_{1}

^{*}

In this paper, we want to make a new type linear piezoelectric motor by mode shape coating or effective electrode surface coating. The mode shape is derived from the mechanical boundary conditions of the linear piezoelectric motor. We only have access to the first three modes of formas, the effective electrode surface coating basis, as well as with the linear piezoelectric motor of normal shape do comparison. Next, we will inspect their gain or axial velocity through theoretical analysis, simulation and experiment. According to the results of the theoretical analysis, we have found that the gain or axial velocity of the linear piezoelectric motors of mode shape is much larger than the linear piezoelectric motors of normal shape. However, according to the results of simulation and experiments, we have found that the gain or axial velocity of the linear piezoelectric motors of mode shape is much greater than the linear piezoelectric motors of normal shape, which is about 1.2 to 1.4 times. The linear piezoelectric motor of mode shape 3 has the fastest axial velocity, which is about -48 mm/s and 48 mm/s under conditions of 180 V
_{p-p} driving voltage, 21.2 kHz driving frequency (the third vibration modal), 25 gw loading and the position of loading or mass at
x = 5 mm & 45 mm respectively. And its axial velocity is about 1.4 times the linear piezoelectric motor of normal shape under the same conditions. Overall, the mode shape coating helps to enhance the gain or axial velocity of the linear piezoelectric motor.

Over the past decade, the linear piezoelectric motors have a very significant development and excellent performance, as shown in

Year | Movement Speed (mm/s) | Thrust or Loading (N) | Driving Frequency (kHz) | Driving Conditions | References |
---|---|---|---|---|---|

2006 | 810 | 100 | 31.5 ~ 31.75 | 150 V_{P−P} | [ |

2007 | 44.8 | 12.85 | 63.4 | 100 V_{P−P} | [ |

2008 | 1280 | 45 | 27.476 | 150 V_{rms} | [ |

2008 | 235 | 21.4 | 36 | 200 V_{rms} | [ |

2009 | 960 | 103 | 23.9 | 200 V_{rms} | [ |

2009 | 1500 | 13 or 140 | 14330 | 80 V_{p−p} or 2 W | [ |

2009 | 286 | 142 | 42.9 | 200 V_{p−p} | [ |

2010 | 1160 | 20 | 25.7 ~ 26.8 | 200 V_{rms} | [ |

2011 | 240 | 29.43 | 34 | 100 V_{P−P} | [ |

2011 | 150 | 30 | 17.9 | 30 V_{p−p} | [ |

2012 | 854 | 40 | 23.59 | 200 V_{rms} or 10.39 W | [ |

2013 | 88 | 1.96 | 41.13 | 16 V_{p−p} | [ |

2013 | 230 | 0.6 | 92.00 | 150 V_{p−p} | [ |

2013 | 928 | 60 | 22.50 | 200 V_{rms} | [ |

2013 | 1527 | 50 | 24.80 | 200 V_{rms} | [ |

2014 | 170 | 1.8 | 136.5 | 100 V_{p−p} | [ |

2014 | 760 | 64.2 | 50.00 | 100 V_{p−p} or 17.4 W | [ |

2014 | 140 | 0.3 | 99.00 | 100 V_{p−p} or 9.3 mW | [ |

2015 | 230 | 0.3 | 174.0 | 80 V_{p−p} | [ |

transducer [_{p−p}, 30 V_{p−p} and 80 V_{p−p} respectively. Due to the linear piezoelectric motors have superior performance, and they have a high use value in the precision mechanical systems. So in addition to the above the linear piezoelectric motors, in any form or performance of the linear piezoelectric motors are worth trying to develop. In addition, we, according to the rotary piezoelectric motor of mode shape, have the advantage of a high rotational speed and excellent performance [

In this study, the linear piezoelectric motor can be regarded as a like simple supported beam with pinned-pinned boundary conditions [

where the dimensionless eigenvalues

And further we can get another function of the effective electrode surface

where the

x/L | ||||||
---|---|---|---|---|---|---|

0.0 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

0.1 | 0.31 | 0.59 | 0.81 | −0.31 | −0.59 | −0.81 |

0.2 | 0.59 | 0.95 | 0.95 | −0.59 | −0.95 | −0.95 |

0.3 | 0.81 | 0.95 | 0.31 | −0.81 | −0.95 | −0.31 |

0.4 | 0.95 | 0.59 | −0.59 | −0.95 | −0.59 | 0.59 |

0.5 | 1.00 | 0.00 | −1.00 | −1.00 | 0.00 | 1.00 |

0.6 | 0.95 | −0.59 | −0.59 | −0.95 | 0.59 | 0.59 |

0.7 | 0.81 | −0.95 | 0.31 | −0.81 | 0.95 | −0.31 |

0.8 | 0.59 | −0.95 | 0.95 | −0.59 | 0.95 | −0.95 |

0.9 | 0.31 | −0.59 | 0.81 | −0.31 | 0.59 | −0.81 |

1.0 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

According to the functions of the effective electrode surface or from Equation (3), we can design a series of linear piezoelectric motors, including the linear piezoelectric motors of normal shape and mode shape, as shown in _{3}) and resonance frequency (fr) or proper loading, we can make linear motion of the piezoelectric motor, as shown in

membrane force, the piezoelectric motor will move to the right. On the contrary, the piezoelectric motor will then move to the left.

In order to distinguish differences of the linear piezoelectric motor of the normal shape and mode shape, we can simplify the equations of motion of 3D structure into the equation of motion of the two 1D structures. One of which is the equation of motion of the axial vibration direction, the other is the equation of motion of the transverse vibration direction as follow:

and

where the membrane force (N_{1}) and of bending moment (M_{1}) unit width are defined as:

and

We can let Equation (1)-(3) & (6) (7) are substituted into Equation (4) & (5), and then after finishing, we can get the equations of motion by axial and transverse displacement of the linear piezoelectric motor of the normal shape and mode shape as follow:

and

According above the Equations (8)-(11), we can get the general solutions of axial and transverse vibration displacement (

and

where the definition of constants or symbols in the above Equations (4)-(15) as follow:

where in above the symbols of A_{11}, A_{n}, c_{11}, D_{11}, e_{31}, f_{i}, f_{n}, h, t, U, u, V_{3}, W, w, x, z, ρ, ω_{i}, ω_{n},

Where above undetermined coefficients can be determined by the following electro-mechanical boundary conditions:

and

where

and

Another the frictional force is defined as follow:

where

Because the functions of mode shape are equal to zero under pinned-pinned boundary conditions as:

So the Equation (30) and Equation (32) can be rewritten as follows:

and

where

Let Equations (25)-(36) are substituted into Equations (12)-(15), we can get the solution of the axial and transverse vibration displacement of the first three modes of the linear piezoelectric motors as follow:

and

Let Equations (35)-(41) are substituted into Equations (12)-(15), we can get the particular solutions of the transverse and axial vibration displacement of the linear piezoelectric motor as follow:

and

Further, we can make Equations (42)-(45) simplified as follows:

and

where

Therefore, we can once again get the equations of elliptical trajectory of the linear piezoelectric motor of normal shape and mode shape at contact points on both sides from above Equations (46)-(49), as follow or shown in

and

At this point we can see the difference in the linear motors of normal shape and mode shape from the solutions of displacement. Then we will be to test their differences or gains by the case study of theoretical analysis, computer simulation and experiments.

We have to understand the differences and gain between the linear piezoelectric motors of the normal shape and mode shape in this study. We will try to verify the differences and gain of the linear piezoelectric motors through theoretical analysis, simulation analysis and experiments. Where in the associated electrical and mechanical properties of the linear piezoelectric motors in special case study is shown in

The first is theoretical analysis in this special case study, which is as follows:

(5-1-1) Try to find the approximate solution of the transverse displacement of the linear piezoelectric motors of the normal shape and mode shape in the special case study. We can take to understand their difference and gain under condition of the normal shape and different mode shape electrode coating.

(5-1-2) Try to find the approximate solution of the axial displacement of the linear piezoelectric motors of the normal shape and mode shape in the special case study. We can further understand the impact of the axial or linear movement under conditions of the normal shape and different mode shape electrode coating or the above modal vibration.

(5-1-3) Try to understand the impact of the axial or linear movement of the linear piezoelectric motors of the normal shape and mode shape under conditions of the different loading (or mass) and positions.

Second is simulation analysis by ANSYS code (its modal analysis under different driving conditions shown in Figures 5-10, its main analytical work as follows:

(5-2-1) Try to find the vibration modal of the linear piezoelectric motors of the normal shape and mode shape in the special case study.

(5-2-2) Try to find the axial (or linear) and transverse displacement from the above vibration modal under condition of net weight.

(5-2-3) Try to find the axial (or linear) and transverse displacement under conditions of different loading, position and steady state.

(5-3-1) Finally the experimental work (its prototype and structure shown in

(5-3-2) Try to find the maximum axial (or linear) speed of linear piezoelectric motors of the normal shape and mode shape under condition of net weight.

(5-3-3) Try to find the maximum axial (or linear) speed of linear piezoelectric motors of the normal shape and mode shape under condition of the different loading (or mass) and position.

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

1) Shown in Figures 13-15, we have found that the change in axial displacement or velocity of more focus and smooth of the linear piezoelectric motors of mode shape with respect to the linear piezoelectric motor of normal shape by theoretical analysis under conditions of the first three vibration modal or driving frequency, 180 V_{p−p} driving voltage, net weight, frequency spacing of 1 Hz and the first three eigenvalues.

2) Shown in _{p−p} driving voltage, net weight, frequency spacing of 1 Hz and the first three eigenvalues.

Physical Name | PZT | Al |
---|---|---|

Relative Permittivity (N.A.) | ε_{11} = ε_{22} = 1730, ε_{33} = 1700 | 0 |

Piezoelectric Stress Constant (V/Nm) | e_{31} = e_{32} = −5.3, e_{33} = 15.8, e_{24} = e_{15} = 12.3. | 0 |

Young’s Modulus (Pa) | c_{11} = c_{22} = 1.2e11, c_{12} = c_{21} = 7.52e10, c_{13} = c_{31} = c_{23} = c_{32} = 7.51e10, c_{33} = 1.11e11, c_{44} = 3.0e10, c_{55} = c_{66} = 2.6e10 | c_{11} = c_{22} = c_{33} = 7E10 |

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

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

Size (mm^{3}) | 50 × 10 × 1.4 | 50 × 10 × 4.0 |

3) Shown in _{p−p} driving voltage, different position of loading or mass, frequency spacing of 1 Hz and the first three eigenvalues.

4) Shown in _{p−p} driving voltage, 25 gw loading and the first three vibration modal or driving frequency.

5) Shown in

6) Shown in

Mode | Fr(kHz)_1D Theory Analysis | Fr(kHz)_3D Simulation Analysis | Fr(kHz)_Experiments |
---|---|---|---|

1 | 2.140 | 2.658 ~ 3.103 | 2.800 ~ 3.200 |

2 | 8.560 | 8.197 ~ 10.419 | 10.800 ~ 11.200 |

3 | 19.260 | 18.574 ~ 21.734 | 20.800 ~ 21.200 |

under conditions of different position of loading or mass, 180 V_{p−p} driving voltage, 25 gw loading and the first three vibration modes or driving frequency. Wherein the linear piezoelectric motor of mode shape 3 having the fastest axial velocity, which is about −48 mm/s and 48 mm/s under conditions of 180 V_{p−p} driving voltage, 21.2 kHz driving frequency (the third vibration modal), 25 gw loading and the position of loading or mass at x = 5 mm & 45 mm respectively. And its axial velocity is about 1.4 times the linear piezoelectric motor of normal shape under the same conditions.

According to the results of theoretical analysis, simulation analysis and experiments, we have found that the change in axial displacement or velocity of the linear piezoelectric motors of mode shape is more focus and smooth than the linear piezoelectric motor of normal shape. We also have found that the gain in axial and transverse displacement or velocity of the linear piezoelectric motors of mode shape is much larger than the linear piezoelectric motor of normal shape. We have further found that the axial velocity of the linear piezoelectric motors of mode shape is much larger than the linear piezoelectric motor of normal shape under condition of eccentric loading or mass. We also have found that the linear piezoelectric motors of normal shape and mode shape cannot move faster, when the state of net weight or the loading (or mass) placed at the center position of the motors. When the loading or mass is farther away from the center, it’s moving faster. And we also have found that the placement of the loading or mass can change the direction of axial movement of the motors. Overall, for all motors, the gain of the axial velocity of the third vibration modal is much larger than the first and second vibration modal. From the above results, we have found that the transverse mode shape coating has a positive effect on the axial displacement. That is, the transverse mode shape coating can make energy more concentrated with respect to the normal shape coating, so that axial displacement can more quickly move.

This study can be finished smoothly, I especially want to thank MOST of Taiwan of ROC sponsor on funding, (Project No.: MOST104-2221-E-230-016).

Jwo MingJou, (2015) A Study on the Linear Piezoelectric Motor of Mode Shape. Open Journal of Acoustics,05,153-171. doi: 10.4236/oja.2015.54013