J. Cent. South Univ. (2013) 20: 393–404

DOI: 10.1007/s11771-013-1500-4

Optimal design of butterfly-shaped linear ultrasonic motor using finite element method and response surface methodology

SHI Yun-lai(时运来), CHEN Chao(陈超), ZHAO Chun-sheng(赵淳生)

State Key Laboratory of Mechanics and Control of Mechanical Structures

(Nanjing University of Aeronautics and Astronautics), Nanjing, 210016, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2013

Abstract: A new method for optimizing a butterfly-shaped linear ultrasonic motor was proposed to maximize its mechanical output. The finite element analysis technology and response surface methodology were combined together to realize the optimal design of the butterfly-shaped linear ultrasonic motor. First, the operation principle of the motor was introduced. Second, the finite element parameterized model of the stator of the motor was built using ANSYS parametric design language and some structure parameters of the stator were selected as design variables. Third, the sample points were selected in design variable space using latin hypercube Design. Through modal analysis and harmonic response analysis of the stator based on these sample points, the target responses were obtained. These sample points and response values were combined together to build a response surface model. Finally, the simplex method was used to find the optimal solution. The experimental results showed that many aspects of the design requirements of the butterfly-shaped linear ultrasonic motor have been fulfilled. The prototype motor fabricated based on the optimal design result exhibited considerably high dynamic performance, such as no-load speed of 873 mm/s, maximal thrust of 27.5 N, maximal efficiency of 43%, and thrust-weight ratio of 45.8.

Key words: linear ultrasonic motor; piezoelectric; optimal design; response surface methodology; finite element method

1 Introduction

As one of excellent solutions for precise positioning applications that require rapid response and high positioning accuracy, linear ultrasonic motors have been rapidly developed because of their attractive characteristics, e.g., direct driving, quick response, high resolution, and off power self-braking. To date, there had been many types of linear ultrasonic motors [1–16]. Among them, the linear ultrasonic motors based on hybrid vibration modes using d₃₃ effect of piezoelectric ceramic were often bolt-clamped motors [3, 5], which had attractive features of large electromechanical conversion efficiency, large force density, etc. These types of linear ultrasonic motors can be used in many application fields such as industry robots, aeronautics and spaceflight. However, to realize the effective ellipse motion trajectory on the driving tip of the stator, the stator dimensions of these types of motors based on hybrid vibration modes must be strictly designed to harmonize the eigen-frequencies of the operation modes. Thus, the optimal design of these types of linear ultrasonic motors is very critical on the performance of the motor. Generally, the ultrasonic motor mainly consists of two parts: stator and rotor/slider. The stator is a piezoelectric actuator that completes the conversion of electrical energy into mechanical vibration energy and thus its dynamic characteristics are the key to the motor performance. Hence, many researchers have done much works on the optimum design of these types of stators. As we know, there is a need for a dynamic model as a basis for the optimization problem of the ultrasonic motors. To date, the main dynamics modeling methods of ultrasonic motors are as follows: the analytical method [17], the equivalent circuit method [18–19], semi-analytical- numerical method, and numerical method (finite element method) [20–22]. Analytical method is generally suitable for theoretical qualitative analysis. The equivalent circuit method can better reflect the nonlinear characteristics of electromechanical energy conversion of the composite stator, which can make up for the lack of analytical method. However, this method is not suitable for the optimization of ultrasonic motor for its shortcoming of posteriori. The semi-analytical- numerical method, which combines the analytical method and numerical method to calculate some of parameters in the analysis process to ensure the accuracy of the calculation, is suitable for solving the more structured problem. The numerical method is a common technique method used in engineering, which can realize higher solution accuracy with little simplifications of the structure. As we know, it is hard to build a unified analytical model of the standing wave linear ultrasonic motors because of the varieties of the motor structures and their operating principles. Furthermore, many factors that influence the performances of the linear ultrasonic motors should be considered. Thus, the numerical method is much suitable for building the structural dynamic model of the stator to solve the optimum problem of the linear ultrasonic motors. Currently, some commercial numerical analysis softwares such as ANSYS and NASTRAN have been used to realize the optimization problem. By using these types of softwares, the parametric study method is often utilized to study the relationship of the structural parameters and the dynamic features of the stator. However, this method is not good for searching the design space of the stator. Response surface methodology (RSM) [23–26], derived from statistical analysis, is a method for constructing global approximations to system behavior based on results calculated at various points in the design space. The strength of the method is in applications where the calculation of the design sensitivity information is difficult or impossible to compute, as well as in cases with noisy functions, where the sensitivity information is not reliable, or when the function values are inaccurate.

Moreover, RSM is more open to the designers who can carry out the experiment design (DOE) and choose the approximate function of RSM based on their needs. Thus, we can effectively explore the design space using RSM in the case of considering many factors.

In this work, finite element method (FEM) combined with RSM were used to realize the goal of obtaining better dynamic features of the butterfly-shaped linear piezoelectric actuator. First, the fundamental configuration and operating principle of the actuator were described briefly and the finite element parameterized model of the stator was built using ANSYS parametric design language (APDL). Second, the optimization process was described in detail.

2 Motor operating principle and optimization strategy

Figure 1 shows the structure of the linear ultrasonic motor. The composite piezoelectric stator is a symmetry structure piezoelectric actuator. Eight rectangular piezoelectric ceramic plates polarized in the thickness direction are clamped between the front-end block and the rear-end block by bolts. Two such piezoelectric ceramic plates with opposite polarization direction are set as one group and a bronze electrode are clamped between the two piezoelectric ceramic plates. The mounting block is used to mount the stator. Here, the symmetric mode and anti-symmetric mode are utilized as operating modes.

The mode shapes (simulation results calculated through ANSYS software) are shown in Fig. 2.

In the symmetric mode (E_s), as shown in Fig. 2(a),two piezoelectric vibrators (called left wing and right wing) simultaneously stretch or shrink and thus result in the motion of the driving feet in the horizontal direction. In the anti-symmetric mode (E_a), as shown in Fig. 2(b), two wings simultaneously stretch or shrink in the reverse direction and thus result in the motion of the driving feet in the vertical direction. If the two operating models are actuated at the same time with a phase difference of 90°, the elliptical motion on the surface of the driving feet can be achieved. Thus, the stator can drive the slider pressed on its driving feet. To efficiently actuate the operating mode, the piezoelectric ceramic components are located at the position of the nodal surface of the two wings.

Fig. 1 Structure of motor

Fig. 2 Operating modes:

Figure 2 shows that there is no elastic deformation of the mounting block while the symmetric mode of the stator is actuated; however, the local bending vibration of the mounting block is excited while anti-symmetric mode of the stator is actuated. To decrease the influence on the vibration modal of the stator while clamping the stator, node points 1 and 3 were taken as the clamping points of the stator.

As described above, hybrid vibration modes were used as operating modes of the stator. Thus, the primary task of the optimization of the stator was to solve the problem of frequency consistency. At the same time, the interference modes were desired to be far from the operating modes. Furthermore, the amplitude of the driving feet was desired to be large enough and the amplitude of the mounting block was desired to be small enough.

Based on these requirements, the basic idea of the optimization is shown in Fig. 3. First, the suitable structure parameters were selected as design variables and the sample points in variable space were chosen using DOE. Second, the infinite element model of the was built based on the sample points using APDL (ANSYS parametric design language), and the mode analysis and harmonic response analysis of the stator were carried out and thus the response values were obtained (such as mode frequency difference, amplitude of driving feet, and amplitude of mounting block). Finally, the sample points and the response values were combined together to build the response surface approximation model, and then the best solution was found using suitable optimization method.

3 Numerical modeling of stator

Figure 4 shows the main structure parameters of the stator. The initial values of the main structure dimensions of the stator are shown in Table 1.

To obtain larger amplitude on the driving feet of the stator, the rear-end block of the stator used phosphor bronze material (QSN6.5) and the front-end block used duralumin material (LY12). The piezoelectric components used PZT-8 and the mounting block used stainless steel (1Cr18Ni9Ti). The material parameters of the metal elastic body and the piezoelectric element of the stator are listed in Table 2. Based on the structure parameters, as listed in Table 1, a 3D model of the stator with 11 248 elements and 14 202 nodes was globally built to perform the modal analysis, as shown in Fig. 5. Here, SOLID5, which has 3-D magnetic, thermal, electric, piezoelectric, and structural field capabilities with limited coupling between the fields and has eight nodes with up to six-degrees-of-freedom at each node, was selected as the element type of the PZTs. Solid45, which has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities and has eight nodes with three-degrees-of-freedom at each node, was selected as the element type of the metal elastic bodies.

Here, the nodes in the black border were selected and used to identify the operation modes in the optimization process, as shown in Fig. 6. The modal identification methods can be referred to Ref. [27].

4 Optimization

4.1 Optimization objectives

To obtain better output performance of the motor and the electromechanical conversion efficiency, as shown in Fig. 3, the optimization design of the motor should satisfy the requirements as follows.

4.1.1 Frequency consistency of two operating modes

As we know, to realize the effective ellipse motion trajectory on the driving tip of the stator, the primary task of the optimization of the stator that utilizes hybrid vibration modes as operating modes is to decrease the frequency difference of the two operating modes. Consequently, the dimensions of the stator based on hybrid vibration modes must be strictly designed to harmonize the eigen-frequencies of the stator. Thus, these design requirements of the stator can be written as

                       (1)

where f_s is the symmetric vibration mode; f_sis the anti-symmetric vibration mode.

Fig. 3 Optimization flow of stator

Fig. 4 Structure of stator

Table 1 Initial dimension of stator (Unit: mm)

Table 2 Material parameters of stator

Fig. 5 Finite element model of stator

Fig. 6 Nodes used for extracting operation modal shapes

4.1.2 Interference modes

The presence of interference modes causes unwanted vibration displacement, thus the output performance of the motor will descend. To solve the problem, the interference modes were desired to be far from the operating modes. This condition can be described as

(2)

where f_distb1 and f_distb2 are the interference mode frequencies.

4.1.3 Amplitude of driving feet

Under a certain pre-pressure, the amplitude of the driving feet was vital for the thrust and speed of the linear ultrasonic motor. At the same time, the vibration needed of the stator presented in this work was only in the xoy plane, not in the z direction. The vibration in the z direction will exacerbate friction damage of the friction interface. Thus, the optimum objectives in this condition can be described as

                        (3)

where U_x, U_y and U_z are the amplitudes of the driving feet in x, y and z directions, respectively.

4.1.4 Location of PZTs and clamping position of stator

As for the piezoelectric composite type of ultrasonic motor, the PZTs should be arranged on the places of maximum mode strain of the stator to gain large electromechanical conversion efficiency. At the same time, the places of the maximum mode strain is also the node planes of the stator, thus the mounting block of the stator was placed in here, and the amplitude of the mounting block was desired to be small enough. Consequently, the vibration displacements of those places were also used as an optimum objective.

.                       (4)

As we know, multi-objective optimization problem was often converted into a single-objective optimization problem. Here, the unified objective method with linear weight was utilized and the objective function can be written as

                            (5)

where F₁=R₁(x), F₂=|1/R₂(x)|, F₃=|1/R₃(x)|, F₄=|1/R₄(x)|, F₅=|1/R₅(x)|, F₆=|1/R₆(x)|, c_i(i=1, 2, ×××, 6) are the weighting coefficients for each sub-goal; F_i(i=1, 2, ×××, 6) are the sub-objective functions. Obviously, the six sub-objective functions are as small as possible. At the same time, the normalization processing of the values of F₁–F₆ is necessary.

4.2 Establishment of objective function response surface model

The parameters of P₁ P₄ P₅ and P₈ were selected as design variables, which can be described as

      (6)

The initial value of the design variables was set to be [11.5 18 7.5 3], and the value range of the design variables was

                          (7)

Using multidisciplinary design optimization software ISIGHT, latin hypercube design was carried out with the four-factor design and 30 groups of experimental sample points were obtained. 30 groups of response value, as given in Table 3, were therefore obtained by using finite element analysis. Based on Eq. (5), the response values of the stator were converted and all the data were normalized, as given in Table 4.

Table 3 Experimental sample points and response values of stator

Table 4 Normalized data of sample points and response values

Based on Table 4, a third-order polynomial was used to obtain the response surface model of the stator. The values of the parameters are presented in Table 5.

1) Approximate function of R₁(x)

             (8)

2) Approximate function of R₂(x)

      (9)

3) Approximate function of R₃(x)

            (10)

4) Approximate function of R₄(x)

                            (11)

5) Approximate function of R₅(x)

           (12)

6) Approximate function of R₆(x)

                    (13)

Table 5 Values of parameters in Eqs.(8)–(13)

The quality of response surface approximation function can be evaluated by using fitting degree evaluation index, such as multiple correlation coefficient R2, modified multiple correlation coefficient , and root mean square error (RMSE). Here, the coefficient was used and the values of of each response surface approximation function were almost 1, which indicates that the fitting degree of the six approximation functions was good enough to predict the response values of the stator based on different structure parameters defined in the design space.

4.3 Objective optimization designs

It can be seen from Table 4 that each optimization sub-objective was converted into dimensionless value and the magnitude was limited to the region [–1, 1]. Based on Eq. (5), different weighting factors were assigned to the different optimization sub-objectives according to the importance and design purpose. Here, simplex method was used for solving the unconstrained minimization problems to find optimization of objective function. The initial value of normalized design variable was set to be [0 0 0 0], and the convergence error was set to be 0.001. The optimization results are shown in Table 6.

Comprehensive comparison optimization results are given in Table 6, the synthesized optimization design effects of the fourth group is the best, and its optimization iteration curves are shown in Fig. 7.

Table 6 Optimization results with different weighting factors

Fig. 7 Optimization iteration curves of objective functions:

To validate the optimization result, the anti- normalization of the fourth group data was first carried out. Then, the modal analysis and harmonic analysis of the stator based on the structure parameters of the fourth group data was carried out using ANSYS software. The analysis results are shown in Table 7. The analysis results indicate that the optimization results obtained from the response surface model is effective. However, the precision of the parameters P₁, P₄, P₅ and P₈ achieves 0.000 1 mm, which will induce the difficulty in manufacturing of the stator. To solve the problem, the value of the parameters P₁, P₄, P₅ and P₈ were therefore rounded to 0.01 mm and recalculated using ANSYS software. The corresponding calculation results are shown in Table 8. Obviously, the optimization results compared with the initial values were significantly improved. The mode frequency difference decreased to be 5.698 Hz; the interference modes were far from the operating modes above 1 000 Hz; and the amplitudes of the driving feet in the x, y and z directions were 2.567, 5.779, and 0.107 1 μm, respectively. However, the amplitude of the clamping position of the stator was 0.944 2 μm, which was larger than the initial value. The main reason was that there was a conflict among the sub-objective functions F₁, F₅ and F₆.

5 Experiment verifications

One prototype motor was machined to the specifications of the final design for testing, as shown in Fig. 8. The mass of the stator was about 60 g, and the overall dimension of the stator was 86 mm×34 mm×8 mm. To find the suitable operating frequency, the frequency response of the stator without pre-load was measured using PSV-300F-B Doppler laser measuring meter, and thus the natural frequencies of the stator were obtained as shown in Fig. 9. The E_s mode frequency was 50.63 kHz, and the Ea mode frequency was 50.61 kHz. The frequency difference was only 20 Hz.

Fig. 8 Prototype of motor

The fixed-frequency test of the stator according to the frequency response results was carried out to measure the amplitude of the driving feet, as shown in Fig. 10.

Driven by 200 V (peak-peak) sinusoidal voltage signal with no phase difference at the frequency of 50.63 kHz, the Es mode was excited and the amplitude of the driving feet in x direction reached 4 μm. Driven by 200 V (peak-peak) sinusoidal voltage signal with a phase difference of 90°at the frequency of 50.61 kHz, the Ea mode was excited and the amplitude of the driving feet in y direction reached 2.5 μm.

Figure 11 shows the load characteristics of the motor under different pre-loads. In the case of the pre-load of 60 N and driving frequency of 50.61 kHz, the no-load speed of the motor was 873 mm/s, the maximum thrust was 16 N, and the maximal efficiency of the motor was 33.4%. In the case of the pre-load of 90 N and driving frequency of 50.61 kHz, the no-load speed of the motor was 723 mm/s, the maximum thrust was 27.5 N, the maximal efficiency of the motor was 43%, and the thrust-weight ratio was 45.8.

Table 7 Optimization results versus ANSYS analysis results

Table 8 Comparison results before and after optimization

Fig. 9 Frequency response curves of stator:

Fig. 10 Fixed-frequency test of stator:

Fig. 11 Load characteristics of motor

6 Conclusions

1) The main problem of empirical solutions is that it may take a long time to complete the design work of linear ultrasonic motors and usually only few people with large experience in the field are involved in this process.

2) The main advantage of applying proposed method is to obtain a systematic procedure to design linear ultrasonic motors and decrease the requirements for design experience of linear ultrasonic motors.

3) The proposed method is more open for designers who can select the response surface model by their special demands.

4) The experimental results show that many aspects of the design requirements of the butterfly-shaped linear ultrasonic motor have been fulfilled. The prototype motor fabricated based on the optimal design result exhibits considerably high dynamic performance, such as no-load speed of 873 mm/s, maximal thrust of 27.5N, maximal efficiency of 43%, and thrust-weight ratio of 45.8.

5) The future work is to carry out the optimization design of the motor including the stator and the slider.

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(Edited by HE Yun-bin)

Foundation item: Projects(51275235, 50975135) supported by the National Natural Science Foundation of China; Project(U0934004) supported by the Natural Science Foundation of Guangdong Province, China; Project(2011CB707602) supported by the National Basic Research Program of China

Received date: 2012–01–09; Accepted date: 2012–08–24

Corresponding author: CHEN Chao, PhD, Associate Professor; Tel: +86–25–84896685; E-mail: shijianchanghe@163.com