# A Lamb Wave Wavenumber-Searching Method for a Linear PZT Sensor Array

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- (2)
- Phase-unwrapping method [42,43]: The excitation signal and the received signal of Lamb wave are separately phase–unwrapped to obtain the phase at the center frequency of the signal. Then, according to the wavenumber definition formula, the wavenumber of the Lamb wave can be calculated using the propagation distance. However, for unknown excitation signal or propagation distance of the Lamb wave, or for complex signal components, this method is difficult to use.
- (3)
- Fourier-transform method [44,45,46]: According to the definition of the wavenumber, the wavenumber of Lamb wave can be obtained by the Fourier transform of the Lamb wave spatial sampling signal. This method generally requires a higher spatial sampling rate and longer spatial sampling length according to the properties of the Fourier transform. Therefore, this method is generally used to process Lamb wave spatial signals collected by scanning laser Doppler vibrometer (SLDV). However, it is inappropriate for use in online damage monitoring.
- (4)
- Spatial-wavenumber filter method [34,35,36,37,47]: A series of spatial-wavenumber filters with different central wavenumbers are used to filter the spatial sampling signal of the Lamb wave. When the scanning wavenumber is equal to the wavenumber of the Lamb wave spatial sampling signal, the Lamb wave spatial sampling signal is able to pass the spatial–wavenumber filter. The wavenumber of the Lamb wave spatial sampling signal is then obtained. The spatial sampling rate of the linear PZT sensor array should satisfy the Nyquist sampling theorem in this method.

## 2. The Lamb Wave Wavenumber-Searching Method

#### 2.1. Theoretic Foundations of the Method

_{a}is the angular frequency of the Lamb wave, k

_{LW}is the wavenumber of the Lamb wave, and φ

_{0}is the initial phase of the Lamb wave.

_{p}is the phase velocity of the Lamb wave.

_{s}is the temporal sampling frequency of the sensor and △t is the time-domain sampling interval of the sensor; then, the spatial sampling interval Δx of the sensor is:

_{s}of the sensor is:

#### 2.2. The Principle of the Method

_{a}, y

_{a}). The direction (angle) and distance of the acoustic source relative to the linear PZT sensor array are θ

_{a}and l

_{a}, respectively. A narrowband-frequency Lamb wave with the central frequency f

_{a}and wavenumber k

_{LW}, respectively, is excited by the acoustic source. The angular frequency of the narrowband–frequency Lamb wave is ω

_{a}= 2πf

_{a}. All PZT sensors in the linear PZT sensor array are used to collect the Lamb wave simultaneously with the temporal sampling frequency f

_{s}, which satisfies the Nyquist sampling theorem f

_{s}> 2f

_{a}. Then, the Lamb wave received signals collected by the linear PZT sensor array in the time-spatial domain can be expressed as Equation (6). The central frequency of the Lamb wave received signal is also f

_{a}, and its wavenumber is k

_{a}= k

_{LW}·cos(θ

_{a}) [34,35,36,37]:

_{1}(0) is the sampling value of PZT 1 at t = 0 (starting time), s

_{1}(t) is the sampling value of PZT 1 at time t, s

_{1}(T) is the sampling value of PZT 1 at time T, s

_{m}(0) is the sampling value of PZT m at t = 0 time, s

_{m}(t) is the sampling value of PZT m at time t, s

_{m}(T) is the sampling value of PZT m at time T, s

_{M}(0) is the sampling value of PZT M at t = 0, s

_{M}(t) is the sampling value of PZT M at time t, and s

_{M}(T) is the sampling value of PZT M at time T.

_{m}(x

_{m}

_{–t}) is the spatial sample value of the spatial sampling position x

_{m}

_{–t}. Then, Equation (7) can be rewritten as:

_{m}(x

_{m}

_{–t}) are sorted according to the spatial sampling position x

_{m}

_{–t}from small to large. If the spatial sampling positions are the same, their spatial sample values are averaged. Finally, the two–dimensional time-spatial-domain Lamb wave received signal S is converted into one-dimensional spatial-domain Lamb wave received signal G:

_{r}is the synthesized spatial sampling position, g(z

_{r}) is the spatial sampling value at the spatial position z

_{r}, and R is the length of the synthesized spatial sampling signal.

_{a}, as shown in Figure 3b. Otherwise, the synthetic spatial sampling signal G′ is messy, as shown in Figure 3c. In Figure 3, the central frequency and wavenumber of the Lamb wave received signal are 50 kHz and 342.5 rad/m. “Signal 1” is the spatial sampling signal of the Lamb wave. “Signal 2” is the Morlet wavelet fitting waveform G″. According to the properties of the Morlet wavelet transform, when the wavenumber of the Morlet wavelet fitting waveform G″ is equal to that of the synthetic spatial sampling signal, the sum of the squared error, e, calculated by Equation (11), between the Morlet wavelet fitting waveform G″ and the synthetic spatial sampling signal G′ is small, as shown in Figure 3d. Otherwise, the sum of the squared error e is large, as shown in Figure 3e.

_{1}, k

_{N}] and the resolution is set to Δk. Then, the sum of squared error e at each searching wavenumber is obtained. Finally, the wavenumber at the minimum value of e is considered the best match to the wavenumber k

_{b}of the Lamb wave received signal.

## 3. Damage Localization Based on the Lamb Wave Wavenumber–Searching Method and Cruciform PZT Sensor Array

_{a}of the damage can then be obtained by the wavenumber of the damage scattering signal projected at the X-axis and Y-axis (No. 1 and No. 2 linear PZT sensor arrays), as studied in detail in the previous references [34,35].

_{a}of the damage scattering signal. The start time t

_{e}of the excitation signal is obtained by the Shannon wavelet transformation of the excitation signal [35,36]. Then, the damage distance l

_{a}can be calculated as Equation (15) using the Lamb wave velocity.

_{1}is the wavenumber of the Lamb wave received signal of the No. 1 PZT sensor array, k

_{2}is that of the No. 2 array.

_{g}is the Lamb wave velocity. This yields the localized damage position (θ

_{a}, l

_{a}).

## 4. Validation Experiment of the Lamb Wave Wavenumber-Searching Method

#### 4.1. Experimental Setup

_{s}of the linear PZT sensor array is:

#### 4.2. Theoretical Wavenumber Calculation

_{0}and S

_{0}modes exist in the Lamb wave at low central frequency (30–70 kHz), and the amplitude of the A

_{0}mode is much higher than that of the S

_{0}mode. Thus, only the wavenumber of the A

_{0}mode is calculated.

_{l}and transverse wavenumber k

_{t}are calculated as:

_{l}and transverse wave velocity C

_{s}are calculated as:

_{0}denotes the horizontal wavenumber, b is one-half the plate thickness, ω is the angular frequency of the Lamb wave, E is Young’s modulus, μ is Poisson’s ratio, and ρ denotes density.

#### 4.3. Typical Signal Analysis

_{m}(t) of PZT A are shown in Figure 6.

_{n}= 100 rad/m is selected as an example. The converted spatial-domain sampling signal s

_{m}(x

_{m}

_{–t}) of each PZT sensor is shown in Figure 7. In Figure 7, the spatial sampling point 0 m of each converted spatial-domain sampling signal refers to the position of the PZT sensor. In other words, the converted spatial-domain sampling signal of each PZT sensor begins at the sensor position.

_{m}(x

_{m–t}) of each PZT sensor is sorted according to the spatial sampling position x

_{m}

_{–t}from small to large. If the spatial sampling positions are the same, their spatial sample values are averaged. Finally, the converted spatial-domain sampling signal of each PZT sensor is synthesized into a series of spatial sampling signals, as shown in Figure 8.

_{n}= 100 rad/m, the Morlet wavelet fitting waveform G″ can be obtained by the Morlet wavelet transformation of the synthetic spatial sampling signal G′, as shown in Figure 9. The sum of the squared error e

_{n}= 6839.9 between the Morlet wavelet fitting waveform and the synthetic spatial sampling signal can be calculated by Equation (11).

_{n}between the Morlet wavelet fitting waveform and the synthetic spatial sampling signal are calculated in turn, as shown in Figure 10.

## 5. Validation of the Damage Localization Method

#### 5.1. Experimental Setup

- (1)
- The group velocity of Lamb wave was measured using the continuous complex Shannon wavelet transformation. The actuator was used to excite Lamb wave propagating on the composite plate, while the three reference PZT sensors were used to acquire the corresponding Lamb wave. For each reference PZT sensor, the group velocity was calculated as c
_{g-}_{Ref 1}= 1569.32 m/s, c_{g-}_{Ref 2}= 1557.29 m/s, and c_{g-}_{Ref 3}= 1485.65 m/s. Then, the average group velocity c_{g}= 1537.42 m/s was obtained and used in damage localization. - (2)
- In the health state of the carbon fiber composite laminate plate, the Lamb wave signals of the cruciform PZT sensor array were acquired as health reference signals, f
_{HR}. - (3)
- Damage is created in each position and the corresponding Lamb wave signals of the cruciform PZT sensor array were acquired as the online monitoring signals, f
_{OM}.

#### 5.2. Damage Localization Validation

_{HR}and online monitoring signals f

_{OM}of the cruciform PZT sensor array are shown in Figure 13 and Figure 14, respectively. The damage scattering signals extracted by subtracting f

_{OM}from f

_{HR}are shown in Figure 15.

_{1}= 121.5 rad/m and k

_{2}= 328.3 rad/m are obtained from the two plots of e.

_{a}= 69.7° relative to the center point of the cruciform PZT sensor array can be obtained using the wavenumbers k

_{1}and k

_{2}. According to Equation (14) and the wavenumbers k

_{1}and k

_{2}, the arrival time t

_{a}= 0.5124 ms of the damage scattering signal could be obtained, as shown in Figure 17. The start time t

_{e}= 0.1032 ms of the excitation signal was obtained by the Shannon wavelet transformation of the excitation signal, as shown in Figure 18.

_{a}= 314.6 mm was calculated according to Equation (15) and the Lamb wave velocity c

_{g}= 1537.42 m/s.

_{D}, y

_{D}) is the actual position of the damage.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Schematics of the Lamb wave wavenumber–searching method. (

**a**) Lamb wave received signals collected by the linear PZT sensor array; (

**b**) The synthetic spatial sampling signal G′ with the wavenumber k = 342.5 rad/m = k

_{a}; (

**c**) The synthetic spatial sampling signal G′ with the wavenumber k = −100 rad/m ≠ k

_{a}; (

**d**) The Morlet wavelet fitting waveform G″ with the wavenumber k = 342.5 rad/m = k

_{a}; (

**e**) The Morlet wavelet fitting waveform G″ with the wavenumber k = −100 rad/m ≠ k

_{a}.

**Figure 5.**Experimental setup for the Lamb wave wavenumber-searching method. (

**a**) Experiment setup; (

**b**) Illustration of the linear PZT sensor array, acoustic sources, and coordinate system.

**Figure 7.**Converted spatial-domain sampling signal s

_{m}(x

_{m}

_{–t}) of each PZT sensor whose searching wavenumber is k

_{n}= 100 rad/m.

**Figure 12.**Experimental system for validating the damage localization method based on the Lamb wave wavenumber-searching method and cruciform PZT sensor array. (

**a**) Experiment setup; (

**b**) Cruciform PZT sensor array; (

**c**) Illustration of the cruciform PZT sensor array placement and damage positions.

**Figure 13.**Health reference signals of the cruciform PZT sensor array. (

**a**) No. 1 PZT sensor array; (

**b**) No. 2 PZT sensor array.

**Figure 14.**Online monitoring signals of the cruciform PZT sensor array. (

**a**) No. 1 PZT sensor array; (

**b**) No. 2 PZT sensor array.

**Figure 15.**Damage scattering signals of the cruciform PZT sensor array. (

**a**) No. 1 PZT sensor array; (

**b**) No. 2 PZT sensor array.

**Figure 16.**The sum of squared error of the cruciform PZT sensor array. (

**a**) No. 1 PZT sensor array; (

**b**) No. 2 PZT sensor array.

Parameter | Value |
---|---|

Density / (kg·m^{-3}) | 2.78×10^{3} |

Elastic modulus / GPa | 73.1 |

Shear modulus / GPa | 28 |

Poisson ratio µ | 0.33 |

**Table 2.**Theoretical wavenumber of A

_{0}mode calculated by the Rayleigh–Lamb equation solving method.

Frequency (kHz) | Theoretical Wavenumber (rad/m) |
---|---|

30 | 261.67 |

35 | 283.63 |

40 | 304.26 |

45 | 323.83 |

50 | 342.53 |

55 | 360.47 |

60 | 377.78 |

65 | 394.54 |

70 | 410.82 |

**Table 3.**The wavenumber of each actuator and central frequency calculated by the Lamb wave wavenumber-searching method.

Signal Source | Frequency (kHz) | Theoretical Wavenumber (rad/m) | Lamb Wave Wavenumber-Searching Method (rad/m) | Wavenumber Error (rad/m) |
---|---|---|---|---|

PZT A | 30 | 261.67 | 259.8 | −1.9 |

35 | 283.63 | 282.3 | −1.3 | |

40 | 304.26 | 302.5 | −1.8 | |

45 | 323.83 | 325.3 | 1.5 | |

50 | 342.53 | 340.4 | −2.1 | |

55 | 360.47 | 358.8 | −1.7 | |

60 | 377.78 | 375.6 | −2.2 | |

65 | 394.54 | 393.1 | −1.4 | |

70 | 410.82 | 408.8 | −2.0 | |

PZT B | 30 | 130.84 | 131.5 | 0.7 |

35 | 141.82 | 142.6 | 0.8 | |

40 | 152.13 | 153.6 | 1.5 | |

45 | 161.92 | 161.7 | −0.2 | |

50 | 171.27 | 171.2 | −0.1 | |

55 | 180.24 | 179.9 | −0.3 | |

60 | 188.89 | 188.2 | −0.7 | |

65 | 197.27 | 196.8 | −0.5 | |

70 | 205.41 | 204.9 | −0.5 | |

PZT C | 30 | −168.20 | −166.3 | 1.9 |

35 | −182.31 | −180.6 | 1.7 | |

40 | −195.57 | −193.7 | 1.9 | |

45 | −208.15 | −206.9 | 1.3 | |

50 | −220.17 | −218.2 | 2.0 | |

55 | −231.71 | −231.0 | 0.7 | |

60 | −242.83 | −242.0 | 0.8 | |

65 | −253.61 | −252.5 | 1.1 | |

70 | −264.07 | −262.1 | 2.0 |

Parameter | Value |
---|---|

0° tensile modulus (GPa) | 135 |

90° tensile modulus (GPa) | 8.8 |

± 45° in-plane shearing modulus (GPa) | 4.47 |

Poisson ratio µ | 0.328 |

Density (kg·m^{-3}) | 1.61 × 10^{3} |

Position Label | Cartesian Coordinates (mm, mm) | Polar Coordinates (°, mm) |
---|---|---|

Ref 1 | (200, 0) | (0.0, 200.0) |

Ref 2 | (200, 200) | (45.0, 282.8) |

Ref 3 | (0, 400) | (90.0, 400.0) |

A | (100, 300) | (71.6, 316.2) |

B | (100, 200) | (63.4, 223.6) |

C | (0, 200) | (90.0, 200.0) |

D | (−100, 200) | (116.6, 223.6) |

E | (−200, 200) | (135.0, 282.8) |

F | (−200, 100) | (153.4, 223.6) |

G | (−200, 0) | (180.0, 200.0) |

Damage Label | k_{n}_{1}(rad/m) | k_{n}_{2}(rad/m) | Arrive Time (ms) | Start Time (ms) | Localized Position (mm, mm) | Actual Position (mm, mm) | Damage Localization Error (mm) |
---|---|---|---|---|---|---|---|

A | 121.5 | 328.3 | 0.5124 | 0.1032 | (109.2, 295.0) | (100, 300) | 10.4 |

B | 163.1 | 299.8 | 0.4094 | 0.1034 | (112.4, 206.6) | (100, 200) | 14.1 |

C | 4.8 | 340.3 | 0.3583 | 0.1034 | (2.8, 195.9) | (0, 200) | 4.9 |

D | −152.4 | 304.7 | 0.3861 | 0.1034 | (−97.2, 194.4) | (−100, 200) | 6.3 |

E | −241.7 | 228.9 | 0.4570 | 0.1034 | (−197.4, 186.9) | (−200, 200) | 13.4 |

F | −309.2 | 151.5 | 0.4001 | 0.1031 | (−205.0, 100.5) | (−200, 100) | 5.0 |

G | −343.1 | −9.3 | 0.3897 | 0.1031 | (−220.0, −6.0) | (−200, 0) | 21.1 |

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**MDPI and ACS Style**

Liu, B.; Liu, T.; Zhao, J.
A Lamb Wave Wavenumber-Searching Method for a Linear PZT Sensor Array. *Sensors* **2019**, *19*, 4166.
https://doi.org/10.3390/s19194166

**AMA Style**

Liu B, Liu T, Zhao J.
A Lamb Wave Wavenumber-Searching Method for a Linear PZT Sensor Array. *Sensors*. 2019; 19(19):4166.
https://doi.org/10.3390/s19194166

**Chicago/Turabian Style**

Liu, Bin, Tingzhang Liu, and Jianfei Zhao.
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https://doi.org/10.3390/s19194166