1
A planar piezoelectric drive with a stepping and a resonant operation mode S. Devos, W. Van de Vijver, K. Decoster, D. Reynaerts, H. Van Brussel Katholieke Universiteit Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, 3001 Heverlee, Belgium.
[email protected]
Abstract Low speeds and smooth movements are difficult to achieve with resonant piezoelectric motors. The presented XY-drive combines the fast positioning capability of resonant motors with a stepping function enabling fine positioning in the nanometer range. A mathematical formulation to obtain the resulting motion from the driving signals to the piezoactuators has been derived for the resonant mode. Experimental results in the resonant mode show that the no-load speed of the slideway corresponds the horizontal speed of the contact point at the highest vertical position of the elliptical motion.
Introduction
The use of piezoceramic resonant linear motors has led to a new generation of high-precision positioning stages(1). Compared to conventional ball screws or lead screws driven by stepper or DC-motors, resonant motors have several advantages: they (1,2) are compact and generate a high torque at low speeds . Furthermore, they are particularly adapted to work in high vacuum and in conditions where no magnetic field is allowed. Many bimodal resonant piezoelectric motors have been designed over (3,4) the last few years, some of which are commercially available . However, this motor type is not able to generate a defined (1) motion without position feedback . Because the propulsion is based on friction between an oscillating piezo-element and a second body, there is no inherent accuracy. The characteristic dynamics of the motor make a very low speed and smooth movements difficult to achieve. On the other hand, several nonresonant piezosteppers were developed, resulting in very accurate (5,6) motors, but with limited speed . This paper presents a new type of XY-drive, that is designed to work in a resonant actuation mode, as well as in a non-resonant stepping mode, by means of four piezoelectric actuators. This way, the speed of a resonant drive is combined with the accuracy of a stepping motor for a virtually unlimited travel in the XY-plane. This paper will focus on the resonant operation mode.
Description of the drive
Fig. 1 The planar piezoelectric drive combining a resonant and a stepping positioning mode.
As shown in Figure 1, the piezoelectric drive consists of two halfellipses that are connected to the fixed frame by leaf springs. Four piezoelectric actuators are mounted inside, able to shift the tip relatively to the fixed frame by bending the leaf springs. A slideway can be actuated by pressing it onto the tip of the drive. This is an (7) extension of the linear drive presented before .
p3
V H
x x
=
horizontal mode
0,2
1 1 1 -1
p1 p3
V
x
H
x
vertical mode
Ahx= Ahy
Av
0,1
10 0º
11
12
13
14
ϕhx=ϕhy
-90 º
(2 )
-180 º 11
10
13
12
14
p4
V H
y y
=
1 1 1 -1
p2 p4
Z
zx
z x = R e [ V x . A v ( f ) . e jϕ v(f) ] (4 ) jϕ hx(f) ] h x( f ).e
x
Σ
X
x
Z
Z
(5 ) y
y
16
Fig. 2 Frequency Response Functions (FRF) between the excitation signals and the motion of the tip for the vertical mode and the two horizontal modes.
z y= R e [V y .A v( f) . e
H
15
frequency [kHz]
x = R e [H x. A
V
16
ϕv
z = z x+ z y
p2
15
frequency [kHz]
(3 )
(1 )
p1
0,3
0
Phase(FRF) [ º]
In the resonant operation mode, the tip performs an elliptical motion to drive the slider that is pushed against the tip. The desired driving direction can be any direction in the XY-plane, therefore the necessary generated motion of the tip is in the plane, determined by this direction and the Z-axis. The speed of the motor can be adjusted by the horizontal amplitude of the elliptical motion and the thrust force can be adjusted by the vertical amplitude of the elliptical motion. To obtain a large vertical and horizontal stroke of the tip with the given limited stroke of the piezoactuators, amplification by mechanical resonance is used. Therefore, the drive is designed to have two horizontal and one vertical eigenmode in the same frequency region. Figure 2 shows Frequency Response Functions (FRF) defined as the ratio of the excitation signals and the corresponding motion of the tip. For a symmetrical structure, the eigenfrequencies and damping ratios of the two horizontal modes will coincide.
Abs(FRF) [10-6m/V]
Driving signals in the resonant operation mode
jϕ v(f)
zy
]
(6 )
y = R e [H y. A
hy(f).
e
jϕ
hy(f)
]
y
Fig. 3 Relation between driving signals and motion of the tip in the resonant operation mode. Proc. of 4th euspen International Conference- Glasgow, Scotland (UK), May-June 2004
Σ
Y X
y
Y
2 When the four piezoactuators excite the three eigenmodes in that frequency region, the tip performs an amplified elliptical motion in space. The vertical eigenmode is excited when two opposing piezoactuators are driven in phase, a horizontal eigenmode is excited when the two opposing piezoactuators corresponding to the horizontal direction are driven in antiphase. This is formulated in Figure 3 by formula (1) for the X-direction and formula (2) for the Y-direction. Where: pi = Pˆi .e j ( 2πf .t +θ i ) is the phasor representation of the excitation voltage to piezoactuator number i for i =1,2,3,4 with
Pˆi = voltage amplitude, θ = phase shift, f = frequency, t = time.
Thus the real excitation voltage to piezoactuator i can be expressed as
Re( pi ) = Pˆi cos(2πft + θ i ) .
Vx is the phasor representation of the excitation signal of the vertical mode by the piezoactuators in X-direction Vy is the phasor representation of the excitation signal of the vertical mode by the piezoactuators in Y-direction Hx is the phasor representation of the excitation signal of the horizontal mode in X-direction by the piezoactuators in X-direction Hy is the phasor representation of the excitation signal of the horizontal mode in Y-direction by the piezoactuators in Y-direction
Working at resonance implies an amplitude amplification and a phase shift of the resulting motion versus the excitation voltage (see Figure 2), which is expressed in phasor notation in formula (3) and (5) in Figure 3 for the vertical mode, as formula (4) for the horizontal mode in X-direction and formula (6) for the horizontal mode in Y-direction. This formulation leads to the following relation between the input voltages to the four piezoactuators and the resulting elliptical motion of the tip:
⎧x = Pˆ1.Ahx ( f ).cos(2πf .t +θ1 + ϕhx ( f )) − Pˆ3 .Ahx ( f ).cos(2πf .t +θ3 + ϕhx ( f )) ⎪ ⎪⎪ y = Pˆ2 .Ahy ( f ).cos(2πf .t + θ 2 + ϕhy ( f )) − Pˆ4 .Ahy ( f ).cos(2πf .t +θ 4 + ϕhy ( f )) ⎨ ⎪z = z x + z y = Pˆ1.Av ( f ).cos(2πf .t + θ1 + ϕv ( f )) + Pˆ2 .Av ( f ).cos(2πf .t +θ 2 + ϕv ( f )) ⎪ ˆ ⎪⎩+ P3 .Av ( f ).cos(2πf .t + θ3 + ϕv ( f )) + Pˆ4 .Av ( f ).cos(2πf .t + θ 4 + ϕv ( f )) This formula shows that the motion of the tip can be adjusted by the amplitudes Pˆ1 , Pˆ2 , Pˆ3 , Pˆ4 and the phases
θ 1 ,θ 2 ,θ 3 ,θ 4 of
the four
piezoactuators.
Experiments in the resonant operation mode Figure 4 shows a measurement of the motion of the tip in the X-Z plane by a laser vibrometer. The piezoactuators in the Y-direction were not excited. The excitation frequency was 13264 Hz and the voltage amplitude was 10V. The different motions were obtained by Fig. 4 Elliptical motion of the tip changing the phase shift θ3. Figure 5 shows that the measured no-load speed of the slider actuated by the motor corresponds to the horizontal speed of the contact point at the highest vertical position of the elliptical motion. The desired speed of the slider can thus be adjusted as well by the amplitude as by the phase of the elliptical motion. This also means that no slip occurs without load. Functional slip occurs when the motor has to carry a load. For example, a speed of 30 mm/s was measured for a traction force of 0.25 N with a preload force of 2N while the no-load speed was 60 mm/s.
Conclusions A planar piezoelectric drive has been built that combines the fast positioning capability of resonant motors with a stepping function enabling fine positioning in the nanometer range. A method to obtain the necessary driving signals to the four piezoactuators for a desired XY-motion has been derived. Experimental results in the resonant operation mode show that the no-load speed of the slideway corresponds the horizontal speed of the contact point at the highest Fig. 5 Relation between the elliptical motion and the vertical position of the elliptical motion. resulting speed of the slider.
Acknowledgements This research is sponsored by the Fund of Scientific Research (FWO)- Flanders, project G0336.98 and G0138.03.
References (1) (2) (3) (4) (5) (6) (7)
Bromme, A., Scheurle,R. (2002): ‘Application of Piezoceramic Ultrasonic Linear Motors in XY-linear Precision Stages’, Proc. of Actuator 2002, 8th International Conference on New Actuators, Bremen, 10-12 June 2002, pp. 506-511. http://www.feinmess.de Ueha, S., Tomikawa, Y. (1993): 'Ultrasonic Motors: Theory and Applications’, Oxford Science Publications. Karasikov, N. , Ganor, Z. (2000): ‘A Novel Non-Magnetic Miniature motor for Ultra High Vacuum Applications’, Nanomotion Ltd. http://www.nanomotion.com Claeyssen, F. et al.: ‘Versatile Ultrasonic Piezo Drive for Direct-Drive Motorization ‘, Proc. of Actuator 2000, 7th International Conference on New Actuators, Bremen, 19-21 June 2000, pp. 262-265. http://www.cedrat.com Versteyhe, M.: ‘Development of an Ultra-stiff Piezostepper with Nanometer Resolution’, Ph. D. Thesis K.U.Leuven, D/2000/7515/15, 2000, ISBN 90-5682-243-2. Henderson, D., Fasick, J. (2000): ‘The Inchworm Piezoelectric Stepping Motor - Advances in Design, Performance and Applications’, Proc. of Actuator 2000, 7th International Conference on New Actuators, Bremen, 19-21 June 2000, pp. 451-455. Devos S., De Volder M., Reynaerts D., Van Brussel H., (2003) ‘A Piezoelectric Drive combining a Resonant and a Stepping Positioning Mode’ Int. Topical Conf. on Precision Engineering, Micro Technology, Measurement Techniques and Equipment, Aachen, 109-112.
Proc. of 4th euspen International Conference- Glasgow, Scotland (UK), May-June 2004
