5.3 Detection and Localization of Breathing Cracks
5.3.1 Estimation of the Crack Orientation Angle
For cracks with unknown orientations, the MCDLA (Section 3.2) will give the approximate location of the cracks. The optimization problem, defined in Section 4.6, handles two variables per crack (the size and the location of cracks). Also, objective functions (Eqn. (4.12)) of the optimization problem involve FEM predicted cracked shaft responses and measured cracked shaft responses. FEM predicted cracked shaft responses can be obtained if crack orientation
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0 0.2 0.4 0.6 0.8
Measurement locations, j
Magnitude
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0 0.2 0.4 0.6 0.8 1
Measurement locations, j
Magnitude
(a) CPFv (b) CPFh
angles are known. In the present section, a method is presented for estimation of the crack ori- entation angle of cracks. It uses the MCDLA, presented in Section 3.2.
Coefficients avII and ahII for a cracked shaft, with a crack at the element number 35 and the crack depth ratio of 0.7, are plotted in Figure 5.5. Other system parameters such as length and diameter of the shaft remain same as in Simulation-I of Table 3-1. However, the noise is not added in the shaft response.
Figure 5.5: Coefficients avII and ahII from responses at the excitation frequency of 100 rad/s.
From Figure 5.5, coefficients avII and ahII have peaks at the location of crack. This indicates presence of a single crack in the shaft. The magnitude of peaks in Figure 5.5 (a) and Figure 5.5 (b) vary with the crack orientation angle. These variations (i.e. the magnitude of the peak ver- sus the crack orientation angle) are plotted in Figure 5.6. The coefficients avII show a sinusoidal variation with the crack orientation angle, while coefficients ahII show a stepwise variation with the crack orientation angle. The coefficients avII attain minimum at angles 23° and (180+23)°
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0.75 0.8 0.85 0.9 0.95 1
Measurement locations, j
Magnitude
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0 1 2 3 4 5 6
Measurement locations, j
Magnitude
(a) avII (b) ahII
(a) avII (b) ahII
whereas maximum at angles (90+23)° and (270+23)°. Also, the coefficient, ahII, changes its sign at angles 23°, (90+23)°, (180+23)°, and (270+23)°.
Figure 5.6: Peak value variations in coefficients at excitation frequency of 100 rad/s for (a)
II
av and (b) ahII
Peak value variations of coefficients avII and ahII with angular position of the shaft at the ex- citation frequency of 50 rad/s is plotted in Figure 5.7. Again the angle at which the coefficient
II
av attains minimum and coefficientahII changes sign remain the same. Hence, it does not de- pend upon the excitation frequency.
Figure 5.7: Peak value variations in coefficients at excitation frequency of 50 rad/s (a) avII and (b) ahII
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0.7 0.8 0.9 1 1.1
Angular position of the shaft (degree)
Magnitude
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-6 -4 -2 0 2 4 6
Angular position of the shaft (degree)
Magnitude
0 100 200 300
0.9985 0.999 0.9995 1
Angular position of the shaft (degree)
Magnitude
0 100 200 300
-0.2 -0.1 0 0.1 0.2 0.3
Angular position of the shaft (degree)
Magnitude
(a) avII (b) ahII
Peaks in coefficients avII and ahII have highest values for the crack orientation angle of 23 , (180 23)
φ = ° + ° and the lowest value for φ =(90+23) , (270° +23)°. From Figure 5.5 (b) and Figure 5.6 (b), coefficientsahII, obtained from the shaft response in the horizontal direction, have positive values at φ =23 , (180° +23)° and negative values atφ = (90+23) , (270° +23)°. This information can be used to find out the orientation of cracks. For this, it is proposed to get the coefficients avII and ahII at regular angular positions of the shaft. This can be obtained by ei- ther rotating the shaft by small incremental angles or by rotating the exciter (by which the shaft is getting the sinusoidal excitation).
(a) (b)
Figure 5.8: Estimation of crack orientation angle ɸ. (a) ɸ = 0°, (a) ɸ ≠ 0°.
Let the shaft is rotated gradually with small increments and the shaft response is measured after each increment. If the crack front is initially aligned with the Y-axis (Figure 5.8 (a)), it would take θ = (90+23)° rotation of the shaft to get the maximum in the coefficients avIII. Whereas if the initial angular displacement of the crack front with respect to Y-axis is ɸ (Figure
5.8 (b)), it would take θ = (90+23- ɸ)° rotation of the shaft to get the minimum in coefficients
III
av . Hence if minimum of the coefficients avIII are obtained after θ rotation of the shaft, initial- ly the crack front must be at an angular position of (90+23- θ)° with respect to the Y-axis.
Since there are two maxima in the coefficients avIII, the two possible values for the initial angu- lar position of the crack would be (90+23- θ)° and (270+23- θ) °. But the shaft response for the two possible initial angular positions would be same. It is explained in Figure 5.9. For θ = 68°, the shaft responses are plotted for ɸ = (90+23- 68 = 45)° and ɸ = (270+23- 68 = 225)°. The two shaft responses are exactly same. Hence, it is expected that with known orientation of the cracks, the optimization problem (Section 4.6) will give the size and accurate locations of the cracks, since the number of variables remained the same.
Figure 5.9: The cracked shaft response for the crack orientation angle ɸ = 45° and ɸ = 225°
(the two responses are same hence they are overlapping).
Finding the orientation of cracks alone is not a very important issue, but with crack orienta- tion unknown, it takes one extra variable (per crack) to be solved to get the crack size infor- mation which is the most vital. The crack model considered for the present case is an open crack, which remains open for all degree of shaft rotation. For actual fatigue cracks, when dis-
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0 0.5 1 1.5
2x 10-3
Shaft length (m)
Magnitude
0 0.5 1
0 0.2 0.4 0.6 0.8
1x 10-4
Magnitude
Shaft length (m)
(a) Vertical response (b) Horizontal response
placements due to weight of the shaft dominates the vibration amplitude, cracks will be breath- ing cracks which open and closes depending upon its angular orientation. When cracks are ful- ly in compression region, the cracks will be fully closed and the stiffness of the shaft element containing the crack would be same as that of the intact shaft. Hence, cracks would go unde- tected by any vibration based condition monitoring system and this can be dangerous. Also when the shaft is partially open and partially closed, the effect of cracks would be less in the dynamics of the shaft motion compared to when cracks are fully open. All the symptoms like the reduction in natural frequencies and the change in model parameters would be less. Be- cause of this small change in system parameters, detection of cracks will be even more diffi- cult. Hence, it is important to get the shaft response at several angular positions. At some of the angular orientation when the crack is open, the effect of crack would be more on dynamics of the shaft which will help the condition monitoring system to find out crack parameters. Hence, taking the shaft response at several angular positions of the shaft, to get the crack orientation angle, is justified.