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

pippi@ifsc.edu.br, cesar@ieee.org

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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