#
Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Figure 2.**Nonlinear effects on high power ultrasonic transducers (

**a**) decrease in the resonance frequency of the transducer with voltage amplitude; and (

**b**) frequency response hysteresis.

## 2. Theoretical Background

#### 2.1. Nonlinear Constitutive Equations

_{0}. This proportionality is also observed in diamagnetic and paramagnetic materials. However, in ferromagnetic materials the B-H curves exhibit hysteresis and saturation. In low field conditions, the Rayleigh law introduces a magnetic permittivity µ depending on the H-field.

_{33}in the polarization direction follows a Rayleigh law, so that:

_{ext}(t), the mechanical displacement follows a sinusoidal law:

_{0}, with the previous state of the system, named X

_{−1}. This linear combination is computed as:

#### 2.2. Nonlinear Model of the Langevin Transducer

_{ext}[17]. Using a pair of symmetric external forces, the barycenter remains unchanged.

_{ext}can be expressed as the mechanical stress T multiplied by the transducer area A:

_{33}and e

_{33}, respectively.

_{0}and X

_{−1}, are considered as known constants, depending only on the angular frequency ω. Replacing Equation (6) in Equation (9) derives:

_{x}and the natural frequency ω

_{0}is given by the following equation:

_{x}

^{2}depends on X, Equation (15) must be solved iteratively. Assuming that all parameters are known, the process for obtaining the amplitude X can be summarized in three steps: First, introduce the value for the previous step, X

_{−1}, and the initial value for the actual amplitude, X, for each angular frequency ω. Second, calculate the ω

_{x}

^{2}value by using Equation (14), and finally calculate the new value of the amplitude X.

_{0}

^{2}, K, β]. The objective function of the minimization is the sum over the frequency spectra of the quadratic error. The error is computed as the mean square root of the difference between the experimental data and the outputs from the models for each parameter set.

^{−1}, C = 4 ms

^{−2}V

^{−1}, ω

_{0}= 167 krad/s, K = −1.5 × 10

^{9}m

^{−1}s

^{−2}, and β = 1. This set of values is the initial condition for the minimization described in the next section. The negative sign for the coefficient K is expected, as the frequency decreases with the amplitude X.

**Figure 5.**Velocity response curve obtained by the model. Here, the input voltage was increased from 1 V to 5 V, with steps of 1 V. Black curves were obtained by increasing the frequency, whereas gray curves are associated with those responses obtained during frequency decreases.

_{0}. In order to obtain this approximation, Equation (15) must be differentiated in respect to ω, which is equivalent to:

## 3. Experimental Results

^{−1}, C = 25.55 m·s

^{−2}V

^{−1}, ω

_{0}= 167.2 krad·s

^{−1}, K = −8 × 10

^{8}m

^{−1}s

^{−2}. For this case, the non-dimensional parameter is β = 5.

**Figure 7.**(

**A**) Adjusted model (continuous line), and (

**B**) experimental data (dots). Black curves are obtained by sweeping the driving frequency up; gray curves are obtained when the driving frequency is swept down.

Experimental | Numerical | |||||
---|---|---|---|---|---|---|

Voltage [V] | Maximum Displacement Amplitude [μm] | Frequency of Maximum Displacement [kHz] | Hysteresis [Hz] | Frequency of Maximum Displacement [kHz] | Maximum Displacement Amplitude [μm] | Hysteresis [Hz] |

1 | 0.19 | 26.580 | 0.25 | 26.577 | 0.20 | 1 |

2 | 0.40 | 26.544 | 7 | 26.543 | 0.41 | 3 |

3 | 0.67 | 26.502 | 12 | 26.510 | 0.61 | 7 |

4 | 0.95 | 26.467 | 15 | 26.477 | 0.82 | 14 |

5 | 1.26 | 26.431 | 20 | 26.444 | 1.03 | 20 |

**Figure 8.**Evaluation of the theoretical model applying Equation (16). Dots are experimental and simulated data taken from Table 1. The continuous gray line is the interpolation of the numerical model, whereas the black dashed line is the linear approximation of the fist three elements in the experimental data.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Alvarez, N.P.; Cardoni, A.; Cerisola, N.; Riera, E.; Andrade, M.A.B.; Adamowski, J.C. Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers. *Actuators* **2015**, *4*, 255-266.
https://doi.org/10.3390/act4040255

**AMA Style**

Alvarez NP, Cardoni A, Cerisola N, Riera E, Andrade MAB, Adamowski JC. Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers. *Actuators*. 2015; 4(4):255-266.
https://doi.org/10.3390/act4040255

**Chicago/Turabian Style**

Alvarez, Nicolás Peréz, Andrea Cardoni, Niccolo Cerisola, Enrique Riera, Marco Aurélio Brizzotti Andrade, and Julio Cezar Adamowski. 2015. "Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers" *Actuators* 4, no. 4: 255-266.
https://doi.org/10.3390/act4040255