Crack Assessment Based on the Use of Severity-Adjusted …
501
1
2
3
4
5
6
Fig. 1 Proposed damage indicating algorithm
The ratio between the frequency shift of a bending vibration mode i and the
corresponding frequency of the healthy beam represents the RFS and is calculated
as:
¯
f i (x, a) =
f i−U − f i−D (x, a)
f i−U
= γ (0, a) · ( ¯
φ
i (x))
2
(3)
It follows from relation (3) the severity for a crack placed at the location x and
producing a known frequency drop as:
¯
f i (x, a)
( ¯
φ
i (x)) 2 = γ (0, a)·
(4)
This finding is used in [9] to assess single cracks in beams. By employing relation 4, we can plot the PSC that considers the RFS and the SNMC. Obviously, the
intersection of the PSC plotted for more vibration modes indicates the crack location,
because here the combination of the SNMC and the damage severity fits the RFS for
all considered modes. Finally, the severity can be found directly from the graphical
representation or by involving relation (4). The algorithm is presented in Fig. 1.
2.2 The Simulation Scheme
For the proposed study, we have considered a beam both in healthy and in damaged
states, with different end constraints. The geometry was modeled with the help of
501
1
2
3
4
5
6
Fig. 1 Proposed damage indicating algorithm
The ratio between the frequency shift of a bending vibration mode i and the
corresponding frequency of the healthy beam represents the RFS and is calculated
as:
¯
f i (x, a) =
f i−U − f i−D (x, a)
f i−U
= γ (0, a) · ( ¯
φ
i (x))
2
(3)
It follows from relation (3) the severity for a crack placed at the location x and
producing a known frequency drop as:
¯
f i (x, a)
( ¯
φ
i (x)) 2 = γ (0, a)·
(4)
This finding is used in [9] to assess single cracks in beams. By employing relation 4, we can plot the PSC that considers the RFS and the SNMC. Obviously, the
intersection of the PSC plotted for more vibration modes indicates the crack location,
because here the combination of the SNMC and the damage severity fits the RFS for
all considered modes. Finally, the severity can be found directly from the graphical
representation or by involving relation (4). The algorithm is presented in Fig. 1.
2.2 The Simulation Scheme
For the proposed study, we have considered a beam both in healthy and in damaged
states, with different end constraints. The geometry was modeled with the help of