Ultrasonic motor with embedded preload mechanism

Accepted Manuscript Title: Ultrasonic motor with embedded preload mechanism Authors: Tomoya Kazumi, Yuta Kurashina, Kenjiro Takemura PII: DOI: Reference:

S0924-4247(18)32062-4 https://doi.org/10.1016/j.sna.2019.02.010 SNA 11242

To appear in:

Sensors and Actuators A

Received date: Revised date: Accepted date:

24 December 2018 5 February 2019 10 February 2019

Please cite this article as: Kazumi T, Kurashina Y, Takemura K, Ultrasonic motor with embedded preload mechanism, Sensors and amp; Actuators: A. Physical (2019), https://doi.org/10.1016/j.sna.2019.02.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Ultrasonic motor with embedded preload mechanism

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Tomoya Kazumia, Yuta Kurashinab,c, Kenjiro Takemurab,*

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School of Science for Open and Environmental Systems, Graduate School of

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Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku,

Department of Mechanical Engineering, Faculty of Science and Technology,

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b

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Yokohama, 223-8522, Japan

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Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan.

c

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Department of Materials Science and Engineering, School of Materials and

Chemical Technology, Tokyo Institute of Technology, 4259 Nagatsutacho,

*

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Midori-ku, Yokohama, 226-8503, Japan

Correspondence to: Kenjiro Takemura. Telephone: +81-45-566-1826; fax:

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+81-45-566-1826; e-mail: [email protected]

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Graphical abstract

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Highlights

We propose an embedded preload mechanism for ultrasonic motor.

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We eliminate the need for external preload mechanism from ultrasonic motors.

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The proposed concept contributes to establish compact and simple ultrasonic motors.

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The maximum velocity and thrust values of the motor are 62.5 mm/s and 0.12 N.

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

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ABSTRACT:

A linear/rotary ultrasonic motor, which is driven using a friction force, generally requires a preload mechanism between the slider/rotor and the stator. The preload mechanism is of considerable importance to the motor’s ability to produce a practical

thrust/torque because the output thrust/torque is dependent on the friction force. However, the preload mechanism generally occupies a relatively large space as a proportion of the space for the entire motor, which may limit motor miniaturization. To

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overcome this limitation, a linear ultrasonic motor has been developed in this work with a slider that has an inherent preloading capability. The ultrasonic motor is composed of

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a phosphor bronze stator with piezoelectric ceramic plates and a U-shaped stainless

steel slider. The stator is inserted into the slider with an interference fit, which means

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that the preload between the slider and the stator is applied mechanically. By selective

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excitation of the second or third resonance vibration modes of the stator, the slider can

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be driven bidirectionally. The maximum velocity, thrust and output power values of the

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proposed motor are 62.5 mm/s, 0.12 N and 1.01 mW, respectively. This simple but effective solution, which eliminates the need for an external preload mechanism, will

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contribute to progress in ultrasonic motor research and design.

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KEYWORDS: Ultrasonic motor, preload mechanism, miniaturization

1. Introduction

An ultrasonic motor, which generally consists of a stator, a rotor/slider and a preload mechanism, uses the resonant vibration of the stator to drive the rotor/slider, i.e., it converts the vibration energy of the stator into the kinetic energy of the rotor/slider via a

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frictional force [1]. When compared with the more widely used electromagnetic motor, an ultrasonic motor offers several preferable features: ultrasonic motors can be driven

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silently, provide high torque at low speeds, offer high stall torque without input electric

power, and deliver quick responses without electromagnetic noise while they are driven

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[2-4]. Because of these advantages, ultrasonic motors are used in, for example,

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autofocus systems for camera lenses. However, the application field of ultrasonic

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motors has been almost entirely limited to these autofocus systems to date because of

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their lifetime limitations and their requirement for a preload mechanism, which is essential to provide the friction force between the rotor and the stator. The preload

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mechanism is generally composed of a coil spring with supporting components [5-9]. While a number of stator/rotor designs [10-15] have been reported, almost no attention

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has been paid to the preload mechanism, which occupies a relatively large volume in a motor when compared with that of the stator/rotor, as shown in Table I [16-19].

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In previous research on preload mechanisms, buckling leaf springs have been used [20]

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and the preload mechanism and the stator were integrated [21] to enable motor miniaturization. In addition, an ultrasonic motor with a permanent magnet [22] or with adjusting bolts [23] has been used to apply the preload another motor that used a coil spring as a slider [24], has also been reported. These research efforts have contributed by improving the volume efficiency of these motors and can also extend the design

possibilities for ultrasonic motors. However, these motors still require complex designs and require further simplification and higher output powers for practical use. Against this background, this research proposes a simple but effective linear ultrasonic motor

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design with a preloading function that is embedded in its stator and slider.

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2. Design

2.1 Entire structure

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Figure 1(a) shows a schematic illustration of the ultrasonic motor that is developed in

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this study. This ultrasonic motor consists of a stator and a slider. The stator is supported

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simply using four metal rods that are connected to a supporting base. The alignment of the stator and the slider is illustrated in Fig. 1(b). The stator has four projections that act

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as contact points against the slider. Holes are drilled at each corner of the stator for pinning. Piezoelectric ceramic plates are bonded to the inner surface of the stator to

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excite the resonant vibrations. To apply a preload between the stator and the slider using an interference fit, the slider is bent inward into a U-shape. This causes the preload to be

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provided by the interference fit when the stator is inserted into the slider.

2.2 Driving principle

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The driving principle of the ultrasonic motor developed in this work is shown in Fig. 2. By applying an AC voltage to the piezoelectric ceramic plates at an appropriate

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frequency, a resonance vibration mode can be excited in the stator. Consequently, the four projections on the stator reciprocate along specific trajectories. When the second

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resonance mode is excited (Fig. 2(a)), projections A and B reciprocate from the −z/−x

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direction to the +z/+x direction. In contrast, projections C and D reciprocate from the

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+z/−x direction to −z/+x direction. The friction force against the slider is then generated

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in the positive x direction. In addition, when the third resonance mode is excited (Fig. 2(b)), projections A and B reciprocate from the −z/+x direction to the +z/−x direction. In

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contrast, projections C and D reciprocate from the +z/+x direction to the −z/−x direction, which results in generation of a friction force in the negative x direction. This motor can

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be driven by one projection on each upper/lower surface of the stator, however, two projections are required on each surface in order to stably support the slider in parallel.

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Therefore, by causing the stator to resonate selectively via the second or third resonance

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modes, we can drive the proposed ultrasonic motor in either direction.

2.3 Stator

The designed stator is shown in Fig. 3(a). The stator is made from hollow phosphor bronze and two piezoelectric ceramic plates (C-213, Fuji Ceramics Co., Ltd., Shizuoka, Japan) are bonded on its inner surface. The stator’s resonant vibration modes are

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confirmed using finite element method software (COMSOL Multiphysics ver. 5.2a, COMSOL Inc., California, U.S.). The physical properties used in this analysis are listed

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in Table II. Figure 3(b) shows the resonant vibration modes that were obtained through an eigenvalue analysis. The resonant frequencies of the second and third resonance

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modes are 23.5 kHz and 43.8 kHz, respectively. Furthermore, it was also confirmed that

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the trajectories of the four projections correspond to the trajectories given in Section 2.2.

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In addition, the results of the eigenvalue analysis confirm that the stator can be

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supported using pins that are fixed at the four corners of the stator because the nodes of each resonance vibration mode are located at the corners.

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To excite the resonant vibration modes of the stator selectively and efficiently, it is necessary to generate repetitive strains at the antinodes with corresponding resonance

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frequencies. From the resonant vibration modes shown in Fig. 3(b), the electrode pattern on the piezoelectric ceramic plate is designed as shown in Fig. 3(c). Because the

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piezoelectric ceramic plate is polarized along the thickness direction, positive electrodes

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and a ground (GND) electrode are formed on both sides of the piezoelectric ceramic plate. Since the phosphor bronze part and the piezoelectric ceramic plate are adhered, the entire stator is grounded. When an AC voltage is applied to the two electrodes for the second mode on the upper and lower piezoelectric ceramic plates, the second resonant vibration mode is then excited. Similarly, when an AC voltage is applied to the

two electrodes for the third mode on the upper and lower piezoelectric ceramic plates, the third resonant vibration mode is excited. The sensor electrode is then used to

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monitor the amplitude of the vibration while driving the motor.

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2.4 Slider

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The slider design is shown in Fig. 4(a). The slider is made from stainless steel

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(X2CrNi18-9) and has a slight bending angle, denoted by , which is defined in the figure. We designed three sliders, designated slider A, slider B and slider C, with

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different bending angles of 0°, 0.5° and 1.0°, respectively. To estimate the magnitude of the preload generated by each slider, we performed a structural analysis using the finite

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element method software. Figure 4(b) and Table III show the magnitudes of the preloads generated when the stator is inserted into the center of each slider. At the

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bending angle of 0° (slider A), the total preload is caused by gravity alone, which is

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only applied on the top surface of the slider. However, for slider B and the slider C, preloads of 1.1 N and 2.3 N are generated, respectively, because of the interference fit between the stator and the slider. With the designed stators, the generated pressure between the slider/stator is not uniform in the contact region, however, the total preload should be constant with each stator. We believe these simple designs with different

bending angles are effective to confirm the concept of this study, i.e., the preloading

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capability of using interference fit.

3. Experimental results and discussion

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3.1 Vibration characteristics of the stator

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We measured the resonance frequencies of the second and third resonant vibration

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modes using a laser Doppler vibrometer (LV-1800, Ono Sokki Co., Ltd., Yokohama,

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Japan). Figure 5(a) shows the relationship between the input frequency and the vibration amplitude of the stator at an input voltage of 100 Vp-p. Note that the amplitude was measured at the antinode position for each resonance mode. The resonance frequencies

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of the second and third resonant vibration modes are 23.8 kHz and 43.6 kHz,

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respectively. The measured resonance frequencies are 1.2% higher for the second mode and 0.5% lower for the third mode when compared with the corresponding values

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obtained via the eigenvalue analysis. This discrepancy may be explained by the fact that the adhesive bond layer between the phosphor bronze and the piezoelectric ceramic plates is not modeled as part of the analysis, and may also be attributed to machining errors.

We confirmed the mode shape at each resonance frequency. Figure 5(b) shows the measurement results for the amplitude distribution on the upper surface of the stator at the input voltage of 100 Vp-p. Note that the sign of the amplitude means that the phase is

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inverted. Because the horizontal axis represents the distance from the stator edge, the excited vibrations correspond obviously to each resonance mode. The stator and the

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slider are not in contact except at the tip of four projections since the height of

projections is 1.0 mm (Fig. 3) and the amplitude of the stator is less than 1.0 μm (Fig.

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5).

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From these results, we calculated the trajectories of projections A and B, as shown in

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Fig. 5(c) and (d). When the second resonant vibration mode is excited, the two

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projections reciprocate from the −z/−x direction to the +z/+x direction. However, when the third resonant vibration mode is excited, they reciprocate from the +z/+x direction to

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the −z/−x direction. Projections C and D reciprocate in a similar manner. These results indicate that the four projections reciprocate successfully on the desired trajectories that

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were described in Section 2.2. Although the two projections on each surface have different trajectories which may conflict each other when driving the motor, they

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alternately push the slider resulting in making the stator velocity stable due to the inertia

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effect.

3.2 Motor performance

A prototype of the proposed ultrasonic motor was fabricated and the experimental system was assembled as shown in Fig. 6. The stator is supported by the four metal rods and the preload between the stator and the slider is provided by the interference fit. The

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slider moves along a linear guide that is mediated using two bearings. We evaluated the motor performances in terms of thrust, velocity and output power for

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each slider. The driving conditions were 100 Vp-p and 23.8 kHz for the second resonant

vibration mode and 100 Vp-p and 43.6 kHz for the third resonant vibration mode. Figure

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7 illustrates the relationship between thrust and velocity and the relationship between

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thrust and output power for all sliders when driven using the second resonant mode. As

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shown in Fig. 7(a), the maximum velocity was highest with slider A at 62.4 mm/s, and

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was lowest at 31.5 mm/s with slider C; meanwhile, the maximum thrust was obtained with slider C at 0.12 N, while the lowest thrust was obtained with slider A at 0.025 N.

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As shown in Fig. 7(b), the output power of slider B was the highest at 1.01 mW, while that of slider A was the lowest at 0.36 mW. From these results, the linear relationship

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between thrust and velocity in the prototype motor was confirmed. In addition, when the magnitude of the preload increased, the thrust also increased while the velocity

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decreased. Furthermore, the output power reached a maximum when the preload was

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1.17 N (slider B). Figure 8 shows the relationship between thrust and velocity along with the relationship

between thrust and output power when the motor is driven using the third resonant vibration mode. The tendencies of the thrust, the velocity and the output power are nearly equivalent to those observed when using the second vibration mode. However,

the maximum thrust, velocity and output power values when using the third resonant modes are lower than those obtained using the second resonant mode, as shown in Figs.

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7 and 8.

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3.3 Discussion

As noted in Section 3.2, the velocity, the thrust and the output power of the ultrasonic

motor will differ depending on whether slider A, slider B or slider C is used. Therefore, the output can be adjusted by varying the magnitude of the preload, i.e., by changing the bending angle, θ, of the slider. However, given that the output power of slider B, which

had a preload magnitude of intermediate value, was the highest, there must be an appropriate preload magnitude for driving of the motor from an output power viewpoint.

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As shown in Table I, the preload mechanism generally accounts for 30% to 50% of the entire motor volume and size. However, the ultrasonic motor that was developed in this

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study, in which the stator and slider have the function of applying the preload, does not

need an external preload mechanism. The supporting base and linear guide are parts of

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the experimental system. For simplification, the base was designed as shown in Fig. 6.

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Since the motor itself is only composed of the slider and the stator, supported by four

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pins, the base and the linear guide can be eliminated from or integrated in a system

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when the motor is installed in a mechanical system. Therefore, the proposed concept of

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integration of the preload mechanism in the stator/slider contributes to easy design and

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fabrication of compact and simple ultrasonic motors.

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4. Conclusions

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A prototype linear ultrasonic motor with an embedded preload mechanism is designed,

fabricated and characterized in this work. The proposed motor is driven bidirectionally by selective excitation of the second and third resonant vibration modes of the stator. The U-shaped slider, which has a slight inward bending angle, easily enables preload generation between the stator and the slider. The maximum velocity, thrust and power

of the motor are 62.5 mm/s, 0.12 N and 1.01 mW, respectively. When the magnitude of the preload increases, the velocity decreases while the thrust increases. This ultrasonic motor, in which the stator and slider have a preload application function,

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does not require an external preload mechanism. Therefore, when compared with traditional ultrasonic motors, it is possible to reduce the overall size of the proposed

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motor.

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Declaration of Interest: none

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Biography

Tomoya Kazumi received his B.E. degree in Mechanical Engineering from Keio University, Yokohama,

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Japan in 2017. He is currently working for his M.E. degree in Open and Environmental Systems in Keio

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University.

Yuta Kurashina received his B.E. degree in Mechanical Engineering and M.E. degree in integrated design engineering from Keio University, Yokohama, Japan in 2012 and 2014, respectively. He also received his Ph.D. degree in open and environmental systems from

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Keio University in 2017. He was with the Department of Mechanical Engineering at Keio University in 2017 as a Research Associate. He has been with the Department of Materials

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Science and Engineering, School of Materials and Chemical Technology at Tokyo Institute of Technology since 2018 where he is currently an Assistant Professor. He received a Best

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Paper Award at the ASME International Mechanical Engineering Congress & Exposition in 2015 and an Audience Award at the JSME Dynamics and Design Conference in 2017, etc. His research interests are on biomedical engineering, ultrasonic transducers, and micro-nano fabrication.

Kenjiro Takemura received his B.E. degree in Mechanical Engineering and M.E. degree in Biomedical Engineering from Keio University, Yokohama, Japan in 1998 and 2000, respectively. He also received his Ph.D. degree in integrated design engineering from Keio University in 2002. He was with the Department of Mechanical Engineering at Keio University in 2002 as a Research Associate, with the Precision and Intelligence Laboratory at Tokyo Institute of Technology as an Assistant Professor from

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2003 to 2008. He has been with the Department of Mechanical Engineering at Keio University since 2008 where he is currently an Associate Professor. He received a JSME Young Engineers Award in 2002 for a research on Multi-DOF ultrasonic motor, a Best Research Paper Award at the 10th International

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Conference on Mechatronics Technology in 2006, a Best Paper Award at the ASME International

Mechanical Engineering Congress & Exposition in 2015, etc. His research interests are on ultrasonic

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transducers, functional fluids, haptic interface, and biomedical devices.

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REFERENCES:

[1] K. Uchino, Piezoelectric Actuators and Ultrasonic Motors, Kluwer Academic

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Publishers, Boston, 1997.

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[2] B. Watson, J. Friend, L. Yeo, Piezoelectric ultrasonic micro/milli-scale actuators, Sensors and Actuators A: Phys. 152, No. 2, 219-233, 2009.

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[3] W. Lee, C. Kang, D. Paik, B. Ju, S. Yoon, Butterfly-shaped ultra slim piezoelectric ultrasonic linear motor, Sensors and Actuators A: Phys. 168, No.1, 127-130, 2011. [4] A. Iino, K. Suzuki, M. Kasuga, M. Suzuki, T. Yamanaka, Development of a self-

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oscillating ultrasonic micro-motor and its application to a watch, Ultrasonics 38, 54-59, 2000.

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[5] V. Dabbagh, Ahmad A.D. Sarhan, J. Akabari, N.A. Mardi, Design and experimental evaluation of a precise and compact tubular ultrasonic motor driven

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by a single-phase source, Precision Engineering 48, 172-180, 2017.

[6] W. Chen, Y. Liu, X. Yang, J. Liu, Ring-type traveling wave ultrasonic motor using a radial bending mode, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 61, No. 1, 197-202, 2014.

[7] Y. Liu, D. Xu, Z. Yu, J. Yan, X. Yang, W. Chen, A novel rotary piezoelectric motor using first bending hybrid transducers, Applied Science 5, No. 3, 472-484, 2015. [8] M. Kurosawa, S. Ueha, Hybrid transducer type ultrasonic motor, IEEE

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Transactions on Ultrasonics, Ferroelectrics and Frequency Control 38, No. 2, 89-92, 1991.

[9] D. Yamaguchi, T. Kanda, K. Suzumori, An ultrasonic motor for cryogenic

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temperature using bolt-clamped Langevin-type transducer, Sensors and Actuators A, Phys. 184, 134-140, 2012.

[10] T. Mashimo, Micro ultrasonic motor using a one cubic millimeter stator, Sensors

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and Actuators A, Phys. 213, 102-107, 2014.

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[11] S. Nakajima, H. Kajiwara, M. Aoyagi, H. Tamura, T. Takano, Study on spherical

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stator for multidegree-of-freedom ultrasonic motor, Japanese Journal of Applied Physics 55, No. 7S1, 2016.

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[12] T. Peng, X. Wu, X. Liang, H. Shi, F. Luo, Investigation of a rotary ultrasonic

2015.

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motor using a longitudinal vibrator and spiral fin rotor, Ultrasonics 61, 157-161,

[13] S. Oh, K. Takemura, Development of robot finger using ultrasonic motors driven by superimposed signal input, IEEE International Ultrasonics Symposium

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Proceedings, 2490-2493, 2014.

[14] H. Sanikhani, J. Akbari, Design and analysis of an elliptical-shaped linear

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ultrasonic motor, Sensors and Actuators A, Phys. 278, 67-77, 2018.

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[15] S. Izuhara, T. Mashimo, Design and evaluation of a micro linear ultrasonic motor, Sensors and Actuators A, Phys. 278, 60-66, 2018.

[16] T. Ishii, H. Yamawaki, K. Nakamura, An ultrasonic motor using thrust bearing for friction drive with lubricant, IEEE International Ultrasonics Symposium Proceedings, 197-200, 2013.

[17] Z. Chen, X. Li, J, Chen, S. Dong, A square-plate ultrasonic linear motor operating in two orthogonal first bending modes, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 60, No. 1, 2013. [18] J. Yan, Y. Liu, J. Liu, D. Xu, W. Chen, The design and experiment of a novel

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ultrasonic motor based on the combination of bending modes, Ultrasonics 71, 205-210, 2016.

[19] K. Chang, M. Ouyang, Rotary ultrasonic motor driven by a disk-shaped ultrasonic

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actuator, IEEE Transactions on Industrial Electronics 53, No. 3, 831-837, 2006. [20] M. Takano, K. Hirosaki, M. Takimoto, S. Ichimura, K. Nakamura, Holding and preloading mechanism using a buckling parallel leaf spring for ultrasonic linear

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motor, Transactions of the Japan Society of Mechanical Engineers Series C 77, No.

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783, 4144-4154, 2011.

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[21] M. Aoyagi, N. Kawashima, M. Ishiguro, An ultrasonic spindle motor built-in

1013-1014, 2006.

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preload mechanism, The Japan Society for Precision Engineering Proceedings,

[22] K. Takemura, T. Maeno, Design and control of an ultrasonic motor capable of

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generating multi-DOF motion, IEEE/ASME Transactions on Mechatronics 6, No. 4, 499-506, 2001.

[23] L. Wang, J. Liu, Y. Liu, X. Tian, J. Yan, A novel single-mode linear piezoelectric

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ultrasonic motor based on asymmetric structure, Ultrasonics 89, 137-142, 2018. [24] T. Mashimo, Miniature preload mechanism for a micro ultrasonic motor, Sensors

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and Actuators A, Phys. 257, 106-112, 2017.

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the stator and slider.

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Fig. 1. The developed ultrasonic motor. (a) Schematic illustration, and (b) side view of

(a) Projection A

Projection B

� �

4

t=

T

t=

2 4

3 4

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1

T

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t=

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Projection C Projection D t= 0

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A

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(b)

1 4

T

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t=

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t= 0

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t=

2 4

t=

3 4

T

T

A

Fig. 2. Driving principle of the ultrasonic motor. (a) Second vibration mode of the Fig. stator 2. Driving principle ultrasonic motor. (a) Second vibration mode of the stator and the loci ofof thethe four projections, (b) third mode of and the stator and and the loci of thevibration four projections, (b) third vibration mode of the stator and the and the loci of the four projections. loci of the four projections.

(a)

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10.0

Phosphor bronze PZT

SOLIDW OR KS 教育用製品 (教育用目的でのみ使用可)

0

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(b)

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1

(c) +

A

+

t 0.50

Third

Second 12.5 17.5

Sensor

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+

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24.0

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Fig. 3. The developed stator. (a) Stator Fig. 3. The stator. with PZT. (b) Steady state of vibration modes with PZT.developed (b) Steady state(a)ofStator vibration modes via obtained via eigenvalue analysis obtained eigenvalue analysis using the finite element method. The resonance using the finite element method. The frequencies the secondofand vibrations resonance of frequencies thethird second and are 23.5 kHz and 43.8 kHz, respectively. third vibrations are 23.5 kHz and 43.8 kHz, with respect to the maximum amplitude. The color bar shows the amplitude normalized respectively. The color bar shows the (c) Electrode patterns on the PZT for excitation of the second and third vibration modes. amplitude normalized with respect to the The entire backamplitude. surface is ground electrode and the center part is used for sensing of the maximum (c) Electrode patterns on the PZT for excitation of "mm") the vibration amplitude. (All scale order is second and third vibration modes. The entire back surface is ground electrode and the center part is used for sensing of the vibration amplitude. (All scale order is "mm" )

(a)

θ

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θ

0

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(b)

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1

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Fig. 4. (a) Design and analysis of the Fig. 4. (a)(b) Design andofanalysis of analysis the slider. (b) Results of structural analysis performed slider. Results structural performed using the finite element using the finite element method. The color bar shows the stress normalized with respect method. The color bar shows the stress to the maximum stress. normalized with respect to the maximum stress.

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Fig. 5. Vibration characteristics of the stator. (a) Relationship between the stator amplitude and the input frequency, and (b) measured vibration distribution of the stator along the upper surface. (c) and (d) Loci of the two projections on the upper surface. (c)

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Trajectories of projection A, and (d) trajectories of projection B.

(a) Supporting Stator

Linear guide Slider

Stator

Slider

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(b)

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base

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6. Evaluation system ultrasonic Fig.Fig. 6. Evaluation system for for the the ultrasonic motor. (a) Experimental setup for the motor. (a) Experimental setup for the ultrasonic motor. (b)(b) Oblique image of the stator and the slider. ultrasonic motor. Oblique image of the stator and the slider.

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Fig. 7. Motor performance while second vibration mode is excited. (a) Relationship

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between thrust and velocity. (b) Relationship between thrust and power.

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Fig. 8. Motor performance while third vibration mode is excited. (a) Relationship

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between thrust and velocity. (b) Relationship between thrust and power.

Table I. Estimated volume/length ratio of preload mechanism relative to the entire motor size based on design and images. Volume

Length

[16]

30%

40%

[17]

30%

50%

[18]

50%

50%

[19]

30%

40%

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Reference

Table II. Physical properties used in the eigenvalue analysis. PZT

Density (kg/m3)

8800

7800

Young’s modulus (GPa)

110

82.0

Poisson ratio (-)

0.33

0.29

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Phosphor bronze

Table III. Relationship between θ and preload calculated by structural analysis. Preload (N)

Slider A

0

0.06

Slider B

0.5

1.17

Slider C

1.0

2.32

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θ (°)