Fatigue crack prediction is a critical aspect of prognostics and health management research. The particle filter algorithm based on Lamb wave is a potential tool to solve the nonlinear and non-Gaussian problems on fatigue growth, and it is widely used to predict the state of fatigue crack. This paper proposes a method of lamb wave-based early fatigue microcrack prediction with the aid of particle filters. With this method, which the changes in signal characteristics under different fatigue crack lengths are analyzed, and the state- and observation-equations of crack extension are established. Furthermore, an experiment is conducted to verify the feasibility of the proposed method. The Root Mean Square Error (RMSE) of the three different resampling methods are compared. The results show the system resampling method has the highest prediction accuracy. Furthermore, the factors affected by the accuracy of the prediction are discussed.

Prognostics and health management (PHM) has recently become a novel engineering research hotspot [

Fatigue crack is an important aspect of PHM research, which significantly involves developing safety features as well as the extension of working time [

Due to complex work environments, however, the growth of fatigue cracks may be easily affected by a plethora of uncertain factors, such as material properties, temperature, unsteady force, and humidity [

With the development of non-destructive monitoring technology, PF, in conjunction with ultrasonic lamb wave detection, may be more suitable for application in engineering. Yang et al. [

The aforementioned studies were based on the standard PF. Following particle propagation, weight computation, resampling, and the sample impoverishment [

Accordingly, this paper proposes an early fatigue microcrack damage prediction method based on the particle filter algorithm. Combined with finite element simulation, the variation in characteristic parameters when the wave passes through different crack lengths is then analyzed, while the damage index (DI) is used to observe crack length. The RMS of three different resampling methods is subsequently compared. Finally, the errors caused by the crack shape are discussed.

The state-space model based on the crack propagation law consists of a state equation and observation equation [_{k−1} is the crack length at time _{k−1} is the uncertainty of crack growth. _{k} is the observation value, which includes the fatigue crack information at the time _{k} refers to uncertainties during testing.

Paris’ law [

where

The specimens are labeled as {L_{j},

Given the crack growth rate _{j}) and _{j}, which corresponds to the specimen _{j} can be obtained through linear fitting.

The uncertainty estimate of crack propagation can be written as

where difference

In order to consider uncertainty during the growth of fatigue crack as well as for simplification [

Among the acquired monitored signals, the increase in the crack can cause signal distortion, energy attenuation, and phase delay [

where _{XY} is the covariance of the two signals, _{X} and _{Y} are the mean square deviations of the two signals,

The corresponding observation is descriptive of the fatigue crack monitoring result, which can be obtained to fit all of the specimen’s DIs under different crack lengths, as shown in _{0}, _{1}, _{2}, _{3}} are the coefficients of polynomial fitting.

The uncertainty of measurement is assumed to be normally distributed and is expressed as _{i} can be expressed as

The standard PF was provided by the findings put forward by Gordon et al. [_{k} | _{1:k}) with the specimen’s observations [_{k} | _{1:k}), as expressed in

However, the pdf _{k} | _{1:k}) is actually difficult to calculate. The basic idea of the PF is to approximate the posterior pdf _{k} | _{1:k}) through a set of particles _{k} | _{1:k}).

When _{k} | _{1:k}) is unknown, the importance density function _{k} | _{1:k}) can then be introduced. Therefore, a posterior estimate can be expressed using

The importance density function can be derived as

The unnormalized weight can be calculated through the iteration of

The weight normalization is shown in

The prognostic result [_{k} is calculated as in

The following describes three resampling methods:

Residual resampling [

_{i} are allocated to the new distribution. Additionally, _{i}} by making _{i} where the probability for selecting _{i} is proportional to

Systematic resampling [

Here,^{−1} denotes the generalized inverse of the cumulative probability distribution of the normalized particle weights.

Multinomial resampling [

Here,

The resampling method involves eliminating particles that have small normalized importance weights and copying upon particles with large weights, setting all the weights to 1/

Step 1. Record the number of loads and crack length during the fatigue crack growth process, calculate the crack growth parameters, and establish the crack growth state equation.

Step 2. Obtain response signals under different fatigue crack conditions via ultrasonic lamb wave detection technology, extract characteristic values, and establish observation equations.

Step 3. Use particle filter algorithm to realize early fatigue microcrack prediction.

In order to ascertain the energy change of lamb waves passing through different crack lengths, numerical experiments are performed using the finite element method (FEM). An aluminum plate with the encastre boundary condition is modeled using the ABAQUS software, as shown in

A through-thickness crack with a width of 0.5 mm was modeled onto the side of the plate. In order to simulate early fatigue cracks, at the center of this crack, another size crack was cut out with a width of 0.1 mm and length varying from 0 to 12 mm, which had an increment of 3 mm, as shown in

According to the dispersion curve of Lamb wave propagating in a 6061-T6 aluminum alloy plate with 5 mm thickness, when the frequency thickness product is less than

Material | Young’s modulus |
Density ^{3}) |
Poisson’s ratio |
---|---|---|---|

Al6061-t6 | 69 | 2700 | 0.33 |

With regard to verify the proposed method, a fatigue test for plate specimens consisting of 6061-t6 aluminum was performed. The thickness of all specimens was 5 mm, which were labeled from L1 to L5. A 5 mm notch was then machined at the specimen’s edge in order to initiate the direction of fatigue crack growth. As illustrated in

The Lamb wave based on the SHM technique was used for fatigue crack detection, which was sensitive to fatigue crack. The change in characteristics was due to discontinuities introduced by the growth of fatigue crack. These discontinuities can disperse and reflect the energy of the original Lamb wave, triggering changes in wave characteristics [

A Multipurpose Servohydraulic Universal Testing Machine (Serises LFV 250 KN) was used to apply fatigue load. Specimen L1 was initially performed to determine the fatigue load with a fracture load of 35 KN. According to the results, a sinusoidal load with a peak value of 10 KN was chosen for specimens L2 to L5. The stress ratio was

During the fatigue test, a digital microscope was used to observe the growth of the crack. When the crack grew to about 1 mm, the specimen was taken down. The system was then used to monitor the fatigue specimen, as shown in

In order to verify the feasibility of the proposed method. L3–L5 specimens were used to establish the state-space model, while L3–L4 specimens underwent prognostic validation. As mentioned in _{0}, _{w} of specimens L3–L5 were calculated [

Specimen | L3 | L4 | L5 | Mean |
---|---|---|---|---|

log _{0} |
−34.2 | −40.38 | −44.67 | −39.75 |

4.436 | 5.482 | 6.3 | 5.406 |

The value of log _{0} and _{0} = −39.75, ^{2} and ^{2}).

By substituting the calculation parameters into

DI was used to quantify the length of the crack, which was the energy attenuation and phase delay when the wave passed through the crack that was directed by the shape of the crack, affecting the accuracy of the observation. The fatigue crack is divided into three stages during growth: initiation, propagation, and fracture [

When the wave passed through a closed crack, wave A passed through directly, whereas wave B passed through the crack tip. The wave received by the piezoelectric plate was a superposition of these two waves. When the wave passes through an open crack, only wave B passed through the crack [

This paper proposes an ultrasonic Lamb wave fatigue damage detection method according to the particle filter algorithm. Here, the change in eigenvalues when the lamb wave passes through the open crack is simulated by the finite element method. Additionally, this experiment analyzes the energy and phase changes of the S0 mode when passing through the crack. Accordingly, the findings of this study demonstrate that as the crack increases, the energy attenuation and DI rise, while the phase delay becomes longer. The simulation is found to be consistent with the results of the experiment. Moreover, the residual resample prediction errors of L3 and L4 are noted to be 0.48 and 0.7, while the systematic resample are 0.46 and 0.66, and the multinomial resamples are 1.94 and 0.75. The RMS under the three different resampling methods are then compared, in which systematic resampling is found to have a smaller prediction error. The results demonstrate that the proposed method is capable of effectively predicting fatigue crack propagation. Finally, the factors that may cause prediction errors are discussed. In view of the corresponding findings, conducting a further analysis on the propagation mode of Lamb waves under different crack types is needed. According to present results, the following conclusions were drawn:

The increase of the growth fatigue crack has an obvious effect on the characteristic parameters of ultrasonic Lamb waves, which can be used to establish a fatigue crack observation model.

The particle filter algorithm can be used well in order to solve the uncertainty in crack propagation and can realize the prediction of the fatigue crack.

The prediction errors of the system resampling algorithm are 0.46 and 0.66, which are smaller than those of the other two resampling algorithms.