Fatigue evaluation of metallic components based on chaotic characteristics of second harmonic generation signal

2.1. Lyapunov exponent

The basic feature of chaotic system is extremely sensitive to initial conditions, tracks generated by two close initial values separate exponentially with time, and the Lyapunov exponent studies this feature [11]. In n-dimensional phase space, the long-term evolution process of an infinitesimal n-sphere with initial condition $p_i\left(0\right)$ is monitored, which the sphere would become an ellipsoid due to locally deformation. The $i$th Lyapunov exponent is defined according to the length of ellipsoidal principal axis $p_i\left(t\right)$ [21]:

The small data quantity method is commonly used to calculate the maximum Lyapunov exponent $\lambda$ from time series [12]. If a time series with embedding dimension $m$ and time delay $\tau$, the reconstructed phase space is:

The nearest neighboring point $Y_{j\mathrm{\text{'}}}$ is searched for each point $Y_j$ and the brief separation is limited, namely:

where $P$ is the average period of time series. For each pair of initial neighboring points, the distance after $i$ discrete steps is:

For each discrete step$i$, $y\left(i\right)$ is the average value of all $\mathrm{l}\mathrm{n}{d}_j\left(i\right)$:

where $q$ is the number of non-zero $d_j\left(i\right)$. The optimal fitting curve of $i-y\left(i\right)$ is gotten using least square method which the slope of straight line is maximum Lyapunov exponent $\lambda$.

2.2. Kolmogorov entropy

Kolmogorov entropy describes the degree of how quickly information produced in nonlinear system and is an important characteristic of attractor which measures the randomness of system. The value of Kolmogorov entropy could reflect the complexity of nonlinear system. A continuous dynamical system in $n$-dimensional phase space, the space is divided into many $n$-dimensional boxes with length $\epsilon$. If a trajectory $x\left(t\right)$ is in the state space and the time interval is selected a small value $\tau$. The joint probability $P\left(i_0,i_1,\dots ,i_d\right)$ indicates the trajectory is in $i_0$ box at initial time, then in $i_1$ box at $t=\tau$, next in $i_2$ box at $t=2\tau$, then in $i_3$ box at $t=3\tau$, and finally in $i_d$ box at the time $t=d\tau$, therefore the definition of Kolmogorov entropy is:

The maximum likelihood method is usually applied to calculate Kolmogorov entropy from time series which is proposed by Schouten [22].

2.3. Correlation dimension

The fractal dimension is another important characteristic of attractor and widely used in quantitative description the nonlinear behavior of system. The correlation dimension is one of them which measures the complexity of attractors [16]. The G-P algorithm is applied to calculate the correlation dimension from time series proposed by Grassberger and Procaccia [23]. In $m$-dimensional phase space, if the distance $d_{ij}$ between two points $y_i$ and $y_j$ is less than a given value $r$, the associated components are called for them [24, 25]. There are $N$ points in this space, the correlation integral $C_n\left(r\right)$ is a proportion of relating vector number:

Here $\theta$ is the Heaviside unit function:

The correlation dimension is defined as:

The double logarithmic relationship between the given value $r$ and $C_n\left(r\right)$ is taken and the curve is fitted using the least squares method, which the slope is correlation dimension $D$.

According to the calculation methods of chaotic characteristics, the corresponding MATLAB programs are written to calculate Lyapunov exponent, Kolmogorov entropy and correlation dimension from SHG signals applying the small data quantity method, maximum likelihood method and G-P algorithm.

4.1. Calculation chaotic characteristics from SHG signals

Based on collected SHG signals, the phase space reconstruction is firstly conducted on time series to calculate characteristic values of attractors, the optimal delay time $\tau$ and embedding dimension $m$ are gotten to reconstruct high-dimensional phase space. For the SHG signal shown in Fig. 6(a), $\tau =$ 7 and $m=$ 10 are obtained using C-C method [26]. In 10-dimensional phase space, the Lyapunov exponent $\lambda$, Kolmogorov entropy $K$ and correlation dimension $D$ of attractor are calculated. Fig. 7(a) and 7(b) display calculation results of Lyapunov exponent and correlation dimension.

Fig. 7Calculation results of: a) the Lyapunov exponent, b) correlation dimension for the SHG signal shown in Fig. 6(a)

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b)

When calculating the Lyapunov exponent from the SHG signal shown in Fig. 6(a), the relationship between discrete step $i$ and $y\left(i\right)$ is gotten as displayed in Fig. 7(a). The least square method is applied to fit the straight part which the slope is Lyapunov exponent, therefore $\lambda =$ 0.2294 is the Lyapunov exponent of SHG signal shown in Fig. 6(a).

When estimating the correlation dimension $D$ of attractor, the double logarithmic relationship between given value $r$ and correlation integral $C_n\left(r\right)$ is taken and curves are fitted using the least square method, then the slope is correlation dimension $D$. Thus in Fig. 7(b), the correlation dimension of SHG signal shown in Fig. 6(a) is $D=$ 1.3627.

4.2. Chaotic characteristics for beams with different fatigue state

Based on SHG signals of 45steel beams, firstly the ultrasonic nonlinear parameter ${\beta }_0$, Lyapunov exponent ${\lambda }_0$, Kolmogorov entropy $K_0$ and correlation dimension $D_0$ of undamaged beams are calculated applying the definition of ultrasonic nonlinear parameter $\beta =A_2/A_1^2$, small data quantity method, maximum likelihood method and G-P algorithm. Then a series of ultrasonic nonlinear parameter ${\beta }_n$ and chaotic characteristics ${\lambda }_n$, $K_n$, $D_n$ are calculated from SHG signals when the fatigue states of beams are 10 %, 20 %, 30 %, 40 %, 50 %, 60 %, 70 % ,80 %, 90 % and 100 %, where the subscript $n$ represents different fatigue state. At last the ultrasonic nonlinear parameter and chaotic characteristics are normalized according to ${\beta }_n/{\beta }_0$, ${\lambda }_n/{\lambda }_0$, $K_n/K_0$, and $D_n/D_0$. In the process of fatigue for beams, the variation trend of nonlinear parameter, Lyapunov exponent, Kolmogorov entropy and correlation dimension are drawn in Fig. 8.

Fig. 8Variation curves of four characteristic values when the fatigue cycles of beams increase from 0 % to 90 %: a) the ultrasonic nonlinear parameter, b) the Lyapunov exponent, c) Kolmogorov entropy, d) the correlation dimension

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b)

c)

d)

As can be seen from Fig. 8, the overall variation tendency of ultrasonic nonlinear parameter, Lyapunov exponent, Kolmogorov entropy and correlation dimension of four beams are similar although some differences are existed among them. With increase of fatigue cycles, the ultrasonic nonlinear parameter and chaotic characteristics increase gradually, which the related increments are about 60 %, 75 %, 10 % and 40 %, respectively. In addition, when the fatigue cycles are more than 70 %, four characteristic values tend to be stable. The variation trend of correlation dimension for fatigued beams is similar to Wang L’s results [16]. Throughout the fatigue process, chaotic characteristics have similar variation trend with ultrasonic nonlinear parameter, and the Lyapunov exponent is the most sensitive to fatigue of beams. The experiment results indicate that chaotic parameters could reasonably characterize and evaluate the fatigue state of beams.

Because the Lyaounov exponent indicates the chaotic degree of system, Kolmogorov entropy measures the randomness of nonlinear system, and correlation dimension represents the complexity of attractor. In the process of fatigue crack extension, the size, quantity, complexity and randomness of cracks and the microstructure of beams have been changing. Due to the beams with prefabricated mechanical gap, the tip of gap would firstly appear microscopic cracks by cyclic loading. The quantity of cracks increases, the size enlarges, and the microstructure of beams would become more complex with cyclic cycles, thus the complexity and randomness of cracks become big, and four characteristic values increase with fatigue cycles. When fatigue cycles are about 70 %, beams have appeared macroscopic crack, the ultrasonic nonlinear parameter, Lyapunov exponent, Kolmogorov entropy and correlation dimension reach a maximum value. When the macroscopic crack has been appeared, the attenuation of ultrasonic would become obvious especially the second harmonic, therefore the ultrasonic nonlinear parameter tends to be stable. At the late stage of fatigue crack extension, the size of cracks becomes bigger, the beam would fracture at any moment. However, the quantity of crack and the complexity of the crack extension path haven’t changed much, the Lyapunov exponent, Kolmogorov entropy and correlation dimension would become stable. Therefore, in the process of fatigue, chaotic characteristics could well evaluate the fatigue state of beams and reveal the complexity and randomness of fatigue crack propagation.

4.3. Chaotic characteristic analysis of used connecting rods

When the driving frequency is 3.8 MHz and the width of driving signal is 8 us, higher harmonic experiments are conducted on three connecting rods with service time of 0h, 296 h and 450 h, respectively, SHG signals are collected by an oscilloscope for analyzing the chaotic characteristics. The SHG signal of connecting rod with 450 h service time is shown in Fig. 6(b), and the corresponding frequency spectrogram is shown in Fig. 9.

Fig. 9Spectrum of SHG signal for used connecting rods when the driving frequency is 3.8 MHz

From the frequency spectrum shown in Fig. 9, there are strong signal at 3.8 MHz and very weak second harmonic at 7.6 MHz which couldn’t be distinguished from nearby signals. This illustrates the nonlinearity of connecting rods is very weak.

Based on SHG signals, the ultrasonic nonlinear parameter and chaotic characteristics of three connecting rods are calculated and listed in Table 1, where the values are the mean of four repeating experiments. The increments of four characteristics values are computed according to $\left(v_{450}-v_0\right)/v_0\mathrm{}$, where $v_{450}$ and $v_0$ indicate the ultrasonic nonlinear parameters and chaotic characteristics at 450h and 0h, respectively.

As shown in Table 1, ultrasonic nonlinear parameters of three connecting rods with 0 h, 296 h and 450 h service time are 1.5206×10^-3V^-1, 1.5215×10^-3V^-1 and 1.5293×10^-3V^-1. The increment of ultrasonic nonlinear parameter is only about 0.57 % when the service time increases to 450 h. According to ref [27], when the connecting rod of diesel engines made in china appears macroscopic cracks, it has worked about 24000 h. Furthermore, the fatigue life is about 80 % when the macroscopic cracks are emerged in actual engineering applications. Thus, the total fatigue life of connecting rods is about 30000 h. Due to the fact that the service time of used connecting rod are 296 h and 450 h respectively, the fatigue life is about 296 h/30000 h = 0.99 % and 450 h/30000 h = 1.5 %, which indicates the accumulation fatigue damage of connecting rods is very weak. Therefore, the ultrasonic nonlinear parameter calculated from the frequency spectrum of SHG signals is very small.

It also can be seen from Table 1, when the service time of connecting rods increases to 450 h, the Lyapunov exponent adds from 0.2227 to 0.2264 which the growth rate is 1.66 %, Kolmogorov entropy increases from 0.5351 to 0.5396 fortifying about 0.84 %, while the correlation dimension augments from 1.1889 to 1.2035 which increases about 1.23 %. It can be concluded that although the fatigue damage degree of connecting rods is very weak, the Lyapunov exponent, correlation dimension and Kolmogorov entropy still monotonously increase with service time. This illustrates the nonlinearity of connecting rods is ever-increasing.

Furthermore, comparing the ultrasonic nonlinear parameter and characteristics values shown in Table 1, the most obvious increment is Lyapunov exponent, then correlation dimension, followed by Kolmogorov entropy, and finally ultrasonic nonlinear parameter. Therefore, chaotic characteristics, especially the Lyapunov exponent, could well evaluate and characterize the earlier cumulative fatigue damage of used connecting rods, the chaos and fractal theory can effectively extract the nonlinearity of SHG signals which may provide an effective analysis method for nonlinear received signals in nonlinear ultrasonic technique.