Damage identification for nonlinear fatigue crack of cantilever beam under harmonic excitation

2.2.1. Contact model

Contact modelling is a highly nonlinear method, which can be used to simulate the fatigue crack process. Contact models are divided into frictional contact model and non-frictional one. When there is friction contact, the friction coefficient $f>0$. Slipping occurs along the contact surface when the tangential force reaches a certain degree. At this time, the tangential force is related to the loading path, and the change process is irreversible. When there is no friction contact, the friction coefficient $f=0$, tangential force and loading path are independent, and the change process is reversible. The results have shown that the friction between crack surfaces presented little influence on dynamic response. Therefore, the friction-free contact model is used to simulate fatigue cracks in this study.

The calculation methods of surface-to-surface contact include penalty function method and extended Lagrange multiplier method. Compared with penalty function method, the stiffness matrix is not easy to be ill-conditioned when using extended Lagrange multiplier method, and the setting of contact stiffness is more reasonable. The contact surface is composed of contactor surface and target surface. According to the contact conditions, the contactor elements cannot penetrate the target surface, but the target elements can penetrate the contactor surface during convergence solution. In the non-friction contact problem, there are three states between the contact node and the target element: (1) there is no contact between them, and the crack is in the open state; (2) there is complete contact between them, and the crack is in the closed state; (3) in the transitional stage between the first two states, the crack is neither fully opened nor fully contacted. Therefore, the non-linear effect of contact element can be simulated through establishing contact pairs at the crack location, as shown in Fig. 2. It can be used to characterize the phenomenon that the stiffness of structure with fatigue crack always changes.

Fig. 2Nonlinear contact model

2.2.2. Bilinear model

Comparative analysis of harmonic response for cracked cantilever beam was conducted between contact model and bilinear model. When the bilinear model is used to simulate the fatigue crack, it can be considered that there are only two states for fatigue crack: opening and closure. Bilinear model of crack was shown in Fig. 3. Structural stiffnesses changes in opening stiffness $k_o$ and closure stiffness $k_c$. When $u\left(t\right)>0$, the crack surface was in open state, and the stiffness of structure was consistent with $k_o$; When $u\left(t\right)\le 0$, the crack surface was in closure state, and the stiffness of structure was consistent with $k_c$.

Fig. 3Bilinear model of fatigue crack

a) Bilinear spring element model

b) Diagram of bilinear stiffness change

2.3.1. Phase spectrum-based damage indicator

Phase spectrum will produce a certain degree of distortion when the damage occurs. The more serious the damage is, the more obvious the distortion presents. The distortion value of phase spectrum is proposed as damage indicator according to the change of the spectrum along the displacement direction, which was expressed in Eq. (1) and illustrated in Fig. 4:

where, $D_d$ is distortion value of phase spectrum; $d_r^{dam}$ is the maximum positive displacement of damaged beam in phase spectrum; $d_l^{dam}$ is the maximum negative displacement of damaged beam in phase spectrum. When beam is intact, $D_d=d_r^{dam}=d_l^{dam}$.

2.3.2. Fourier spectrum-based damage indicator

When a damaged structure is subjected to non-linear vibration, the spectrum energy of the structure is no longer concentrated at the excitation frequency. Moreover, it generates a certain degree of super/sub-harmonic components. The magnitude of the harmonic component is positively correlated with the damage degree. The amplitude energy at the excitation frequency alters when the damage degree changes. Assuming $H_h$ is the sum of amplitude energy of all generated super/sub-harmonic components, and $H$ is the sum of energy in all frequency domains including amplitude energy at excitation frequency. $H_h$ and $H$ can be calculated by:

where $h_i$ is the amplitude of $i$th super/sub-harmonic component, $H_{ff}$ is the amplitude at excitation frequency. $h_i$ and $H_{ff}$ was expressed in Fig. 5.

Fig. 4Index value in phase spectrum

Fig. 5Index value in Fourier spectrum

Since the occurrences and magnitudes of super/sub-harmonic components are obviously correlated with the structural damage severity. Super/sub-harmonic amplitudes ratio $h_{\mathrm{\Delta }}$was proposed as damage indicator, which can be expressed by:

where, $h_{\mathrm{\Delta }}$ represents the proportion of super/sub-harmonic amplitudes in the total Fourier-spectrum energy due to the breathing effect of fatigue crack. When beam is intact, $h_{\mathrm{\Delta }}=0$.

3.2.1. Effect of excitation frequency

In order to investigate that the influence of nonlinear fatigue crack on harmonic vibration response of the beam, acceleration time history, phase spectrum and Fourier spectrum of the intact structure and damaged one ($p=$ 0.07, $s=$ 0.2) under harmonic excitation with three frequencies ($\eta =$ 1/3, 1/2, 2, $\eta =$ excitation frequency / natural frequency) are calculated, as shown in Figs. 7-9.

Fig. 6Natural frequency of cracked cantilever beam

Fig. 7Harmonic vibration response for intact and cracked beam with η= 1/3

a) Acceleration time history

b) Phase spectrum

c) Fourier spectrum

As observed from Figs. 7-9, fatigue damage makes acceleration time history of cantilever beam oscillate and distort. The central axis of acceleration curve shifts, and the sawtooth shape appears at the edge of the curve. For the phase spectrum, the displacement-velocity curve of the intact structure presents a standard elliptical shape, while the curve is deformed, drifted and expanded when the structure is damaged with fatigue crack. Meanwhile, there are small irregular grooves and subsidence circles. The degree of distortion under different excitation frequencies is different. For Fourier spectrum, the most obvious non-linear characteristic is that the spectrum energy is no longer concentrated in the excitation frequency, and energy redistribution occurs. The harmonic components occur at the positions of the integer or one-fold integer times of excitation frequency. The super-harmonic component is generated at resonance of $\eta =$1/3 and $\eta =$1/2, while the sub-harmonic component is generated at $\eta =\mathrm{}$2.

Fig. 8Harmonic vibration response for intact and cracked beam with η= 1/2

a) Acceleration time history

b) Phase spectrum

c) Fourier spectrum

Fig. 9Harmonic vibration response for intact and cracked beam with η= 2

a) Acceleration time history

b) Phase spectrum

c) Fourier spectrum

3.2.2. Effect of damage severity

In analyzing the influence of damage severity on the harmonic vibration response of cantilever beam, different damage severities ($s=$ 0.1, 0.2, 0.3, 0.4, 0.5) were considered. Fig. 10 presents the acceleration time history curve of cantilever beam with different crack depths under harmonic loading with $\eta =$1/3. As observed from the results, the curve presents migration phenomenon, and the degree of acceleration oscillation increases slightly with the increasing of damage severity when the crack occurs at the same location. Meanwhile, the sawtooth shape distortions begin to appear at the lower edge of the curve at $s=$ 0.3. The sawtooth shape distortion appears slightly at the upper edge of the curve at $s=$ 0.4. At $s=$ 0.5, the sawtooth shape distortions of the upper and lower edges of the curve were all obvious. The same trends were also found for other conditions ($p=$0.27, 0.47, 0.67, $\eta =$ 1/2, 2). This phenomenon can reflect the non-linear vibration characteristics of beam with fatigue crack. The greater the damage severity, the more obvious the nonlinear vibration characteristics. Therefore, the degree of distortion in acceleration time history curve can be used to characterize the effect of damage severity on vibration response of structures with fatigue cracks.

Fig. 10Acceleration time history of cracked beam with different damage severity at p= 0.07, η= 1/3

a)$s=\text{0.1}$

b)$s=\text{0.2}$

c)$s=\text{0.3}$

d)$s=\text{0.4}$

e)$s=\text{0.5}$

For the effect of damage severity on phase spectrum of cracked beam, Fig. 11 presents the variations of displacement versus velocity under different damage severities ($s=$0.2, 0.3, 0.4, 0.5) at $p=$ 0.07 and $\eta =$1/2. As observed from this figure, phase spectrum curves no longer keep the regular ellipse shapes after the structure is damaged.

Fig. 11Phase spectrum of cracked beam with different damage severity at p=0.07, η= 1/3

a)$s=\text{0.2}$

b)$s=\text{0.3}$

c)$s=\text{0.4}$

d)$s=\text{0.5}$

Fig. 12Fourier spectrum of cracked beam with different damage severity at p=0.27, η=2

a)$s=\text{0.2}$

b)$s=\text{0.3}$

c)$s=\text{0.4}$

d)$s=\text{0.5}$

The centers of circles shift to the positive direction of displacement, and the curves distort. The displacement-velocity distribution domain increases. The deeper the damage depth is, the more obvious the non-linear phenomenon presents. When the damage severity is $s=$ 0.3, a small groove appears in the curve; when $s=$ 0.4, the groove evolves into an irregular shape of a small circle; when $s=$ 0.5, the area of the small circle obviously enlarges. The change phenomena were also applied for other cases. Therefore, the change of damage depth has a great influence on the harmonic vibration characteristics of non-linear damaged structures. The more serious the damage is, the more significant the phase spectrum distortion is.

Fig. 12 shows the Fourier spectrum of beam under different damage severity ($s=$0.2, 0.3, 0.4, 0.5) at $p=\text{0.27}$, $\eta =\text{2}$. It can be seen that there are obvious sub-harmonic components at half of the excitation frequency. When the damage severity is small, the magnitude of subharmonic component is weak compared with the amplitude of excitation frequency. When the damage severity increases, the magnitude of subharmonic component increases correspondingly, and becomes much larger than the corresponding magnitude of excitation frequency. When the damage severity is large enough ($s=\text{0.4}$, $s=\text{0.5}$), a certain degree of super-harmonic component appears at its integer multiple of the excitation frequency. These phenomena show that the magnitude of sub-harmonic component is the main characteristic of power spectrum under high frequency harmonic excitation, and these characteristics are enough to characterize the influence of damage severity on the harmonic response characteristics of non-linear fatigue cracked beams. the more serious the damage is, the greater the impact exhibits.

3.2.3. Effect of damage location

To demonstrate the influence of damage location on harmonic vibration response of non-linear cracked structures, crack locations $p=$0.07, 0.27, 0.47, 0.67 were considered. Fig. 13 shows the acceleration time history curve of different crack positions under harmonic excitation frequency $\eta =$1/3 and relative damage severity $s=\text{0.4}$.

Fig. 13Acceleration time history of cracked beam with different damage locations at s=0.4, η= 1/3

a)$p=\text{0.07}$

b)$p=\text{0.27}$

c)$p=\text{0.47}$

d)$p=\text{0.67}$

It can be observed from the results that when the damage location is near the fixed end ($p=\text{0.07}$), the curve contains obvious sawtooth distortion, and there is unequal distribution at both ends of the acceleration zero line. With the crack away from the fixed end, the sawtooth shape in the curve gradually weakens, and the unequal distribution of the curve is no longer obvious. Therefore, the influence of crack location on the structural vibration response can be reflected by the acceleration time-history signal curve for a cantilever beam with fatigue crack. When the crack is near the fixed end, its non-linear characteristics are obvious. When the crack is away from the fixed end, its non-linear characteristics are weakened.

Fig. 14 illustrates the phase spectra of different crack locations under harmonic excitation of relative excitation frequency $\eta =$1/2 and damage severity $s=\text{0.4}$. It can be concluded that the shape of the curve in the phase spectrum changes with crack position. When the crack is near the fixed end ($p=$ 0.07), the phenomenon of curve distortion and migration is the most obvious. Irregular small circles are sunk inward, and the displacement value of the curve is the largest. When the position of the crack changes to the free end, the phenomenon of curve distortion and migration is weakened, and the irregular small circles gradually become smaller ($p=\text{0.27}$) until disappear ($p=$ 0.47, 0.67). And the displacement value of the curve also decreases. It can be found from phase spectrum that the changes of fatigue crack location cause the variation of the harmonic vibration characteristics of cracked beam. The more near to the fixed end the damage is, the more obvious the distortion of phase spectrum exhibits.

Fig. 14Phase spectrum of cracked beam with different damage locations at s=0.4, η= 1/2

a)$p=\text{0.07}$

b)$p=\text{0.27}$

c)$p=\text{0.47}$

d)$p=\text{0.67}$

Fig. 15 shows the Fourier spectrum of cracked beam at different crack locations when the relative damage degree $s=\text{0.5}$ and the relative frequency of harmonic excitation $\eta =$1/2. The analysis shows that when the crack is near the fixed end ($p=\text{0.07}$), the excited super-harmonic order is more. Corresponding amplitude of the first harmonic component is relatively larger, which occupies the main position in the total amplitude energy. When the crack is away from the fixed end and close to the free end, the amplitude of the harmonic component decreases gradually, and the second-order harmonic amplitude nearly disappears. When the position of the crack changes, the amplitude of the first harmonic component is less than the amplitude of the excitation frequency, which indicates that the non-linear effect of the structure caused by the fatigue crack is weak. Therefore, the farther the crack is from the fixed end, the smaller the impact of fatigue crack on the vibration response of the structure presents.

Fig. 15Fourier spectrum of cracked beam with different damage locations at s=0.5, η= 1/2

a)$p=\text{0.07}$

b)$p=\text{0.27}$

c)$p=\text{0.47}$

d)$p=\text{0.67}$

3.3.1. Distortion value of phase spectrum, ${\mathit{D}}_{\mathit{d}}$

The evolutions of $D_d$ with damage severity and damage location are illustrated in Fig. 16 and 17, respectively. For cantilever beam, when the damage location is fixed, $D_d$ increases obviously with the increase of damage severity. And when the damage severity is fixed, the closer the damage is to the fixed end, the larger the value of $D_d$ is. The value of $D_d$ can reflect the damage severity of cracked beam, and the magnitude of $D_d$ is positively correlated with the damage severity. When the damage severity is taken as variable, the change degree of $D_d$ at $p=\text{0.07}$ is the most obvious and it is the weakest at $p=\text{0.67}$. when the damage location is taken as a variable, the change degree of $D_d$ at $s=\text{0.5}$ is the most obvious and it is the weakest at $s=\text{0.1}$. It can be seen that $D_d$ is effective for the identification of non-linear fatigue crack in cantilever beam.

Fig. 16Change curve of distortion value of phase spectrum (Dd) with damage severity

a)$\eta =\mathrm{}$1/3

b)$\eta =$1/2

c)$\eta =$ 2

Fig. 17Change curve of distortion value of phase spectrum (Dd) with damage location

a)$\eta =\mathrm{}$1/3

b)$\eta =$1/2

c)$\eta =$ 2

3.3.2. Proportion of super/sub-harmonic amplitudes, ${\mathit{h}}_{\mathbf{\Delta }}$

The variation curves of the proportion of super/sub-harmonic amplitudes ($h_{\mathrm{\Delta }}$) versus damage severity and damage location are illustrated in Figs. 18 and 19, respectively. As observed from the results, $h_{\mathrm{\Delta }}$ increases with the increase of damage severity when the damage location and excitation frequency are fixed. For the influence of excitation frequency, $\eta =$1/2 presents more obvious effect on $h_{\mathrm{\Delta }}$ at all damage locations. Moreover, the influence of damage location on $h_{\mathrm{\Delta }}$ decrease with the increase of damage location away from the fixed end. The results reveal that $h_{\mathrm{\Delta }}$ can be used to reflect the damage evolution of fatigue crack in cantilever beam.