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TRANSCRIPT
Resonant Modes of Tubular Shells and Their Use
in Ultrasonic Linear Motors Rafael Siva Pippi, Cesar Ramos Rodrigues
Instituto Federal de Santa Catarina – IFSC, Universidade Federal de Santa Maria - UFSM
[email protected], [email protected]
Abstract- The association of two or more resonance modes of
tubular shells on a new type of ultrasonic linear motor is discussed in this paper. Finite element analysis showed that tubular geometries provide several resonance modes with very
close frequencies, indicating that axisymmetrical traveling waves could be generated as a composition of two or more them. Finite element method (FEM) simulations were employed to investigate
resonance modes in tubular structures in a range of dimensions, and how strain distributes along the tubes in resonance. We found, for a tube with a length of 55mm long, and diameter
30mm, that excitation of 14th and 15th modes yields traveling waves with radial and longitudinal amplitudes of 2.8μm and 10.2μm respectively, when excited with two PZT rings driven
with 35Vrms at 64.876kHz.
I. INTRODUCTION
One the most important applications of piezoelectricity is
mechanical actuation devices. There are several ways to use
the piezoelectric effect to mechanical actuation, originating
different classes of actuators with different of characteristics.
According to their driving conditions, piezoelectric actuators
could be classified as [1]: 1- Quasi-static, or 2- resonant.
Resonant actuators are called ultrasonic motors (USMs), and
can be further divided into standing-wave and traveling wave
motors. In both cases, the excitation of a piezoelectric
material, produces deformation waves in a structure (named
vibrator or stator), impelling the points on its surface to
describe elliptical movements. While, in standing wave
motors the spatial distribution of deformation waves remains
static, as suggests the name, in traveling wave ultrasonic
motors it moves along the vibrator. Traveling wave motors
need two phase-shifted sinusoidal voltage driving sources to
generate a traveling deformation wave on the stator surface.
Traveling wave motors can be designed to produce linear,
rotary, or combined (screw) movements [2] with high thrust,
and micropositioning capacity.
According to Fu [3], the classifications for LUSM (linear
ultrasonic motor) can be further divided, depending on the
type of waves created on the stator. Standing wave LUSMs
can be classified as [4], [5] single-foot and multi-feet [6] by
the number of foot. Running wave LUSM can be classified as
straight-girder [6], [7], [4], [5] and circle-girder. LUSM can
be classified as self-running and not self-running by oscillator
vibration. They can also be classified as single-model and
multi-model depending on the direction of movement in the
ellipse. Concerning the vibration direction of LUSM, it can
be classified as out-plane vibration, when vibrations are
perpendicular to the plane of stator, and in-plane vibration
when they are parallel.
In the most of LUSM, movement is obtained when the
stator is excited with Langevin vibrators producing
longitudinal and flexural traveling waves [4], [5] that impel
the moving part.
A revision of piezoelectric motors using tubular structures
shows that they are mainly employed in angular USMs [8],
[9], [10], [11] [12] . An exception is presented by Xu [13],
that build a motor using tubular piezoceramics with
helicoidally patterned electrodes. The proposed structure
generates screw-type displacements on a hollow metallic
cylinder employed as a rotor.
Linear motors using on tubular geometry are less common.
The most prominent is the inchworm motor, a standing-wave
piezoelectric motor [14], [15] patented by May [16].
No reports of axisymmetrical traveling wave LUSM using
external tubular stators were found in the literature.
In this study we investigate resonant vibration modes in
hollow cylinder structures, and show how to combine them to
obtain axisymmetrical traveling waves for a new type of
LUSM.
II. DESCRIPTION OF PROTOTYPE AND
PRINCIPLE OF PERATION
The stator of the motor (Fig. 1(a)) consists of a phosphor
bronze tube with two externally fixed PZT-4 rings. The
properties of these materials are defined in the Appendix.
The stator has its inner side toothed, in order to enlarge the
displacements [17], [18] resulting from traveling waves.
The position of rings along the tube is fundamental to the
effectiveness of movement production. The vibration in the
tube is a linear superposition of an infinite number of mode
shapes. The amplitude of deformations transversal to the
thickness of the tube can be calculated as [19], [20]:
1
1 2 21
2
2 2 21
( ) ( )( , ) ( )
( ) ( )( )
2
n n
n n
n n
n n
x Ly x t F sin t
f f
x LF sin t
f f
∞
=
∞
=
Φ ⋅Φ= ⋅ ω ⋅ +
−
Φ ⋅Φπ+ ⋅ ω − ⋅−
(1)
2010 9th IEEE/IAS International Conference on Industry Applications- INDUSCON 2010 -
978-1-4244-8010-4/10/$26.00 ©2010 IEEE
Fig. 1. Stator of the motor. (a) 3D view and details of the geometry (b)
where: F1 and F2 are the excitation forces due to the action of
PZT rings, Φn(x) is the normalized shape factor and fn is the
natural frequency of the vibration mode n. L1 e L2 are the
position where the PZT apply forces on the pipe.
The contribution of each excitation source on a particular
resonant mode can be estimated by using the modal
participation factor (MPF) [20]. The MPF depends mainly on
frequency and position of vibration sources (L1, L2) on the
tube, and can be calculated as:
2 2
( )n
n
LMPF
f f
Φ=
−(2)
The nearest from the maxima of a vibration mode the
vibrator is positioned, the more efficiently this mode is
excited. The same is valid for frequency: as the excitation
frequency move away from a particular mode, its contribution
to vibration decreases significantly [20].
The principle of movement production on a tubular motor
suggests the classic rotary traveling wave ultrasonic motor
[17], [18], except by the geometry. A traveling wave is
generated on the internal surface of the stator as a result
addition of two standing waves, in this case, excited by the
PZT rings. When a deformation wave travels on the stator,
the particles on its surface describe an elliptical trajectories
having opposite direction to the traveling wave. If another
part is pressed against the stator it will be driven by tangential
forces generated in the interface layer.
III. FINITE ELEMENT ANALISYS
A modal analysis with Finite Element Method (FEM) with
ANSYS® was performed for identification of resonant modes,
on a metal tube excited with piezoelectric ceramic rings. Free
edges were used as boundary conditions for all simulations.
The conditions for simulations of diameter and length
variations are shown in Fig. 1(b). In simulations where the
diameter of the tube was varied, the difference between outer
and inner diameters of the ceramic ring has been fixed to
8mm. While the length (L) was swept, from a simulation run
to another, positions of piezoceramic rings were kept at
L1=(L-L2)= x mm from the edges.
Fig. 2 shows the behavior from ninth to fifteenth
longitudinal natural modes of the stator plotted against: (a)
the tube length (L), and (b) the diameter (D).
The diameter is fixed at 17mm in the first plot and length is
kept at 55mm in the second. Fig. 2(a) and (b) shows that all
resonant modes decrease [8] when the length increases, and
some of them tends to become closer in frequency. The
closeness of these modes is convenient for obtaining traveling
waves, since it favors the excitation of two or more modes
with high MPFs. On the other hand, the frequencies of all
first seven natural modes also decrease [8] if the diameter
increases, and the frequencies of all modes become closer.
IV. EXPERIMENTS
Several runs of modal analysis were performed in order to
analyze the behavior of resonant frequencies in a range of
diameters and lengths of the tube. We looked for dimensions
having longitudinal resonant modes with close frequencies,
allowing the excitation of both with high MPFs. An important
condition for generating traveling waves and linear
movements from this joint excitation of modes, is no radial
modes of resonance in the vicinity of chosen longitudinal
modes. The occurrence of radial modes tends to clamp the
rotor, lowering the efficiency of energy conversion and
scratching the slider.
The condition above is fulfilled for the modes identified
with the arrow in Fig. 2(b). Longitudinal and radial strain
distributions for these modes were obtained with PZT rings
driven with 35Vrms. Fig. 3(a) shows maxima of longitudinal
vibration amplitudes (8μm) obtained for the 14th
mode, and
for the 15th
mode (4μm).
Radial strain distributions from both modes are depicted in
Fig. 3(b). The 14th
mode produces nearly constant amplitudes
of 2μm along the tube, and the 15th
mode imposes larger
deformations next to the extremities, about 6μm.
Fig. 2. Variation of ninth to fifteenth longitudinal resonance modes with (a) tube length (for D=17mm), and (b) diameter (for L=55mm).
By adding longitudinal and radial vibration amplitudes
from these two modes, the plots on Figs. 4(a) and 4(b) are
respectively obtained. The amplitudes are normalized to the
maximum value (7.626μm).
In order to enhance the slider propulsion by the stator, teeth
must be placed on the inner surface of the stator, at maximum
vibration points. From Fig. 4(a), the longitudinal maxima can
be identified at 9, 12, 13, 18, and 19mm from the edges of the
tube. On the other hand, maxima of radial vibrations are at 1,
6, 19, 20, 21, and 24mm from the edges. Thus, larger
displacements can be obtained placing teeth at 19mm from
the edges of the stator.
The Fig. 5 shows the elliptical movement of the points on
the top of teeth placed at 19mm from each edge of a tubular
stator with length of 55mm and diameter of 30mm, when
submitted to vibrations generated by PZT-4 rings, excited
with 35Vrms, 64876Hz. If the reader follows the plot
sequence, imagining a slider horizontally placed right below
the radial deformation line of 4μm, he (or she) will “see” both
Fig. 3. (a) Longitudinal and (b) radial deformations in the inner surface of the tube.
ellipses pushing the slider to the left. If otherwise, opposite
forces were produced upon the slider, the mechanical
coupling between stator and slider would degrade, increasing
wearing of both pieces. The tooth 1 will push first, at 30°, and
the tooth 2 later, at 240°. This elliptical movement can be
observed along the entire stator surface, but larger
deformations, and stronger movements, occur near the
maxima of combined strain. Maxima for radial and
longitudinal vibrations are respectively 2.8 μm and 10.2 μm.
Larger longitudinal strain favors the production of axial
movements on the slider.
A 3-D plot of the deformation profile along the tube is
shown in Fig. 6. The arrows indicate the points where the
elliptical movements plotted in Fig. 5 are obtained. Other
maxima are visible, but the longitudinal deformations at these
points are small, or on the opposite direction. Placing teeth at
these places would spoil the movement production.
(a)
(b)
Fig. 4. Longitudinal (a) and radial (b) levels of vibration amplitudes from the 14th e 15th modes excited simultaneously.
Fig. 5. Elliptical trajectory of the points at the top of the teeth: moving to the same direction.
Fig. 6. Traveling-wave composed from the 14th and 15th vibrational modes.
V. CONCLUSION
Resonant vibration modes and conditions for exciting them
aiming the generation of traveling waves in a metallic tube
were investigated with FEM. Simulations were performed for
tubes with lengths between 21 and 65mm, and diameters
ranging from 12 to 30mm. As the frequencies of resonant
modes become closer from each other, two modes can be
simultaneously excited with a high MPF.
Transient analysis shows that longitudinal traveling waves
are produced if ceramic rings are placed near the deformation
wave maxima, and excited at the vicinity of the resonant
frequency of the chosen modes.
ACKNOWLEDGMENT
The authors would like to thanks to HTMG International,
American Piezo Representative in Brazil, for donation of
ceramics. This research is partially supported by CAPES.
APPENDIX
Material properties of the used PZT-4 ceramic.
Density: ρ=7500 Kg/m3.
Stiffness matrix:
13.2 7.1 7.3 0 0 0
7.1 13.2 7.3 0 0 0
7.3 7.3 11.5 0 0 0
0 0 0 3.0 0 0
0 0 0 0 2.6 0
0 0 0 0 0 2.6
c = x1010
N/m2.
Piezoelectric matrix:
0 0 4.1
0 0 4.1
0 0 14.1
0 0 0
0 10.5 0
10.5 0 0
e
−
−
= C/m2.
Dieletric matrix:
804.6 0 0
0 804.6 0
0 0 659.7
rε =
Material properties of the phosphor bronze.
Density: ρ=8900 Kg/m3.
Youg modulus: E=11.2x1010
N/m2.
Poisson ratio: σ=0.35.
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