# Attenuation Analysis of Lamb Waves Using the Chirplet Transform

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## Abstract

Guided Lamb waves are commonly used in nondestructive evaluation to monitor plate-like structures or to characterize properties of composite or layered materials. However, the dispersive propagation and multimode excitability of Lamb waves complicate their analysis. Advanced signal processing techniques are therefore required to resolve both the time and frequency content of the time-domain wave signals. The chirplet transform (CT) has been introduced as a generalized time-frequency representation (TFR) incorporating more flexibility to adjust the window function to the group delay of the signal when compared to the more classical short-time Fourier transform (STFT). Exploiting this additional degree of freedom, this paper applies an adaptive algorithm based on the CT to calculate mode displacement ratios and attenuation of Lamb waves in elastic plate structures. The CT-based algorithm has a clear performance advantage when calculating mode displacement ratios and attenuation for numerically-simulated Lamb wave signals. For experimental data, the CT retains an advantage over the STFT although measurement noise and parameter uncertainties lead to larger overall deviations from the theoretically expected solutions.

### Keywords

Dispersion Curve Group Delay Window Function Individual Mode Lamb Wave## 1. Introduction

Ultrasonic waves are often used in nondestructive testing to evaluate the integrity of structural components, as well as to determine material properties of composite or layered materials. In various disciplines such as civil or aerospace engineering, multimode, dispersive guided waves such as Lamb waves have been applied, see Chimenti [1] for an overview. However, complicated signal analysis is the trade off for their versatility. In fact, the main challenges to process Lamb wave signals are due to their very characteristics. Firstly, dispersion phenomena require a resolution of the frequency content of a Lamb wave signal over time which is inherently compromised by the uncertainty principle. Secondly, Lamb waves are multi-modal, which means that interferences between individual modes complicate the allocation of energy and displacement related quantities to a specific mode of excitation. A powerful technique to address these difficulties are time-frequency representations, see for an overview Niethammer et al. [2]. To improve results obtained with conventional methods like the STFT or WT, Hong et al. [3] developed an advanced algorithm based on the STFT using window functions that approximated the group delay of each mode of propagation individually. Kuttig et al. [4] further refined this approach by using the chirplet transform as a generalized TFR, which allows for higher order approximations of the group delay. Encouraged by these advances in signal processing, this paper further explores the potential of CT-based methods for dispersive wave analysis. The problem considered is to extract displacement and energy-related quantities of individual Lamb wave modes in elastic plates, a problem which is relevant for several NDE applications. Variations of the energy associated with a particular mode can for example be used to localize notches in plates by means of a correlation technique, see [5]. Ultrasonic attenuation describes the amplitude decay of wave modes due to energy leakage or the geometry of a specimen. Geometric spreading of Lamb waves in plate structures was examined by Luangvilai et al. [6] using the STFT. Energy leakage in absorbing plates was studied by Luangvilai et al. [7] to determine attenuation coefficients using a refined STFT algorithm. While STFT-based techniques analyze multimode time-domain signals as a whole, the CT-based algorithm uses basis functions specially adjusted to the dispersion relation of each mode of propagation. Physical quantities like displacement or energy can thus be allocated more consistently to individual modes. Since the shaping of the basis functions depends on the knowledge of the dispersion relation for a given set-up, this work considers both numerically simulated ([6]) and experimentally generated ([5]) time-domain signals of Lamb waves in an aluminum plate to evaluate the robustness of the CT-based algorithm as well as its performance.

The paper is organized as follows: first a general definition of the CT is given before describing its use in NDE applications to resolve the time-frequency content of dispersive wave signals by means of an adaptive model-based algorithm. Section 3 contains a description of the candidate NDE problem. The results for mode displacement ratios and geometric spreading of both theoretically and experimentally generated wave signals are presented in Section 4. The concluding remarks of Section 5 outline possibilities to apply the presented technique to other NDE applications.

## 2. The Chirplet Transform and Its Use in Dispersive Wave Analysis

The chirplet transform has been introduced as a generalized time-frequency representation by Mann and Haykin [8]. The basis function can be adjusted by means of shift, shear, and scaling operators, resulting in a five-dimensional parameter space for the energy density which comprises as projections the respective densities obtained from a short-time Fourier transform (time and frequency shift) and a wavelet transform (time shift and scaling).

### 2.1. Definition of the Chirplet Transform

- Time shift:
- Frequency shift:
- Scaling:
- Time shear:
- Frequency shear:

A more detailed discussion of TFRs used for dispersive wave analysis can be found, for example, in [2].

### 2.2. Adaptive Algorithm Based on the Chirplet Transform

For ease of visualization, only subspaces of the five-dimensional parameter space of the chirplet transform are considered. In a fashion analogous to the STFT and its energy density representation, the spectrogram, the time-frequency plane is chosen to analyze dispersive waves. According to the definition of the energy density Open image in new window , the squared amplitude of a time-domain signal recording particle displacement is proportional to the energy of the incident Lamb wave, but comprises contributions of all modes of propagation. The objective is to identify energy or displacement related components of individual modes in regions of sufficient mode separation. To that end, the energy content of the time-domain signal is averaged in the time-frequency plane over a region around every point of the dispersion curve of a particular mode using a specially designed window function. In the case of the STFT, time and frequency shift operations result in a region of averaging that approximates the group delay of an individual mode of propagation with zeroth order, whereas the CT-based algorithm as described by Kuttig et al. [4] additionally uses time shearing resulting in higher order approximations. Note that the dispersion curves for a given system depend on the material properties—in the example of a single aluminum plate, its elastic modulus and density as well as its thickness—which determines the robustness of the CT-based algorithm.

## 3. Problem Setting

and Open image in new window is the plate thickness. The numerical solution of (17) for an aluminum plate is shown in Figure 1 up to frequencies of 10 MHz. The Open image in new window - and Open image in new window - mode are well separated in the frequency ranges 0–1.8 MHz and again between about 2-3 MHz. The same holds for the Open image in new window -mode between 2-3 MHz and 4-5 MHz and for Open image in new window -mode between 3.5–5 MHz, so that the evaluation will be restricted to the first two symmetric ( Open image in new window ) and antisymmetric ( Open image in new window ) modes in these frequency ranges. Consider two different time-domain signals: firstly, numerically simulated data was taken from Luangvilai et al. [6] to have an undisturbed signal for performance evaluation. The authors used normal mode expansion to simulate the out-of-plane displacement field for particles on the plate surface excited by a point-like source for source-receiver distances between 50 mm and 90 mm. Secondly, real measurement data of the out-of-plane velocity field at the surface of an aluminum plate acquired by Benz et al. [5] was available to determine the robustness of the proposed signal processing technique. The experimental setup in this case consisted of an aluminum plate of dimensions 100 mm Open image in new window 100 mm Open image in new window 1 mm and a noncontact, point-like laser measurement and detection system. A laser source was used to generate Lamb waves in the aluminum plate for different source-receiver distances ranging from 50 to 150 mm.

For each of the two data sets, the mode displacement ratios for selective modes are determined as a means to detect material irregularities, for example, for notch localization as in [5]. Apart from that, the amplitude decay over time of individual modes is analyzed as it contains information about the geometry of the specimen. Geometric spreading is given by the quotient Open image in new window for two propagation distances Open image in new window . Such a normalized measure for geometric spreading is chosen since the effect of the excitation source on the energy density will be canceled out.

## 4. Results

Average deviation from theoretical mode displacement ratio in %.

displacement ratio | |||||||
---|---|---|---|---|---|---|---|

numerical signal | CT (in %) | 9.49 | 67.89 | 8.46 | 10.54 | 4.00 | 12.19 |

STFT (in %) | 18.60 | 686.36 | 14.65 | 51.33 | 66.28 | 10.24 | |

experimental signal | CT (in %) | 173.73 | 46.96 | 343.88 | 35.75 | 272.29 | 484.40 |

STFT (in %) | 115.36 | 83.06 | 223.07 | 52.99 | 289.76 | 385.70 |

where Open image in new window and Open image in new window is defined as the inner product for functions [10], Open image in new window . This metric will be used to measure the mean absolute deviation of quantities extracted with the introduced signal processing techniques from the theoretical solution. Note that the adaptive CT-based algorithm only computes energy densities in frequency regions where individual modes are sufficiently separated, that is, when the 3 Open image in new window -region of averaging does not intersect with any other mode. The performance measure for both the STFT- and the CT-based method will therefore be restricted to these regions only. Table 1 confirms that the CT-based results for the numerically simulated signal deviates much less from the theoretical solution compared to the ones obtained from the STFT.

Average deviation in % from theoretical geometric attenuation.

distance | ||||||||
---|---|---|---|---|---|---|---|---|

CT | STFT | CT | STFT | CT | STFT | CT | STFT | |

Synthetically generated signal | ||||||||

80/90 mm | 0.31 | 3.64 | 1.65 | 2.54 | 1.71 | 9.08 | 5.61 | 19.96 |

70/90 mm | 0.43 | 6.40 | 2.80 | 3.62 | 1.99 | 11.78 | 8.39 | 24.37 |

60/90 mm | 0.49 | 8.48 | 4.65 | 5.20 | 6.23 | 24.37 | 13.19 | 24.09 |

50/90 mm | 0.24 | 10.12 | 1.79 | 7.80 | 6.56 | 28.91 | 38.22 | 32.26 |

40/90 mm | 0.28 | 14.04 | 3.14 | 7.28 | 6.33 | 28.26 | 132.4 | 40.03 |

Experimentally generated signal | ||||||||

120/150 mm | 1.91 | 4.16 | 15.03 | 22.78 | 17.01 | 20.91 | 32.92 | 18.7 |

90/150 mm | 7.26 | 14.95 | 15.83 | 22.08 | 9.63 | 20.63 | 24.63 | 16.38 |

60/150 mm | 2.50 | 16.48 | 9.71 | 26.91 | 13.56 | 32.68 | 67.88 | 14.46 |

50/150 mm | 3.44 | 19.21 | 3.12 | 14.56 | 10.78 | 31.37 | 76.49 | 5.29 |

## 5. Conclusions

The main goal of this paper is to evaluate the potential of the chirplet transform for dispersive wave analysis. The problem of associating displacement or energy related quantities to individual modes of propagation is of interest in nondestructive evaluation. The theoretical advantage of the proposed method, that is, tailoring regions of averaging to individual modes based on the dispersion relation, becomes apparent when analyzing numerically simulated Lamb wave signals traveling in an aluminum plate. Extracting displacement ratios and geometric spreading for individual modes of propagation succeed with high accuracy in regions with sufficient mode separation. This strongly indicates that the CT-based algorithm can achieve a better performance than more conventional approaches like the spectrogram. The potential to extract displacement and energy-related quantities associated with a particular mode of a dispersive wave therefore qualifies it as a versatile tool in NDE applications. As a model-based approach, the CT based algorithm uses information about the dispersion relation. Since the dispersion relation in turn depends on the material properties and geometry of the specimen, precise knowledge about experimental set-up is a prerequisite to obtain reliable results with this technique. Consequently, the level of accuracy is considerably lower when applied to the experimentally generated data, also for the STFT-based approach. Improving robustness properties as well as algorithmic efficiency remains a goal of future research to make the CT-based technique more easily available and applicable for quantitative nondestructive evaluation.

## Notes

### Acknowledgment

The Deutscher Akademischer Austausch Dienst (DAAD) provided partial support to F. Kerber.

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