 rpm. Measurement of the rotor lateral position was done via an orthogonal pair of position-sensitive laser diode pairs from LMI/Selcom that were mounted on a rigid structure surrounding the rotor (see Fig. 9). A total of 88 tests were conducted with two variants of the above setup. VII. ANALYSIS Using the data gathered from experiments, a comparison will be made between actual and simulated data. Three representative transient (spin-up) cases will be presented for comparison with the bond graph simulation data. A. Formulation Comparison To gain confidence in the formulation procedures, the two modeling techniques are compared using an identical set of initial conditions for a simulated experiment of the experimental rotor spinning up through its critical speed. As can be noted, the

5 278 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005 Fig. 9. Photograph of final test setup. TABLE II SIMULATION INITIAL CONDITIONS FOR MODEL COMPARISON Fig. 11. Plot of radial deflection of shaft versus time of both Hamiltonian and bond graph model simulations. TABLE III 3600-RPM TEST Fig. 12. Plot of experimental versus simulation data of x-center position for 3600 rpm test. physics. This provides verification for both techniques, giving a check on the respective model formulations. Fig. 10. Plot of center position trace versus time of both Hamiltonian and bond graph simulations. state equations resulting from the Hamiltonian derivation and the Lagrangian bond graph derivation are slightly different from each other. The following simulation comparison should serve to verify these derivations. The conditions in Table II were used in the simulation comparison. The following results show the Hamiltonian simulation results plotted with the Lagrangian bond graph model results. The first plot (Fig. 10) is of the trace of the rotor center. The second plot (Fig. 11) is of the radial displacement as seen rotating with the rotor. The simulation results are almost identical. Computed values are approximately 0.2% of each other. So, even if the formulation produced state equations that may have seemed different, they are in fact similar and produce similar results for the same B. Comparison With Experimental Data The first case presented, spin-up to 3600 rpm, is the only spin-up test presented that is conducted at subcritical speed. Table III shows the conditions for this test, and Fig. 12 plots the results. The lighter color is the bond graph simulation data and the darker is the experimental. There are three main differences between the test and simulation results. The main and most obvious difference is the amplitude of vibration. There seems to be a resonance at about 3500 rpm that does not match the bump-test resonance for the direction of the experimental rotor, but is close to the resonant frequency of the direction, which is 3540 rpm. It is possible that the resonant amplitude of vibration is coupling into the direction from the direction, and the fact that the resonant peaks are occurring roughly at the orthogonal direction resonance is evidence. This test is a good example of how the ideal case and the actual case differ because of the assumption of a symmetric system. The third difference between

6 CAMPOS et al.: ROTORDYNAMIC MODELING USING BOND GRAPHS 279 TABLE IV 4200-RPM TEST INITIAL CONDITIONS Fig. 14. Plot of experimental versus simulation data of x-center position for 4400 rpm test. TABLE V 4400-RPM TEST INITIAL CONDITIONS Fig. 13. Plot of experimental versus simulation data of y-center position for 4200 rpm test. the simulation and the experiment is that the start-up vibration amplitudes are very different. The simulation shows hardly any start-up displacement amplitude, whereas the experiment shows comparatively large displacement amplitude. It is hypothesized that this is due to the supporting structure giving during rapid acceleration of the offset mass. The simulation assumes an infinitely stiff supporting structure, so start-up displacement amplitudes due to high accelerations cannot occur. Since the differences in amplitude for this test were so different from the simulation data, it is only presented as a qualitative comparison. The next studied test took place at speed of 4200 rpm, which was just above the critical speed of the shaft-rotor system in the direction ( 64 Hz or 3840 rpm). This test gave an opportunity to observe transition through critical speed and to observe the qualitative shape of the amplitude envelope. Table IV contains conditions for the 4200-rpm test, and Fig. 13 plots the results. There was a s delay ( ) between when the data input was initiated and when motion began in the flywheel for the experiment, so the data had to be shifted in time by in order to compare with the of the simulated data. The results were typical of those that crossed the critical speed with a characteristic envelope shape. The shape and amplitude was found to be highly sensitive to the damping ratio and the stiffness. This was observed while varying the values of the stiffness and damping coefficient in the model. The experimental data shows an envelope that precedes the predicted critical speed of 64 Hz. This envelope appears to be centered at about 54 Hz. This phenomenon is due to one of two things; either it is a coupling in the shaft displacement amplitude from the direction, as discussed previously, due to the asymmetric nature of the stiffnesses, or something in the supporting structure has a resonance at this particular speed. Excluding this precritical-speed amplitude peak, the simulation generally follows the shape and form of the critical-speed amplitude envelope. There is some slight difference in the drop-off and in the transition between envelopes in the experiment which does not exist in the simulation. The overall performance, given adjusted values, gave a 7% difference between measured and adjusted values of stiffness and a 38% dif- ference between the average measured and adjusted values for the damping coefficient. It should be noted that this adjusted value of the damping coefficient was still within the range of measured values between 0.01 and The final case to be studied as representative of the series was conducted at 4400 rpm (see Fig. 14), the highest speed tested. Testing conditions were as follows in Table V. There was a s delay ( ) between when the data input was initiated and when motion began in the flywheel for the experiment. Again, the data had to be shifted in time by in order to compare with the simulated data. As in the 4200-rpm test, there is good agreement between the model and the experiment. Again, we do not have the pre-critical-speed envelope centered at the same rpm as in the 4200-rpm case, which was 54 Hz 2 Hz. The transition between envelopes also occurs at about the same rpm as in the previous case. The overall performance for the given adjusted values gave a 6% difference between measured and adjusted values of stiffness and again a 38% difference between the average measured and adjusted values for the damping coefficient, keeping in mind the previous case s discussion on the value of damping coefficient. VIII. CONCLUSION The primary goal of this study was to create an experimentally validated bond graph model of the Jeffcott rotor. This was done successfully. Three main points on the performance of the Lagrangian Bond Graph model are as follows. There is excellent agreement ( 99.8 ) between the Hamiltonian model and the Lagrangian bond graph model. The transient case for subcritical speeds shows poor agreement with the experimental data, but is due to modeling assumption of ideal Jeffcott rotor conditions, i.e., structural resonances not accounted for in the supporting structure. The transient case for the spin-up through critical speed shows very good agreement ( 97 ) between the experiment and simulation in the amplitude and shape of the resonant amplitude envelope if the resonant envelope that

7 280 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 1, JANUARY 2005 does not exist in the simulated data is excluded. Recalling again that the assumption of an ideal Jeffcott rotor is used for the model, which is being compared to a nonideal simple shaft-rotor system. The major modeling issue to come out of this study is that using the Hamiltonian method to analytically model this system proved extremely tedious, which indicates that there could have been a better approach chosen as the baseline to the bond graph model. It should also be noted that most rotor/shaft systems of greater than two degrees of freedom will act like the two-degrees-of-freedom case (i.e., the Jeffcott rotor) as observed by Dimentberg [15] in his canonical text on flexural vibrations of rotating shafts. So, it should be concluded that the bond graph model constructed should readily be able to describe rotor systems of greater degrees of freedom. Because of the bond graph s innate ability to connect systems and multiple energy domains, the current bond graph model is well suited for use in more complex systems, thus making it a robust and adaptable element for modeling larger systems where multiple energy domains and multiple systems are interacting. REFERENCES [1] J. M. Vance, Rotordynamics of Turbomachinery. New York: Wiley, 1988, pp [2] R. G. Loewy and V. J. Piarulli, [SVM-4] Dynamics of Rotating Shafts. Washington, D.C.: The Shock and Vibration Information Center, U.S. Department of Defense, 1969, pp [3] D. Childs, Turbomachinery Rotordynamics Phenomena, Modeling, and Analysis. New York: Wiley, 1993, pp [4] R. G. Loewy and V. J. Piarulli, [SVM-4] Dynamics of Rotating Shafts. Washington, D.C.: The Shock and Vibration Information Center, U.S. Department of Defense, 1969, pp [5], [SVM-4] Dynamics of Rotating Shafts. Washington, D.C.: The Shock and Vibration Information Center, U.S. Department of Defense, 1969, p. 2. [6] J. M. Vance, Rotordynamics of Turbomachinery. New York: Wiley, 1988, p. 17. [7] R. G. Loewy and V. J. Piarulli, [SVM-4] Dynamics of Rotating Shafts. Washington, D.C.: The Shock and Vibration Information Center, U.S. Department of Defense, 1969, pp [8] J. M. Vance, Rotordynamics of Turbomachinery. New York: Wiley, 1988, p. 21. [9] R. Neptune, Class Notes on Hamilton s Method, Dept. Mech. Eng., The University of Texas at Austin, [10] J. M. Vance, Rotordynamics of Turbomachinery. New York: Wiley, 1988, p [11] J. J. Beaman and H. M. Paynter, Modeling of Physical Systems, 1993, unpublished, pp [12] D. C. Karnopp, D. L. Margolis, and R. C. Rosenberg, System Dynamics: A Unified Approach, 2nd ed. New York: Wiley, 1990, pp [13] M. Hubbard, Whirl dynamics of pendulous flywheels using bond graphs, J. Franklin Inst., vol. 308, no. 4, pp , Oct [14] J. J. Beaman and H. M. Paynter, Modeling of Physical Systems, 1993, unpublished, pp [15] F. M. Dimentburg, Flexural Vibrations of Rotating Shafts. London, U.K.: Buttersworth, 1961, p. 43.