# Experimental Configuration to Determine the Nonlinear Parameter β in PMMA and CFRP with the Finite Amplitude Method

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Setup

#### 2.2. Materials Description

#### 2.3. Variables

**Excitation level**. This is the excitation energy sent from the transducer to the hydrophone, going through the specimen (PMMA or CFRP). The signal voltage is produced in the wave generator (Agilent 33250 A) and amplified 27.5 dB by the amplifier (Amplifier Research 150A 100B). Three excitation levels were considered: 320 mV, 240 mV and 160 mV. This choice is based on the previous experience for generating nonlinearity.**Frequency**. It is generated in the transmitter transducer. It depends on the type of transducer. The transducer used has a central frequency of around 5 MHz, so it was decided to do a frequency sweep around this central frequency. This sweep was done as follows: from 4 MHz to 7 MHz increasing that frequency by 0.1 MHz. It was expected to get a accurate information about the correct frequency.**Distance**. The distance was varied between specimen-hydrophone, while maintaining a fixed distance between the transducer-specimen. This last distance is established because of the effects of the near field. The distance specimen-hydrophone is varied from 0.5 mm to 50.5 mm. The step is defined in 1 mm. This movement can be automatized because of the mechanical arm of the immersion tank controlled in MATLAB with the correspondent libraries for controlling the step-by-step motors.**Specimen thickness**. This is important data since the specimen thickness is inversely proportional to the nonlinear parameter $\beta $. This was measured with a gauge to ensure this variable with more accuracy.**Sampling**. In the sampling, the acquisition card was adjusted with a number of points for each cycles by which the sampling frequency was integer. This was necessary because if this was not done, the FFT in MATLAB was not well done, and may have problems like aliasing and leakage. With this adjustment these problems were avoided.**Window variation**. This variable is the time window in which the ultrasonic wave arrives to the hydrophone until a certain number of cycles. Different windows for each frecuency were got. This was done because it was necessary to adjust the number of points analysed by the acquisition card. It was taken the number of points divided by 10 (the number of points that represent the wave) and it was got the number of cycles that the windows are able to capture. The wave region varies in the distance, so the variance was established with an estimation of the retardment with the wave arriving at the hydrophone.**Wave region**. The first part of the wave was analysed in that window, trying to avoid undesired interferences which distort the value of the non-linear parameter $\beta $. This region depends on the material, for PMMA it was analysed until 150 cycles, nevertheless in the CFRP laminates it was analysed the firsts 30 cycles.**Hydrophone sensitivity**. In order to obtain a value of efficient beta (with water and material), it is necessary to obtain the value of the fundamental and second harmonic and each one has a different value of frequency (double), this was corrected with the sensitivity curve of the hydrophone, and each harmonic was treated with different value of this sensitivity.**Alignment**. It is important the correct alignment between transducer and hydrophone because a little misalignment causes variations in the nonlinear parameter $\beta $.

#### 2.4. Theoretical Foundations to Determine the Real Parameter $\beta $ Considering Geometric and Viscous Attenuation

#### Solution

#### 2.5. Semi-Analytical Approach

#### 2.6. Specimen Attenuation

## 3. Results

#### 3.1. PMMA Results

#### 3.1.1. Excitation Level

#### 3.1.2. Frequency

#### 3.1.3. Cycles and Distance

#### 3.1.4. Selected Parameters

#### 3.2. CFRP Results

#### 3.2.1. Excitation Level

#### 3.2.2. Frequency

#### 3.2.3. Cycles and Distance

#### 3.2.4. Selected Parameters

#### 3.3. Application of the Semi-Analytical Approach with PMMA Specimen

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PMMA | Polymethylmethacrylate |

CFRP | Carbon Fiber Reinforced Polymer |

FAM | Finite Amplitude Method |

NEWS | Nonlinear Elastic Wave Spectroscopy |

NRUS | Nonlinear Resonant Ultrasound Spectroscopy |

NWMS | Nonlinear Wave Modulation Spectroscopy |

DAE | Dynamic Acousto-Elasticity |

NDT | Nondestructive Testing |

SAW | Surface Acoustic Waves |

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**Figure 2.**Polymethylmethacrylate material (PMMA) and carbon fibre reinforced polymer (CFRP) specimens.

**Figure 4.**Semi-analytical approach used to extract the nonlinear material’s properties from the measurements.

**Figure 5.**Determination of the $\beta $ parameter and geometric attenuation in water layer 2 without specimen.

**Figure 7.**Determination of the fundamental and second harmonic in B by propagating this values from C to B.

**Figure 9.**Plots beta versus distance specimen-hydrophone. (

**a**) Frequency of 5 MHz and Energy of 320 mV; (

**b**) Frequency of 5.5 MHz and Energy of 320 mV; (

**c**) Frequency of 6 MHz and Energy of 320 mV.

**Figure 10.**Box-plot beta versus frequencies from 4 MHz to 7 MHz in steps of 0.1 MHz and using 50 cycles. Mean and standard deviation values are computed considering different distances specimen-hydrophone: in (

**a**) distance 0–10 mm, in (

**b**) distance 10–20 mm and in (

**c**) distance 20–30 mm.

**Figure 11.**Box-plot beta versus cycles using a frequency of 5.9 MHz. Mean and standard deviation values are computed considering different distances specimen-hydrophone: in (

**a**) distance 0–10 mm, in (

**b**) distance 10–20 mm and in (

**c**) distance 20–30 mm.

**Figure 12.**Plots beta versus distance specimen-hydrophone. (

**a**) Frequency of 4 MHz and Energy of 320 mV; (

**b**) Frequency of 5 MHz and Energy of 320 mV; (

**c**) Frequency of 6 MHz and Energy of 320 mV.

**Figure 13.**Box-plot beta versus frequencies from 4 MHz to 7 MHz in steps of 0.1 MHz for 4 cycles. Mean and standard deviation values are computed considering different distances specimen-hydrophone: in (

**a**) distance 15–20 mm, in (

**b**) distance 20–25 mm and in (

**c**) distance 25–30 mm.

**Figure 14.**Box-plot beta versus cycles using a frequency of 5.8 MHz. Mean and standard deviation values are computed considering different distances specimen-hydrophone: in (

**a**) distance 15–20 mm, in (

**b**) distance 20–25 mm and in (

**c**) distance 25–30 mm.

**Figure 15.**Semi-analytical approach used to extract the nonlinear material’s properties from the measurements in PMMA specimen.

**Figure 16.**Determination of the $\beta $ parameter and geometric attenuation in water layer 2 without specimen.

**Figure 18.**Determination of the fundamental and second harmonic in B by propagating this values from C to B.

Material | Excitation Level (mV) | Frequency (MHz) | Cycles | Distance (mm) |
---|---|---|---|---|

PMMA | 320 | 5.9 | 50–80 | 10–30 |

Material | Excitation Level (mV) | Frequency (MHz) | Cycles | Distance (mm) |
---|---|---|---|---|

CFRP | 320 | 5.8 | 4 | 25–30 |

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**MDPI and ACS Style**

Callejas, A.; Rus, G.
Experimental Configuration to Determine the Nonlinear Parameter *β* in PMMA and CFRP with the Finite Amplitude Method. *Sensors* **2019**, *19*, 1156.
https://doi.org/10.3390/s19051156

**AMA Style**

Callejas A, Rus G.
Experimental Configuration to Determine the Nonlinear Parameter *β* in PMMA and CFRP with the Finite Amplitude Method. *Sensors*. 2019; 19(5):1156.
https://doi.org/10.3390/s19051156

**Chicago/Turabian Style**

Callejas, Antonio, and Guillermo Rus.
2019. "Experimental Configuration to Determine the Nonlinear Parameter *β* in PMMA and CFRP with the Finite Amplitude Method" *Sensors* 19, no. 5: 1156.
https://doi.org/10.3390/s19051156