A mode conversion-based algorithm for detecting rectangular notch parameters in plates using Lamb waves

Abstract

In this paper, an algorithm for identifying rectangular notch parameters as damage in a plate using Lamb waves is presented. In this algorithm, a combination of pulse-echo and pitch-catch methods is used. The method is divided into two steps: notch localization and notch geometry detection. The bases for this algorithm are mode conversion and scattering phenomena because of interaction of Lamb wave modes with defects. The method is applied to some numerical examples, and the results show that it can successfully identify all of rectangular notch parameters, i.e., its location, depth, and width.

1 Introduction

Damage detection in structures using Lamb waves is an ultrasonic technique that is usually used to detect defects in thin structures. Lamb wave damage detection techniques have become very common in industries for structural health monitoring (SHM) purposes because of their numerous advantages. One of their advantages is that they can be easily generated into a structure using small pieces of piezoelectric ceramics. Another important advantage of Lamb waves is their ability for global inspection. Someone can inspect the health of a large area of a structure using a limited number of sensors and actuators without moving them along the structure. Therefore, an in situ inspection would be possible [1, 2]. The purpose of SHM projects is not only the detection of large defects and structural failures, but also the detection of damage in the early stage of defect growth before they could reach a dangerous level for the structure [3, 4].

Cracks and notches are the common types of damage in engineering structures. Therefore, many researchers have studied the effect of these types of damage on the variations in Lamb wave propagation in structures [5]. Alleyne and Cawley [6] studied the interaction of Lamb waves with a V-shape notch. Rucka [7, 8] studied damage detection in plates and bars using wave propagation. Giurgiutiu et al. [9] studied crack detection and Lamb wave propagation in beams and plates. Mirahmadi and Honarvar [10] presented a signal processing method to detect several notches in structures using Lamb waves. Atashipour et al. [11] detected notch location and depth in thick beams using Lamb wave propagation properties.

One of the most important steps for damage detection in structures using Lamb waves is to detect damage severity. For this purpose, the study of Lamb wave interaction with defects is necessary. Alleyne and Cawley [12] used FEM and 2D-FFT to obtain reflection and transmission coefficients of Lamb wave modes from a notch. Lowe et al. [13] obtained the reflection and transmission coefficients of Lamb wave modes from rectangular notches in the plate. Gunavan and Hirose [14] presented a mode-exciting method to determine scattering coefficients of Lamb waves in a plate. Strom et al. [15] investigated scattering of Lamb wave modes by corrosion pits in a plate. Kim and Roh [16, 17] divided the scattering of Lamb wave modes by a rectangular notch into three independent processes and extracted scattering diagrams for each process. Using those diagrams, the reflection and transmission coefficients of the fundamental Lamb modes were computed.

In the previous damage detection studies, the damage detection problem was defined as the detection of damage location and/or its depth. Although the scattering coefficients of Lamb waves are obtained for different kinds of damage, no common algorithm for geometry detection of any existing damage type has been presented yet. In this paper, a novel algorithm for detection of damage geometry is presented. The damage is considered to be a rectangular notch. The presented algorithm allows a user to identify not only the damage location, but also its depth, $d$, and width, $g$, as shown in Fig. 1. This algorithm is based on mode conversion of Lamb waves. Moreover, a combination of both pitch-catch and pulse-echo methods is used to extract damage features. The damage detection procedure is based on scattering coefficients of Lamb wave modes.

Fig. 1

Rectangular notch geometry in a waveguide

In the following sections, first, the procedure of notch localization is presented. After that, an algorithm is introduced for detecting notch geometry. Finally, some examples are presented to validate this method.

2 Lamb waves

Lamb wave particle displacement in structures is the superposition of guided longitudinal and transverse shear waves. Lamb waves consist of two groups of waves, symmetric $(S_i)$ and anti-symmetric $(A_i)$, and each can propagate independently in waveguides. To find the propagation specifications for symmetric and anti-symmetric modes, the Rayleigh–Lamb equation is used as follows [18]:

$$\begin{aligned} \frac{\tan \,\,ph}{\tan \,\,qh}&= - \frac{(k^{2}-q^{2})^{2}}{4\,pqk^{2}},\end{aligned}$$

(1)

$$\begin{aligned} \frac{\tan \,\,ph}{\tan \,\,qh}&= - \frac{4\,pqk^{2}}{(k^{2}-q^{2})^{2}}, \end{aligned}$$

(2)

where $h$ is the half of thickness of a plate $(H/2)$, $k$ is the wavenumber, $p$ and $q$ are given as follows:

$$\begin{aligned} p^{2}&= \frac{\omega ^{2}}{c_p^2 }-k^{2},\end{aligned}$$

(3)

$$\begin{aligned} q^{2}&= \frac{\omega ^{2}}{c_s^2 }-k^{2}, \end{aligned}$$

(4)

where $\omega$ shows the excitation frequency and $c_p$ and $c_s$ indicate pressure and shear wave velocity, respectively. Eqs. (1) and (2) are related to symmetric and anti-symmetric modes, respectively. The dispersion curve of Lamb waves can be obtained using Eqs. (1) and (2) and is shown in Fig. 2. As it can be seen from the Fig. 2, for low-frequency excitations, only $S_0$ and $A_0$ modes are existent, which are the lowest order symmetric and antisymmetric Lamb modes, respectively. As many practical uses of Lamb waves in SHM deal with lower frequency excitation, in this paper, we study the “$f\times H$” domain that only $S_0$ and $A_0$ can be propagated in the structure.

Fig. 2

Phase velocity dispersion curve of aluminum plate [10]

2.1 Mode conversion

Mode conversion phenomenon of Lamb waves is defined as energy transformation from one Lamb wave mode to another [18]. In the study of mode conversion in a waveguide, the geometry of waveguide is important. If the waveguide geometry is symmetric, the mode conversion is only possible in the same mode group and it could not occur between symmetric and antisymmetric Lamb wave modes. However, if the waveguide is asymmetric, the mode conversion would also occur between mode families.

On the other hand, defects in a waveguide could cause mode conversions. As the defects in structures occur without any control, they tend to make the geometry of the waveguide asymmetric. A good example is a transverse crack in a structure. The presence of crack causes mode conversion in the structure both in the mode families and in between them. According to the above explanations, it can be inferred that if in a symmetric waveguide, the mode conversion between symmetric and antisymmetric modes could occur, a defect should be present in the structure. This point can be used in damage detection for simple structures, like uniform beams and plates. For example, the mode conversion phenomenon is shown in Fig. 3, which is due to the presence of a crack in the waveguide. In the Fig. 3, the incident wave is $S_0$ and it is assumed that only the fundamental modes ($S_0$ and $A_0$) can be propagated in the waveguide. It can be seen that although the incident wave is $S_0$, the reflected and transmitted waves contain both $S_0$ and $A_0$ modes. It is noted that according to the principle of conservation of energy, the sum of reflected and transmitted wave powers is equal to the power of incident wave.

Fig. 3

Scattering of Lamb wave modes by interaction with a crack

2.2 Lamb wave power

The power of nth mode Lamb wave, $C_n^2$, can be obtained, as follows [16]:

$$\begin{aligned} C_n^2 =\frac{i\omega }{4}\int \limits _{-h/2}^{h/2} {\begin{array}{l} \left\{ \left[ U_n (k_n ,y) S_n^*\left( k_n^*,y\right) +V_n (k_n ,y) T_n^*\left( k_n^*,y\right) \right] \right. \\ -\left. \left[ S_n^ (k_n ,y) U_n^*\left( k_n^*,y\right) +T_n (k_n ,y) V_n^*\left( k_n^*,y\right) \right] \right\} dy \\ \end{array}} , \end{aligned}$$

(5)

where the superscript asterisk means a complex conjugate. Moreover, $U^{n}$, $V^{n}$, $S^{n}$, and $T^{n}$ are the modal functions, as defined below:

$$\begin{aligned} U^{n}&= ik_n\cosh (\gamma y)+\delta B\cosh (\delta y), \nonumber \\ V^{n}&= \gamma \sinh (\gamma y)-i k_n B\sinh (\delta y) \nonumber \\ S^{n}&= -(2\gamma ^{2}+(\omega /c_s )^{2})\cosh (\gamma y)+2i k_n \delta B \cosh (\delta y) \nonumber \\ T^{n}&= 2i k_n \gamma \sinh (\gamma y)+(2k_n^2 -(\omega /c_s )^{2}) B \sinh (\delta y) \end{aligned}$$

(6)

where $U^{n}$ and $V^{n}$ are related to the longitudinal and transverse displacements, respectively, while $S^{n}$ and $T^{n}$ are related to the normal and shear stresses, respectively. Furthermore, $B$, $\gamma$, and $\delta$ are as follows:

$$\begin{aligned}&\displaystyle B=\frac{\left( 2k_n^2 -(\omega /c_s )^{2}\right) \cosh \gamma }{2ik_n \delta \cosh \delta }\end{aligned}$$

(7)

$$\begin{aligned}&\displaystyle \gamma =\sqrt{k_n^2 -(\omega /c_p )^{2}} , \delta =\sqrt{k_n^2 -(\omega /c_s )^{2}} \end{aligned}$$

(8)

This damage detection procedure is based on the power of Lamb wave modes. In the following section, we introduce the scattering coefficients based on Lamb wave power.

2.3 Scattering coefficients

The amplitudes of reflected and transmitted modes are related to notch dimensions. Therefore, by defining scattering coefficients, the magnitude of scattered modes can be related to the damage dimensions, and therefore, the damage geometry can be detected.

The reflected and transmitted coefficients of Lamb waves can be defined as the ratio of scattered amplitudes of reflected and transmitted modes to the amplitude of incident mode. If the power of Lamb modes be used instead of their amplitude, the scattering coefficients would be independent of the measurement direction of mode amplitudes. Therefore, more general coefficients could be obtained. Several studies have been done in order to obtain the scattering coefficients [12–17].

In this paper, the method presented by Kim and Roh [16] is used to compute the scattering coefficients. The reflected/ transmitted coefficients are defined as the square root of the ratio of reflected/ transmitted mode power to the incident mode power. Using the method presented in [16], the scattering coefficients of Lamb waves, caused by a rectangular notch, are computed. The coefficient diagrams obtained for a 5 mm aluminum plate are shown in Fig. 4. The excitation frequency of the wave is 100 kHz. The coefficients are computed for notch depths from 0 to 50 % of the plate height, percentage of $d/H$, and the notch width $(g)$ is considered to be 0 to 10 mm. These parameters are shown on the horizontal axes. The magnitude of scattering coefficient is shown on the vertical axis for the related notch geometry. As long as we use $S_0$ mode, as the incident wave, only the scattering diagrams for this incident mode are presented here. In the scattering coefficients, the first character, i.e., either R or T, shows the reflection or transmission coefficients, respectively. The second character represents the incident mode to the notch. “S” represents $S_0$, and “A” represents $A_0$ mode. Finally, the third character represents the scattered mode under consideration. For example, RSA means the reflection coefficient of $A_0$ mode when the incident mode to the notch is $S_0$. These scattering diagrams form the basis for notch geometry identification algorithm and will be used in the related section.

Fig. 4

Scattering coefficient diagrams for an 5 mm aluminum plate at 100 kHz excitation frequency, a TSS, b TSA, c RSS, d RSA

3 Notch localization procedure

Detection of damage location is the first step in a damage detection plan. The location of notch can be identified using either pulse-echo or pitch-catch method. Although different damage localization methods have been presented in the previous research projects [19], in this article, a damage localization algorithm based on mode conversion phenomenon is proposed.

Consider a plate as shown in Fig. 5. A sensor is installed at the end of inspection zone of the plate. An incident mode is excited in the plate, and after interacting with the notch, two modes are collected by the sensor, because of mode conversion phenomenon. For example, if the $S_0$ mode is generated into the structure, both $S_0$ and $A_0$ modes can be observed in the signals collected by the sensor. Using the difference between arrival times of $S_0$ and $A_0$ modes, someone can compute the distance of the notch from the sensor, $D$, as follows:

$$\begin{aligned}&\displaystyle \Delta t = t_{A_0 } -t_{S_0 } =D . \left( {\frac{1}{V_{A_0 } }-\frac{1}{V_{S_0 } }} \right) ,\end{aligned}$$

(9)

$$\begin{aligned}&\displaystyle D=\frac{V_{A_0 } . V_{S_0 } }{V_{S_0 } -V_{A_0 } } \Delta t \end{aligned}$$

(10)

Fig. 5

Sensor placement for notch localization in a plate

In the Eqs. (9) and (10), $V_m$ represents the group velocity of mode ‘m’, $\Delta t$ is the time difference between $S_0$ and $A_0$ modes received at the sensor, and $D$ is the distance of the notch from the sensor. It is noted that whether $S_0$ or $A_0$ is excited as the incident mode, the $S_0$ mode would arrive first at the sensor, as its velocity is more than that of $A_0$ mode. Someone can extract the group velocities of modes from the dispersion curves and plot Eq. (10) theoretically. This diagram can be used in notch localizing problems for the related structure. Based on the present method, the damage localization diagram is shown in Fig. 6 for a 5 mm aluminum plate.

Fig. 6

Notch localization diagram for a 5 mm aluminum plate

4 Notch geometry identification

In this section, an algorithm is presented to extract the notch dimensions using scattering coefficients. The method is based on a combination of pulse-echo and pitch-catch methods, as the notch geometry cannot be extracted in a single step using only one of these methods.

Consider a uniform plate with a notch as is shown in Fig. 7. Two sensors are used in the plate in order to collect the Lamb waves at the beginning and the end of the plate. When a mode is excited in the structure as an incident wave, we can compute scattering coefficients from the existing sensors. From the sensor1, the power of incident and reflected modes can be computed, which is known as pulse-echo method. On the other hand, the transmitted modes and their power can be obtained from the sensor2, which is known as pitch-catch method. Therefore, the reflection and transmission coefficients can be computed. The theoretically obtained coefficients shown in the Fig. 4 are pure scattering coefficients, i.e., they are the pure effect of notch on the scattering of Lamb waves. However, the power, computed from the sensors, contains another effect that should be differentiated from the effect of notch on the Lamb wave. It should be noted that Lamb wave modes would be attenuated in the waveguides. If someone could neglect the attenuation, the power reduction in Lamb modes would be considered only as the effect of notch. This assumption would cause erratic results in the computation of scattering coefficients, as well as in the geometry identification of the notch. For this reason, someone should omit the effect of wave attenuation from the received wave powers.

Fig. 7

Sensors placement in plate for notch geometry identification

4.1 Lamb wave attenuation

The wave attenuation phenomena would decrease the amplitude and power of the wave in a waveguide. Wave attenuation is the result of the distance a wave would propagate. Therefore, it is necessary to consider the attenuation effect in the formulations. The attenuation of Lamb wave amplitude can be estimated as follows [20]:

$$\begin{aligned} A_2 =A_1 e^{(-k_i \Delta x)} \end{aligned}$$

(11)

where $A_1$ and $A_2$ are the wave amplitudes at the locations $x_1$ and $x_2$, respectively. $\Delta x$ is the propagation distance from point $x_1$ to $x_2$. In Eq. (11), $k_i$ represents the attenuation coefficient and can be computed using [20]:

$$\begin{aligned} k_i =\frac{\omega \zeta }{c_g }, \end{aligned}$$

(12)

where $\omega$ is the excitation frequency, $\xi$ is the damping of the structure, and $c_g$ is the group velocity of the wave. As the group velocity of Lamb wave modes is usually different for a single excitation frequency, their attenuation coefficients would be different, according to Eq. (12).

A power decay coefficient is defined for attenuation. In the waveguides having constant thickness, the power is proportional to the square of wave amplitude. Therefore, according to Eq. (11), we define the power attenuation coefficient, $D_\zeta$ as:

$$\begin{aligned} D_\zeta ^2 =\frac{P_2 }{P_1 }=\left( {\frac{A_2 }{A_1 }} \right) ^{2}=e^{(-2k_i \Delta x)}, \end{aligned}$$

(13)

where $P_i$ is the power of wave at the location $x_i$. This coefficient will be used later to eliminate the effect of attenuation from the scattering coefficients.

4.2 Computation of scattering coefficients

Consider a plate having a notch as shown in Fig. 7. Furthermore, consider that the mode $\alpha$, which can be either $S_0$ or $A_0$, is excited as incident wave. The power of incident mode computed from sensor1 is shown as $P_{incident}^{\alpha }$. The scattered reflected modes are collected by sensor1 and the transmitted modes by sensor2. The power of received mode $\beta$ by sensor1 is introduced as $P_{reflected}^\beta$ and by sensor2, as $P_{transmitted}^\beta$. According to the type of scattering coefficient, which is computed, the received mode $\beta$ can be either $S_0$ or $A_0$.

According to Sect. 2.3, in this study, the reflected/ transmitted coefficients are defined as the square root of the ratio of reflected/ transmitted mode power to that of incident mode. The power of scattered modes arrived at the sensors1 and sensor2 is computed using Eqs. (14) and (15), respectively:

$$\begin{aligned} \sqrt{P_{reflected}^\beta }&= D_{\xi 1}^\alpha \cdot R\alpha \beta \cdot D_{\xi 1}^\beta \cdot \sqrt{P_{incident}^\alpha }\end{aligned}$$

(14)

$$\begin{aligned} \sqrt{P_{transmitted}^\beta }&= D_{\xi 1}^\alpha \cdot T\alpha \beta \cdot D_{\xi 2}^\beta \cdot \sqrt{P_{incident}^\alpha } \end{aligned}$$

(15)

In Eqs. (14) and (15), $D_{\xi 1}^m$ is the power attenuation coefficient of mode $m$ for distance between sensor1 and the notch, and $D_{\xi 2}^m$ represents the power attenuation coefficient of mode $m$ for distance between the notch and sensor2. Eq. (14) is used whenever the reflection coefficient is desired, while Eq. (15) is used whenever the transmission coefficient is requested. Therefore, we can reformulate Eqs. (14) and (15) in order to compute the scattering coefficients as follows:

$$\begin{aligned} R\alpha \beta&= \frac{\sqrt{P_{reflected}^\beta }}{D_{\xi 1}^\alpha \cdot D_{\xi 1}^\beta \cdot \sqrt{P_{incident}^\alpha }},\end{aligned}$$

(16)

$$\begin{aligned} T\alpha \beta&= \frac{\sqrt{P_{transmitted}^\beta }}{D_{\xi 1}^\alpha \cdot D_{\xi 2}^\beta \cdot \sqrt{P_{incident}^\alpha }}. \end{aligned}$$

(17)

As it can be seen from Eqs. (16) and (17), for computing coefficients precisely, it is necessary to know the location of notch as precise as possible. This indicates the importance of the notch localization step, as is presented in Sect. 3.

4.3 The proposed mode conversion-based algorithm

In the previous sections, a computational procedure for scattering coefficients was introduced. Using these coefficients, we propose a notch geometry identification algorithm. Either $S_0$ or $A_0$ mode could be generated into the structure, as the incident wave. According to Fig. 3, four coefficients can be obtained as the result of incident wave interaction with the notch and mode conversion phenomenon. In this algorithm, we need three of these four coefficients to complete the identification procedure for notch geometry. The reason is that using two coefficients could lead to nonunique solutions. On the other hand, using all four coefficients should not be necessary but could have computational burden. This is investigated in the following examples.

When a scattering coefficient is computed, it would be compared with the corresponding diagram in Fig. 4. Therefore, a set of solution would be obtained for this coefficient as a “solution curve.” The same procedure is repeated for two remaining coefficients. Therefore, three solution curves could be obtained, containing the solution for the problem, i.e., the notch depth ratio, % d/H, and width, $g$, for the corresponding coefficient. Finally, we plot these three curves in a single diagram. As long as the solution for the problem could satisfy each curve, the intersection of these curves would be the unique solution for the problem. Our experience with this algorithm shows the possibility of intersection in more than one point for every two curves. That is why we use three coefficients in our method.

5 Examples

In this section, two examples are modeled in $\hbox {ABAQUS}^{\circledR }$ software [21] and the presented method is used to detect the notch parameters in the models.

As the first example, consider a notched plate with the thickness of 5 mm as shown in Fig. 7. The plate is made of aluminum with the longitudinal and shear velocities of 6400 and $3170\,m/s$, respectively. In order to investigate the effect of wave attenuation in the examples, we assume a damping of 0.001 for this plate. The inspection length of the plate is set to be 500 mm, and the parameters of existing notch are presented in Table 1.

Table 1 Notch parameters of example 1

We use a 5-cycle Hanning-windowed sinusoidal tone burst with a central frequency 100 kHz to excite $S_0$ mode as the incident wave into the structure. The problem is simulated in $\hbox {ABAQUS}^{\circledR }$ software. The resulting signals from sensor1 and sensor2 are shown in Fig. 8. The signals are normalized using the maximum amplitude of incident wave. It should be mentioned that at the sensor locations, the axial displacements of a point at the top surface of the plate are considered as the output signals. In practice, these signals can be gathered using piezoelectric patches. Moreover, using a laser vibrometer, these signals can be easily obtained and the Lamb wave modes can be differentiated.

Fig. 8

Signals collected from a sensor1 and b sensor2 in example 1

At first, we should identify the notch location. For this purpose, we use the signal obtained from sensor2. In this signal, the occurrence of mode conversion phenomenon is obvious, and $S_0$ and $A_0$ modes can be observed and differentiated from each other. To compute the difference between the time arrivals of received modes, the scale-average wavelet power (SAP) is defined and used [19] as follows:

$$\begin{aligned} SAP^{2}(n)=\frac{1}{M}\sum _{i=1}^M {\left| {CWT(a_i ,n)} \right| ^{2}} . \end{aligned}$$

(18)

In Eq. (18), CWT is the continuous wavelet transform of the signal, $a$ is the scale, $M$ is the largest scale during CWT, and $n$ represents the number of sampling point. This energy spectrum for Fig. 8b is shown in Fig. 9a. For obtaining the spectrum, the 9th complex Gaussian function is used as the mother wavelet function in Eq. (18). Using this spectrum, the time difference is obtained as $\Delta t = 4\times 10^{-5}$. Using this $\Delta t$ in the diagram of Fig. 6, the notch location is predicted as 258.4 mm far from sensor2. As it can be seen, this prediction is accurate enough.

Fig. 9

a Energy spectrum of received wave in sensor2, b notch localization diagram in example 1

Fig. 10

Intersection of result curves to obtain the notch geometry of example 1

In order to identify the notch geometry, we should use Eqs. (16) and (17). Using the received signals from the sensor1 and sensor2 (Fig. 8), the wave mode powers are computed. The procedure of computation of Lamb mode power was represented in Sect. 2.2 as well as in more detail in [22]. The calculated mode powers are as follows:

$$\begin{aligned} P_{incident}^{S_0}=1; \quad P_{reflected}^{S_0}=0.0562; \quad P_{transmitted}^{S_0}=0.7491; \quad P_{transmitted}^{A_0}=0.0402; \end{aligned}$$

(19)

Note that these powers are normalized with respect to the power of incident mode. Moreover, someone needs to compute the attenuation coefficients. According to Eq. (12), considering $\zeta =0.001,\;f=100\,kHz,\;c_g^{S0}=5375.4\,m/s$, and $c_g^{A0}=2934.1\,m/s$ for this plate, the attenuation coefficients are:

Table 2 Final results of damage detection in example 1

Table 3 Notch parameters of example 2

Fig. 11

Damage detection diagrams of example 2, a received signal in sensor1, b received signal in sensor2, c energy spectrum of Fig. 11b, d notch localization diagram

$$\begin{aligned} k_{S0}=0.1167; \quad k_{A0}=0.2131 \end{aligned}$$

(20)

Note that the group velocities of $S_0$ and $A_0$ modes can be obtained from the aluminum dispersion curves (Fig. 2) for $f=100\,kHz$ and $H=5\,mm$. Therefore, having the received mode powers, the attenuation coefficients, as well as notch location, we can obtain the scattering coefficients as follows:

$$\begin{aligned} TSS&= \frac{\sqrt{P_{transmitted}^{S_0 } }}{D_{\xi 1}^{S_0 } . D_{\xi 2}^{S_0 } . \sqrt{P_{incident}^{S_0 } }}=0.9175 ,\end{aligned}$$

(21)

$$\begin{aligned} RSS&= \frac{\sqrt{P_{reflected}^{S_0 } }}{\left( {D_{\xi 1}^{S_0 } } \right) ^{2}. \sqrt{P_{incident}^{S_0 } }}=0.2510 ,\end{aligned}$$

(22)

$$\begin{aligned} TSA&= \frac{\sqrt{P_{transmitted}^{A_0 } }}{D_{\xi 1}^{S_0 } . D_{\xi 2}^{A_0 } . \sqrt{P_{incident}^{S_0 } }}=0.2175 . \end{aligned}$$

(23)

Intersecting the plane of coefficients TSS, TSA, and RSS by the diagrams shown in Fig. 4a,b and c, respectively, we obtain the solution curves, as shown in Fig. 10. Using Fig. 10, the notch geometry can be identified. It is noticed that these three curves may not intersect precisely at a single point, because of computational, simulation, or experimental errors. Therefore, we take the centroid of the triangle formed by the intersection of the three curves, as the intersection point. As it can be seen, the notch depth ratio and its width are identified as 29.89 % and 3.95 mm, respectively. Therefore, all of the notch parameters are identified with acceptable accuracy using the present algorithm. The results are tabulated in Table 2.

As another example, the same plate with the notch parameters, as presented in Table 3, is considered. Since the algorithm procedure was explained in the previous example, in this example, only the results are presented. The resulting signals, energy spectrum, and solution curves are shown in Figs. 11 and 12. The obtained mode powers and scattering coefficients are presented in Table 4. Finally, the final results are tabulated in Table 5.

The results show that the algorithm can successfully predict notch parameters. In both examples, the notch location, depth, and width are predicted with a good accuracy. The presented algorithm can predict not only the notch location, but also the notch dimensions. This method can be used and developed in order to predict the geometry of different kinds of damage.

Fig. 12

Intersection of result curves to obtain the notch geometry of example 2

Table 4 Mode powers and scattering coefficients of example 2

Table 5 Final results of damage detection in example 2

6 Conclusion

In this article, an algorithm for detection of rectangular notch parameters in plates was presented. The algorithm could successfully detect not only the notch location, but also its geometry, containing its depth and width. The present algorithm was mainly based on mode conversion phenomenon of Lamb waves. Detection of notch location was done using time difference between arrival scattered modes at a sensor in a pitch-catch method. For detecting notch geometry, scattering coefficients were computed using the power of Lamb wave modes. Although in this work the present algorithm was used for detecting rectangular notch geometry, it could also be used for other kinds of damage. By computing the scattering coefficients of any kind of damage and obtaining the scattering coefficient diagrams for its geometry variations, this algorithm could be used and developed to detect any damage geometry.