Signature analysis of cracked cantilever beam

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

ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 1, Number 1, May - June (2010), © IAEME

105

SIGNATURE ANALYSIS OF CRACKED CANTILEVER

BEAM

Sharad V. Kshirsagar

Asst. Professor, Mechanical Engineering Department

Sinhgad College of Engineering

Pune, E-mail: [email protected]

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

International Journal of Advanced Research in Engineering

and Technology (IJARET), ISSN 0976 – 6480(Print)

ISSN 0976 – 6499(Online) Volume 1

Number 1, May - June (2010), pp. 105-117

© IAEME, http://www.iaeme.com/ijaret.html

IJARET

© IAEME



106

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)

where [ ]( )eM is the element mass matrix, [ ]( )e

wcK is the element stiffness matrix,

( ) ( )etF is the element external force vector, ( ) ( )e

tq 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)

where ( ) ( )e

c tq is the nodal degrees of freedom of the cracked element, the

subscript c represents the crack and [ ]( )e

cK is the stiffness matrix of the cracked element

and is given as

[ ]( ) [ ][ ]( ) [ ]Tee

c TCTK1−

= ------------------------------------------- (3)

with [ ]( ) [ ]( ) [ ]( )e

c

eeCCC += 0 ------------------------------------------ (4)



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where [ ]( )eC0 is the flexibility matrix of the uncracked beam element, [ ]( )e

cC is the

flexibility matrix of the crack, and [ ]( )eC 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,

( ) tF is the assembled external force vector, and ( ) tq is the assembled vector of nodal

dofs of the system.

Let the force vector be defined as

( ) ,jwteFtF = ----------------------------------------------- (6)

where w is the forcing frequency, F is the force amplitude vector (elements of

which are complex quantities) and 1−=j . Thus, the response vector can be assumed as

( ) ,jwteqtq = ------------------------------------------------ (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:

[ ] [ ]( ) .2 FqKMw =+− ------------------------------------------ (8)

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/m

3 and the Poisson’s ratio (µ) is 0.33.



108

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 Ωt --------------------------------------------- (10)

where:

umax = maximum displacement

i = square root of -1

Ω= imposed circular frequency (radians/time) = 2πf

f = imposed frequency (cycles/time)

t = time

Φ = displacement phase shift (radians)

Fa = Fmax ei ψ ei Ωt-------------------------------------------(11)

where:

Fmax = force amplitude

ψ = force phase shift (radians)

The dependence on time (eiΩt) 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 Mode 1 Mode 2 Mode 3 Crack

case LC / ha / 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.

0.5

0.6

0.7

0.8

0.9

1

0.1 0.3 0.5 0.7 0.9

Crack Depth Ratio

Fre

qu

en

cy R

ati

o

1st

Mo

de

Figure 4 Fundamental natural frequency ratio in terms of crack depth for various crack

positions lC / (1→15/16; 2→9/16; 3→7/16; 4→5/16; 6→2/16).

1

2

3

4

5 6



112

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.3 0.5 0.7 0.9

Crack Depth Ratio

Fre

qu

en

cy R

ati

o

2n

d M

od

e

Figure 5 Second natural frequency ratio in terms of crack depth for various crack

Positions lC / (1→15/16; 2→2/16; 3→5/16; 4→6/16; 5→7/16; 6→10/16).

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.3 0.5 0.7 0.9

Crack Depth Ratio

Fre

qu

en

cy R

ati

o

3rd

Mo

de

Figure 6 Third natural frequency ratio in terms of crack depth for various crack positions

lC / (1→2/16; 2→15/16; 3→9/16; 4→14/16; 5→13/16).

0.20

0.40

0.60

0.80

1.00

0.0625 0.3125 0.5625 0.8125

Crack Location Ratio

Fre

qu

en

cy R

ati

o

1st

Mo

de

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.

3

1

2

4

5

6

1

3

2

5 4



113

0.30

0.50

0.70

0.90

0.0625 0.3125 0.5625 0.8125

Crack Location RatioF

req

ue

nc

y R

ati

o

2n

d M

od

e

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.

0.30

0.50

0.70

0.90

0.0625 0.3125 0.5625 0.8125

Crack Location Ratio

Fre

qu

en

cy R

ati

o

3rd

Mo

de

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 lC / = 11/16 for all crack

depths.

iv. The third natural frequency was almost unaffected for the crack locations ( lC / = 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



114

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 lC / =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).

0

5

10

15

20

25

30

35

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Crack Depth

Are

a R

ati

o

0.0625 0.2500 0.3125 0.5000 0.5625

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



115

0

2

4

6

8

10

12

14

0.0625 0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

Crack Location

Are

a R

ati

o

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.



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Signature analysis of cracked cantilever beam