Performances estimation of a rotary traveling wave ultrasonic motor based on two-dimension analytical model

Ultrasonics 39 (2001) 115±120

www.elsevier.nl/locate/ultras

Performances estimation of a rotary traveling wave ultrasonic motor based on two-dimension analytical model Yang Ming *, Que Peiwen Department of Instrumentation, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai 200030, People's Republic of China Received 10 April 2000; accepted 29 August 2000

Abstract The understanding of ultrasonic motor performances as a function of input parameters, such as the voltage amplitude, driving frequency, the preload on the rotor, is a key to many applications and control of ultrasonic motor. This paper presents performances estimation of the piezoelectric rotary traveling wave ultrasonic motor as a function of input voltage amplitude and driving frequency and preload. The Love equation is used to derive the traveling wave amplitude on the stator surface. With the contact model of the distributed spring-rigid body between the stator and rotor, a two-dimension analytical model of the rotary traveling wave ultrasonic motor is constructed. Then the performances of stead rotation speed and stall torque are deduced. With M A T L A B computational language and iteration algorithm, we estimate the performances of rotation speed and stall torque versus input parameters respectively. The same experiments are completed with the optoelectronic tachometer and stand weight. Both estimation and experiment results reveal the pattern of performance variation as a function of its input parameters. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Ultrasonic motor; Analytical model; Performances estimation

1. Introduction Traveling wave ultrasonic motor is a new type of actuator, which is characterized by high torque at low rotational speed, simple mechanical design and good controllability. It provides a high holding torque even if no power is applied. The motor is operated near the resonance frequency of the stator [1]. How the performances are a€ected by the input parameters such as input voltage, driving frequency and the preload on the rotor, is an interesting question for the high e€ectiveness application and the control of ultrasonic motor. To answer this question, various researchers have undertaken theoretical and experimental studies in recent years [2±7]. Apart from the elastic contact model between the rotor and stator, three methods have been proposed to estimate motor performances. The ®rst one employs ®nite element method, where no electrical parameters are considered. The

*

Corresponding author. Tel.: +86-21-62932810. E-mail address: [email protected] (Y. Ming).

second one uses energy method. However, it may be too complicated to estimate motor performances under di€erent driving frequency. The third one employs electrical equivalent circuit method. Nevertheless, it ignores the variation of the input voltage. Therefore, no method is suitable to estimate motor performances under di€erent driving frequency and input voltage. The purpose of the present paper is to propose an analytical model for the estimation of the motor performances under di€erent input parameters. In Section 2, a two-dimension analytical model of the ultrasonic motor is constructed with the forced response of the stator produced by the piezoceramics bonded under the stator and with the distributed spring-rigid body contact model between the stator and the rotor. Especially, the performances of rotation speed and stall torque under di€erent input parameters are deduced from the analytical model. Then we simulate the rotation speed and stall torque of ultrasonic motor as a function of input voltage and driving frequency and preload. In Section 3, we describe the experiments and their results. The comparison of the simulation and experiment results is also discussed in this section.

0041-624X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 0 ) 0 0 0 5 3 - 6

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2. Performance estimation Usually traveling wave ultrasonic motors can be made a very simple structure. Fig. 1 illustrates a rotary piezoelectric traveling wave ultrasonic motor. The stator is composed of a teethed steel plate with piezoceramic elements bonded onto the underside for exciting vibration. When traveling waves are developed on the surface of the stator, points on the stator surface move in retrograde elliptical motions. The rotor must be pressed against the stator in order to couple the vibrator motion of the stator into the rotational motion of the rotor. Usually, the contact part between the stator and rotor can be divided into three sections. In the ®rst section, stator surface movement speed is larger than that of the rotor, in which the friction drive between the stator and rotor propels the rotor movement. In the second section, the stator surface movement speed is equal to the rotor movement speed. There is an adhesion between the stator and rotor in this section. In the third section, the stator surface movement speed is less than that of the rotor, in which the friction drive impedes the movement of the rotor. The rotor movement speed is determined by the superposition of all these friction forces. 2.1. The amplitude of traveling wave Once the ultrasonic motor is manufactured, its vibration mode and resonant frequency are determined. When studying the performances versus input voltage amplitude and driving frequency, we may ignore the parameters that a€ect the resonant frequency. Considering the resonant frequency may be tracked by driving circuit, we investigate the performance as a function of the preload through the contact variation between the rotor and stator. Then we model the stator as an annular plate and the piezoelectric ceramics as an excitation source without mass and sti€ness. A disturbance will excite the various natural modes of a circular plate in various amounts. The amount of the participation of each mode in the total dynamic response is de®ned by the modal participation factors. When the initial conditions are set to zero, and the pi-

Fig. 1. Construction of the rotary traveling wave ultrasonic motor.

ezoelectric ceramics excitation varies harmoniously with time, we may write excitation as qi …r; h; t† ˆ Qi …r; h†ejxt :

…1†

When the driving frequency is near the k resonant frequency, and is far from the other resonant frequency, the transverse de¯ection of the stator surface in the direction 3, u3 …r; h; t†, can be written [8] Fk R…r† cos…kh†ej…xt /k † r  h i

u3 …r; h; t† ˆ x2k

1

…x=xk †

2

2

ˆ f …q†R…r† cos …kh†ej…xt where 1 Fk ˆ qhNk Z aZ Nk ˆ b

f …q† ˆ x2k

Z

0

a

b 2p

Z

2p 0

‡ 4n2k …x=xk † /k †

2

;

…2†

Q3 …r; h†R…r† cos …kh†r dr dh; 2

‰R…r† cos …kh†Š r dr dh;

Fk r ; h i 1

…x=xk †

2

2

‡ 4n2k …x=xk †

2

n  r  r o R…r† ˆ Jk ak ‡ Ck Ik ak ; a a Ck ˆ

a2k Jk …ak † ‡ …1 a2k Ik …ak † …1

r†fak Jk …ak † r†fak Ik …ak †

k 2 Jk …ak †g : k 2 Ik …ak †g

nk is called the modal damping coecient, /k is phase lag, x is the driving frequency, xk is the resonant frequency, ak is the frequency constant, r is the PoissonÕs ratio, b is inner radius and a is outer radius, h is the halfthickness of the plate, q is the mass density of the material, Jk is Bessei functions, Ik is modi®ed Bessel functions. From Eq. (2), we may obtain e, the amplitude of traveling wave on the stator surface e ˆ f …q†R…r†

…3†

2.2. Two-dimension analytical model We have obtained the amplitude of traveling waves, which take place on the surface of the stator. What we are interested in, however, is how much power can be delivered to a load. To answer these questions, we focus on the vibration recti®cation achieved through frictional coupling between the stator and rotor. In the ultrasonic motor studied, the friction material is Nylon; the stator is made of steel. Considering the sti€ness of the stator is much larger than that of friction material in the rotor, we take the contact condition between the rotor and stator as a distributed spring-rigid contact model.

Y. Ming, Q. Peiwen / Ultrasonics 39 (2001) 115±120

117

2.3. Algorithm of performance estimation Substituting Eqs. (2)±(4) into Eq. (5), we have Z ca Z p / … sin h sin /† nlf …q†Kf cb

/

XrŠR…r†r2 dh dr

 sign‰Vs …r; h† dX ‡ dX; ˆJ dt Fig. 2. Contact model between rotor and stator.

In pursuit of motor performance estimation as a function of its input parameters, coulomb friction model is utilized, that is, l, is a constant. Fig. 2 depicts the model between the stator and the rotor, where frictional material has a thickness of hf and a distributed sti€ness of Kf . The friction material is modeled, as a series of springs placed in parallel along the circumference of the rotor. That is, the displacement of the rotor at any position, h, is una€ected by the piece next to it at h ‡ Dh. If the area of contact for a given preload per wave, FN , extends from / to p /, We take the preload per wave to be related to the pressure distribution by [9] DF …r; h† ˆ Kf e… sin …h†

sin …/††;

…4†

where Kf ˆ Ef d=hf e… sin …h† sin …/†† is the displacement of each spring at any location in the region of contact, d is the contact width of the friction material, hf is the thickness of friction material, Ef is the elasticity modulus of friction material. If J denotes moment of inertia, X denotes the rotation speed of the rotor, d denotes the torque coecient, we have Z Z n DF …r; h†lr2 sign‰Vs …r; h† XrŠdr dh M r

ˆJ

h

dX ‡ d; dt

…5†

where n is the number of the traveling wave, l is the dynamic friction coecient between the stator and the friction material, M is the load torque 8 Vs …r; h† Xr > 0; < 1; sign‰Vs …r; h† XrŠ ˆ 0; Vs …r; h† Xr ˆ 0; : 1; Vs …r; h† Xr < 0: Tangential component of velocity on the stator surface, which acts the rotor on any region of the stator, may be written [10] Vs …r; h† ˆ

nxhe sin …kh r

xt†:

Eq. (5) is the two-dimension analytical model.

…6†

M …7†

where cb, ca are inner and outer contact radius between the rotor and stator. We denote Z ca Z p / T ˆ nlf …q†Kf … sin h sin /† cb

/

XrŠR…r†r2 dh dr:

 sign‰Vs …r; h†

Substituting Eqs. (2) and (3) into T, we have T ˆ

R a R 2p

lnKf Z 

Q3 …r; h†sign‰Vs …r; h† XrŠR…r† cos…kh†r dr dh r  h i2 2 2 2 2 Nk qhxk 1 …x=xk † ‡ 4nk …x=xk † Z p / … sin h sin /†R…r†r2 dh dr: …8†

b

ca

cb

0

/

We can not solve Eq. (7) directly due to its nonlinearity. We have to eliminate its nonlinearity. Therefore the algorithm can be written as follows: (1) For a certain preload, FNi , the contact area, /i to p /i , can be determined through Z p /i Z cb DFr dh dr: …9† FNi ˆ n /i

ca

(2) When the /i is determined, we need a /j , where the movement speed on stator begins to be not less than that of the rotor. In the case, the contact area can be divide into 2nn sections. nn is an integral constant. We assume /j ˆ /i ‡ …n

1†…p=2

/i †=nn;

…10†

where n is an integral, and may take values from one to nn. First we make n ˆ 1, we can get a /j from Eq. (10). Then we can obtain friction drive torque, Tj , from Eq. (8). (3) According to a Tj , we can solve Eq. (7). The rotation speed can be expressed as: Xj ˆ Xmj …1

e

t=tr

†;

…11†

where tr ˆ J =d, Xmj ˆ

Tj

M d

:

…12†

(4) If the rotation speed of the rotor, Xmj , is larger than that of the stator surface at the /j , we take n ˆ n ‡ 1 and then go to step (2). If the rotation speed of the rotor, Xmj , is not larger than that of the stator surface at

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Y. Ming, Q. Peiwen / Ultrasonics 39 (2001) 115±120

the /j , which means that the rotation speed, Xmj , is the result of Eq. (7) under a certain preload FN at this moment, we write Xmj as Xm and Tj as T. (5) Once the rotation speed, Xm , is determined, we can obtain no-load steady rotation speed: X0 ˆ

T : d

…13†

From Eqs. (12) and (13), we can get load performance:   T M M ˆ X0 1 Xm ˆ : …14† d T When ultrasonic motor is stalled, that is X ˆ 0, and dX=dt ˆ 0, we can have stall torque Mmax ˆ T :

Table 1 The physical parameters of the rotary traveling wave ultrasonic motor Thickness 2h Inner radius b Outer radius a Contact inner radius cb Contact outer radius ca Friction material modulus of elasticity Ef Friction material thickness hf Piezoceramic thickness hp PZT-4 transverse d31 Dynamic friction coecient between steel and nylon l Frequency constant ak Torque damping coecient d Modal damping coecient nk

1.3 mm 2 mm 10 mm 5.5 mm 10 mm 254 kg/mm2 0.5 mm 0.5 mm 1.2E 7 mm/v 0.35 4.8 6E 6 1E 5

…15†

For a given preload, the rotation speed versus voltage amplitude and driving frequency can be calculated from Eqs. (13) and (14). Also we obtain performances of stall torque versus voltage amplitude and driving frequency using Eq. (15). When preload is changing, we utilize iteration algorithm, steps (1)±(5), to obtain di€erent steady rotation speed and stall torque at the constant voltage and driving frequency.

3. Experiment and simulation results

Fig. 3. Arrangements of the piezoceramics in the stator.

3.1. Equipment and methods Some experiments have been carried out to prove validity of the performances estimation. The rotation speed is acquired with optoelectric tachometer. The standard weight is used to get the stall torque of the motor. The screw pitch multiplied by the number of the screw pitch applied by the nut equals the spring elastic deformation. Then preload is determined by the spring elastic coecient multiplied with the spring elastic deformation. Changing the amplitude of the input voltage and its driving frequency, we can regulate the rotation speed and stall torque. Changing the number of the screw pitch applied by the nut, we can get di€erent preload. Resonant frequency is one of the most important parameters in the ultrasonic motor. However, when the ultrasonic motor is made, vibration mode is determined, and its resonant frequency is constant. Therefore, when the performances versus the input voltage and driving frequency are estimated, we take the resonant frequency according to the experimental result. For the convenience of the comparison between the motor performance estimation and experiment results, the input data for the estimation such as the voltage and driving frequency variation and preload equals to the values in the experiment condition of the ultrasonic motor. The physical parameters of the motor, which we

Fig. 4. Vibration mode of the stator.

studied, are illustrated in Table 1. Fig. 3 shows the polarity arrangement of the piezoceramics in the stator. The vibration mode of the stator is illustrated in Fig. 4. 3.2. Results Fig. 5 shows speed performance versus its input voltage. In Fig. 5, the preload is 1.72 N, the resonant

Y. Ming, Q. Peiwen / Ultrasonics 39 (2001) 115±120

Fig. 5. Rotation speed versus voltage.

frequency is 103.2 kHz, and the driving frequency is 103 kHz. The relation between the stall torque and input voltage is shown in Fig. 6. In the case, the preload is 1.72 N, the resonant frequency is 103.2 kHz, and the driving frequency is 102.8 kHz. In Figs. 5 and 6, both rotation speed and stall torque is proportional to the amplitude of the voltage. Fig. 7 shows the rotation speed versus its driving frequency variation. In the case, the preload is 1.71 N, the resonant frequency is 103.1 kHz, and the voltage amplitude is 61 V. The relation between stall torque and driving frequency is showed in Fig. 8, where resonant frequency is 102.7 kHz, the preload is 1.69 N, and the voltage amplitude is 54 V. In Figs. 7 and 8, both stead rotation speed and stall torque shows a gradual increase up to the resonant frequency. In Fig. 9, resonant frequency is 103.2 kHz, the preload is 1.72 N, the input voltage is 54 V, and the driving frequency equals 103 kHz. Fig. 9 shows the load performance of ultrasonic motor, which is similar to that of direct-current machine.

119

Fig. 7. Rotation speed versus driving frequency (Hz).

Fig. 8. Stall torque versus driving frequency (Hz).

Fig. 9. Load performance.

Fig. 6. Stall torque versus voltage.

In Figs. 10 and 11, driving frequency is 103 kHz, and the voltage is 61 V. Fig. 10 indicates that the stall torque is directly proportional to the preload. From Fig. 11, we can ®nd that the steady rotation speed increases with the rise in preload before it is less than certain preload, and decreases with its rise after it is larger than the certain value.

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Y. Ming, Q. Peiwen / Ultrasonics 39 (2001) 115±120

4. Conclusion

Fig. 10. The curve of rotation speed and pressing force.

In this paper, we have proposed a two-dimension analytical model of ultrasonic motor using Love equation and distributed spring-rigid body contact model between the rotor and stator. Then the performances of ultrasonic motor as a function of input parameters are deduced and simulated. The experiment results can be interpreted in terms of the simulation results. Both simulation and experiment results provide us an insightful understanding about that how the input parameters contribute to the performances of the rotary traveling wave ultrasonic motor. When the analytical model is constructed, we have taken the resonant frequency of ultrasonic motor as a constant. Therefore, the two-dimension analytical model is not suitable to estimate the performances under di€erent structural parameters due to the resonant frequency variation of ultrasonic motor. For further understanding performances of ultrasonic motor as a function of all its parameters, we are trying to examine the relation between the performances of ultrasonic motor and structure parameters. References

Fig. 11. The curve of stall torque and pressing force.

3.3. Discussion From Figs. 5±8, we can ®nd the coincidence of rotation speed performances between the estimation and the experiment is better than that of the stall torque performance. The reason may be that the error produced by standard weight is larger than that by the optoelectric tachometer. In Figs. 10 and 11, there is too much error between the experiment and estimation results. This implies that our contact model is preliminary, and needs much improvement for the condition of changing preload. As indicated in previous simulation and experiment results, there must exist an optimal preload and an optimal driving frequency, where the output power of ultrasonic motor reaches its maximum.

[1] T. Sashida, An Introduction to Ultrasonic Motors, Oxford Science Publication, 1993. [2] T. Maeno, T. Tasukimoto, A. Miyake, Finite-element analysis of the rotor/stator contact in a ring-type ultrasonic motor, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 39 (6) (1992) 668±674. [3] P. Hagedorn, J. Wallaschek, Traveling wave ultrasonic motors, part I: Working principle and mathematical modeling of the stator, J. Sound Vibr. 155 (1) (1992) 31±46. [4] N.W. Hagood, A.J. Mcfarland, Modeling of a piezoelectric rotary ultrasonic motor, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 42 (2) (1995) 210±224. [5] H. Hirata, S. Ueha, Design of a traveling wave type ultrasonic motor, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 42 (2) (1995) 225±231. [6] H. Hirata, S. Ueha, Characteristics estimation of a traveling wave type ultrasonic motor, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 40 (4) (1993) 402±406. [7] M. Aoyagi, Y. Tomikawa, Simpli®ed equivalent circuit of ultrasonic motor and its application to estimation of motor characteristics, Jpn. J. Appl. Phys. Part 1 34 (5B) (1995) 2752± 2755. [8] W. Soedel, Vibrations of Shells and Plates, Marcel Dekker, New York, 1981, pp. 199±213. [9] M. Kurosawa, S. Ueha, Eciency of ultrasonic motor using traveling wave, J. Acoust. Soc. Jpn. 44 (1) (1988) 40±46. [10] Suganonobiwa, Ultrasonic wave motor, Jpn. J. Horol. Soc. 24 (1995) 63±79.