Vibration Analysis Of Cantilever Shaft With Transverse Cracks
R.K Behera, D.R.K. Parhi, S.K. Pradhan1, and Seelam Naveen Kumar2
Dept. of Mech Engg.
N.I.T., Rourkela,769008
1
Dept. of Mech. Engg
P.C.E., Rourkela Rourkela, 769008
2
Dept. of Mech. Engg,
N.I.T, Rourkela, 769008
ABSTRACT
It has been observed since long that, the dynamic behavior of a structure changes due to
presence of crack. Scientific analysis of such phenomena can be utilized for fault diagnosis
and detection of cracks in structures. In this paper attempts have been made to detect the
cracks of a mild steel cantilever shaft. Theoretical expressions have been developed for
determining natural frequencies and mode shapes for elastic cantilever shaft having 'two'
cracks using flexibility influence coefficients and local stiffness matrix. The numerical
results for the beams having ‘no’ crack, 'single' crack and ‘two’ cracks are compared. Mode
shapes have been plotted for relative crack depth 0.5 for 'single' crack and 0.5 each for 'two'
cracks respectively. It is observed from the numerical results that, there are appreciable
changes in vibration characteristics of the cantilever shaft with and without cracks which
can be utilized for multi crack identification of structures.
INTRODUCTION
Since the dynamic behavior of structure changes due to presence of crack, identification
and location of cracks are required in structural design. The frequencies of natural
vibrations, amplitude of forced vibrations and areas of dynamic stability change due to
existence of such cracks [1-6].An analysis of these changes make it possible to identify the
magnitude and location of the crack. This information enables us to determine the degree of
sustainability of the structural element and the whole structures.
LOCAL FLEXIBILITY OF A CRACKED SHAFT UNDER BENDING AND AXIAL
LOADING
The presence of transverse surface cracks of depth a 1 and a 2 at a distance L1 and L 2
respectively from the fixed end on a shaft of diameter ‘D’ introduces a local flexibility Fig.
1, which can be defined in a matrix form. The geometry of the cracked section is shown in
Fig. 2. The cantilever shaft is subjected to axial force P1 and bending moment P2 which
gives coupling with the longitudinal and transverse vibration motion.
y
U1
U2
U3
dξ
a
d -b
o
L1
Y1
L2
Y2
Y3
η
b
h
R
ξ
L
ξ
Fig. 1: Beam Model
Fig. 2: Geometry of cracked section
Using the available expressions for stress intensity factors, Castigliano’s theorem and strain
energy release rate the compliance matrix can be obtained. The local stiffness matrix can be
obtained by taking the inversion of compliance matrix.
The stiffness matrices for relative crack position β and γ are obtained as:
−1
′′
′ 
C12
 k 11
 and K ′′ = 
′
C ′22 
k ′21
′
′   C11
k 12
=
k ′22  C ′21
′
 k 11
K′ = 
k ′21
′′   C11
′′
k 12
=
′  C ′21
′
k ′22
′′ 
C12

′ 
C ′22
−1
ANALYSIS OF VIBRATION CHARACTERISTICS OF A CRACKED SHAFT
A cantilever shaft of length ‘L’ and radius 'R' with “two” crack depths ‘ a 1 ’at a distance
‘ L1 ’ and crack depth ‘ a 2 ’at a distance‘ L 2 ’ from the fixed end is considered (Fig.1) .If T
is the period of vibration, by substituting x = x / L , U = U / L , Y = Y / L , t = t / T and
β = L1 / L , γ = L 2 / L the system can be derived with the help of equations for longitudinal
and transverse vibration in non dimensional form [2].
2
2
∂ 2 Ui ∂x = ∂ 2 U i c u ∂t
2
4
2
and ∂ 4 Y i ∂ x = ∂ 2 Y i c y ∂ t
2
(1)
where i = 1 for 0 ≤ x ≤ β , i = 2 for β ≤ x ≤ γ , i = 3 for γ ≤ x ≤ 1 ,
c u = c u T / L , and c y = c y T / L2 ,
The normal function for the system can be defined as
u 1 ( x ) = A1 cos(k u x ) + A 2 sin( k u x )
(2)
u 2 ( x ) = A 3 cos(k u x ) + A 4 sin( k u x )
(3)
u 3 ( x ) = A 5 cos(k u x ) + A 6 sin( k u x )
(4)
y1 ( x ) = A 7 cosh(k y x ) + A 8 sinh(k y x ) + A 9 cos(k y x ) + A10 sin(k y x )
(5)
y 2 ( x ) = A11 cosh(k y x ) + A12 sinh(k y x ) + A13 cos(k y x ) + A14 sin(k y x )
(6)
y 3 ( x ) = A15 cosh(k y x ) + A16 sinh(k y x ) + A17 cos(k y x ) + A18 sin(k y x )
(7)
Where x = x / L, u = u / L, y = y / L, t = t / T
1
1
1
k u = ωL / c u , c u = (E / ρ) 2 , k y = (ωL2 / c y ) 2 , c y = (EI / µ) 2 , µ = Aρ
A i , (i = 1,18) are the constants to be determined from boundary conditions[ 7]
NUMERICAL ANALYSIS
The mode shapes for no crack, single crack and two cracks are plotted. They are compared
in order to observe the change in mode shapes. For small relative crack depth it is difficult
to notice the change in mode shape. However for large crack depth the change in mode
shapes are quite substantial. The mode shapes for relative crack depth of 0.5 are shown in
2.5
3
2
2
A m plitude
Amplitude
Fig. 3–Fig. 11
1.5
1
0.5
1
0
-1 0
20
40
60
80
100
-2
0
-3
0
20
40
60
80
100
B eam P osition
Beam Position
Fig. 3: First mode of transverse vibration
a/D=0.5, L1 / L = 0.125
cracked ,
uncracked
Fig. 5: Third mode of transverse vibration
a/D = 0.5, L1 / L = 0.125
cracked ,
uncracked
2.5
1
2
0
-1
0
20
40
60
80
10 0
Amplitude
A m plitude
2
1.5
1
-2
0.5
-3
0
0
B eam P osition
1.5
1
20
40
60
80
100
Amplitude
Amplitude
1.5
0
60
80
100
Fig. 6: First mode of transverse vibration
a 1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L 2 / L = 0.25
cracked,
uncracked
2.5
-0.5
40
Beam Position
Fig. 4: Second mode of transverse vibration
a/D = 0.5, L1 / L = 0.125
cracked
uncracked
0.5
20
0.5
0
-0.5 0
-1.5
-1
-2.5
-1.5
Beam Position
Fig. 7: Second mode of transverse vibration
a 1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L 2 / L = 0.25
cracked
uncracked
20
40
60
80
100
Beam Position
Fig. 10: Second mode of longitudinal vibration
a1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L2 / L = 0.25
cracked,
uncracked
2
1
1
0
-1 0
20
40
60
80
100
Amplitude
1.5
0.5
0
-0.5 0
-2
-1
-3
-1.5
Fig.8: Third mode of transverse vibration
a 1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L 2 / L = 0.25
cracked ,
uncracked
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
20
40
60
80
100
Beam Position
Beam Position
Amplitude
Amplitude
3
80
100
Beam Position
Fig. 9: First mode of longitudinal vibration
a 1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L 2 / L = 0.25
cracked,
uncracked
Fig. 11: Second mode of longitudinal vibration
a1 / D = 0.5, a 2 / D = 0.5, L1 / L = 0.125, L2 / L = 0.25
cracked,
uncracked
CONCLUSION
It is observed from the numerical results that, there are appreciable changes in
vibration characteristics of the cantilever shaft with and without cracks which can be
utilized for multi crack identification of structures
NOMENCLATURE
A= cross-sectional area of shaft
Ai , i =1,18 = unknown co-efficients of matrix A
a 1 , a 2 =depth of crack
b = half the width of the crack
B1 = vector of exciting motion
C ij = elements of the compliance matrix
R(D/2)=Radius of shaft
E = young’s modulus of elasticity
Fi , i = 1,2 = experimentally determined function
h=height of rectangular strip
I = moment of inertia of shaft section
i,j =variable
J= strain energy release rate
k ij =Local flexibility matrix element
L = length of shaft
L 1 , L 2 = location of first and second crack from fixed end
Pi , i = 1,2 = axial force (i= 1 ), bending moment (i= 2 )
u i , i = 1,2 = normal functions (longitudinal) u i ( x )
U i , i = 1,2 =longitudinal vibration, U i ( x, t )
x, y = co-ordinate of the shaft
Y 0 = amplitude of the exciting vibration
Yi , i = 1,2 = normal functions (transverse) y i ( x )
Greek symbols
ω = natural circular frequency
β , γ = relative crack locations
λ= πη / 2h
µ = Aρ
ν =poison ratio
ξ = coordinate at the cracked surface
ρ =mass density of the shaft
ξ1 = a 1 D = relative crack depth
ξ 2 = a 2 D = relative crack depth
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subjected to
follower and vertical loads. International Journal of Solids and
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3.
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