vibration paper 2

7/29/2019 Vibration Paper 2

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Modelling of transverse vibration of beam of linearlyvariable depth with edge crack

T.D. Chaudhari, S.K. Maiti*

Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, 400 076, India

Received 7 September 1998; received in revised form 31 January 1999

Abstract

In this paper modelling of transverse vibration of a beam of linearly variable depth and constant

thickness in the presence of an `open' edge crack normal to its axis has been proposed using the concept

of a rotational spring to represent the crack section and the Frobenius method to enable possible

detection of location of the crack based on the measurement of natural frequencies. The method can

also be used to solve the forward problem. A number of numerical examples are presented involving

cantilever beams to show the eectiveness of the method for the inverse problem. The error in the

prediction of crack location is less than 2% and size is around 10% for all locations except at the xed

end. Crack sizes 1050% of section depth have been examined. # 1999 Elsevier Science Ltd. All rights

reserved.

Keywords: Vibration based detection of crack; Vibration modelling of cracked beam; NDT of crack; Finite elementanalysis

1. Introduction

In many applications non-uniform beams nd applications to satisfy special functional

requirements and achieve a better distribution of strength and weight. The analysis of

transverse free vibration of beams of variable cross-section (without any crack) has received a

Engineering Fracture Mechanics 63 (1999) 425445

0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

P I I : S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 2 9 - 6

www.elsevier.com/locate/engfracmech

* Corresponding author. Tel.: +91-22-578-2545; fax: +91-22-578-3480.

E-mail address: [email protected] (S.K. Maiti)


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considerable attention in the literature. These eorts date back to the time of Kirchho in

1879. Rao [1] has used the Galerkin method to determine the fundamental frequency of a

cantilever beam of variable depth and thickness. He has compared the results with Martin's [2].

Using the Bessel function approach Conway and Dubil [3] analysed the truncated wedge beam

problem. Gaines and Volterra [4] have used the RayleighRitz method to calculate the upper

and lower bounds for the three lowest natural frequencies of transverse vibration of cantileverbeams of variable cross-section. They have considered the eects of both the rotary inertia and

shear, and compared the results obtained by neglecting these eects. They have presented

results for beams in the form of cones and truncated cones, as well as beams of constant cross-

section.

Carnegie and Thomas [5] have considered the complex geometry of a turbine or compressor

blade as a taper beam of constant thickness. They have solved the related EulerBernoulli

equation by reducing it to a number of simultaneous linear algebraic equations by the nite

dierence method. Wang [6] has obtained solutions for transverse vibration of a tapered beam

in terms of generalized hypergeometric functions by the method of Frobenius for solving

EulerBernoulli type equation with variable coecients. Mabie and Rogers [7] have considered

free vibrations of nonuniform cantilever beams with an end support and have solved the

governing equation through numerical integration. Naguleswaran [8] has obtained solutions

directly for wedge and cone beams employing the method of Frobenius. He has presented the

rst three natural frequencies in a number of cases with dierent boundary conditions. It is not

clear whether the Frobenius method can be employed to solve the free vibration of a similar

beam with an `open' crack normal to its axis. Open is used to mean that the crack remains

always open during the vibrations of the beam.

When a crack develops in a component it leads to a change in its stiness, damping [9], etc.

These factors have been used as the basis to estimate the location and size of the crack. Rizos

et al. [10] have presented a method for detection of both crack location and size through the

study of transverse vibration for a slender beam like component of uniform section. Whilemodelling they have used the concept of a rotational spring to represent a normal edge crack.

This replacement makes use of the fact that the crack causes only a local jump in slope of the

beam. For the detection, it is only necessary to measure the amplitude of vibration at any two

arbitrary locations along the beam. Liang et al. [11] have given a scheme which is very similar,

but, according to them it is necessary to measure the three lowest transverse natural

frequencies. A review about these modellings and their practical exploitations for detection of

crack location and size are presented by Dimarogonas [12].

The method based on the rotational spring appears to be very attractive for a beam like

component. The method permits an analytical formulation of the inverse problem and a

precise detection of crack location. Nandwana and Maiti [13,14] have shown the suitability ofthe method for stepped beams, multiply supported beams, inclined edge cracks, etc. Extension

of this approach to cover beams of variable section depth with normal edge cracks will permit

a solution to the inverse problem.

The present study has derived motivation from these issues. The objective here is to present

an analytical method for the study of transverse vibration of cantilever beams of linearly

varying section depth and constant thickness with an open crack normal to its axis. The

method permits a solution to both the forward and inverse problems. That is, the method

T.D. Chaudhari, S.K. Maiti / Engineering Fracture Mechanics 63 (1999) 425445426


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Fig. 1. (a) Actual beam geometry. (b) Representation by rotational spring.

T.D. Chaudhari, S.K. Maiti / Engineering Fracture Mechanics 63 (1999) 425445 427


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allows a determination of the natural frequencies given the spring stiness. It also enables a

determination of the crack location and size given the natural frequencies. The method's

eectiveness for the detection of the location and size of a crack is shown through a number of

numerical examples.

2. Formulation

For a wedge beam as shown in Fig. 1, the vibration in the x-y plane is of interest here.

Introducing dimensionless parameters X xal, YX yxal and a truncation factor a=h1/h2,

where L=(1 a )l, local mass per unit length and exural rigidity can be expressed as follows:

mx

x

l

ml 1

EIx

x

l

3EIlX 2

The bending moment M(x ) and shear force V(x ) are given by

Mx EIxd2yx

dx 23

Vx dMx

dx4

Non-dimensional moment M(X) and shear force V(X), can be written as follows:

MX Mxl

EIl X3D2YX 5

VX Vxl2

EIl DX3D2YX 6

where D is the dierential operator.

The associated EulerBernoulli equation

d2

dx 2

EIx

d2yx

dx 2

! mxo 2yx 0

can be converted to a convenient non-dimensional form as given below

D2X3D2YX l4

EIlmlo2XYX 0 7



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iXeX D2X3D2YX mXYX 0 8

where

m O2 mlo2l4

EIlX 9

The above equation is valid for a full wedge beam as well as a truncated wedge beam. For a

truncated wedge beam the frequency parameter Om can be written following Naguleswaran [8],

in the form

Om 1 a2O,

whereO is the frequency parameter of the full wedge beam.

The boundary conditions for a cantilever are as follows:

Fixed end iXeX X 1: Deflection YX 0 10

Slope DYX 0

Free end iXeX X a: Moment D2YX 0 11

Shear Force D3YX 0X

2.1. Direct solution of mode shape equation

Eq. (8) is a fourth order dierential equation. Its general solution can therefore be written in

terms of four linearly independent solutions. Because of the sharp corner, there is a stress

singularity at the tip of a full wedge beam, i.e. X=0. Under the circumstances, this type of an

equation can be solved by the method of Frobenius. The trial solution, according to this

method, can be written in the form of the following power series [8]

YX,C In0

an1CXCkn 12

where C is an undetermined exponent. Naguleswaran [8] has shown that a viable value of k is2. Therefore

YX,CIn0

an1CXC2n a1CX

C a2CXC2 F F F 12a

Substituting Y(X,C) into Eq. (8)



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C 1C2C 1a1CXC1 C 3C 22C 1a2C ma1CX

C1

In0

C 2n 5C 2n 42C 2n 3an3C man2CXC2n3 0X 13

Since the right hand side is 0, the coecients of all the exponential terms must vanish. Thisgives a set of equations connecting constants a1C, a2C, etc., and the constant C. That is,

C 1C2C 1a1C 0

a1C C 3C 22C 1a2Cam

C 2n 5C 2n 42C 2n 3an3C man2C 0 for n 0,1,2, F F F IX 13a

If a2(C) is assumed to be 0, then all other constants a1(C), a3(C), etc., become zero. This

therefore gives the trivial solution. For a nontrivial solution a2(C) is taken as 1 and other

constants a3(C), a4(C), a5(C), etc., are obtained through recurrence relation in terms of C.After substituting all these constants in Eq. (12) the mode shape equation is obtained

YX,C C 3C 22C 1XCam XC2 In0

an3CXC2n4X 14

Consequently the left hand side of Eq. (8) can be reduced to the form

D2X3D2YX,C mXYX,C C 3C 22C 12C2C 1XC1amX 15

Since the left hand side of Eq. (15) is zero, therefore

C 3C 22C 12C2C 1 0X 16

This gives the possible values of C:

C 0, 0, 1, 3, 2, 1, 1, 2X

ForC= 3, it is seen from Eq. (13a) that an+3(C) becomes singular. Hence this value of C is

not acceptable. Solutions Y(X,C) corresponding to C= 2 and C=0 are linearly related. The

same is true for C= 1 and C=+1. Only the roots C= 1 and C= 2 are free of any such

problems and are acceptable. Hence the two independent mode shapes which corresponds to

C= 1 and C= 2, are given by

YX, 1 XmX3

4 32 2

m2X5

6 52 4 4 32 2 F F F X 17

YX, 2 1 mX2

3 22 1

m2X4

5 42 3 3 22 1 F F F X 18

These are the rst and second solutions of the mode shape Eq. (8).



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The third and fourth solutions are determined as follows. By dierentiating both sides of Eq.

(15) with respect to C, the following equation is obtained

D2

X3D2dYX,C

dC

! mX

dYX,C

dC

d

dC

4C 3C 22C 12C2C 1

XC1

m

5X 19

The right hand side of Eq. (19) vanishes when C=0, 1, 2. Therefore it is obvious thatdYX,C

dC

!C1

and

dYX,C

dC

!C2

are also solutions of Eq. (8)

dYX,C

dC C 3C 22C 3 C 2C 12C 5XCam C 3

C 22C 1XC ln Xam XC2 ln XIn0

an3CXC2n4Cn3X,C 20

where

Cn3X,C Cn2X,C 1

C 2n 5

2

C 2n 4

1

C 2n 3X

In particular for n 0, c3X,C c2X,C 1

C5 2

C4 1

C3, where c2(X,C)=ln X.

Substituting C= 1 and C= 2 the other two solutions are obtained

YX, 1 2 1X1

m

X ln X mX3

4 32 2! ln X 1

4

2

3

1

2! F F F 21

YX, 2 ln X

mX2

3 22 1

! ln X

1

3

2

2

1

1

! F F F 22

These are again linearly independent, and are therefore, the third and fourth solutions of the

governing Eq. (8). Hence the general solution of the mode shape Eq. (8) can be written in the

form

YX C1YX, 1 C2YX, 2 C3YX, 1 C4Y

X, 2X 23

The four arbitrary constants C1, C2, C3 and C4 are obtained from the boundary conditions.

3. Formulation for beam with crack

A crack oriented normally to the axis of a slender beam can be assumed to produce only a

local change in slope without altering the overall mode shape substantially from that of the



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corresponding uncracked beam. This change can then be modelled by introducing a rotational

spring [11] at the crack location. For the convenience of vibration analysis the beam can be

split into two beams interconnected by a rotational spring (Fig. 1).

The solutions for the two segments, which are nothing but two taper beams, can be written

independently using Eq. (23). That is,

Y1X C1Y1X, 1 C2Y1X, 2 C3Y1X, 1 C4Y1X, 2

for the left hand segment, and

Y2X C5Y2X, 1 C6Y2X, 2 C7Y2X, 1 C8Y

2X, 2

for the right hand segment.

The continuity of displacement, moment and shear forces at the crack location (say,

X=b=L1/l) and jump in the slope can be written in the following form

Y1X Y2X 24

D2Y1X D2Y2X 25

D3Y1X D3Y2X 26

DY1X 1

KD2Y1X DY2X 27

where K=(Ktl/EI) is the non-dimensional stiness of the rotational spring. The other

boundary conditions are

YX 0 and DYX 0 for X 1 28

MX 0 and VX 0 for X aX 29

From the above eight conditions (2429) the following characteristic equation of free vibration

of the beam is obtained.



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In short

j D j 0

Explicit form of vDv follows.

D2Y1a, 1 D2Y1a, 2 D

2Y1a, 1 D2Y1a, 2 0 0 0

aD3Y1a, 1 aD3Y1a, 2 aD

3Y1a, 1 aD3Y1a, 2 0 0 0

0 0 0 0 Y21, 1 Y21, 2 Y21, 1

0 0 0 0 DY21, 1 DY21, 2 DY21,

Y1b, 1 Y1b, 2 Y1 b, 1 Y

1b, 2 Y2b, 1 Y2b, 2 Y

2b,

D2Y1b, 1 D2Y1b, 2 D

2Y1b, 1 D2Y1b, 2 D

2Y2b, 1 D2Y2b, 2 D

2Y2b,

3D2Y1b, 1 3D2Y1b, 2 3D

2Y1b, 1 3D2Y1b, 2 3D

2Y2b, 1 3D2Y2b, 2 3D

2Y2b,

bD3Y1b, 1 bD3Y1b, 2 bD

3Y1b, 1 bD3Y1b, 2 bD


3Y2b,

KDY1b, 1 KDY1b, 2 KDY1 b, 1 KDY

1b, 2 KDY2b, 1 KDY2b, 2 KDY

2b,

D2Y1b, 1 D2Y1b, 2 D

2Y1 b, 1 D2Y1 b, 2

Alternatively

K j D2 j

j D1 j

where the explicit forms of vD1v and vD2v are as follows

j D1 j

D2Y1a, 1 D2Y1a, 2 D

2Y1a, 1 D2Y1a, 2 0 0 0


3Y1a, 1 aD3Y1 a, 2 0 0 0

0 0 0 0 Y21, 1 Y21, 2 Y

0 0 0 0 DY21, 1 DY21, 2 DY

Y1b, 1 Y1b, 2 Y1b, 1 Y

1b, 2 Y2b, 1 Y2b, 2 Y

D2Y1b, 1 D2Y1b, 2 D

2Y1b, 1 D2Y1b, 2 D

2Y2b, 1 D2Y2b, 2 D

2

3D2Y1b, 1 3D2Y1b, 2 3D

2Y1 b, 1 3D2Y1 b, 2 3D

2Y2b, 1 3D2Y2b, 2 3D


3Y1 b, 1 bD3Y1 b, 2 bD


3Y

DY1b, 1 DY1b, 2 DY1b, 1 DY

1 b, 2 DY2b, 1 DY2b, 2 D

j D2 j

D2Y1a, 1 D2Y1a, 2 D

2Y1a, 1 D2Y1a, 2 0 0 0


3Y1 a, 1 aD3Y1a, 2 0 0 0

0 0 0 0 Y21, 1 Y21, 2 Y

0 0 0 0 DY21, 1 DY21, 2 DY

Y1b, 1 Y1b, 2 Y1b, 1 Y

1b, 2 Y2b, 1 Y2b, 2 Y

D2Y1b, 1 D2Y1b, 2 D

2Y1 b, 1 D2Y1b, 2 D

2Y2b, 1 D2Y2b, 2 D

2

3D2Y1b, 1 3D2Y1b, 2 3D

2Y1b, 1 3D2Y1b, 2 3D

2Y2b, 1 3D2Y2b, 2 3D




3Y

D2Y1b, 1 D2Y1b, 2 D

2Y1 b, 1 D2Y1b, 2 0 0 0


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4. Methodology for crack detection

Eq. (30) can be used straightaway to study the free vibration of a taper beam with a crack.

Eq. (31) can serve as a basis for detection of crack location or solving the inverse problem. In

this case, for a given problem, it is necessary to measure or compute the rst three transverse

natural frequencies of the beam with a crack and the corresponding uncracked beam. For eachmode, a variation of K with crack location b is obtained using Eq. (31). Since the stiness of

Fig. 2. Typical nite element discretisations.



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Table 1

Natural frequencies of truncated cantilever beam of linearly variable depth

Actual Natural frequencies (Hz) Predicted

Location Size o1 o2 o3 Location Stiness Size

b a/h b % Error K a/h % Error

Truncation factor (a )=0.1

uncracked 377.65 1192.78 2572.47

0.5 0.294 371.97 1167.23 2543.26 0.5 0.0 32.0 0.280 4.76

0.5 0.393 366.66 1145.08 2519.32 0.5 0.0 16.0 0.379 3.56

0.5 0.5 357.02 1108.58 2481.32 0.5 0.0 8.25 0.486 2.8

0.6 0.296 367.71 1183.33 2487.40 0.609 0.9 25.0 0.288 2.7

0.6 0.5 344.62 1164.57 2337.32 0.6 0.0 7.0 0.483 3.4

0.7 0.298 363.09 1189.24 2513.35 0.702 0.2 20.0 0.297 0.33

0.7 0.397 351.02 1189.05 2472.61 0.702 0.2 10.0 0.4 0.75

0.95 0.359 333.13 1091.12 2420.51 0.948 0.2 7.95 0.388 7.93

0.95 0.5 304.95 1044.26 2362.65 0.948 0.2 4.32 0.487 2.540.99 0.299 360.18 1145.44 2485.83 0.985 0.5 21.4 0.245 18.1

0.99 0.399 344.28 1104.39 2416.70 0.985 0.5 10.9 0.334 16.4

0.99 0.5 320.15 1052.78 2340.29 0.985 0.5 5.82 0.431 13.7

1.0 0.3 366.11 1159.84 2506.13 0.995 0.5 33.0 0.198 34.0

1.0 0.4 354.83 1127.25 2445.59 0.993 0.7 18.2 0.264 34.0

1.0 0.5 337.17 1082.99 2372.21 0.992 0.8 7.90 0.381 23.8


uncracked 351.62 1271.54 2925.31

0.5 0.288 348.36 1230.03 2923.00 0.495 0.5 38.0 0.273 5.2

0.5 0.384 345.32 1195.26 2920.83 0.496 0.4 20.0 0.363 5.4

0.5 0.5 338.93 1132.12 2916.80 0.495 0.5 10.0 0.474 5.20.6 0.292 344.38 1242.13 2862.73 0.595 0.5 28.0 0.289 1.03

0.6 0.389 338.04 1219.69 2820.60 0.598 0.2 15.0 0.379 2.5

0.6 0.5 326.23 1181.62 2753.86 0.6 0.0 7.5 0.492 1.6

0.95 0.399 307.34 1156.18 2751.79 0.948 0.2 8.9 0.388 2.58

0.95 0.5 280.17 1106.83 2690.90 0.949 0.1 5.82 0.457 8.52

0.99 0.254 332.22 1202.19 2781.68 0.985 0.5 23.1 0.250 1.73

0.99 0.359 316.10 1155.09 2700.80 0.985 0.5 11.8 0.339 5.56

0.99 0.5 292.07 1097.61 2613.03 0.986 0.4 6.30 0.437 12.4

1.0 0.3 339.09 1222.67 2814.41 0.995 0.5 37.0 0.197 34.3

1.0 0.4 328.23 1185.09 2742.11 0.993 0.7 17.5 0.284 29.0

1.0 0.5 310.48 1135.18 2656.46 0.993 0.7 13.0 0.324 35.2


uncracked 334.81 1349.34 3219.74

0.5 0.279 333.60 1312.10 3135.43 0.5 0.0 45.0 0.269 3.6

0.5 0.371 332.46 1279.66 3070.15 0.498 0.2 22.0 0.369 0.54

0.5 0.5 329.61 1207.05 2945.99 0.5 0.0 10.5 0.488 2.4

0.6 0.291 330.31 1292.89 3201.31 0.602 0.2 38.0 0.267 8.2

0.6 0.381 326.42 1254.36 3200.14 0.602 0.2 20.0 0.356 6.5

(continued on next page)



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the spring is assumed to be independent of the mode, the location where three such curves

intersect gives the crack location and the spring stiness. In order to get a common

intersection point of the three K vs b curves, it has been shown [14] that the modulus of

elasticity E, which goes as input to relation (31) for each mode, is calculated using the zero

setting procedure. This is why the computation/measurement of uncracked beam natural

frequencies is compulsory. The crack size is then obtained using the relationship between

stiness K and crack size a.

4.1. Numerical studies

A beam with the following geometrical data has been chosen: length 240 mm, width 12 mm

and depth at the xed end 20 mm. The depth at the free end is varied; six cases, 2, 4, 6, 8, 10

and 12 mm, are considered. The corresponding six truncation factors are a=0.1, 0.2, 0.3, 0.4,

0.5 and 0.6 respectively. The material data employed include: density r=7860 kg/m3, Poisson

ratio g=0.3 and modulus of elasticity E=210 GPa. A number of crack sizes ranging from 10

to 50% of section depth and four dierent locations have been examined. The natural

frequencies of the cracked as well as the uncracked beams are computed using a nite element

program [15]. For discretization mostly eight-noded quadrilateral elements are employed [16].Two sample discretizations considered are as shown in Fig. 2. The change in frequency due to

this change in discretization is not that signicant. For most of the remaining case studies only

the second discretization has been employed. The natural frequencies so computed are given in

Table 1 and Table 2.

The variations of rotational spring stiness K with crack location b are obtained from Eq.

(31) are shown (Figs. 35) for a few cases for each of the six truncation factors. It must be

emphasized that forty terms in the expansion (12) are considered in all case studies for

Table 1 (continued)




0.6 0.5 318.09 1184.19 3197.86 0.602 0.2 10.0 0.466 6.8

0.8 0.393 306.63 1338.97 3069.11 0.785 1.5 10.0 0.418 6.4

0.8 0.5 286.23 1338.35 2975.87 0.782 1.8 5.0 0.533 6.6

0.95 0.298 309.98 1271.60 3105.16 0.948 0.2 20.0 0.290 2.81

0.95 0.398 290.60 1227.41 3049.97 0.948 0.2 10.1 0.389 2.33

0.95 0.5 263.78 1177.38 2990.86 0.95 0.0 5.5 0.489 2.22

0.99 0.291 314.02 1262.49 3041.28 0.985 0.5 25.4 0.255 14.9

0.99 0.381 297.27 1210.63 2951.91 0.986 0.4 13.0 0.345 13.6

0.99 0.5 272.80 1149.03 2857.61 0.987 0.3 6.98 0.442 11.4

1.0 0.3 321.68 1287.29 3080.46 0.992 0.8 41.5 0.199 33.6

1.0 0.4 310.15 1245.29 2998.83 0.995 0.5 23.5 0.263 34.2

1.0 0.5 292.59 1190.94 2903.94 0.992 0.8 11.4 0.363 27.4



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Table 2

Natural frequencies of truncated cantilever beam of linearly variable depth





uncracked 322.83 1415.36 3486.97

0.5 0.1 322.82 1413.47 3474.91 0.494 0.6 370.0 0.101 1.0

0.5 0.2 322.72 1409.42 3438.85 0.497 0.3 110.0 0.188 6.25

0.5 0.308 322.61 1401.29 3369.86 0.5 0.0 41.0 0.301 2.27

0.5 0.383 322.47 1391.76 3295.07 0.502 0.2 25.0 0.373 2.61

0.5 0.5 322.08 1366.62 3155.94 0.505 0.5 15.0 0.455 9.0

0.6 0.1 322.55 1402.95 3448.26 0.6 0.0 350.0 0.094 6.0

0.6 0.167 322.15 1391.79 3432.13 0.6 0.0 140.0 0.151 9.58

0.6 0.306 320.63 1349.32 3373.22 0.592 0.8 32.0 0.310 1.31

0.6 0.389 318.99 1308.46 3321.45 0.595 0.5 24.5 0.348 10.50.6 0.5 315.35 1229.17 3424.24 0.598 0.2 9.8 0.495 1.0

0.7 0.1 322.02 1403.36 3454.07 0.7 0.0 305.0 0.093 7.0

0.7 0.143 321.28 1397.52 3450.19 0.7 0.0 150.0 0.135 5.59

0.7 0.304 315.93 1357.18 3424.24 0.7 0.0 31.0 0.293 3.62

0.7 0.393 310.49 1320.40 3399.57 0.7 0.0 18.0 0.372 5.34

0.7 0.5 300.14 1259.09 3358.20 0.7 0.0 9.0 0.483 3.4

0.8 0.1 321.22 1408.64 3440.24 0.8 0.0 270.0 0.093 7.0

0.8 0.125 320.40 1408.33 3431.69 0.8 0.0 170.0 0.118 5.6

0.8 0.3 309.17 1404.32 3318.43 0.8 0.0 30.0 0.280 6.67

0.8 0.396 298.37 1400.39 3218.71 0.8 0.0 15.0 0.379 4.29

0.8 0.5 280.68 1394.01 3070.23 0.798 0.2 8.0 0.481 3.8


uncracked 313.93 1478.10 3710.39

0.6 0.3 313.61 1456.09 3551.35 0.6 0.0 45.0 0.289 3.67

0.6 0.5 312.74 1398.41 3215.78 0.6 0.0 12.5 0.485 3.0

0.7 0.3 310.63 1402.14 3667.56 0.7 0.0 38.0 0.291 3.0

0.7 0.4 309.03 1370.70 3654.46 0.7 0.0 23.5 0.359 10.2

0.7 0.5 302.33 1259.87 3608.43 0.7 0.0 10.8 0.483 3.4

0.8 0.3 303.59 1443.48 3555.43 0.798 0.2 38.0 0.273 9.0

0.8 0.4 297.16 1425.75 3481.61 0.798 0.2 20.0 0.363 9.2

0.8 0.5 280.68 1383.58 3320.60 0.8 0.0 8.0 0.511 2.2


uncracked 306.85 1538.13 3935.17

0.65 0.259 306.81 1534.37 3889.03 0.65 0.0 65.0 0.260 0.38

0.65 0.346 306.77 1530.83 3845.45 0.65 0.0 30.0 0.367 6.0

0.65 0.5 306.65 1518.58 3688.92 0.65 0.0 15.0 0.478 4.4

0.75 0.275 304.62 1467.52 3835.64 0.75 0.0 53.0 0.267 2.9

0.75 0.367 302.83 1414.92 3779.99 0.75 0.0 29.0 0.350 4.63

0.75 0.5 297.95 1300.52 3668.50 0.75 0.0 12.5 0.485 3.0

(continued on next page)



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evaluating vD1v and vD2v. Quadruple precision based computation has been found to be veryuseful and is recommended. The three curves intersect clearly at a location, which indicates the

possible crack position. The error in the prediction of location is not more than 2%

considering all the cases presented. The method does not pose any problem for locating crack

at the xed end too.

The inverse procedure can be easily applied to determine the spring stiness to represent acrack of a given size. The spring stiness can also be computed from the fact that the change

in strain energy DU of the beam in response to change in geometry arising out of origination

of a crack is related [17]

DUM2

2Kt

where M is the bending moment at the crack section. The loading must be the same in the case

of both cracked and virgin geometries.

In order to compute the stiness through the above relation for example for the cantilever

beam, deections d2 and d1 with and without the crack have been obtained here by the nite

Table 2 (continued)




0.8 0.281 301.63 1470.35 3903.51 0.798 0.2 52.0 0.262 6.76

0.8 0.375 296.95 1419.80 3892.98 0.798 0.2 28.0 0.346 7.73

0.8 0.5 286.45 1324.88 3871.08 0.8 0.0 13.0 0.468 6.4

Table 3

Comparison of rotational spring stiness obtained by vibration and energy methods

a b a/h K

Vibration method Energy method

0.3 0.5 0.5 10.5 10.53

0.3 0.6 0.5 10.0 8.78

0.3 0.8 0.5 5.0 6.520.3 0.95 0.5 5.5 5.44

0.3 0.99 0.5 6.98 6.92

0.3 1.0 0.5 11.4 11.34

0.4 0.5 0.5 15.0 12.23

0.4 0.6 0.5 9.8 10.23

0.4 0.7 0.5 9.0 8.76

0.4 0.8 0.5 8.0 7.63



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Fig. 3. Plots of stiness vs crack location for truncation factor of (a) 0.1 and (b) 0.2.



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element calculations when a load P is acting at the beam end, K1 is then calculated noting that

DUP

2d2 d1 and M PlcrX

where lcr is the distance between load point and crack locations. The non-dimensionalised

stinesses (K KtlaEI) so obtained are compared in Table 3 and Fig. 6. Although at somespecic locations (b=0.6 and 0.8 in Fig. 6(a); b=0.5 in Fig. 6(b)), the dierence in the

Fig. 6. Comparison of rotational spring stiness obtained by vibration and energy methods. (a) a=0.3 and a/h=0.5.

(b) a=0.4 and a/h=0.5.



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stinesses obtained by the two methods is up to 30%, there is good agreement in the overall

sense. More importantly, there is excellent agreement around b=1.0.

The crack size can be obtained from [18]

Kbh2E

72pIaah2faah32

where f(a/h ) is obtained using results of reference [19]

faah 0X6384 1X035aah 3X7201aah2 5X1773aah3 7X553aah4

7X332aah5 6X799aah6 6X9956aah7 20X094aah8 22X1145aah9

6X93aah10 16X115aah12 33

b and h are the beam thickness and depth respectively. A truncated form of Eq. (33) is given in

[18].

The eect of truncation of this polynomial on the prediction of crack size is presented in

Table 4. Results of Tables 1 and 2 correspond to full expansion. The maximum error in

predicting crack size is around 10% for all b excluding b=1.0; for b=1.0 the error is about

35%. Since the stiness computed independently by the vibration and energy methods are in

excellent agreement (Table 3 and Fig. 6(a)) around location b=1.0, the high error in crack size

prediction may be partly due to the inadequacy of the formula (Eq. (32)), which is mainly

proposed for beams of uniform depth and constant thickness.

Table 4

Eect of number of terms in K vs a/h relationship on crack size computation

Actual Crack size computed with various number of terms

b a/h 3 terms %error 4 terms %error 6 terms %error


1.0 0.3 0.195 35.0 0.196 34.6 0.198 34.0

1.0 0.4 0.257 35.7 0.265 33.7 0.264 34.0

1.0 0.5 0.361 27.8 0.384 23.2 0.381 23.8


1.0 0.3 0.196 34.6 0.198 34.0 0.197 34.31.0 0.4 0.275 31.2 0.284 29.0 0.284 29.0

1.0 0.5 0.311 37.8 0.325 35.0 0.324 35.2


1.0 0.3 0.197 34.3 0.200 33.3 0.199 33.6

1.0 0.4 0.256 36.0 0.263 34.2 0.263 34.2

1.0 0.5 0.345 31.0 0.365 27.0 0.363 27.4



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5. Conclusions

A method of modelling of transverse vibration of a beam of variable depth with an open

edge crack has been presented by combining the Frobenius method and the representation of a

crack in a beam by a rotational spring to take care of the local jump in slope. The modelling

enables solution of both the forward and inverse problems. The accuracy of the method inrespect of detection of crack location is particularly encouraging. This can be exploited for

detection of crack location by knowing the changes in its natural frequencies. The main

conclusions of the study are as follows:

1. An analytical solution for the study of vibration of taper beams with crack normal to its

axis has been presented.

2. The Frobenius method has been combined with the modelling based on rotational spring for

a crack in a linearly variable depth beam having constant thickness. This method can help

to solve both the forward and inverse problems.

3. While solving the inverse problem, the method predicts the location of a crack quite

accurately. The maximum error in the prediction of the location considering all the casesstudied is about 2%. The scheme can be employed to locate an unknown crack in a taper

beam.

4. The error in predicting the crack size is around 10% for all b cases except b=1.

References

[1] Rao JS. The fundamental exural vibration of a cantilever beam of rectangular cross-section with uniformtaper. The Aeronautical Quarterly 1965;16:13948.

[2] Martin AI. Some integrals relating to the vibration of a cantilever beam and approximation for the eect of

taper on overtone frequencies. The Aeronautical Quarterly 1956;7:1208.

[3] Conway HD, Dubil JF. Vibration frequencies of truncated-cone and wedge beams. Transactions of the

American Society of Mechanical Engineers, Journal of Applied Mechanics (series E) 1965;32:9324.

[4] Gaines JH, Volterra E. Transverse vibrations of cantilever bars of variable cross-section. Journal of Acoustical

Society of America 1966;39:6749.

[5] Carnegie W, Thomas J. Natural frequencies of long tapered cantilevers. The Aeronautical Quarterly

1967;18:30920.

[6] Wang HC. Generalised hypergeometric function solutions on the transverse vibration of a class of non-uniform

beams. ASME Journal of Applied Mechanics 1967;34:7028.

[7] Mabie HH, Rogers CB. Transverse vibrations of tapered cantilever beams with end support. Journal of

Acoustical Society of America 1968;44:173941.

[8] Naguleswaran S. A direct solution for transverse vibration of Euler Bernoulli wedge and cone beams. Journal

of Sound and Vibration 1994;172:289304.

[9] Adams RD, Walton D, Flitcroft JE, Short D. Vibrations testing as a nondestructive test tool for composite ma-

terials. Composite Reliability ASTM STP 1975;580:15975.

[10] Rizos PF, Aspragathos N, Dimarogonas AD. Identication of crack location and magnitude in a cantilever

beam from the vibration modes. Journal of Sound and Vibration 1990;138:3818.



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[11] Liang RY, Choy FK, Hu J. Detection of cracks in beam structures using measurements of natural frequencies.

Journal of the Franklin Institute 1991;328(4):50518.

[12] Dimarogonas AD. Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics

1996;55(5):83157.

[13] Nandwana BP, Maiti SK. Detection of location and size of a crack in stepped cantilever beams based on

measurements of natural frequencies. Journal of Sound and Vibration 1997;203(3):43546.

[14] Nandwana BP, Maiti SK. Modelling of vibration of beam in presence of inclined edge and internal crack forits possible detection based on frequency measurements. Engineering Fracture Mechanics 1997;58(3):193205.

[15] Maiti SK. Finite element package for stress and vibration analysis. MED-SKM-TR-96-01, Mechanical Eng.

Dept., Indian Institute of Technology, Bombay 1996. p. 874.

[16] Bathe KJ. Finite element procedures in engineering analysis. New Delhi: Prentice Hall of India, 1990.

[17] Nandwana BP. On foundation for detection of crack based on measurement of natural frequencies. PhD

Thesis, Mechanical Engineering Department, Indian Institute of Technology, Bombay (India) 1997.

[18] Ostachowicz WM, Krawkczuk M. Analysis of the eect of cracks on the natural frequencies of a cantilever

beam. Journal of Sound and Vibration 1991;150:191210.

[19] Anifantis N, Dimarogonas A. Stability of columns with a single crack subjected to follower and vertical loads.

International Journal of Solid and Structures 1983;19:28191.


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