500
G.-R. Gillich et al.
based on natural frequency measurements have proven to be reliable and economically feasible for detecting, locating and evaluating damages [2–4]. The main argument is that the presence of cracks has a negative impact on the stiffness, decreasing
the structure’s capacity to store energy and changing its dynamic properties [5].
Cracks that are present in structures have a transversal form, but can become more
complex as they branch and propagate through the material [6]. Studies have shown
that every crack has its own signature, so it is possible to determine the location of
a crack by knowing the frequency drop it produces for several bending vibration
modes and the modal energy distribution, i.e., the modal curvature evolution along
the beam.
The aim of the present paper is to study the precision of detecting and locating
cracks by plotting curves resulted from the RFS obtained by measurements that are
adjusted with the SNMC for several vibration modes. The resulted curves are, in fact,
the PSC plotted for the severity of the real damage. It is shown in this paper that the
intersection of such curves belonging to the first four out-of-plain vibration modes
accurately indicates the crack position and severity for transverse cracks.
2 Materials and Methods
2.1 The Developed Method to Assess Cracks
The model predicting the frequency of the bending modes for a beam with a transverse
crack, f i-D , developed in previous studies [7], is defined by the frequency of the
healthy beam f i-U , a term ( ¯
φ
i (x))
2 that is the square of the normalized modal
curvature (SNMC) and the damage severity γ (0, a). The relation of the damaged
beam’s frequency takes the form:
f i−D (x, a) = f i−U
1 − γ (0, a) · ( ¯
φ
i (x))
2
(1)
The SNMC for a crack located at distance x from the left end of the beam indicates
the energy stored by the healthy beam in an infinitesimal slice dx in which the crack is
positioned. The SNMC is different for the different vibration modes and constraints.
The damage severity γ (0, a) is the function representing the highest stiffness
decrease due to a crack of depth a, which for the fixed-free as well as for the fixedfixed beams occurs for the crack located at the fixed end. It can be derived using the
beam deflection in healthy and damaged state and is similar, for a given beam cross
section and damage, for all boundary conditions [8].
The frequency decrease is found from the following mathematical relation:
f i = f i−U − f i−D (x, a) = f i−U · γ (0, a) · ( ¯
φ
i (x))
2
(2)
G.-R. Gillich et al.
based on natural frequency measurements have proven to be reliable and economically feasible for detecting, locating and evaluating damages [2–4]. The main argument is that the presence of cracks has a negative impact on the stiffness, decreasing
the structure’s capacity to store energy and changing its dynamic properties [5].
Cracks that are present in structures have a transversal form, but can become more
complex as they branch and propagate through the material [6]. Studies have shown
that every crack has its own signature, so it is possible to determine the location of
a crack by knowing the frequency drop it produces for several bending vibration
modes and the modal energy distribution, i.e., the modal curvature evolution along
the beam.
The aim of the present paper is to study the precision of detecting and locating
cracks by plotting curves resulted from the RFS obtained by measurements that are
adjusted with the SNMC for several vibration modes. The resulted curves are, in fact,
the PSC plotted for the severity of the real damage. It is shown in this paper that the
intersection of such curves belonging to the first four out-of-plain vibration modes
accurately indicates the crack position and severity for transverse cracks.
2 Materials and Methods
2.1 The Developed Method to Assess Cracks
The model predicting the frequency of the bending modes for a beam with a transverse
crack, f i-D , developed in previous studies [7], is defined by the frequency of the
healthy beam f i-U , a term ( ¯
φ
i (x))
2 that is the square of the normalized modal
curvature (SNMC) and the damage severity γ (0, a). The relation of the damaged
beam’s frequency takes the form:
f i−D (x, a) = f i−U
1 − γ (0, a) · ( ¯
φ
i (x))
2
(1)
The SNMC for a crack located at distance x from the left end of the beam indicates
the energy stored by the healthy beam in an infinitesimal slice dx in which the crack is
positioned. The SNMC is different for the different vibration modes and constraints.
The damage severity γ (0, a) is the function representing the highest stiffness
decrease due to a crack of depth a, which for the fixed-free as well as for the fixedfixed beams occurs for the crack located at the fixed end. It can be derived using the
beam deflection in healthy and damaged state and is similar, for a given beam cross
section and damage, for all boundary conditions [8].
The frequency decrease is found from the following mathematical relation:
f i = f i−U − f i−D (x, a) = f i−U · γ (0, a) · ( ¯
φ
i (x))
2
(2)