Signature analysis of cracked cantilever beam

International Journal of Advanced in Engineering and Technology (IJARET)
International Journal of Advanced Research Research in Engineering
and Technology (IJARET), ISSN 0976 – 6480(Print) IJARET
ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME
ISSN 0976 – 6499(Online) Volume 1
Number 1, May - June (2010), pp. 105-117 © IAEME
© IAEME, http://www.iaeme.com/ijaret.html

SIGNATURE ANALYSIS OF CRACKED CANTILEVER
BEAM
Sharad V. Kshirsagar
Asst. Professor, Mechanical Engineering Department
Sinhgad College of Engineering
Pune, E-mail: sharadkshirsagar@gmail.com

Dr. Lalit B. Bhuyar
Mechanical Engineering Department
Prof. Ram Meghe Institute of Technology & Research
Badnera, Maharashtra

ABSTRACT
Beams are more widely used in the machine-structures. Fatigue-type of loading of
such engineering parts is likely to introduce cracks at the highly stressed regions and lead
to damage and deterioration during their service life. Cracks are a main cause of
structural failure. Once a crack is initiated, it propagates and the stress required for
propagation is smaller than that required for crack initiation. After many cycles operating
stresses may be sufficient to propagate the crack. The crack propagation takes place over
a certain depth when it is sufficient to create unstable conditions and fracture take place.
The sudden failure of components is very costly and may be catastrophic in terms of
human life and property damage. Forced vibration analysis of a cracked cantilever beam
was carried out and the results are discussed in this paper. An experimental setup was
designed in which a cracked cantilever beam excited by an exciter and the signature was
obtained using an accelerometer attached to the beam. To avoid non-linearity, it was
assumed that the crack remain always open.
Keywords: Crack detection, forced vibrations, signature analysis.
1. INTRODUCTION
Literature on Fault detection and condition monitoring was focused on the
vibration-based method which can be classified into modal-based and signature-based
methods. In modal based techniques data can be condensed from the actual measured

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International Journal of Advanced Research in Engineering and Technology (IJARET)

quantities like resonant frequencies, mode shape vectors and quantities derived from
these parameters for the crack detection [1, 3, 4, 6].
In signature based methods the vibration signature of cracked machinery structure
can be useful for the fault diagnosis and condition monitoring. Thus, the development of
crack detection methods has received increasing attention in recent years. Among these
techniques, it is believed that the monitoring of the global dynamics of a structure offers
favorable alternative if the on-line (in service) damage detection is necessary. In order to
identify structural damage by vibration monitoring, the study of the changes of the
structural dynamic behavior due to cracks is required for developing the detection
criterion. [2, 5, 7-13].
2. GOVERNING EQUATIONS OF FORCED VIBRATION
The equation of motion for the beam element without crack can be written as
follows from [14]:

-------------------------------- (1)
(e ) (e )
where [M ] is the element mass matrix, [K wc ] is the element stiffness matrix,

{F (t )}(e ) is the element external force vector, {q(t )}
(e )
is the element vector of nodal

degree of freedoms and t is the time instant. The subscript wc represents without crack,
the superscript e represents element and dot represents the derivative with respect to the
time. The crack is assumed to affect only the stiffness. Hence the equation of motion of a
cracked beam element can be expressed as

------------------------------- (2)
(e )
where {qc (t )} is the nodal degrees of freedom of the cracked element, the
(e )
subscript c represents the crack and [K c ] is the stiffness matrix of the cracked element
and is given as
−1
[K c ](e ) = [T ][C ](e ) [T ]T ------------------------------------------- (3)
with [C ](e ) = [C0 ](e ) + [Cc ](e ) ------------------------------------------ (4)

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(e ) (e )
where [C0 ] is the flexibility matrix of the uncracked beam element, [Cc ] is the
(e )
flexibility matrix of the crack, and [C ] is the total flexibility matrix of the cracked
beam element.
Equations of motion of the complete system can be obtained by assembling the
contribution of all equations of motion for cracked and uncracked elements in the system.
Then the system equation of motion becomes

--------------------------------------- (5)
where [M ] is the assembled mass matrix, [K ] is the assembled stiffness matrix,
{F (t )} is the assembled external force vector, and {q(t )} is the assembled vector of nodal
dofs of the system.
Let the force vector be defined as

{F (t )} = {F }e jwt , ----------------------------------------------- (6)
{}
where w is the forcing frequency, F is the force amplitude vector (elements of
which are complex quantities) and j = −1 . Thus, the response vector can be assumed as

{q(t)} ={q}ejwt, ------------------------------------------------ (7)
{}
where q is the response amplitude vector and their elements are complex
quantities. Using Eqs. (6) and (7) for modal frequency, the system governing equation as
follows:
(− w [M ] + [K ]){q}= {F }. ------------------------------------------ (8)
2

For a given system properties (i.e. [M ] and [K ] the response can be simulated

from Eq. (8) corresponding to a given force F . {}
3. SIMULATION
In the finite element simulation, a cantilever beam with rectangular edge crack is
considered. The length and cross-sectional area of the beam are 800 mm, and 50x6 mm2,
respectively. As for the material properties the modulus of elasticity (E) is 0.675 1011
N/m2, the density (ρ) 27522.9 kg/m3 and the Poisson’s ratio (µ ) is 0.33.

107


3.1. Generation of Cracked Beam Model
A 8-node three-dimensional structural solid element under SOLID 45 was
selected to model the beam. The beam was discretized into 11859 elements with 54475
nodes. Cantilever boundary conditions modeled by constraining all degrees of freedoms
of the nodes located on the left end of the beam. APDL PROGRAMMING is used to
create 135 cracked beam models by varying the crack depth from 5 mm to 45 mm and
crack location from 50 mm to 750 mm. Figure 1 show the finite element mesh model of
the beam generated in Ansys (12).

Figure 2 Finite element mesh model.
3.2 Harmonic Analysis
Full Solution Method, Reduced Solution Method, Mode superposition Method are
the methods to be used to solve the harmonic equation.
Mode Superposition Method is used to solve in the current analysis.
[M] {ü} + [C] {ú} + [K] {u} = {Fa} __________________________(9)
where:
[M] = structural mass matrix
[C] = structural damping matrix
[K] = structural stiffness matrix
{ü} = nodal acceleration vector
{ú} = nodal velocity vector
{u} = nodal displacement vector

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{Fa} = applied load vector
All points in the structure are moving at the same known frequency, however, not
necessarily in phase. Also, it is known that the presence of damping causes phase shifts.
Therefore, the displacements may be defined as:
{u} = {umax ei Φ }ei t --------------------------------------------- (10)
where:
umax = maximum displacement
i = square root of -1
= imposed circular frequency (radians/time) = 2πf
f = imposed frequency (cycles/time)
t = time
Φ = displacement phase shift (radians)
{ Fa } = {Fmax ei ψ }ei t-------------------------------------------(11)
where:
Fmax = force amplitude
ψ = force phase shift (radians)
The dependence on time (ei t) is the same on both sides of the equation and may
therefore be removed. Figure 2 show the boundary condtions for harmonic analysis.

Figure 2 Boundary conditions for harmonic analysis.

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4. EXPERIMENTATION
A number of carefully designed experiments were carried out on a Cantilever
Beam. Figure 3 shows the components of this experimentation. Vibration signals were
collected for both uncracked and several cracked beam conditions. The excitation
frequency was set at approximately 40 Hz. With the sensor mounted on the beam at free
end, vibration signals were measured for various fault conditions by on-line monitoring
when beam was under stationary excitation. Table 1 show the comparison of the
experimental results with the simulated results.

Figure 3 Experimental setup
Table 1 Comparison of simulated and experimental results
Crack Crack Mode 1 Mode 2 Mode 3
case C/L a/h Simulated Expt. Simulated Expt. Simulated Expt.
0.1 0.9901 1 0.9945 1 0.9973 1
1 1/16
0.2 0.9614 0.97 0.9792 0.9867 0.9899 0.9904
0.1 0.9935 1 0.9999 1 0.9987 1
2 3/16
0.4 0.8929 0.9118 0.9972 1 0.9769 0.9856
0.3 0.9636 0.9708 0.9863 0.99 0.9589 0.9604
3 5/16
0.4 0.9315 0.9433 0.9747 0.9780 0.9265 0.9394
0.2 0.9917 0.9987 0.9804 0.9890 0.9962 1
4 7/16
0.3 0.9805 0.9898 0.9559 0.9623 0.9912 1
0.2 0.9944 1 0.9767 0.9901 0.9999 1
5 8/16
0.5 0.9537 0.9611 0.8477 0.8602 0.9986 1
0.2 0.9990 1 0.9857 0.9945 0.9753 0.9790
6 11/16
0.4 0.9951 1 0.9363 0.9456 0.9046 0.9200
7 14/16 0.5 1 1 0.9917 0.9989 0.9514 0.9654

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5. RESULTS AND DISCUSSIONS
Before the experiments were carried out, the first three natural frequencies of the
beams were simulated by FEA. From the results obtained, it was decided that using a
frequency range upto 1.2 kHz for experimental measurements would be sufficient to
include the first three natural frequencies.
The frequency response functions obtained were curve-fitted. The simulated data
from the curve-fitted results were tabulated and plotted in the form of frequency ratio
(ratio of the natural frequency of the cracked beam that of the uncracked beam) versus
the crack depth ratio (a/h) [the ratio of the depth of a crack (a) to the thickness of the
beam (h)] for various crack location ratios (C/L) (ratio of the location of the crack to the
length of the beam).
Figure 4 to 6 show the plots of the first three frequency ratios as a function of
crack depths for some of the crack positions considered for each set of boundary
conditions (fifteen locations for each set of boundary conditions). Figure 7 to 9 shows the
frequency ratio variation of three modes in terms of crack position for various crack
depth.

1
1
Frequency Ratio

0.9 2
1st Mode

0.8 3
0.7 4
0.6 6
5
0.5
0.1 0.3 0.5 0.7 0.9
Crack Depth Ratio

Figure 4 Fundamental natural frequency ratio in terms of crack depth for various crack
positions C / l (1→15/16; 2→9/16; 3→7/16; 4→5/16; 6→2/16).

111


1
1
0.9 2
3

Frequency Ratio
0.8

2nd Mode
0.7 4
0.6 5
0.5
0.4 6
0.3
0.1 0.3 0.5 0.7 0.9
Crack Depth Ratio

Figure 5 Second natural frequency ratio in terms of crack depth for various crack
Positions C / l (1→15/16; 2→2/16; 3→5/16; 4→6/16; 5→7/16; 6→10/16).

1 1
0.9 2
Frequency Ratio
3rd Mode

0.8
0.7 3
5
0.6 4
0.5
0.4
0.1 0.3 0.5 0.7 0.9
Crack Depth Ratio

Figure 6 Third natural frequency ratio in terms of crack depth for various crack positions
C / l (1→2/16; 2→15/16; 3→9/16; 4→14/16; 5→13/16).

1.00
Frequency Ratio

0.80
1st Mode

0.60
0.40
0.20
0.0625 0.3125 0.5625 0.8125
Crack Location Ratio
0.1 0.3 0.5 0.7 0.9

Figure 7 1st Mode frequency ratio in terms of crack position for various crack depths.

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Frequency Ratio
0.90

2nd Mode
0.70

0.50

0.30
0.0625 0.3125 0.5625 0.8125
0.1 0.3 0.5 0.7 0.9

Figure 8 2nd Mode frequency ratio in terms of crack position for various crack depths.
Frequency Ratio

0.90
3rd Mode

0.70

0.50

0.30
0.0625 0.3125 0.5625 0.8125
0.1 0.3 0.5 0.7 0.9

Figure 9 3rd Mode frequency ratio in terms of crack position for various crack depths.

From the results and plots the following observations were made for al1 the cases
considered:
i. Natural frequencies were reduced due to presence of crack.
ii. Effects of cracks were high for the small values of crack location ratio
iii. The second natural frequency was greatly affected at the C / l = 11/16 for all crack
depths.
iv. The third natural frequency was almost unaffected for the crack locations ( C / l = 2/16
and 8/16); the reason for this influence was that the location of nodal point was located
at that point on the beam.
v. Due to shifts in the nodal positions (as a consequence of cracking) of the second and
the third modes, the changes in the higher natural frequencies depended on how close
the crack location was to the mode shape nodes. Consequently, it was be observed from

113


the results that the trend of changes in the second, and the third frequencies are not
monotonic, as we have in the first natural frequency.
vi. From the results obtained, it is observed, for example, that when the crack depth ratio is
0.9, the third natural frequency was comparatively much less affected than the first and
second frequencies for a crack located at C / l =8/16 but, it is highly affected for other
crack locations. This could be explained by the fact that decrease in frequencies is
greatest for a crack located where the bending moment is greatest. It appears therefore
that the change in frequencies is not only a function of crack depth and crack location,
but also of the mode number.
vii. For various cases considered, the frequencies decreased rapidly with the increase in the
crack depths for all three modes. As stated earlier, the decrease in the fundamental
natural frequency was greatest when the crack occurred closer to the fixed point. This
could be explained by the fact that the bending moment was the largest at that point
(where the amplitude of the first mode shape is greatest) for the first mode, thereby,
resulting in a greater loss of bending stiffness due to crack. However, the second and
third modes were less affected at this location. The frequencies decreased by about
23.7% & 10.2%, and 49.7% for the first, second and third modes, respectively, as the
crack grew to half of the beam depth (for crack at 50 mm from fixed end).

35

30

25
Area Ratio

20

15

10

5

0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Crack Depth
0.0625 0.2500 0.3125 0.5000 0.5625

Figure 10 Area under the frequency response curve as a function of crack position.

114


14
12
10

Area Ratio 8
6
4
2
0
0.0625 0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375
Crack Location

0.4 0.5 0.7 0.9

Figure 11 Area under the frequency response curve as a function of crack depth.
CONCLUSIONS
Based on the experimental data, and plots, and the observations above, numerous
inferences could be made such as follows:
a) For of the cases considered, the dopes of frequency ratio versus crack depth curves
were very small for small crack depth ratios. This implies that small cracks have
little effects on the sensitivities of natural frequencies. Hence, using only results
based on frequency changes alone for identifying cracks in most practical problems
may be misleading as it is very unlikely to have large cracks.
b) For a particular mode, the decrease in frequency and change in mode shape become
noticeable as the crack grew bigger.
c) For a given crack depth ratio, the location of the crack greatly affects the dynamic
response of the cracked beam.
d) Investigating the mode of vibration at some crack location may indicate a pure
bending mode for small crack depth ratios, but, as the crack grows in size. The
ending mode may contain a significant influence of longitudinal vibration mode
also (occurrence of coupling).
e) Fatigue crack alters the local stiffness which changes dynamic response. From Fig.
10 -11 it is seen that area under the frequency response curve can be used as one of
the elements of crack detection.

115


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Signature analysis of cracked cantilever beam

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