Steps in a shaft are provided to fulfill different requirements such as mounting the gears, sprockets, pulleys and bearings. This causes stress concentration in the shaft near the steps.
Occurrence of cracks is more favorable near the region of stress concentration. This necessi- tates the study of detecting cracks in stepped shaft, especially the situation when crack is near a step, since these (step) locations are more favorable for the occurrence of cracks.
0 5 10 15 20 25 30
35 35.2 35.4 35.6
Generation, i Mean value of β2
0 5 10 15 20 25 30
0.6 0.62 0.64 0.66 0.68 0.7 0.72
Generation, i Mean value of β4
(c) Second crack location, β2 (d) Second crack size, β4
Figure 5.22: A stepped shaft with two cracks.
In a numerically simulated example, a stepped shaft (shown in Figure 5.22) of 0.01m diame- ter and 1 m length is considered. Although the MCDLA (Section 3.2) works with responses measured at several frequencies, for explaining the working of the algorithm for stepped shaft, initially a single frequency forcing is used for the shaft excitation and also the measurement noise is not considered. Next, the identification examples are taken with responses at several frequencies and noise added in it (to mimic the actual experimentation).
In the first example, a stepped shaft is considered with a single crack. The step in the shaft is considered near the 7th measurement location while the crack is taken near the 12th measure- ment location. Hence, for this case the crack is located far from the step. The size of step is given in Figure 5.22. The crack depth ratio and the crack orientation angle for the crack are taken to be α1 =0.6 and φ1 = °0 , respectively. Coefficients acvI and awcvI , obtained at 50 Hz of excitation frequency, are plotted in Figure 5.23(a). Normalized coefficients, avII, are plotted in Figure 5.23(b). It is evident from Figure 5.23(a) that the effect of step is not there in the nor- malized coefficients. From Figure 5.23(a), since the slope discontinuity at the location of step is almost same in both the shafts (intact as well as the cracked) its effect is canceled during normalization. But at the location of the crack, only the cracked shaft has the slope discontinui- ty which gives a peak in the normalized coefficients at the location of crack. For the applica-
tion of the algorithm to real cracked shaft, presence of step in the shaft would be known; hence it can be taken into account while modeling the shaft for getting the intact shaft response.
(a) Coefficients acvI (-) and awcvI (--) (b) coefficients, avII
Figure 5.23: Quadratic coefficients for the first example (step near 7th location, one crack of crack depth ratio 0.6 near the 11th measurement location).
In the second example, two cracks are considered. While keeping all other parameters same as in the above example, the second crack is chosen to be very near to the step. Resulting val- ues of coefficients acvI and aIwcv are plotted in Figure 5.24(a), and normalized coefficients avII are plotted in Figure 5.24(b). For the crack located far from the step, the normalization is simi- lar to previous example. For the crack near the step, the slope discontinuity in both shafts are not same and hence the slope discontinuity of the cracked shaft is not canceled out completely while normalization. Hence, it gives a peak near the location of the crack. In the above two ex- amples the crack orientation angles for the two cracks are taken to be zero. Next, the above two examples are presented with measurement noise added in the shaft response.
2 4 6 8 10 12 14 16 18 20
3 4 5 6 7 8 9x 10-4
Measurement locations, j
Amplitude
2 4 6 8 10 12 14 16 18 20
0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
Measurement locations, j
Amplitude
(a) Coefficients acvI (-) and awcvI (--) (b) coefficients, avII
Figure 5.24: Quadratic coefficients for the second example (step near 7th location, two cracks of crack depth ratios 0.6 each and located near the 7th and11th measure- ment location.
In Figure 5.23 and Figure 5.24, a single excitation frequency is used for explaining the working of the algorithm. In actual case, the measurement of shaft responses would be contam- inated with measurement noise. Hence, to mimic the actual experimentation, 1% noise (Ap- pendix D) is added in the shaft response. In presence of measurement noise, the presence of cracks is not evident from the plot of normalized coefficients at a single frequency. It is evident from the plot of normalized coefficients in Figure 5.25. Hence, the identification algorithm us- es shaft response at several frequencies to reduce the effect of noise in the measurements. Now the crack probability functions are obtained using several excitation frequencies (5, 10, …, 110 Hz).
2 4 6 8 10 12 14 16 18 20
3 4 5 6 7 8
x 10-4
Measurement locations, j
Amplitude
2 4 6 8 10 12 14 16 18 20
0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
Measurement locations, j
Amplitude
(a) Coefficients, avII for the first example with one crack.
(b) Coefficients, avII for the second example with two cracks.
Figure 5.25: Normalized quadratic coefficients for (a) the first example (b) the second ex- ample.
Crack probability functions for the first example (crack near measurement location 12 and step near the measurement location 7) are given in Figure 5.26(a). The crack orientation angle for the crack is taken to be 10°. For the second example (first crack near the measurement loca- tion 7 and the second near the measurement location 12 with step in the shaft near the meas- urement location 7) is given in Figure 5.26(b). Crack orientation angles for the first and the se- cond crack are taken to be 10° and 45°, respectively. For both the cases the identification algorithm identifies the presence of cracks properly.
2 4 6 8 10 12 14 16 18 20
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Measurement locations, j
Amplitude
2 4 6 8 10 12 14 16 18 20
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Measurement locations, j
Amplitude
(a) CPF for the with one crack. (b) CPF for the second example with two cracks.
Figure 5.26: Crack probability functions for (a) first example (b) second example.
The MCDLA presented in Section 3.2 is based on finding the slope discontinuity in the elas- tic line of the shaft, caused by the presence of crack in the shaft. Although the presence of step in the shaft is also producing a slope discontinuity but its effect is neutralized during normali- zation because of similar slope discontinuity in the intact shaft. The special normalization technique used in the algorithm is capable of keeping only that the slope discontinuity which is arising from the crack and it is in contrast with techniques based upon the wavelet transform which will show a slope discontinuity at the location of step also.
Up to now, in Chapters 3, 4, and 5, the MCDLA (Section 3.2) and the MCLSA (the optimi- zation problem, Section 4.6) are tested with the response of a cracked shaft obtained from the numerical simulation (FE modeling of the cracked shaft). The measurement noise is added in the shaft response to mimic the actual shaft response. In the next chapter, working of the MCDLA is tested with the measurements of the forced response from a real shaft. The proce- dure of experimentation on the real shaft and the results are presented in next chapter.
2 4 6 8 10 12 14 16 18 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Measurement locations, j
Amplitude
2 4 6 8 10 12 14 16 18 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Measurement locations, j
Amplitude