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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 3, MAY 2003

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Torque-Based Control of Whirling Motion in a Rotating Electric Machine Under Mechanical Resonance Kaoru Inoue, Member, IEEE, Shigeru Yamamoto, Associate Member, IEEE, Toshimitsu Ushio, Member, IEEE, and Takashi Hikihara, Member, IEEE

Abstract—The major objectives of this paper are to clarify the influence of a whirling motion on the electrical characteristics of a rotating electric machine and to propose a method to control the whirling motion under mechanical resonance. First, it is shown that the hysteretic jump phenomena exist in the rotating speed and the power consumption of the motor due to the whirling motion around the natural frequency. The jump of the rotating speed means that a continuous change of output power in a generator can not be achieved under mechanical resonance. The jump of the power consumption gives a large fluctuation to other rotating electric machines connected to the same power source. Second, we propose a torque-based control method that can suppress the whirling motion by controlling a torque. The control input is determined based on a stability condition for a time-varying system that is represented by a second-order vector differential equation with time-varying coefficient matrices. Stability of the system can be checked by the qualitative characteristics of its coefficient matrices. Finally, the effectiveness of our control method is discussed by both a simulation and an experiment. Index Terms—Hysteretic jump phenomena, rotating electric machine, second-order vector differential equation, torque-based control, whirling motion.

NOMENCLATURE Coordinate axis for horizontal direction. Coordinate axis for vertical direction. Origin . Center of the shaft. Center of gravity. Horizontal rotor displacement. Vertical rotor displacement. Phase of the rotation of rotor. Phase of the whirling motion. rotor displacements. rotating speed of the rotor. Manuscript received June 1, 2001; revised June 19, 2002. Manuscript received in final form December 16, 2002. Recommended by Associate Editor F. Svaricek. This work was supported in part by the Ministry of Education, Science, Sports, and Culture of Japan under a Grant-in-Aid for Encouragement of Young Scientists 121615. K. Inoue is with the Department of Electrical Engineering, Doshisha University, Kyoto 610-0321, Japan (e-mail: [email protected]). S. Yamamoto and T. Ushio are with the Division of Systems and Human Science, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan (e-mail: [email protected]; [email protected]). T. Hikihara is with the School of Electrical Engineering, Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2003.810368

Input voltage to dc motor. Power consumption of dc motor. Eccentricity. Proportional factor between and torque. Rotor mass. Damping coefficient in , directions. Shaft stiffness. Rotor inertia. Damping coefficient of the rotation. Free parameter for designing controller. Natural frequency. Field of real numbers. I. INTRODUCTION

R

OTATING electric machines mainly consist of a rotor and an electrical circuit to translate electrical energy to mechanical one, and vice versa. A motor and a generator are typical examples of the rotating electric machine. The rotor in the rotating electric machine vibrates due to a centrifugal force which is proportional to its eccentricity. This vibration is called whirling motion. It also appears in a cyclone separator, a flywheel energy storage system, and others that are coupled systems of a rotating electric machine and a load (or a drive). The rotor has resonant points with its natural frequencies. The whirling motion becomes large under mechanical resonance [1]. The dynamical behavior of the whirling motion has been studied with the Jeffcott rotor model [2]–[5]. Most of these analytical studies are based on an assumption of a constant rotating speed, which implies that a torque is not influenced by the dynamical behavior of the rotor [2], [3]. Hence, the electrical dynamics is completely separated from the dynamics of the rotating electric machine in the analysis. Therefore, the influences of the whirling motion on both the rotating speed of the rotor and the electrical dynamics of electrical rotating machines are important problems [6], [7]. This research topic belongs to both mechanical and electrical engineering but it has not been well understood. Based on the problems mentioned above, we have proposed a flexible rotor system in order to discuss the influences of the whirling motion on the rotating speed in the rotating electric machine under mechanical resonance [8], [9]. We have investigated the behavior of the system by both a simulation and an experiment. The results show that the hysteretic jump phenomena and

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the chaotic fluctuation appear in the rotating speed due to the whirling motion. The jump of the rotating speed means that a continuous change of output power in the generator can not be achieved. The increase of the whirling amplitude causes the destruction of the rotor. Hence, some appropriate control methods are required to suppress the whirling motion. Some conventional methods suppressing the whirling motion by using electromagnets have been proposed [10], [11]. In most of these methods, stability is discussed based on a model with constant rotating speed. For the rotating electric machines, however, stability must be discussed with a model taking into account the influences of the whirling motion on both the rotating speed and the electrical dynamics. In this paper, we investigate the influences of the whirling motion on not only the rotating speed but also the electrical dynamics of the rotating electric machine by using a flexible rotor system. It is clarified that the hysteretic jump phenomena appear both in the rotating speed and the power consumption due to the whirling motion around the natural frequency. The cause of the hysteretic jump phenomena is discussed. Then, we propose a torque-based control method of the whirling motion. The torque of the rotating electric machine is controlled to suppress the whirling motion instead of using electromagnets. Stability is discussed with a model of the flexible rotor system taking into account the influences of the whirling motion on both the rotating speed and the power consumption. The control input is determined based on a stability condition for a time-varying system represented by a second-order vector differential equation with time-varying coefficient matrices. Stability of the system can be checked by the qualitative characteristics of its coefficient matrices [12]–[14]. The effectiveness of our control method is discussed by both a simulation and an experiment. As a result, it is shown that our method can suppress the whirling motion and its influences on both the rotating speed and the electrical dynamics of the rotating electric machine. This paper is organized as follows. In Section II, the flexible rotor system is introduced and its behavior is discussed. Section III presents a torque-based control method of the whirling motion and shows sufficient conditions for stability in a closed-loop system. We will apply the control method to the experimental system in Section IV. Finally, concluding remarks are given in Section V. II. NONLINEAR PHENOMENA A. Flexible Rotor System We introduce a flexible rotor system to analyze the rotor dynamics and the influences of the whirling motion on both the rotating speed and the electrical dynamics of a rotating electric machine under mechanical resonance. This system has a coupled structure with a dc motor, a flexible rotor, and a dc generator as shown in Fig. 1. The dc generator is used as a sensor to measure the rotating speed. The flexible rotor is modeled by the Jeffcott rotor [3], [5]. This rotor consists of a rigid massive imbalanced disk and a flexible shaft of negligible mass that supports the disk. Both ends of the shaft are supported by rigid bearings. The gyroscopic moments associated with the disk are neglected. The qualitative rotor dynamics below

Fig. 1. Flexible rotor system and its coordinate system.

Fig. 2.

Experimental system.

the second critical speed can be discussed by using the Jeffcott rotor model [2]. Fig. 1 also shows the coordinate system of the flexible rotor. , , , and deSymbols , , , , note the origin, the center of the shaft, the center of gravity, the eccentricity, the rotor displacements, the rotating speed of the rotor, the phase of the rotation of the rotor, and the phase of the whirling motion, respectively. B. Experimental System for Analysis We investigate the behavior of the flexible rotor system under mechanical resonance by an experiment. Fig. 2 shows the experimental system. The rotor consists of a flexible shaft made of piano wire and a plastic disk which is 48.8 mm in diameter, 3.9 mm in thickness, and 2.7 g in mass. The disk is mounted at the middle of the shaft whose diameter, length, and mass are 3.0 mm, 612.0 mm, and 33.9 g, respectively. Inherently, the rotor has the eccentricity because the density distribution of the material is not uniform. The rotor is connected to the motor and the generator by brass joints. There is a small gap between the centers of the motor (generator) shaft and the rotor shaft. This gap also gives the massive imbalance to the rotor. We adjusted the rotor and the joints to set the eccentricity caused by both the gap and the inherent density distribution on the same direction from the when line segment becomes origin . We settled of parallel to the horizontal axis . The natural frequency the rotor is 230 rad/s. The dc motors with rating 42.2 W are used for both the motor and the generator. The characteristic of these motors between the input voltage to the dc motor and the rotating speed is linear.

INOUE et al.: TORQUE-BASED CONTROL OF WHIRLING MOTION

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TABLE I PARAMETERS SETUP

There is a strong relationship among the behavior of , , . References [2], [3] investigated the dynamics based on and an assumption of a constant rotating speed, which implies that a torque is not influenced by the whirling motion. Our results clarified that the whirling motion has a strong influence on the torque. is decreased. The We can also observe the jumps when V, while they occur at V jumps occur at is increased. Hence, the hysteretic jump phenomena when exist in the system dynamics. The cause of the jump phenomena will be discussed in Section II-E. The growth of the whirling motion causes the destruction of the rotor. The jump of the rotating speed means that a continuous change of output power in the generator can not be achieved under mechanical resonance. The jump of the power consumption gives a large fluctuation to other rotating electric machines connected to the same power source. Therefore, it is required that an appropriate control method suppresses these nonlinear phenomena. D. Mathematical Model Fig. 3.

System behavior under mechanical resonance.

The input voltage to the dc motor is employed as the input to the system. The vertical rotor displacement , the rotating speed , and the power consumption of the motor are observed when the input voltage is swept monotonically. in Experimental data is synchronously sampled at every rotation. C. System Behavior Under Mechanical Resonance Fig. 3 shows steady-state diagrams of the rotating speed , the vertical rotor displacement , and the power consumption of the motor, respectively, where the input voltage is increased and decreased with constant rate of 13 mV/s. The rotor is in displacement has only negative values since the phase advance of . As is increased, increases proportional to and is sufficiently small below . Around , however, becomes large and the increment of becomes small. The power consumption increases according to the increase of . A load torque against the dc motor becomes large due to the whirling motion so that increase. Then, we the increment of becomes small and at the same input can observe jump phenomena in , , and V. After the jumps, the whirling motion bevoltage comes sufficiently small again and increases proportional to . The load torque becomes small because becomes small so decrease. that

be the rotor mass, the shaft stiffness, the Now let inertia, the damping coefficient in both and directions, the damping coefficient of the rotation, and the proportional and the drive torque of the dc motor. Then factor between a mathematical model of the flexible rotor system is given as follows [15]:

(1) where

(2) Table I shows the values of the parameters. These values were identified from experimental data. E. Analysis of Hysteretic Jump Phenomena We analyze the cause of the observed jump phenomena by using the mathematical model (1). in the mathematical model (1), the equations in Let the steady-state of the rotating speed are derived as follows: (3) (4) (5)

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a stability condition for a time-varying system represented by a second-order vector differential equation with time-varying coefficient matrices. The flexible rotor system can be rewritten with as follows:

(9)

B. Fundamental Theorem Fig. 4.

A system represented by a second-order vector differential equation with constant coefficient matrices , , and of the form

Relationship between input voltage and rotating speed.

The differential equations (3) and (4) can be solved with respect to and , respectively: (6) (7) Substituting (6) and (7) into (5), the relationship between is given as

and

(8) and for the parameters in The relationship between Table I is illustrated in Fig. 4. There are two folds that cause the jump phenomena. Equation (8) has two solutions at the points and , and three solutions exist in the interval where the middle solution is unstable and the others are stable. Hence, the bifurcations at the points and are a saddle-node bifurcation which causes the jump phenomena. III. A CONTROL METHOD A. Torque-Based Control This section presents a method to control the whirling motion by controlling a torque. Some conventional methods use electromagnets to suppress the whirling motion [10], [11]. In the methods, stability is discussed based on a model with a constant rotating speed. For the rotating electric machines, however, stability must be discussed with a model taking into account the influences of the whirling motion on both the rotating speed and the electrical dynamics. On the other hand, our method controls the torque to suppress the whirling motion instead of using the electromagnets. Fig. 4 shows that an uncontrolled steady state is trapped when increases. However, it is desirable that around the steady state always lies on another stable solution because is small and changes proportional to . Hence, we use inducing a torque to accelerate a control input voltage from the stable solution around to another stable one. In Section III-B, the control input voltage is determined based on

(10) system, where , , and , and is called an . Stability of the system can be checked by the qualitative characteristics of its coefficient matrices. The equiof (10) is asymptotically stable if all librium point , coefficient matrices are symmetric positive definite ( , ) [12]. The positive definiteness of has in [13]. A result for time-varying cobeen relaxed to , , and has been presented in efficient matrices for all , but we relax it to [14]. It is assumed that . Our result is useful for stability analysis of a system represented by a couple of the first and the second-order differential equations. In [15], we have derived a sufficient condition for stability in a certain class of MDK systems of the form (11) is a constant matrix with and both where and are continuous, bounded, and timewith varying coefficient matrices. Let and . of (11) is Theorem 1: The equilibrium point uniformly stable if the following conditions 1) and 2) hold, also globally uniformly exponentially stable if conditions 1), 2), and 3) hold, for all . , , 1) . , , , . 2) is independent of (i.e., ). 3) Proof: Define a constant matrix and matrices as

(12)


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with a free parameter so that the coefficients of the closed-loop system satisfy the conditions in Theorem 1. When we use (18), the closed-loop system is given by

From condition 1), we obtain

(19) (13)

where

, , and both and Condition 2) implies are nonsingular matrices. as state variables, we can Choosing rewrite (11) as the following state equation: (20) (14) The equilibrium point of (14) is only the origin is nonsingular. Define a Lyapunov function with

because

(15)

D. Stability of Equilibrium Point We discuss the stability of the equilibrium point of (19) with . Since , we have , which implies that the condition 1) in Theorem 1 holds. Matrices , , and are symmetric and positive definite. and are equivalent to

is radially unbounded and decrescent because .

(21) (22)

Then we have respectively, because a necessary and sufficient condition for the positive definiteness of a matrix is that all the leading principal minors are positive [17]. Thus, (21) and (22) imply that the condition 2) in Theorem 1 holds. Hence, (21) and (22) imply that the equilibrium point of (19) is uniformly stable. E. Uniform Ultimate Boundedness (16) Therefore, the equilibrium point of the system (11) is uniformly stable. is independent of , so is (i.e., ). If When the system is uniformly completely observable, the equilibrium point is globally uniformly exponentially stable [16]. There such that exist positive constants , ,

is nonzero constant, the closed-loop system (19) When has no equilibrium points because of the time-varying coeffiand . However, we can show the unicient matrices form ultimate boundedness [18] of the error system between the closed-loop and a stable reference system given by (23) (24) This reference system is ideal because the whirling motion and the jump of the rotating speed never occur. The error system becomes

(17)

is the state-transition matrix (See Appendix I). where Hence, the system is uniformly completely observable for all . Therefore, the equilibrium point is globally uniformly exponentially stable. C. Closed-Loop System We introduce a control input voltage

(25) , , . where The following theorem gives a sufficient condition for the uniform ultimate boundedness of the error system. Theorem 2: The error system (25) is uniformly ultimately such that bounded if there exists a constant (26)

(18)

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where

We can easily check the condition (26) by using the following lemma. such that Lemma 1: If there exists a constant (34)

(27) (35)

Proof: Define the following matrices: then

and . Proof: See Appendix II. IV. EXPERIMENTS

(28)

Choosing rewrite the error system (25) as

as state variables, we can (29)

A. Experimental Setup In this section, we apply the proposed torque-based control method to the experimental system and discuss its effectiveness. Nms/rad . The control law (18) is implemented for can be rewritten as Then the control input voltage

with

Define a Lyapunov function

(36) (30) Then we have

(31) where

We obtain . Define the constant as

(32) and

as

because . Hence,

is constant. We also have becomes (33)

Furthermore, the Lyapunov function unbounded and decrescent because

is radially

. Therefore, the error system (25) is uniformly ultimately bounded.

is chosen as a positive constant, the damping coeffiWhen cient of the rotation increases. We will discuss the effectiveness of our control method by comparing the diagrams showing the and of the system without control relationship between (Fig. 4) and the closed-loop system. We use the values of parameters shown in Table I. In the experiment, however, is used for gain tuning so that we determ. The unmodeled dynamics cause the mined difference of values in . These parameters satisfy the equations rad/s. Also there exists the constant (21) and (22) for satisfying (34) and (35) for rad/s. Hence, these parameters satisfy the conditions of Theorems 1 and 2. Thus, stability is guaranteed theoretically. Fig. 5 shows the experimental system setup. The rotating speed , the phase , the vertical rotor displacement , and the horizontal rotor displacement are measured simultaneously. is reset to zero in every rotation when line segThe phase becomes parallel to the horizontal axies (refer to ment Fig. 1). The velocities of the rotor displacements and are calculated by the computer. The A/D and D/A converters have 12-bit resolution in a range of 10 V to 10 V. The sampling is 2 ms. and are added by the dc period to calculate amplifier and are provided to the dc motor. The experiment is performed to accelerate from the point of C to D where is 1.8 V in Fig. 3. B. Simulation and Experimental Results Fig. 6 shows a simulation result. Before the control is applied, is around 225 rad/s corresponding to the point C in Fig. 3 and is large. After the control is applied, approaches to 340 rad/s that corresponds to the point D in Fig. 3. The rotor displacement increases as soon as the control is applied. However, after the


Fig. 5.

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Experimental system setup.

Fig. 7.

Time series of experimental result.

(a)

Fig. 6. Time series of simulation result.

elapse of 0.1 s, it begins to decrease. The control input voltage oscillates and converges to around 1.8 V according to the decrease of . In the simulation, our control method can suppress the whirling motion under mechanical resonance. Fig. 7 shows an experimental result. After the control is approaches to the point D and decreases. This applied, experimental result coincides qualitatively with the simulation while one. However, we can observe the fluctuation of it approaches to the point D. The fluctuation is caused by

(b) Fig. 8. Photographs of experiment. (a) Before control is applied. (b) After control is applied.

higher order vibration modes. Fig. 8(a) and (b) show the rotor vibration before and after applying the control, respectively. Two black boxes in the figures are the laser displacement sensors for measuring the rotor displacements. Before the control is applied, the whirling motion is large. On the other

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condition for stability in a certain class of time-varying systems. The control method was implemented to an experimental system. The experimental result shows the effectiveness of the control method. We also analyzed the relationship between the input voltage to the motor and the rotating speed in the closed-loop system. It was clarified that the torque-based control method eliminates the jump phenomena around the natural frequency. It is noted that this control gives new steady solutions around the natural frequency to eliminate the jump phenomena instead of stabilizing the unstable solutions. APPENDIX Fig. 9.

Relationship between V and q of closed-loop system.

PROOF OF UNIFORM COMPLETE OBSERVABILITY hand, Fig. 8(b) shows that the control makes the whirling motion small. Thus, it is confirmed that the proposed torque-based control method can suppress the whirling motion.

The state variable is have s.t.

C. Analysis of Closed-Loop System and of the We investigate the relationship between closed-loop system (19) by using the method in Section II-E. and The equation which shows the relationship between of the closed-loop system is derived as follows:

for (38)

and there exists

such that, for all (39) and the ob-

We denote the state-transition matrix by servability grammian is given by

(40)

(37) Fig. 9 shows the relationship. The solid and the broken lines indicate the steady-states of the closed-loop system (37) and the uncontrolled system (8), respectively. The broken line has two folds that cause the hysteretic jump phenomena, while the solid line has no folds. Hence, the torque-based control method eliminates the jump phenomena around . This result means is also eliminated by the control because that the jump of there is a strong relationship among the behavior of , , and discussed in Section II-C. It is noted that this control gives new steady solutions around the natural frequency to eliminate the jump phenomena instead of stabilizing the unstable solutions.

satisfying . We

because

(41) From (39), we obtain

(42) of the matrix

Hence, there exists a minimum eigenvalue satisfying

(43) V. CONCLUSION In this paper, we have discussed the influence of a whirling motion on both the rotating speed and the electrical characteristics of a rotating electric machine, and have proposed a torque-based control method of the whirling motion under mechanical resonance. It was observed that the hysteretic jump phenomena exist in the rotating speed and the power consumption due to the whirling motion. We clarified that the jump phenomena occur due to a saddle-node bifurcation. The proposed control method can suppress the whirling motion by controlling the torque without using any electromagnets. The control input voltage is determined based on a sufficient

We also define constants

and

as (44) (45)

respectively. Then we have

(46)


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Similarly, there exists a maximum eigenvalue that

such

Moreover, (54) is also derived from

and (55)

(47) ACKNOWLEDGMENT

Then we have (48)

with constant

defined by (49)

From (46) and (48), there exist constants , , and such that, for all

,

(50)

Therefore, the system is uniformly completely observable. PROOF OF LEMMA 1 Suppose that satisfies (34). We have because . We also have . Then, is derived by Schur complement formula [19]. Similarly, we obtain (51) because

The authors acknowledge Prof. S. Ohashi of Kansai University and Dr. S. Hasegawa of Hakodate Mirai University for their helpful support in performing the experiments. K. Inoue wishes to thank Prof. T. Kato of Doshisha University for his helpful comments.

. We define

as

(52) beSuppose that satisfies (35). Then, we obtain and (32). We partition cause of the negative coefficient of as follows: the matrix

(53) Using Schur complement formula, a necessary and sufficient can be written as condition for the positive definiteness of and (54)

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Kaoru Inoue (M’02) was born in Osaka, Japan, in 1973. He received B.S. and M.S degrees from Kansai University, Osaka, Japan, in 1996 and 1998, respectively, and the Ph.D. degree from Osaka University, Osaka, Japan, in 2001. He was a Research Fellow of the Japan Society for the Promotion of Science from 2000 to 2001. He joined the Department of Electrical Engineering of Doshisha University in 2001, where he is now a Research Associate. His research interests include analysis and control of nonlinear phenomena in physical systems. Dr. Inoue is a Member of IEEJ, IEICE, and ISCIE.

Shigeru Yamamoto (M’95–A’96) was born in Osaka, Japan, in 1964. He received the B.S., M.S., and Ph.D. degrees from Osaka University, Osaka, Japan, in 1987, 1989, and 1996, respectively. He has been with the Department of Systems and Human Science, Osaka University, since 1998, where he is currently an Associate Professor. He was a Research Associate of the Department of Mechanical Engineering for Computer-Controlled Machinery from 1989 to 1994, and the Department of Systems Engineering (the old name of the Department of Systems and Human Science) from 1994 to 1998, respectively, at the same university. From 1998 to 2000, he was an Assistant Professor of the same department. His current research interests include identification for robust control, hybrid systems, and controlling chaos.

Toshimitsu Ushio (S’84–M’85) received the B.S., M.S., and Ph.D. degrees from Kobe University, Kobe, Japan, in 1980, 1982, 1985, respectively. He was a Research Assistant with the University of California, Berkeley, in 1985. From 1986 to 1990, he was a Research Associate with Kobe University, and became a Lecturer at Kobe College in 1990. He joined Osaka University, Osaka, Japan, as an Associate Professor in 1994, and is currently a Professor. His research interests include nonlinear oscillation and control of discrete-event systems. Dr. Ushio is a Member of SICE and ISCIE.

Takashi Hikihara (M’85) was born in Kyoto, Japan, in 1958. He received the B.E. degree from Kyoto Institute of Technology in 1982 and the M.E. and Dr. E. degree from Kyoto University in 1984 and 1990, respectively. He is currently a Professor with Kyoto University. His research interests include nonlinear dynamics in electrical circuits, systems, and magneto-mechanical structure. He is also interested in the development of magnetic levitation system. Dr. Hikihara is a Member of IEICE Japan, IEE Japan, and APS. Since 1998, he has been an Associate Editor of the Transactions of IEICE and IEE Japan.