_{1}

^{*}

In this paper, we will try to find a universal theoretical model and approximate solutions which can be applied to both mode shape and normal shape actuators and sensors, and which can be predicted the gain of the first three modes of the mode shape and normal shape actuators and sensors, finally through computer simulation analysis to validate. In order to prove the feasibility of the theory and as well as convenient to use on the electro-mechanical engineering, we will try to simplify the three-dimension structure problem into an one-dimension structure problem. Furthermore we will design one kind of bimorph type piezoelectric cantilever beam, so that it can be used as with the actuator and sensor simultaneously, but also conducive to the theory and simulation analysis. As for the simulation analysis, we will use the ANSYS code.

Due to the piezoelectric materials (such as PZT or PVDF) have the advantages of fast response and high actuating force, so they are very suitable made for sensors and actuators to sense and control the vibration of the flexible structures, especially to sense or control the low frequency vibration, because the high-frequency vibration is very easy to be absorbed by the structure itself. So that the most piezoelectric sensors and actuators are used to sense and control the low frequency vibration of flexible structures. And since the design concept and theory of the modal sensors and actuators have been proposed by C.K., Lee [

However, the theoretical model is only applicable to modal sensors and actuators, but it can’t be applied on the normal shape sensors and actuators. If the theoretical model of modal sensors and actuators used in the normal shape sensors and actuators, we could not find the correct approximate solution. That is, so far, we still can’t find one kind of universal equation to simultaneously describe or predict the behavioral of the mode shape and normal shape sensors and actuators, or to compare the gain or to distinguish the difference between the mode shape and normal shape sensors and actuators from past studies.

In this paper, we will try from the original actuator and sensor equation, and electromechanical boundary conditions to find one kind of universal theoretical model and reasonable approximate solutions to compare the gain and difference between the mode shape and normal shape sensors and actuators. Furthermore we will also through the analysis results of computer simulation to verify the correctness and feasibility of the universal theoretical model and approximate solutions.

In order to understand the differences between the mode shape and normal shape actuators and sensors, we just start from the one-dimension actuator equation of bimorph type piezoelectric cantilever beam. The main contribution of this equation comes from the bending effect, so the membrane effect can be completely ignored. Therefore, the equation can be simplified to as follows [

where the moment M_{1} can be divided into two terms of the mechanical

As for the constants in the above Equations (1) and (2) can be defined as follow:

wherein above the symbols of c_{11}, D_{11}, h, w, t, x, z and

Furthermore the electrical moment

And

wherein above the symbols of FP_{n}, l, ^{th} mode shape, length, natural angular velocity (natural frequency), mode shape eigenvalues, mode shape constant and mode shape function.

Let Equations (2)-(7) substituted into Equation (1), and after finishing, we can get a non-homogeneous partial differential equation of motion with transverse displacement of mode shape actuator as follows:

When the surface is uniformly coated and normal polarization with the bimorph type piezoelectric cantilever beam, that is, the function of effective electrode surface of Equation (7) can be redefined as:

Then, the second derivative of electrical moment becomes zero.

So for the normal shape actuator, Equation (8) can be simplified as:

In order for the mode shape and normal shape actuator equations can be applied to structures of different lengths, we can try to make Equation (8) and Equation (11) become the dimensionless equations:

And

where the dimensionless mode shape constant can be redefined as:

where the solution form of Equation (12) and Equation (13) can be expressed as:

And

Furthermore let the homogeneous solution of Equation (16) and Equation (17) are assumed to be as follows:

As for the particular solution of Equation (15) is assumed to be as follows:

where the symbols of

We can take Equation (17) into Equation (12) and Equation (13), then after finishing, we can get a dimensionless homogeneous four-order ordinary differential equation, as:

where the dimensionless natural eigenvalue of mode shape is defined as:

And the i^{th} natural resonance frequency of piezoelectric beam multilayer or mode shape or normal shape actuator can be derived by Equation (20), as:

We can further get a dimensionless homogeneous solution from Equation (19)

In addition, we can take Equation (18) into Equation (12), so get the undetermined coefficient as:

where the relationship of the fourth-order differential function and zero-order function is:

So Equation (23) can be rewritten as:

Thus we can get the solution of the undetermined coefficient.

where

At this point, we can find the general solution of mode shape and normal shape actuator, as:

And

Next, we can find the special solutions of the bimorph type piezoelectric cantilever beam from the electro- mechanical boundary conditions as follow:

And

Or

And

where the dimensionless mode shape function of mode shape actuator under clamped-free boundary conditions as:

And the dimensionless eigenvalues

And

According to the electro-mechanical and clamped-free boundary conditions, we can determine the constants of Equation (27) and Equation (28), as follow:

And

where the constants of mode shape actuator are defined as

And

where the second-order and third-order derivative mode shape function on the free boundary as follow:

And

As for the constants of normal shape actuator were defined as

And

which let

Until now, we can get a dimensionless general solution of the mode shape and normal shape actuator as follow:

And

Due to the high frequency vibrations can easily be absorbed by the structure itself, so we are only interested in low-frequency vibration. And in order to understand the difference between the mode shape and normal shape actuators, we analyzed only for the first three resonant modes of the structures. And according to Equation (34) and Equation (35), we can get the eigenvalues and parameters of the first three modes, shown as

Mode | ||
---|---|---|

1 | 1.8751 | 0.7341 |

2 | 4.6941 | 1.0185 |

3 | 7.8548 | 0.9992 |

get a dimensionless transverse displacement of the mode shape and normal shape piezoelectric stator under steady state and the same driving conditions as shown in

Another according to Equation (40) and Equation (41), we can get the second and third derivative of mode shape function of the first three modes of mode shape actuator relative to the unit length of structure, shown as

So far, Equation (46) of mode shape actuator appeared to be consistent with the particular solution of previously papers [

Furthermore, we can make Equation (45) of normal shape actuator simplifies to

where the dimensionless mode shape function of normal shape actuator is defined as

And the dimensionless parameters

According to Equation (46) and Equation (47), we can get the ratio of dimensionless transverse displacement

of mode shape and normal shape actuator under conditions of steady state, constant driving voltage and the same bending stiffness constant per unit width as follows:

In order to facilitate understanding of the difference between mode shape and normal shape actuator, we can

set the above ratio at the free end, that is,

fore Equation (50) can again be expressed as

For the bimorph type piezoelectric cantilever beam under condition of constant electric potential, the sensor or current equation per unit length and width of the mode shape and normal shape sensor can be expressed as [

And we can further take Equation (46) and Equation (47) into Equation (52) respectively, let Equation (52) be divided into two types of sensor or current equations as follow:

And

where the first derivative of dimensionless mode shape function of the first three modes can be obtained from clamped-free boundary conditions, or known from

where the first derivative of dimensionless mode shape function of the mode shape and normal shape sensor can

be expressed as follow:

And

Since the current is proportional to voltage under condition of the same load R_{L}, So Equation (55) can also be expressed the ratio of dimensionless voltage of the mode shape and normal shape sensor as follows:

In order to understand the differences between mode shape and normal shape actuators and sensors by theory and simulation analysis, we specially design a series of one-dimension bimorph type piezoelectric cantilever beams, including the mode shape and normal shape actuators and sensors, shown as

Wherein the step of theoretical analysis is as follows:

(1-1) Using different frequency spacing

(1-2) Selecting the minimum frequency spacing

As for the step of simulation analysis is as follows:

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

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

(2-3) Solving of the mode shape and normal shape actuators and sensors respectively, including setting boundary conditions of electro-mechanical, as shown

(2-4) Post-processing of the mode shape and normal shape actuators and sensors respectively, includes processing the first three modes, the maximum deformation or electric potential, as shown Figures 11-13.

Physical Name | Physical Quantities |
---|---|

Size of Single Layer | |

Relative Permittivity | |

Piezoelectric Stress Constants | |

Young’s Modulus | |

Density | |

Poisson Ratio |

According to the results of theory analysis, we found:

1) Under condition of the frequency spacing of 100 Hz, the maximum dimensionless transverse displacement of the second modal of the normal shape and mode shape actuators is smaller than the first and the third modals’, as shown in

2) Under condition of the frequency spacing of 10 Hz, the maximum dimensionless transverse displacement of the normal shape actuators is inversely proportional to the modal, as shown in

Mode | Actuators | Sensors | ||||||
---|---|---|---|---|---|---|---|---|

NSA (m/m) | MS1A (m/m) | MS2A (m/m) | MS3A (m/m) | NSS (V/V) | MS1S (V/V) | MS2S (V/V) | MS3S (V/V) | |

1 | 2.20 | 2.39 | 0.09 | 0.03 | 8.62 | 6.17 | 2.11 | 2.01 |

2 | 1.79 | 0.01 | 2.14 | 0.04 | 40.25 | 0.04 | 48.08 | 2.31 |

3 | 6.86 | 0.00 | 0.01 | 13.30 | 422.45 | 0.00 | 0.30 | 820.33 |

Mode | Actuators | Sensors | ||||||
---|---|---|---|---|---|---|---|---|

NSA (m/m) | MS1A (m/m) | MS2A (m/m) | MS3A (m/m) | NSS (V/V) | MS1S (V/V) | MS2S (V/V) | MS3S (V/V) | |

1 | 217.08 | 188.79 | 0.09 | 0.03 | 487.88 | 764.39 | 2.09 | 2.01 |

2 | 33.59 | 0.01 | 37.77 | 0.04 | 739.25 | 0.04 | 848.82 | 2.29 |

3 | 21.85 | 0.00 | 0.01 | 44.32 | 1348.82 | 0.00 | 0.30 | 2374.72 |

der condition of the frequency spacing of 10 Hz, as shown in

3) Under condition of the frequency spacing of 1 Hz, the maximum dimensionless transverse displacement of the second modal of the normal shape and mode shape actuators is larger than the first and the third modals’, as shown in

4) Overall, in addition to the modal 1 or the first modal and under condition of the frequency spacing of 100 Hz, the gain or ratio of the second and third modal of the mode shape actuators and sensors are better than normal shape actuators and sensors, as shown in

5) In terms of the resonance frequency, the approximate solutions through theoretical derivation are consistent with the analysis results by computer simulations, as shown in

6) Under conditions of the simulation analysis of static, steady state and modal types, the first modal of the normal shape and mode shape actuators and sensors are the same gain or ratio, as shown in

7) Furthermore, the gain or ratio of the second and third modal of the mode shape actuators and sensors are better than the normal shape actuators and sensors, as shown in

According to the results of the theory and simulation analysis, on the whole, we found the first modal of the

Mode | Actuators | Sensors | ||||||
---|---|---|---|---|---|---|---|---|

NSA (m/m) | MS1A (m/m) | MS2A (m/m) | MS3A (m/m) | NSS (V/V) | MS1S (V/V) | MS2S (V/V) | MS3S (V/V) | |

1 | 256.00 | 222.94 | 0.09 | 0.03 | 898.94 | 576.12 | 2.09 | 2.01 |

2 | 301.21 | 0.01 | 350.85 | 0.04 | 6637.87 | 0.04 | 7885.37 | 2.29 |

3 | 138.78 | 0.00 | 0.01 | 333.99 | 8562.97 | 0.00 | 0.30 | 20606.27 |

Frequency Spacing | Actuators | Sensors | ||||
---|---|---|---|---|---|---|

MS1A.NS1A (m/m) | MS2A.NS2A (m/m) | MS3A.NS3A (m/m) | MS1S.NS1S (V/V) | MS2S.NS2S (V/V) | MS3S.NS3S (V/V) | |

1 Hz | 0.87 | 1.16 | 2.40 | 0.64 | 1.19 | 2.41 |

10 Hz | 0.87 | 1.12 | 2.03 | 0.64 | 1.15 | 2.03 |

100 Hz | 1.08 | 1.20 | 1.94 | 0.72 | 1.20 | 1.94 |

normal shape and mode shape actuators and sensors are the same gain. Or in other words, the design concept of the first modal of mode shape actuator or sensor is not necessarily better than of the first modal of normal shape actuator or sensor, or even worse. However, the gain of the second and third modal of the mode shape actuators

and sensors are better than the normal shape actuators and sensors under any operating states. Most importantly, we have to find a universal theoretical model and approximate solutions in this paper, which can predict the gain of the first three modes of normal shape and mode shape actuators and sensors, and through the analysis results

Mode | Theory (Hz) | NS (Hz) | MS1 (Hz) | MS2 (Hz) | MS3 (Hz) | NS Theory | MS1 Theory | MS2 Theory | MS3 Theory |
---|---|---|---|---|---|---|---|---|---|

1 | 359 | 359 | 359 | 360 | 360 | 1.000 | 1.000 | 1.003 | 1.003 |

2 | 2253 | 2245 | 2247 | 2247 | 2247 | 1.001 | 0.999 | 0.999 | 0.999 |

3 | 6308 | 6256 | 6267 | 6262 | 6264 | 0.992 | 0.994 | 0.993 | 0.993 |

Analysis Type | NSA (m) | MS1A (m) | MS2A (m) | MS3A (m) | MS1A.NSA (m/m) | MS2A.NSA (m/m) | MS3A.NSA (m/m) |
---|---|---|---|---|---|---|---|

Static & Steady State | 5.18E−14 | −1.52E−13 | −9.57E−11 | −1.74E−10 | 2.93 | 1846.88 | 3353.80 |

Mode | NSS (V) | MS1S (V) | MS2S (V) | MS3S (V) | MS1S:NSS (V/V) | MS2S:NSS (V/V) | MS3S:NSS (V/V) |
---|---|---|---|---|---|---|---|

1 | 9 | 45 | 643 | 686 | 4.82 | 68.70 | 73.37 |

2 | 330 | 1506 | 25131 | 26925 | 4.56 | 76.13 | 81.56 |

3 | 2325 | 11946 | 194692 | 209525 | 5.14 | 83.75 | 90.13 |

of computer simulation to confirm.

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).