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Superposition Principle applied to Natural Frequency Changes
of a Beam with Multiple Cracks
ZENO-IOSIF PRAISACH
“Eftimie Murgu” University of Resita
P-ta Traian Vuia 1-4, 320085 Resita
ROMANIA
GILBERT-RAINER GILLICH
“Eftimie Murgu” University of Resita
P-ta Traian Vuia 1-4, 320085 Resita
ROMANIA
[email protected] http://www.uni-resita.eu/FH-Sites/uem/index.php?id=1556
PENTRU-FLORIN MINDA
“Eftimie Murgu” University of Resita
P-ta Traian Vuia 1-4, 320085 Resita
ROMANIA
NICOLETA GILLICH
“Eftimie Murgu” University of Resita
P-ta Traian Vuia 1-4, 320085 Resita
ROMANIA
Abstract: - The paper presents the phenomenon that appears in beams with multiple damages in respect to natural
frequencies. Damaged beams with different boundary conditions were analyzed by the authors, both in analytical way
and using the finite element method. The considered damages are open cracks, affecting the whole width of the beam
and have various levels of depth. In this paper the case of a cantilever beam with multiple damages is presented; the
results were compared with those obtained by experiments. The investigation lead to the conclusion that the
superposition principle can be used to find out the frequency changes of a beam with multiple cracks, when the effect
of each crack is known.
Key-Words: - vibration, natural frequency, finite element method, superposition principle, damages
1 Introduction
Damage detection using natural frequency shifts is
largely presented in literature [1], [2], [3] and [4]. The
methods based on frequency change can be classified in
two groups: methods limited to damage detection and
methods destined to detect, locate and quantify damages.
Literature reviews, [3] and [5], affirm that all methods
based on natural frequency changes belong to the second
group are model-based, typically relying on the use of
finite element models. These models can be categorized
into three main categories: local stiffness reduction;
discrete spring models; and complex models [6].
Damages influence the dynamic behavior of
structures, changing their mechanical and dynamic
characteristics such as natural frequencies, mode shapes,
damping ratio, and stiffness or flexibility. These most
common features that are used in damage detection are
identified from measured response time-histories (most
often accelerations or strains) or spectra of these time-
histories [7], [8] and [9].
It is known that all approaches fit particular cases,
given by boundary conditions or by damage location.
Our research intended to find a general method, able to
detect, locate and evaluate the severity of open cracks in
all types of beams.
Recent Advances in Signal Processing, Computational Geometry and Systems Theory
ISBN: 978-1-61804-027-5 233
While the aim of the research was to develop a
simple method to detect, localize and quantify the
severity of damages with the least equipment possible,
bending vibrations were considered. In our research we
have studied the natural frequency changes due to
damages on cantilever, simple supported and double
clamped beams, for a larger number of vibration modes.
In previous papers the analysis was performed using
analytic calculus and the finite element method (FEM),
for the undamaged beam and for the beam having one
crack placed one-by-one in a lot of locations along it
[10] and [11].
2 Numeric and analytic investigation The real analyzed beam was a steel one, having the
following geometrical characteristics: length L = 1000
mm; width B = 50 mm and height H = 5 mm.
Consequently, for the undamaged state the beam has the
cross-section A = 250·10-6 m2 and the moment of inertia
I = 520.833·10-12
m4. The material parameters of the
specimens are: mass density ρ = 7850 kg/m3; Young’s
modulus E = 2.0·1011
N/m2 and Poisson’s ratio µ = 0.3.
Three types of beams were analyzed: cantilever
beam, double clamped beam and simple supported
beam. Figure 1 presents the cantilever beam in damaged
state, while figure 2 and 3 present the double clamped
and simple supported beam respectively.
Fig. 1. Cantilever beam
Fig. 2. Double clamped beam
Fig. 3. Simple supported beam
The beams were analyzed, both in the undamaged
and damaged case, using analytic and the finite element
methods (FEM). The 3D beam was meshed by 2 mm
elements for the undamaged beam and the same mesh
but with finer elements in the vicinity of damage for the
damaged beams. The first ten natural frequencies of the
weak-axis bending modes for the undamaged beams
were determined; the values are presented in table 1, for
cantilever beam, in table 2 for double clamped beam and
in table 3 for simple supported beam.
Afterwards, a series of damages placed separately
one after the other on 190 locations along the whole
length of the beam were modeled. We selected
uncomplicated damage geometry, easy to reproduce on
the real structure by saw cuts, with the constant width of
2 mm and 9 levels of depth, reducing the cross-section
by 8, 17, 25, 33, 42, 50, 58, 67 and 75% respectively.
Table 1
Vibration
Mode
i
Natural
frequency
fi [Hz]
Vibration
Mode
i
Natural
frequency
fi [Hz]
1 4.08986 6 347.4518
2 25.6266 7 485.4578
3 71.7545 8 646.5624
4 140.6275 9 830.7827
5 232.5200 10 1038.1089
Table 2
Vibration
Mode
i
Natural
frequency
fi [Hz]
Vibration
Mode
i
Natural
frequency
fi [Hz]
1 26.099 6 486.421
2 71.927 7 647.735
3 140.991 8 832.150
4 233.071 9 1039.671
5 348.206 10 1270.255
Table 3
Vibration
Mode
i
Natural
frequency
fi [Hz]
Vibration
Mode
i
Natural
frequency
fi [Hz]
1 11.444 6 411.960
2 45.779 7 562.200
3 103.010 8 734.707
4 183.140 9 930.335
5 286.150 10 1149.068
For all the resulting damage cases, the first ten
natural frequencies were determined.
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ISBN: 978-1-61804-027-5 234
Figure 4 present analytical results, [10] and [12], in
form of network, for the first four modes of damaged
cantilever beams. The free end of the beam is on the left
side of the figures.
Fig. 4. Frequency shift for damaged cantilever beam
Fig. 5. Frequency shift for damaged double clamped
beam
Figure 5 present analytical results, in form of
network, for the first four modes of damaged double
clamped beams and figure 6 the analytical results for the
damaged simple supported beam.
Fig. 6. Frequency shift for damaged simple supported
beam
In these pictures (fig. 4, 5 and 6), the vertical axis
represents the natural frequency [Hz], the first horizontal
axis represents the length of the beam [%] and the
second horizontal axis represents the depth of the
damage [%].
3 Description of the method For all ten vibration mode, at each location of the
damage and for each level of damage depth we consider
the relative shift in frequency, given with formula (1):
100),(
),( ⋅∂−
=∂∆
U
iUi
f
xffxf [%] (1)
where,
Uf - represent the natural frequency for the
undamaged beam;
ixf ),( δ - represent the natural frequency for the
damaged beam, with damage at location x and damage
depth δ, for vibration mode No. i.
Recent Advances in Signal Processing, Computational Geometry and Systems Theory
ISBN: 978-1-61804-027-5 235
Representing the relative shift in frequency versus
vibration mode i, is obtained the relative shift in
frequency tendency, or the “genetic algorithm” which
permits to localize and assess the damage on the beam.
Fig. 7. Relative shift in frequency tendency
Considering a cantilever beam with a damage located
at 0.5L (L is the length of the beam) from the clamped
end, the damage having the depth 0.5h (h is the height of
the beam), the relative shift in frequency for the first ten
vibration modes can be represented like in the figure 7.
Fig. 8. Relative shift in frequency tendency
Then, considering a cantilever beam with a damage
located at 0.892L from the clamped end and a depth
0.5h, the relative shift in frequency for the first ten
vibration modes can be represented like in the figure 8.
4 Superposition principle We consider a cantilever beam having simultaneously
two damages like that presented in the section above. By
using the FEM there are obtained the natural frequencies
for the first ten vibration modes. Afterwards, using
relation (1) one can obtain the relative shift in frequency
for those modes.
Figure 9 represents the relative shift in frequency
tendency for cantilever beam with damage at 0.5L
(continuous black line), relative shift in frequency
tendency for cantilever beam with damage at 0.892L
(dashed gray line), relative shift in frequency tendency
for cantilever beam with the two damages (continuous
gray line) and overlapping line (dot black line) that
represents the sum of relative shift in frequency
tendency for damage located at 0.5L and 0.892L.
It is obvious that the FEM analysis for the beam with
two damages present the same the shift in frequency like
the sum of the shifts in frequency for the beam having
the two damages independently. Consequently, the
superposition principle can be used.
Fig. 9. Superposition principle applied on the cantilever
beam with two damages
In the second example it is considered a cantilever
beam with three damages, located as follows: the first
damage located at 0.77L from the clamped end, having a
depth of 0.25h, the second damage located at 0.351L
having a depth of 0.5h and third damage located at 0.5L
from the clamped having a depth of 0.5.h.
Fig. 10. Relative shift in frequency tendency
Recent Advances in Signal Processing, Computational Geometry and Systems Theory
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The relative shift in frequency tendency for the
cantilever beam with damage at 0.77L and 0.25h depth
is presented in figure 10.
The relative shift in frequency tendency for the
cantilever beam with damage at 0.351L and 0.5h depth
is presented in figure 11, and the relative shift in
frequency tendency for the cantilever beam with damage
at 0.5L and 0.5h depth is presented in figure 7.
Fig. 11. Relative shift in frequency tendency
Considering a cantilever beam with three damages,
located in the same position and the same damage depth
as the above mentioned, by using FEM analyses there
are obtained the natural frequencies for the first ten
vibration modes. Using relation (1) one can obtain the
relative shift in frequency for this modes.
Representing the relative shift in frequency tendency
(fig. 12) for the FEM analyzed beam with three damages
(continuous black line) and making the sum of the
relative frequency shift of each of the three damage
cases (dashed black line) it can be observed that two
lines overlaps.
Fig. 12. Superposition principle applied on the cantilever
beam with three damages
This is a new confirmation that the superposition
principle can be involved in the analysis of beams with
multiple cracks.
4 Conclusion The method presented in the paper, applicable to beams
with open cracks, is based on certain phenomena
characteristic to the dynamic behaviour of beams,
highlighted as a result of several analytical, numerical
and experimental studies developed by the authors.
Analyzing the figures 4, 5 and 6 it can be observed
that, for all types of supporting beams, at each vibration
mode, there are locations on the damaged beam that
natural frequency remains unchanged, irrespective of the
damage depth, respectively the relative shift in
frequencies are zero. These locations correspond to the
inflexion points (IP) from the mode shape, presented in
figure. 13.
Fig. 13. Fifth vibration mode, double clamp beam. Mode
shape's critical point versus the frequencies changes for
the damaged beam
This conclusion is valid also for the case of multiple
cracks, by applying the superposition principle. For
example, in the up mentioned case, with the beam with
three damages (fig. 12), at vibration mode 7, the relative
shift in frequency has zero value because in all the three
cases, at seventh vibration mode, with beam with single
damage at 0.77L from clamped end (fig. 10) and damage
at 0.351L from clamped end (fig. 11), and damage at
0.5L from clamped end (fig. 7), the relative shift in
frequency has zero value.
Analyzing figures 4, 5 and 6 it can be observed that,
for beams with all types of support, at each vibration
mode, there are locations on the damaged beam where
natural frequency exhibits local minima, amplified by
the depth of the damage, respectively the relative shift in
frequencies have local maxima.
These locations correspond to the maximum points
(MP) and minimum points (mP) from the mode shape,
like it is presented in figure 13.
Recent Advances in Signal Processing, Computational Geometry and Systems Theory
ISBN: 978-1-61804-027-5 237
However, using the superposition principle, it s
possible to describe the dynamic behavior of the beam
with multiple damages, in terms of the changes in
frequency for several vibration modes.
Acknowledgements The authors gratefully acknowledge the support of the
Managing Authority for Sectoral Operational
Programme for Human Resources Development
(MASOPHRD), within the Romanian Ministry of
Labour, Family and Equal Opportunities by co-financing
the project “Excellence in research through postdoctoral
programmes in priority domains of the knowledge-based
society (EXCEL)” ID 62557 and “Investment in
Research-innovation-development for the future
(DocInvest)” ID 76813.
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Recent Advances in Signal Processing, Computational Geometry and Systems Theory
ISBN: 978-1-61804-027-5 238