Sensors and Actuators A 105 (2003) 247–254

Dynamic analysis of cylindrical ultrasonic motor under external excitation

Xiaopeng Wanga, Chong Jin Onga,∗, Chee Leong Teoa, Sanjib K. Pandaba Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

b Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Received 9 September 2002; received in revised form 16 April 2003; accepted 17 April 2003

Abstract

This paper studies the dynamic behavior of Cylindrical Ultrasonic Motor (CUSM) under the external excitation of two sinusoidal voltagesignals. Such a study is useful for designing effective servo control scheme for the CUSM. In particular, it is important to understand how theoutput speed is related to the three possible control inputs—voltage amplitude, driving frequency and phase difference of the two excitationvoltage signals. In this paper, the Timoshenko beam model and mode-summation method are used to analyze the dynamic response of thelead zirconate titanate (PZT) stator tube of the CUSM under external excitation. Then, the input–output relations between the speed outputand control inputs are established. Finally, the validity of this theoretical analysis is verified experimentally.© 2003 Elsevier B.V. All rights reserved.

Keywords: Cylindrical Ultrasonic Motor; Timoshenko beam model; Mode-summation method; Servo control

1. Introduction

The Cylindrical Ultrasonic Motor (CUSM) developed atthe National University of Singapore utilizes a monolithiclead zirconate and titanate (PZT) tube as its stator. Like othertypes of Ultrasonic Motor (USM), CUSM has many attrac-tive properties that conventional electromagnetic motors donot possess, such as low-speed high-torque characteristics,quiet operation, high holding torque, quick response and lowelectromagnetic emission[1].

Past publications have focused on the fabrication as-pects[2–5] and kinematic analysis[6] of the CUSM. Theiremphases are mainly on the structural optimization of theCUSM. Timoshenko beam model is used in[2] to analyzethe vibration of the stator tube of the CUSM. However, itsmain purpose is to find the resonant frequency and modeshape of the stator, and little attention is paid to the rela-tions between the output speed and various inputs to theCUSM. Such relations are important as they are needed fordesigning effective servo control[8] for the CUSM. Thework presented here addresses this issue. In particular, thiswork looks at the relations between the output speed of theCUSM and its three possible control inputs—voltage am-plitude, driving frequency and phase difference of the twoexcitation voltage signals.

∗ Tel.: +65-68742217; fax:+65-67791459.E-mail address: mpeongcj@nus.edu.sg (C.J. Ong).

The rest of this paper is arranged as follows.Section 2introduces the structure of the CUSM. InSection 3, the Tim-oshenko beam model and the mode-summation method areused to analyze the dynamic response of one phase of thestator PZT tube of the CUSM under external sinusoidal volt-age excitation.Section 4gives the two-phase analysis of theCUSM, in which the speed–voltage and speed–frequencyrelations are established for a 90 phase difference. Theoperating condition for non-90 phase difference is alsodiscussed in this section.Section 5 gives a brief sum-mary of the theoretical analysis. The experimental resultsthat verify the theoretical analysis are given inSection 6.Finally, Section 7 summarizes the results obtained inthis paper.

2. Structure of CUSM

Fig. 1 shows three CUSMs with outer diameters of 2,5 and 10 mm, respectively. All of them are constructed ina similar fashion as shown inFig. 2. One brass cap ‘3’is glued on each end of the PZT stator tube ‘4’. In themotor’s operation, the two rotors ‘2’ rotate together withthe shaft ‘1’. The spring ‘6’ provides the preload, which isnecessary to generate the frictional force between the statorand rotors. This construction is similar to that reported in[3–5]. The only major difference is in the fabrication of thestator tube. In[3–5], PZT thin film is deposited on a titaniumtube, which functions as the stator of the CUSM. In this

0924-4247/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0924-4247(03)00127-4

248 X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254

Fig. 1. CUSMs of different sizes.

Fig. 2. Cross-section of CUSM.

work, a monolithic PZT tube is used as the stator. Such amonolithic PZT stator construction was first reported in[7].The monolithic PZT tube can produce larger torque than thetitanium tube with thin PZT film.

As shown inFig. 3, four electrodes are evenly pasted onthe outer surface of the PZT tube. A common inner electrodeis pasted on its inner surface. The four outer electrodes aredivided into two groups, i.e. phases A–A′ and B–B′. Thepolarizing direction of the PZT tube is aligned from theouter surface to the inner surface along the radial directionas shown. In the CUSM’s operation, these two phases areexcited by two separate sinusoidal voltage signals with aspecific phase difference.

Fig. 3. Cross-section of PZT tube.

3. Dynamic analysis of phase A–A′

3.1. Bending of PZT Tube

Suppose a DC voltage is applied to phase A–A′ across Aand A′. The equivalent circuit of the two capacitors of phaseA–A′ is shown inFig. 4. Due to the transverse piezoelectriceffect, side A expands and side A′ contracts. Consequently,the PZT tube undergoes a static deformation, in which thetwo ends of the tube bend towards the contracting side A′,as shown inFig. 5.

The deformation in the PZT tube is caused by the stressgenerated internally by the transverse piezoelectric effect ofPZT material. In the case shown inFig. 5, the expansion ofside A and the contraction of side A’ is equivalent to theeffect on the tube due to an externally applied uniformlydistributed bending momentMZ as shown.

For simplicity, we neglect the hysteresis effect of piezo-electric material, and assume that the stress generatedwithin the piezoelectric material is linearly proportionalto the strength of the externally applied electric field, andaccordingly the applied voltage. Thus, the bending mo-mentMZ can be expressed as a linear function of inputvoltage,

MZ(t) = µVapp(t) = µVampsin ωt (1)

whereMZ(t) is the time-varying uniformly distributed bend-ing moment induced by the applied voltageVapp(t) acrossphase A–A′. Vapp(t) is a sinusoidal function of time withamplitudeVamp and frequencyω with µ being a constant ofproportionality depending on the inherent properties relat-ing to the material and geometry of the PZT tube, such asits piezoelectric strain constant, Poisson’s ratio, modulus ofelasticity, and shape of cross-section, etc.[9]

Fig. 4. Equivalent circuit for phase A–A′.

Fig. 5. Bending moment generated for phase A–A′.

X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254 249

3.2. Timoshenko beam model for cantilever beam

In [6], the stator PZT tube is modelled as a free–free beam.However, in this work, each half of the PZT tube inFig. 5is modelled as a uniform cantilever beam, and two suchcantilever beams are joined together in the middle. Due tothe symmetrical setup, we will concentrate on the cantileverbeam on the right only in the following analysis. There aretwo reasons why the cantilever beam model is preferred.Firstly, if the free–free beam model is used, we need toclamp the stator tube at its two nodal points in practicalapplications. Nevertheless, the locations of these two nodalpoints are not easily determined. Secondly, even if these twonodal points can be found accurately, holding the stator tubethis way will inevitably affect the vibration of the stator tubebecause the cross-sections at the two nodal points also haveangular movements.

In contrast, if the cantilever beam model is used, we willnot have the above problems. When the PZT tube is clampedas shown inFig. 5, the cross-section in the middle of thetube remains stationary during the operation of the CUSM.Also, we will not have difficulty in locating the nodal points.

It is well known that the Bernoulli–Euler beam model isadequate for analyzing the lower-mode lateral vibration ofbeams whose cross-sectional dimensions are small in com-parison with their lengths (typically, the ratio of 0.1 is usedas a guideline). However, the PZT tube discussed here hasan outer diameter of 10 mm and a length of 30 mm. In thiscase, the Bernoulli–Euler beam model is inadequate. Hence,in line with other analysis[6], the Timoshenko beam modelis used.

The Timoshenko beam model takes into account the rotaryinertia and shear deformation of the beam. The bendingvibration of a uniform beam, such as the PZT tube, can bedescribed by two coupled equations for the total deflectiony and the bending slopeψ as[11]

EI∂2ψ

∂x2+ kAG

(∂y

∂x− ψ

)− J

∂2ψ

∂t2= 0 (2)

m∂2y

∂t2− kAG

(∂2y

∂x2− ∂ψ

∂x

)= 0 (3)

whereE is the modulus of elasticity,G the shear modulus,I the area moment of inertia of cross-section,A the area ofcross-section,J the rotary inertia,m the mass per unit lengthandk the numerical shape factor for the cross-section.

We wish to develop a dynamical model for the total de-flection y in (2) and (3) under the external excitation(1)using the mode-summation method (Section 3.3). To do so,we need to find the normal modes for total deflectiony andbending slopeψ. The derivations of the frequency equationsand normal modes under different boundary conditions arepresented in detail in[12]. In the following, we give onlythe main results for the case of cantilever beam.

When the beam vibrates laterally in itsith naturalmode, the total deflection and shear deformation can be

expressed as

y(x, t) = Yi(x)ejωit, ψ(x, t) = Ψi(x)e

jωit (4)

whereYi(x) is theith normal mode of total deflectiony(x, t)and Ψi(x) is the ith normal mode of shear deformationψ(x, t). Substitute(4) into (2) and (3), and we obtain thefollowing two equations

−ω2i JΨi = EIΨ ′′

i + kAG(Y ′i − Ψi) (5)

−ω2i mYi = kAG(Y ′′

i − Ψ ′i ) (6)

The general solutions ofYi(x) andΨi(x) can be found as

Yi(x)=C1 coshbiαix+ C2 sinhbiαix

+C3 cosbiβix+ C4 sinbiβix (7)

Ψi(x)=C′1 sinhbiαix+ C′

2 coshbiαix

+C′3 sinbiβix+ C′

4 cosbiβix (8)

where

αi = 1√2

−(r2 + s2)+

[(r2 − s2)2 + 4

b2i

]1/2

1/2

βi = 1√2

+(r2 + s2)+

[(r2 − s2)2 + 4

b2i

]1/2

1/2

b2i = m

EIω2i , r2 = I

A, s2 = EI

kAG

The boundary conditions for cantilever are:

Y(0) = 0, Ψ(0) = 0, at clamped end (9)

Ψ ′(L) = 0, Y ′(L)− Ψ(L) = 0, at free end (10)

where primes forY and Ψ represent differentiation withrespect to the length of the beam.

Solving the eigenvalue problem using the general solu-tions (7) and(8), and the boundary conditions(9) and(10)leads to the following frequency equation from whichbi (interms of natural frequencyωi) for the ith vibrational modecan be determined

2+ [b2i (r

2 − s2)2 + 2] coshbiαi cosbiβi

− bi(r2 + s2)

(1 − b2i r

2s2)1/2sinhbiαi sinbiβi = 0 (11)

Consequently, the correspondingith normal mode can beobtained as

Yi(x)=D[ coshbiαix− λiτiδi sinhbiαix

− cosbiβix+ δi sinbiβix] (12)

Ψi(x)=H

[coshbiαix+ θi

λiτisinhbiαix

− cosbiβix+ θi sinbiβix

](13)

250 X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254

whereD andH are correlated constants and

λi = αi

βi

τi = α2i + r2

α2i + s2

δi = (1/λi) sinh biαi − sin biβiτi coshbiαi + cosbiβi

and

θi = − λi sinh biαi + sin biβi(1/τi) coshbiαi + cosbiβi

In addition, it can be shown that the normal modesYi(x)andΨi(x) satisfy the following condition of orthogonality[10,13]∫ L

0[mYi(x)Yj(x)+ JΨi(x)Ψj(x)] dx =

0 for i = j

Mj for i = j

(14)

whereMj is defined as the generalized mass. This conditionis necessary in the mode-summation method discussed next.

3.3. Mode-summation method

The exact expression for the forced vibration of the PZTtube is not easily found because it is a continuous systemwith an infinite degrees of freedom. The modal analysismethod, also known as mode-summation method[10], pro-vides a convenient way to approximate the dynamic behav-ior of such a system subject to external excitations.

In the case of CUSM, the external excitation takes theform of a uniformly distributed bending momentMZ(t) in(1). Hence, the two coupled Timoshenko beamEqs. (2)and (3)can be rewritten as

EI∂2ψ

∂x2+ kAG

(∂y

∂x− ψ

)− J

∂2ψ

∂t2= MZ(t) (15)

m∂2y

∂t2− kAG

(∂2y

∂x2− ∂ψ

∂x

)= 0 (16)

Following from(4), the beam’s response under the externalexcitationMZ(t) is given by two infinite sum of its normalmodes multiplied by the corresponding generalized coordi-natesqi’s

y(x, t) =∑i

Yi(x)qi(t) (17)

ψ(x, t) =∑i

Ψi(x)qi(t) (18)

whereYi(x) andΨi(x) are the normal modes found in(12)and(13). Using(17) and(18) in (15) and(16), we obtain

J∑i

qiΨi =∑i

qi[EIΨ ′′i + kAG(Y ′

i − Ψi)] −MZ(t) (19)

m∑i

qiYi =∑i

qi[kAG(Y ′′i − Ψ ′

i )] (20)

Note that the right sides of(5) and(6) are the coefficients ofthe generalized coordinatesqi in the forced vibrationEqs.(19) and (20), so that we can rewrite(19) and(20) as

J∑i

qiΨi = −∑i

qiω2i JΨi −MZ(t) (21)

m∑i

qiYi = −∑i

qiω2i mYi (22)

Multiplying these two equations byYi dx andΨi dx, respec-tively, and adding, we obtain

∑i

qi

∫ L

0(mYiYj + JΨiΨj)dx

+∑i

qiω2i

∫ L

0(mYiYj + JΨiΨj)dx

= −MZ(t)

∫ L

0Ψj(x)dx (23)

and from(14), we have

qj + ω2j qj = −MZ(t)

Mj

∫ L

0Ψj(x)dx, j = 1,2, . . . (24)

whereMj is the generalized mass defined in(14), andωjis the natural frequency for thejth vibrational mode of thebeam.

As we intend to operate the CUSM near its first naturalfrequency, we can further simplify(17) and(18) into

y(x, t) =∑i

Yi(x)qi(t) ≈ Y1(x)q1(t) (25)

ψ(x, t) =∑i

Ψi(x)qi(t) ≈ Ψ1(x)q1(t) (26)

Consequently,(24) becomes

q1 + ω21q1 = −MZ(t)

M1

∫ L

0Ψ1(x)dx = εVampsinωt (27)

with ε = −(µ/M1)∫ L

0 Ψ1(x)dx, where the last equalityfollows from (1).

The above expression is an undamped second-order sys-tem with sinusoidal external excitation. However, there arealways dampings in real systems. Thus, we assume the ex-istence of the generalized viscous damping force expressedas

Fd = c q1 (28)

wherec is a constant of proportionality. Consequently, thesecond-order system in(27) becomes

q1 + c

M1q1 + ω2

1q1 = εVampsin ωt (29)

which is a damped second-order system. The value of theconstantC can be determined experimentally. The dynamic

X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254 251

responses of the PZT tube with respect to different kindsof excitations can be found from this second-order dynamicmodel.

The particular solution to(29) is a steady-state oscillationof the same frequencyω of the excitation,

q1(t) = εVamp/(M1ω21)√

[1 − (ω/ω1))2]2 + [2ζ(ω/ω1)]2sin(ωt − ϕ)

(30)

in which the damping factorζ = c/(2M1ω1) and the phaselag ϕ = arctan((2ζ(ω/ω1))/(1 − (ω/ω1)

2)).In Eq. (30), we can see how the first mode generalized

coordinateq1(t) is related to the amplitudeVamp and fre-quencyω of the applied sinusoidal voltage signal. Clearly,the amplitude ofq1(t) is proportional toVamp. The ampli-tude ofq1(t) is maximum when the excitation frequencyωis around the first mode natural frequencyω1.

With Eqs. (25) and (30), we can easily specify the lateralvibration of each point along the central axis of the beam.For example, if we are interested in the lateral vibration ofthe point at the end of the central axis of the beam of lengthL, its vibration can be expressed as

yL(t) = Y1(L)q1(t) = γ sin(ωt − ϕ) (31)

where

γ = Y1(L)εVamp/(M1ω

21)√

[1 − (ω/ω1)2]2 + [2ζ(ω/ω1)]2(32)

The amplitude of the lateral vibration of the PZT tubeduring the CUSM’s operation is experimentally measuredto be only of the order of a few hundred nanometers. Ac-cordingly, the rotational movement of every cross-sectionof the PZT tube is very small. Thus, for each point on anycross-section of the beam, we can practically ignore thelateral displacement caused by the rotational movement ofthat cross-section. Under this assumption, each point on anycross-section of the beam has the same lateral displacementin theY direction. Hence, the lateral vibration of each pointon the end of the beam is the same and can be described by(31).

4. Two-phase analysis

From the above analysis, it suffices to consider the lateralvibration of the center of the end surface of the PZT tubeonly. When both phases A–A′ and B–B′ are excited simulta-neously by sinusoidal voltage signals with the same ampli-tude and a specific phase difference, the lateral vibrationalcomponents of the center of the end surface due to the exci-tations on phase A–A′ and phase B–B′ are perpendicular toeach other in a plane normal to the axis of the stator tube.Without the loss of generality, we assume the phase shiftϕ

in (31) to be zero and also drop the subscriptL of y. Let thephase difference between the input voltage signals to phase

Fig. 6. Trajectories of center of end surface with differentφ’s.

A–A′ and phase B–B′ be denoted byφ. Then the motion ofthe center of the end surface can be described in A–A′ andB–B′ coordinates as:

yA–A ′(t) = γ sin ωt (33)

yB–B′(t) = γ sin(ωt − φ) (34)

The trajectories of the center of the end surface with dif-ferent phase differences are shown inFig. 6. Point O is theoriginal center of the stator tube when no excitation volt-age is applied,γ is the vibrational amplitude of both phasesA–A′ and B–B′ as given by(31)and(32). As shown, the tra-jectory of end surface center is a perfect circle when phasedifferenceφ = 90. The trajectory becomes an ellipse when0 < φ < 90 (the case whenφ = 45 is shown). Whenφ = 0, this trajectory becomes a straight line.

We now consider the motions of the end surface center ofthe stator beam in two cases—the phase differenceφ = 90;andφ = 90.

4.1. Phase difference φ = 90

When the phase differentφ = 90, the motion of the endsurface center of the stator tube is a perfect circle, as given by

y2A–A ′ + y2

B–B′ = γ2 (35)

As shown inFig. 7, vA–A ′ = yA–A ′ is the velocity ofthe end surface center C due to the excitation applied onphase A–A′; vB–B′ = yB–B′ is the velocity of C due to theexcitation applied on phase B–B′, andvC is the instantaneousvelocity of C, which is the vector sum ofvA–A ′ andvB–B′ .

When phase differenceφ = 90, velocityvC has only cir-cumferential velocity component and a constant magnitude

vC =√v2

A–A ′ + v2B–B′

=√γ2ω2 sin2ωt + γ2ω2 sin2(ωt − 90)

= γ ω (36)

252 X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254

Fig. 7. Motion of center of end surface (φ = 90).

Accordingly, inFig. 5, we can visualize that the bent cen-tral axis L′ of the stator tube rotates around the originalcentral axisL with an angular velocityω. With this visual-ization, the motions of center point C and the contact pointP between the stator and rotor on the end surface are shownin Fig. 8.

In the right half ofFig. 8, the inner circle is the trajectoryof the center point C; the outer circle is the trajectory ofthe contact point P between the stator and rotor. Note thatthese two circles are not drawn to scale. The inner circleis much smaller than the outer one. If we denote the outerradius of the PZT tube asR, the distance from the contactpoint P to the original center O isR−γ. Typically, we haveγ R, thus we can approximateR− γ asR, as shown inFig. 8.

It is not difficult to realize that the points P, O and Care always on a straight line. As there is no torsion in thestator tube, the instantaneous velocityvC of end surfacecenter C and the instantaneous velocityvP of the point onthe stator at the contact point P are the same both in di-rection and magnitude(36). The estimated angular speedΩrotor of the rotor under the assumption that no slippagetakes place between the stator and rotor can be expres-sed as

Ωrotor = 2πvP

R− γ≈ 2π

vC

R= 2π

γ ω

R(37)

As shown in Fig. 8, the contact point P between thestator and rotor is constantly moving in the directionvR.It rotates with an angular speedω and in a direction op-

Fig. 8. Motion of contact point (φ = 90).

Fig. 9. Motion of contact point (0 < φ < 90).

posite to the moving speedvP of the point on the sta-tor at the contact point P. InFig. 8, the moving directionof the contact point is denoted byvR; the moving direc-tion of the point on stator at the contact point is denotedby vP.

Another conclusion we can make fromFig. 8 is thatthe rotating direction of the rotor can be controlled by thephase differenceφ between the two excitation voltage sig-nals. As we can tell fromFig. 7, if the excitation volt-age signal to phase A–A′ leads that to phase B–B′ by 90,the rotation of that center point C is counterclockwise; ifthe excitation voltage signal to phase A–A′ lags that tophase B–B′ by 90, the rotation of that center point C isclockwise.

4.2. Phase difference φ = 90

Now we consider what happens if the phase differenceφ

is not 90. In a particular case inFig. 9, phase A–A′ leadsphase B–B′ by a phase difference 0 < φ < 90. In this case,the trajectory of the contact point P is still approximately acircle (because the vibrational amplitude of the stator tubeis very small withγ R) and moving in the directiondenoted byvR. However, the trajectory of the center pointC now becomes an ellipse. Points P, O and C are still in astraight line as shown. The instantaneous speedvC of endsurface center C and the instantaneous speedvP of the pointon the stator at the contact point P are still the same both indirection and magnitude.

Note that now the speedvP of the point on stator at thecontact point P consists not only of circumferential compo-nentvP1 but also radial componentvP2. Moreover, the mag-nitude of the circumferential componentvP1 is not a con-stant anymore. Both this time-varying circumferential ve-locity componentvP1 and the radial velocity componentvP2result in a lot of slippage between the stator and rotor, andhence result in substantial amount of heat generated, whichis undesirable in the operation of the CUSM. Consequently,it is not recommended to operate the CUSM using a non-90phase difference.

X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254 253

5. Summary

Now let us summarize the conclusions we make from theabove qualitative analysis.

• The output speed of the CUSM is linearly related to theamplitude of the sinusoidal excitation voltage signals. Thisis a desirable property if we were to design the servocontroller using the voltage amplitude as the control input.

• The output speed of the CUSM is related to the frequencyof the sinusoidal input voltage signals through(32) and(37). The output speed of the CUSM is maximum whenthe driving frequencyω is around the first mode naturalfrequencyω1 of the PZT tube. However, this nonlinearrelation may make the driving frequency not a preferredcontrol input.

• It is desirable to fix the phase difference between the twodriving voltages at 90 in order to reduce slippage andheat losses. Besides, the rotating direction of the CUSMcan be controlled by driving the CUSM with a+90 or−90 phase difference.

However, the above analysis also has the following assump-tions.

• No slippage between the stator and rotor is assumed whenphase differenceφ = 90. There is no guarantee that thisassumption is true in the actual operation of the CUSM.

• The existence of the rotors is not considered. The presenceof the rotors may influence the vibration of the PZT tube.Unfortunately, this interaction is too complicated to bemodelled.

In summary, the above analysis provides valuable insightinto how the CUSM works and how the output speed isrelated to the three possible control inputs. We now discussthe experimental results and the suitability of each input ascontrol to the CUSM.

6. Experimental results

We can verify the previous theoretical analysis experi-mentally. In the experiments conducted by the authors, theCUSM is driven by a two-phase DC–AC inverter whoseDC-Link voltage is 20 V. Other experimental parameters aregiven inTable 1.

Table 1Experimental parameters

DC-Link voltage 20 VDriving frequency 45.5 kHzPhase difference 90Preload 2 NSampling frequency 1000 HzEncoder resolution 20,000 pulses per revolution

Fig. 10. Average steady-state speed vs.voltage amplitude.

As shown by(37)and(32), the angular speedΩrotor of theCUSM is linearly dependent on the amplitudeVamp of theexcitation voltage signal.Fig. 10 shows the actual relationbetween the average steady-state speed and amplitude of theexcitation voltage signal obtained through experiment. Asshown, the average speed is approximately linearly relatedto voltage amplitude. From the control point of view, thelinear relation and absence of the dead band in output speedare favorable for the speed servo control design.

Eqs. (37) and (32)also suggest that the angular speed ismaximum when the driving frequencyω is around the firstmode natural frequencyω1 of the PZT stator tube of theCUSM.

Fig. 11shows the relation between the actual steady-statespeed and the driving frequency of the excitation voltage.As shown, the experimental result is quite consistent withthe prediction made from the theoretical analysis. From thefigure, it does not seem to be easy to control the angularspeed of the CUSM through driving frequency due to thesevere nonlinearity.

Fig. 11. Average steady-state speed vs. driving frequency.

254 X. Wang et al. / Sensors and Actuators A 105 (2003) 247–254

Fig. 12. Average steady-state speed vs. phase difference.

Analysis inSection 4.2indicates that it is difficult to pre-dict how the angular speed of the CUSM is related to thephase difference between the two sinusoidal excitation sig-nals in a principled way due to the slippage that takes placewith a non-90 phase difference.

However, we are able to obtain this relation through ex-periment. InFig. 12, the phase difference is changed from0 to 360 with 10 increments. As shown in the figure, theaverage speed can be controlled rather smoothly via phasedifference control. However, we can observe dead bands inthe output speed when phase difference is close to 0, 180,360.

In Fig. 12, we can also see how the angular speed canbe reversed by changing the phase difference from+90 to+270 (equivalent to−90) as mentioned inSection 4.1.

7. Conclusion

In this paper, the PZT stator tube of the CUSM is mod-elled as two cantilever beam under external bending momentexcitation. By using the Timoshenko beam model and themode-summation method, the relations between the outputspeed and amplitude/frequency of the two excitation volt-age signals are established for a 90 phase difference. It isfound that, theoretically, the output speed is a linear functionof the amplitude of the excitation sinusoidal voltage signal,and the output speed takes maximum value when the driv-ing frequency is around the first natural frequency of thePZT stator tube. The theoretical analysis also shows that a

lot of slippage takes place when a non-90 phase differenceis used.

Experimental results indicate that the results from the the-oretical analysis are quite consistent with the actual oper-ation of the CUSM. The results obtained provide valuableguidelines for the servo control design of the CUSM—it isdesirable to design the servo control scheme using the volt-age amplitude as the main control input while driving themotor at its natural frequency with a 90 phase difference,and the rotating direction of the rotors can be reversed bychanging the phase difference from+90 to−90.

References

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