_{1}

In this paper, we try to use the coating of effective electrode surface and change the direction of polarization to design the mode shape piezoelectric motors of the first three modes. We also com-pare the gain of the mode shape piezoelectric motors with respect to the normal shape piezoelectric motor, including rotational speed, loading ability, torque, phase angle conversion and efficiency. According to the results of theoretical and simulation analysis, we have found that the gain of the mode shape piezoelectric stators are larger than the normal shape piezoelectric stator on average. According to the results of experiments, we found that the gain of the rotational speed, loading ability, torque, driving phase angle conversion and efficiency of the mode shape (MS1 - 3) piezoelectric motors are higher than the normal shape piezoelectric motor (NS) under driving condition of the second vibration mode. Also, the gain of the rotational speed and loading ability of the mode shape 2 (MS2) piezoelectric motor are higher than other shapes piezoelectric motors (NS, MS1 and MS3) under driving condition of the second vibration mode. The used maximum rotational speed of the mode shape 2 (MS2) piezoelectric motor is up to 946 rpm under conditions of 180 V
_{p-p} driving voltage, 10.7 kHz driving frequency, 0o driving phase angle and 13.0 gw net weight. The maximum loading ability and torque of the mode shape 2 (MS2) piezoelectric motor is respectively 451 gw and 0.91 mkgw-m under conditions of 180 V
_{p-p} driving voltage, 10.7 kHz driving frequency, 0o driving phase angle and 173 rpm rotational speed. And the gain of efficiency (output power) and maximum loading ability (torque) of the mode shape 2 (MS2) piezoelectric motor are respectively 2.28 and 1.54 with respect to the normal shape piezoelectric motor under conditions of 180 V
_{p-p} driving voltage, 10.7 kHz driving frequency and 0o driving phase angle. According to the results of the experiments, we have finally found that the piezoelectric motors (NS and MS1 - 3) can be driven only by the second vibration mode because the stator can produce elliptical motion and allows the rotor to generate orientation rotation. However, the first vibration mode can allow the rotor to be rotated very fast but it can’t make the rotation of the rotor orientation. Furthermore, we also found that the rotor can’t rotate by the third vibration mode because its vibration energy is absorbed by the structure itself, so causing the rotor stagnation.

The main object for study is a mode shape piezoelectric motor and its comparison object is the normal shape piezoelectric motor in this paper. Its appearance is similar to the tubular piezoelectric motors or ultrasonic motors. For comparison, the found experimental data are listed in _{p-p} driving voltage and 69.5 kHz driving frequency. And the maximum torque, output power and efficiency are 1.8 mNm, 60 mW, and 25% respectively. The following year, a smaller tubular piezoelectric motor has been proposed for publication [_{p-p} driving voltage and 131.8 kHz driving frequency. And the maximum torque, output power and efficiency are 0.5 mNm, 45 mW and 16% respectively. Then again the next year, another smaller tubular piezoelectric motor is proposed [_{p-p} driving voltage and 70.0 kHz driving frequency. But the maximum torque and loading are only 25 nNm and 0.5 mN respectively. In 2005, a miniature tubular piezoelectric motor is published [_{p-p} driving voltage and 67.0 kHz driving frequency. But the maximum torque is only 1.6 μNm. In 2006, another miniature tubular piezoelectric motor is proposed [_{p-p} driving voltage and 49.4 kHz driving frequency. And the maximum torque is 3.1 Nm. In 2007, another one miniature tubular piezoelectric motor is published [

References | Speed (rpm or mm/s) | Torque (mNm) or Loading (mN) | Voltage (V_{p-p}) | Frequency (kHz) | Size (mm) |
---|---|---|---|---|---|

[ | 573 | 1.8 | 120 | 69.5 | D2.4 ´ L10.0 |

[ | 430 | 0.5 | 100 | 131.8 | D1.6 x L6.0 |

[ | 3850 | 2.5E−5 or 0.5 | 40 | 70.0 | D0.8 ´ L2.2 |

[ | 4000 | 1.6E−3 | 25 | 67.0 | D4.0 ´ L5.0 |

[ | 380 | 3.1 | 200 | 49.4 | L/D = 3.5 |

[ | 9600 | 5.5E-3 | 30 | 314.5 | D4.0 ´ L5.0 |

[ | 2075 | 420 | 140 | 62.0 | D3.0 ´ L12 |

[ | 400 | 0.3 | 100 | 23.5 | D6.6 ´ L25.4 |

[ | 11 | 500 | 150 | 19.7 | D12.0 ´ L55.0 |

[ | 208 | 3.6E−3 or 5.0 | 100 | 97.0 | D5.0 ´ L15.0 |

speed is up to 9600 rpm under conditions of 30 V_{p-p} driving voltage and 314.5 kHz driving frequency. But the maximum torque is 5.5 μNm. In 2009, a metal tube type piezoelectric motor is applied to push the optical lens [_{p-p} driving voltage and 62.0 kHz driving frequency. And the maximum loading is 420 mN. In 2011, a single vibration mode tubular piezoelectric motor is published [_{p-p} driving voltage and 23.5 kHz driving frequency. But the maximum torque is only 0.3 mNm. Next year, a rotary-linear piezoelectric motor is proposed [_{p-p} driving voltage and 19.7 kHz driving frequency. But the maximum torque is up to 500 mN. Recently, a tubular piezoelectric motor is proposed [_{p-p} driving voltage and 97.0 kHz driving frequency. But the maximum torque and loading are only 3.6 μNm and 5.0 mN. At this point, we can find that the tubular piezoelectric motors have characteristics of high speed and low torque or low speed and high torque. However, if the torque or loading of the tubular piezoelectric motor is too small, then a big speed is still not being applied to for optical or electro-mechanical systems. Therefore, the appropriate composition, size, torque, loading ability and rotational or movement speed of the tubular piezoelectric motor must be carefully considered.

In this paper, the composition and operation principle of the normal shape and mode shape piezoelectric motors as shown Figures 1-3. Wherein the composition of the normal shape or mode shape piezoelectric motors which includes a piezoelectric stator and a rotor. The piezoelectric stator is composed of a metal tube and four piezoelectric ceramics which have the normal shape and mode shape effective electrode surface, as shown

As for the coating method of effective electrode surface of the mode shape piezoelectric ceramics are based on the vibration modes of the cantilever beam or piezoelectric stator, shown as

where the dimensionless eigenvalues

and

As for the effective electrode surfaces

Because of the high frequency vibrations are easily absorbed by the structure itself, so we only focus on the first three vibration modes and effective electrode surfaces of the mode shape piezoelectric ceramics, as shown

The equation of motion of the normal shape and mode shape piezoelectric stator can be obtained from the previous paper as follows [

x/L | FP_{1} | FP_{2} | FP_{3} | |||
---|---|---|---|---|---|---|

0.0 | 0.00 | 0.00 | 0.00 | 2.00 | 2.00 | 2.00 |

0.1 | 0.03 | 0.19 | 0.46 | 1.70 | 1.00 | 0.50 |

0.2 | 0.13 | 0.60 | 1.21 | 1.50 | 0.10 | −0.80 |

0.3 | 0.27 | 1.05 | 1.51 | 1.20 | −0.60 | −1.30 |

0.4 | 0.46 | 1.37 | 1.05 | 0.90 | −1.20 | −0.90 |

0.5 | 0.68 | 1.43 | 0.04 | 0.70 | −1.40 | 0.00 |

0.6 | 0.92 | 1.18 | −0.95 | 0.50 | −1.40 | 1.10 |

0.7 | 1.18 | 0.63 | −1.31 | 0.30 | −1.10 | 1.50 |

0.8 | 1.45 | −0.14 | −0.79 | 0.10 | −0.60 | 1.20 |

0.9 | 1.72 | −1.05 | 0.46 | 0.00 | −0.20 | 0.50 |

1.0 | 2.00 | −2.00 | 2.00 | 0.00 | 0.00 | 0.00 |

where

and

where the definition of constants in the above Equation (5)-(6) as follow:

where in above the symbols of

Furthermore we can find the general solution of mode shape and normal shape piezoelectric stator from the previous paper as follow [

and

where above constants can be determined by the following electro-mechanical boundary conditions:

and

Let Equation (12) and Equation (13) are substituted into Equation (14) and Equation (15), we can get the transverse displacement solution of the first three vibration modes of free end of the normal shape and mode shape piezoelectric stators on xz-plane as follow:

and

where the eigenvalues

Similarly, we can get the solution of vertical displacement of the first three vibration modes of the normal shape and mode shape piezoelectric stators at of free end on xy-plane as follow:

and

Under steady-state operating mode, above Equation (16-19) can be rewritten as follow:

and

where

In order to get an elliptical trajectory of the normal shape and mode shape piezoelectric stators at the free end, we can make driving phase angle difference 90˚ or use a different driving waveforms under different sides of piezoelectric stators. Therefore, above Equations (23)-(24) can be rewritten again as follow:

and

Follow by, we can get an elliptical trajectory of the normal shape and mode shape piezoelectric stators at the free end on the zy-plane from Equations (20)-(21) and Equations (24)-(25), shown as

If we further consider the effect of axial or horizontal vibration on the normal shape and mode shape piezoelectric stators, then above equations can be rewritten as again:

and

where the axial or horizontal vibration displacement of the normal shape and mode shape piezoelectric stators as follow:

and

wherein

So far, we can get an oval ball of the normal shape and mode shape piezoelectric stators at the free end, as follow:

and

In fact,

In this paper, we can get the gain of vertical and transverse displacement or velocity (

or

and

or

So for, we can predict the gain of the normal shape and mode shape piezoelectric stators from Equations (34)-(37) under the same driving conditions.

In addition, the conversion efficiency of the normal shape and mode shape piezoelectric stators or motors can be defined as:

where the definition of the input power as:

and the definition of the output power as:

where

In this paper, we have to understand the differences and gain between the normal shape and mode shape piezoelectric motors or stators, in particular, through theoretical analysis, simulation analysis and experiments to validate. And the material properties are used in the analysis and the experimental procedure, as shown in

wherein the step of the theoretical analysis is as follows:

(5-1-1) Using the simple approximate solution of one-dimensional spatial structure to do analysis, and only do a comparative analysis of dimensionless vibration displacement or velocity by Equations (38-39).

(5-1-2) Using the minimum frequency spacing (

As for the simulation analysis procedure is as follows:

(5-2-1) Modeling of the mode shape and normal shape piezoelectric stators respectively, including select element type, enter the physical properties, as well as coordinate system conversion, as shown

(5-2-2) Meshing of the mode shape and normal shape piezoelectric stators respectively, including select the most sophisticated cutting of mesh or select the smart size 1, as shown

(5-2-3) Simulation Analysis: Solving of the mode shape and normal shape piezoelectric stators respectively, including setting boundary conditions of electro-mechanical, as shown

(5-2-4) Post-processing of the mode shape and normal shape piezoelectric stators respectively, includes processing the first three modes, the maximum deformation or electric potential, as shown Figures 10-13.

As regards the experimental step is as follows:

(5-3-1) Cutting the piezoelectric ceramics with the effective electrode surface, and making the piezoelectric stators of normal shape and mode shape, shown as

(5-3-2) Using the dual channel arbitrary function generator (Model: A-303, AA Lab. Systems Ltd. Co.) and

Dimension _ Material Name | 1D_PZT | 2D_PZT | 3D_PZT | 1D_Al | 2D_Al | 3D_Al |
---|---|---|---|---|---|---|

Relative Permittivity (N.A.) | ε11 = 1730 | ε11 = ε22 = 1730, ε33 = 1700 | ε11 = ε22 = 1730, ε33 = 1700 | 0 | 0 | 0 |

Piezoelectric Stress Constant (V/Nm) | e31 = −5.3 | e31 = e32 = −5.3 | e31 = e32 = −5.3, e33 = 15.8, e24 = e15 = 12.3. | 0 | 0 | 0 |

Young's Modulus (Pa) | c11 = 1.2e11 | c11 = c22 = 1.2e11 | c11 = c22 = 1.2e11, c12 = c21 = 7.52e10, c13 = c31 = c23 = c32 = 7.51e10, c33 = 1.11e11, c44 = 3.0e10, c55 = c66 = 2.6e10 | c11 = 7E10 | c11 = c22 = 7E10 | c11 = c22 = c33 = 7E10 |

Density (kg/m^{3}) | 7600 | 7600 | 7600 | 2700 | 2700 | 2700 |

Poisson Ratio (N.A.) | 0 | 0.33 | 0.33 | 0 | 0.35 | 0.35 |

Size (mm^{3}) | 50 × 6 × 0.7 | 50 × 6 × 0.7 | 50 × 6 × 0.7 | 50 × 6 × 6 | 50 × 6 × 6 | 50 × 6 × 6 |

the power amplifier (Model: AFG-3022, Tektronix Co.) to drive the normal shape and mode shape piezoelectric motors, shown as

(5-3-3) Using the digital tachometer (Model: RM-1501, TES Electrical Electronic Co.) and the sound level meter (Model: TES 1350A , TES Electrical Electronic Co.) to measure the rotational speed and noise of the normal shape and mode shape piezoelectric motors, shown as

(5-3-4) Using the loading test weight and rotors to test the loading ability of the normal shape and mode shape piezoelectric motors, shown as

copper. Its size and weight are D15 mm ´ H10 mm and 13 gw. Due to the material of metal tube of the piezoelectric stator is aluminum, and the material of rotor is copper. So the dynamic friction coefficient of metal tube and rotor is about 0.22 - 0.27 or

According to the results of theoretical analysis, simulation analysis and experiments, we found:

1) Under conditions of theoretical analysis of one-dimensional (1D) approximation solution and using the

minimum frequency spacing of 1Hz, the gain of the maximum dimensionless vibration displacement or velocity of the all mode shape 1 - 3 (MS1 - MS3) piezoelectric stators are far than the normal shape (NS) piezoelectric stator, as shown

Mode | G1_Th. | G2_Th. | G3_Th. | G1_1D_Si. | G2_1D_Si. | G3_1D_Si. | G1_2D_Si. | G2_2D_Si. | G3_2D_Si. | G1_3D_Si. | G2_3D_Si. | G3_3D_Si. | G1_Ex. | G2_Ex. | G3_Ex. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 19.70 | 0.00 | 0.00 | 1.10 | 1.00 | 0.80 | 1.30 | 1.30 | 1.30 | 1.20 | 1.30 | 1.00 | N.A. | N.A. | N.A. |

2 | 0.00 | 3.30 | 0.00 | 1.20 | 0.90 | 0.80 | 1.40 | 1.40 | 1.40 | 1.20 | 1.30 | 1.30 | 1.19 | 1.31 | 1.29 |

3 | 0.00 | 0.00 | 20.10 | 1.40 | 1.10 | 0.80 | 1.50 | 1.50 | 1.50 | 1.10 | 1.50 | 1.50 | 0.00 | 0.00 | 0.00 |

Note: Th.: Theoretical Analysis; Si.: Simulation Analysis; Ex.: Experiments; N.A.: Represents the rotation direction of the rotor can not be determined.

the largest gain which is about 20.06. Followed by, the gain of the mode shape 1 (MS1) piezoelectric stator is about 19.70. Furthermore is the gain of the mode shape 2 (MS2) piezoelectric stator is about 3.33.

2) Under conditions of using the 1D material properties to simulate the 3D piezoelectric stators, the gain of the maximum dimensionless vibration displacement or velocity of the first three modes of the mode shape 1 (MS1) piezoelectric stator is larger than other type (NS, MS2 and MS3) piezoelectric stators. Where the gain of the third vibration mode of the mode shape 1 (MS1) piezoelectric stator with respect to the normal shape (NS) piezoelectric stator is about 1.40, as shown in

3) Under conditions of using the 2D material properties to simulate the 3D piezoelectric stators, the gain of the maximum dimensionless vibration displacement or velocity of the all mode shape 1 - 3 (MS1 - MS3) piezoelectric stators are larger than the normal shape (NS) piezoelectric stator. Where the gain of the orthogonal modes of the mode shape 1 - 3 (MS1 - 3) piezoelectric stators with respect to the normal shape (NS) piezoelectric stator are 1.30, 1.40 and 1.50, as shown in

4) Under conditions of using the 3D material properties to simulate the 3D piezoelectric stators, the gain of the maximum dimensionless vibration displacement or velocity of the all mode shape 1 - 3 (MS1 - MS3) piezoelectric stators are larger than the normal shape (NS) piezoelectric stator on average. Where the gain of the orthogonal modes of the mode shape 1 - 3 (MS1 - 3) piezoelectric stators with respect to the normal shape (NS) piezoelectric stator are 1.20, 1.30 and 1.50, as shown in

5) According to the results of experiments, shown as

tion mode of the mode shape 1 - 3 (MS1 - 3) piezoelectric stators with respect to the normal shape (NS) piezoelectric stator are 1.19, 1.31 and 1.29, as shown

6) Shown as in _{p-p} driving voltage, 13 gw net weight, maximum loading and different dynamic friction coefficient (_{p-p} driving voltage, maximum loading,

7) Shown as in _{p-p} driving voltage, maximum loading, different dynamic friction coefficient (_{p-p} driving voltage, maximum loading,

8) Shown as in

9) Shown as in

10) Shown as in

11) Shown as in _{p-p}) and 13gw net weight. And rotational speed of the mode shape piezoelectric motor is higher than the normal shape piezoelectric motor on average. Where the r maximum rotational speed of the mode shape 2 (MS2) piezoelectric motor is about 946 rmp under conditions of 180 V_{p-p} driving voltage and 13gw net weight.

Pout of Net Weight (mW) | Pout of Maximum Loading (mW) | Efficiency ( | Gain | |||||
---|---|---|---|---|---|---|---|---|

Items | ||||||||

NS | 2.27 | 2.79 | 8.30 | 10.18 | 0.03% | 0.04% | 1.00 | 1.00 |

MS1 | 2.70 | 3.32 | 17.22 | 23.25 | 0.07% | 0.09% | 2.08 | 2.28 |

MS2 | 2.99 | 3.67 | 18.95 | 23.26 | 0.07% | 0.09% | 2.28 | 2.28 |

MS3 | 2.94 | 3.60 | 14.66 | 18.00 | 0.06% | 0.07% | 1.77 | 1.77 |

Note: Input Power_P_{In} = 26 Watts Under Condition of 180 V_{p-p} Driving Voltage. The Maximum Loading of the NS, MS1, MS2 and MS3 is 292 gw, 398 gw, 451 gw and 345 gw respectively. The Driving Frequency of the Second Virbration Mode of the NS, MS1, MS2 and MS3 is 10.3 kHz, 10.3 kHz, 10.7 kHz and 10.8 kHz respectively.

Torque (mkgw-m) | Gain of Torque (mkgw-m/mkgw-m) | |||
---|---|---|---|---|

Items | ||||

NS | 0.48 | 0.59 | 1.00 | 1.00 |

MS1 | 0.66 | 0.81 | 1.36 | 1.36 |

MS2 | 0.74 | 0.91 | 1.54 | 1.54 |

MS3 | 0.57 | 0.70 | 1.18 | 1.18 |

According to the results of theoretical analysis, we have found that the gains of the mode shape piezoelectric stators are larger than the normal shape piezoelectric stator. And we also found that the mode shape piezoelectric stators can be effectively separated from the modal vibration. However, we have also found that the normal shape piezoelectric stator does not have this ability in theory. According to the results of simulation analysis, we have also found the gain of the mode shape piezoelectric stators can still appear in nonorthogonal vibration modes. Because the results of theoretical analysis are solutions in an ideal approximation, the results of simulation analysis are closer to reality. Therefore, the real results eventually must be confirmed through experiments. According to the experimental results, we have moreover found that the rotor can be rotated only by the second vibration mode. Here the gain of the rotational speed, loading ability and torque of the mode shape 2 (MS2) piezoelectric motor is higher than other shapes piezoelectric motors (NS, MS1 and MS3) under driving condition of the second vibration mode. And the gains of the rotational speed, loading ability, torque, driving phase angle conversion and efficiency of the mode shape (MS1 - 3) piezoelectric motors are higher than the normal shape piezoelectric motor (NS) under driving condition of the second vibration mode. In addition, we have also found that the first vibration mode allows the rotor to rotate very fast, but we can’t determine the direction of rotation of the rotor. However, it can’t make the rotation of the rotor orientation. Furthermore, we also found

that the rotor cannot be rotated by the third vibration mode because its vibration energy is absorbed by the structure itself, so causing the rotor stagnation. At last, we have also found that the noise spectrum can help us quickly find the best vibration mode position of the normal shape and mode shape piezoelectric motors. When the noise level or decibel increases, the piezoelectric motors are entering into best vibration mode.

This study can be finished smoothly, I especially want to thank MOST of Taiwan of ROC sponsor on funding, (Project No.: MOST103-2221-E-230-007).