Structure design of linear ultrasonic motor with a laminated U-shaped stator based on modal analysis

2.1. Theory of the motor

The uniform rectangular thin sheet has mainly three basic vibration modes. The first is retractable mode in-plane, the second is bending mode in-plane and out-of-plane, and the third is torsional mode in-plane. The other modes are made of the mentioned three basic modes. With mode in-plane as the operation mode of linear ultrasonic motor, the structure can be simple [12].

The ultrasonic motor can operate normally when the resultant movement of the particles on the surface of stator or rotor is elliptical motion. The quality of the elliptical orbit will directly affect the output performance of the linear ultrasonic motor [13]. Generally, lower order of vibration mode should be selected to acquire larger mechanical output force. The operation mode of the ultrasonic motor designed in this paper is retractable mode and bending mode in plane of stators.

Piezoelectric ceramic pieces will excite the stators to enable them to achieve the work condition of the first-order longitudinal vibration and the second-order bending vibration. Longitudinal vibration mainly causes the horizontal displacement to drive the movement of the guide rail, while bending vibration mainly causes vertical displacement to appear intermittent contact and separation between the stators and the guide rail. Under the function of pre-compression, the superposition of the two modes enables the stators to rub with the guide rail intermittently, and then the guide rail is driven by the friction to form rectilinear motion. When the phase difference between the two modes is 90°, an elliptical motion orbit will be formed at the contact points.

2.2. Modal analysis on the motor

To learn the basic vibration characteristics of the stator structure, modal analysis was carried out on the stator structure by using its resonance feature.

Under the condition of linear time invariant, the vibration equation of a continuous structure system is:

where $\left[\mathbf{M}\right]$ is the mass matrix of the system. $\left[\mathbf{C}\right]$ is the damping matrix of the system. $\left[\mathbf{K}\right]$ is the stiffness matrix of the system. $\left\{\mathbf{x}\left(t\right)\right\}$ is the displacement response vector of the system. $\left\{\mathbf{f}\left(t\right)\right\}$ is the excitation force vector of the system.

The finite element vibration equation of the system without considering damping is:

Its solution can be assumed as the following form:

where $\left\{\mathbf{\varphi }\right\}$ is the $n$th order vector. $\omega$ is the vibration frequency of vector $\left\{\mathbf{\varphi }\right\}$. $\theta$ is initial phase angle, which can be determined according to the initial conditions. Substitute Eq. (3) into Eq. (2) to obtain the follows:

$n$ characteristic solutions $\left({\omega }_1^2,{\varphi }_1\right)$,$\left({\omega }_2^2,{\varphi }_2\right)$,…,$\left({\omega }_n^2,{\varphi }_n\right)$ can be acquired by solving the mentioned equation, where characteristic values ${\omega }_1$, ${\omega }_2$,…, ${\omega }_n$ represent $n$ inherent frequencies of the system, characteristic vectors $\left\{{\varphi }_1\right\}$, $\left\{{\varphi }_2\right\}$,…, $\left\{{\varphi }_n\right\}$, represent $n$ inherent vibration modes of the system.

The following orthogonal characteristic exists between the inherent vibration modes of the structure and the mass matrix $\left[\mathbf{M}\right]$ as well as the stiffness matrix $\left[\mathbf{K}\right]$:

where $\left[\mathbf{\Lambda }\right]$ and $\left[\mathbf{\Phi }\right]$ are referred to the matrix of inherent frequencies and the matrix of inherent vibration modes, respectively.

3.1. Laminated U-shaped stator

The stator structure was designed to the following type which was made of two parts of metal elastomer and piezoelectric ceramic pieces, where the metal elastomer was U-shaped structure [14] which was made of beryllium copper through wire cutting. Meanwhile, flexible clamps with bolt holes were cut on both sides of the U-shaped elastomer [15]. The stators were fixed on the base by bolts. The rotors were driven to move by the drive foot which was the top point of the U-shaped stator [16]. Piezoelectric ceramic pieces were pasted on the vibration rods which was two feet of the U-shaped stator, and the piezoelectric material was PZT8.

The selected operation mode is the first-order longitudinal mode and the second-order bending mode. Firstly, the finite element model of the laminated U-shaped stator was built by ANSYS software, then modal analysis was carried out and parameters of the single stator were optimized by means of parameter optimization function as the following. The height of the drive foot was the variable parameter, while other structure parameters were unchanged. The structure would have an optimal parameters when the work frequencies of the first-order longitudinal vibration modal and the second-order bending vibration modal were very close, and there was no other modal interference near this frequency. The optimal parameters were shown in Fig. 1, and the geometry model of the laminated U-shaped stator was shown in Fig. 2.

The first-order longitudinal mode and the second-order bending mode of the laminated U-shaped stator were obtained by the finite element model as shown in Fig. 3 and Fig. 4.

The frequency of the first-order longitudinal vibration and the second-order bending vibration was obtained as shown Table 1. The difference value between the two operation frequency was 192 Hz, which could meet the requirement of frequency consistency.

Fig. 1Parameters of the laminated U-shaped stator after optimization

Fig. 2Diagram of the geometry model for the laminated stator

Fig. 3Diagram of the first-order longitudinal mode

Fig. 4Diagram of the second-order bending mode

3.2. Integrated design of the motor structure

The workbench of common ultrasonic motor can be purchased directly. However, the structure of the laminated motor was special. In this paper, a precise workbench was designed based on the advantages of linear ultrasonic motor as shown in Fig. 5. It had the following characteristics. Firstly, there was no mechanical friction between the workbench and the guide rail as well as the drive foot. Secondly, the inherent frequency of the workbench was far from the operation frequency of the motor to avoid resonance. The detailed structure parameters of the workbench was shown in Fig. 6, and the final real ultrasonic motor was obtained through processing and assembling as shown in Fig. 7.

Fig. 5The workbench of the laminated motor

Fig. 6Structural parameters of the workbench

Fig. 7Diagram of the real ultrasonic motor

3.3. Experimental verification of the simulation analysis

In order to verify the feasible of the simulation analysis, the swept-sine modal test by means of HP35670A which was a dynamic signal analyzer was carried out on the real laminated ultrasonic motor. The experiment results were shown in Fig. 8, it could be seen from the figure that the frequency of the first-order longitudinal vibration and the second-order bending vibration were 58.33 kHz and 58.49 kHz, respectively. They were consistent with the simulation results, which showed that the simulation analysis was feasible.

Fig. 8Experimental results of the swept-sine modal test

a) The frequency of the first-order longitudinal vibration

b) The frequency of the second-order bending vibration

4.1. Experiment design

The mechanical performance test of the laminated motor and single U-shaped motor was completed by the experimental facility as shown in Fig. 9. The driving platform of the motor is composed of signal generator and power amplifier, and the motor performance test system is composed of motor, force sensor and oscilloscope. The driving platform of the motor controls the output mechanical performance of the motor by adjusting the amplitude, phase and frequency of the input signals in certain range. The output performances such as the no-load velocity, acceleration and output force of the motor can be measured by the test system. Meanwhile, the influence of the pre-compression force on the mechanical performance of the motor can be researched by the force sensor.

The operation frequency had been obtained by simulation and experiment, so the test frequency of the motor was controlled in the range of 57.5 kHz-59.5 kHz, hence, the motor performance could be ensured to be in a good operation status.

Fig. 9Experimental facility of the mechanical performance test

a) Ultrasonic motor

b) Signal generator

c) Power amplifier

d) Dynamometer

e) Oscilloscope

f) Winanal software

4.2. Experimental results and analysis

In order to decide the most appropriate operation frequency of the laminated ultrasonic motor, we tested the mechanical performance of the laminated ultrasonic motor under different operation frequency as shown in Fig. 10.

Fig. 10The mechanical performance of the laminated ultrasonic motor

a) The maximum velocity with the change of pre-compression force

b) The maximum acceleration with the change of pre-compression force

c) The maximum output force with the change of pre-compression force

It could be seen from Fig. 10 that the mechanical performance such as velocity, acceleration and output force had an obvious change with the operation frequency. In addition, the laminated ultrasonic motor would achieved a better performance in 58.5 kHz, because this frequency was more close to the frequency of the first-order longitudinal vibration and the second-order bending vibration, which were the main operation modal. Moreover, the mechanical performance of the laminated ultrasonic motor had a larger change with the increase of the pre-compression force. There should be an optimum numerical of pre-compression force, and its value should be determined by different application condition. In order to verify the advantages of the laminated ultrasonic motor, the optimal mechanical performance was compared with that of the single U-shaped motor as shown in Fig. 11 and Fig. 12.

Fig. 11Comparison of the maximum velocities between two types of ultrasonic motor

Fig. 12Comparison of the maximum output force between two types of ultrasonic motor

It could be seen from Fig. 11 and Fig. 12 that the mechanical performance of the laminated was better than that of the single U-shaped motor, and the detailed description was shown in the following. Under the operation frequency of 58.5 kHz, the maximum output force of the single U-shaped motor was approximate 13 N, and its maximum velocity was 145 mm/s. However, under the same operation frequency, the maximum output force of the laminated motor was approximate 18 N, and its maximum velocity was 200 mm/s. The maximum output force of the laminated motor increased by approximate 40 % than that of the single U-shaped motor, while the maximum velocity increased by 38 %. This result was satisfactory, which showed that the mentioned design of the laminated ultrasonic motor was feasible.