Effect of Crack on Natural Frequency1

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Journal of Sound and Vibration (1991) 150(2), 191-201

N LYSIS OF THE EFFECT OF CR CKS ON THE

N TUR L FREQUENCIES OF C NTILEVER BE M

W. M.

OSTACHOWICZ AND

M.

KRAWCZUK

Polish Aca demy of Sciences, Institute of Fluid Flow Mach inery, MI. Gen. J. Fiszera 14,

80-952 Gdafisk, Poland

(Received 2 March 1990, and in revised,form

15

October 1990)

A method of analysis of the effect of two open cracks upon the frequencies of the natural

flexural vibrations in a cantilever beam is presented. Tw o types of cracks are considered:

double-sid ed, occurring in the case of cyclic loadings, and single-sided, whi ch in principle

occur as a result of fluctuating loadings. It is also assumed that the cracks occur in the first

mode of fracture: i.e., the opening mode. An algorithm and a numerical example a re

included.

1. INTRODUCTION

Cracks found in structural elements have various causes. They may be fatigue cracks that

take place under service conditions as a result of the limited fatigue strength. They m ay

also be due to mechanical defects, a s in the case of turbine blades of jet turbine engines.

In these engines the cracks are caused by sand and small stones sucked from the surface

of the runway. Another group involves cracks which are inside the material: they are

created as a result of manufacturing processes.

Cracks present a serious threat to proper performance of machines. Most of the failures

of presently used equipm ent are due to material fatigue. For this reason, m ethods mak ing

early detection and localization of cracks po ssible have been the subject of many investiga-

tions, which have been carried out in num erous research institutes throughout the

world [l-9].

A crack which occurs in a structural element causes some local variations in its stiffness,

which affect the dynam ics of the whole structure to a considerable degree. The frequencies

of natural vibrations, amp litudes of forced vibrations and areas of dynamic stability change

due to the existence of such cracks [l--9]. An analysis of the changes makes it possible to

identify the cracks w ithout disengag ing all of the system.

Up to now investigations have been concentrated on the analysis of the effect of a single

crack on the dynam ics of simple structures, such as shafts, beam s and frames. In wha t

follows here an analysis of the effect of two open cracks upon the frequencies of the natural

flexural vibrations in a cantilever beam is presented (see Figure 1).

Two types of cracks are to be considered: open double-sided, occurring in the case of

cyclic loadings, and open single-sided, which in principle occur a s a result of fluctuating

loadings. It is also assumed that the crack is in the opening mode (first mode of fracture;

see Figure 2).

2. CALCULATION OF THE EQUIVALENT STIFFNESS IN THE CRACK

2.1. OPEN DOUBLE SIDED CRACK

The open double-sided crack is illustrated in Figure 3. In actual p ractice, cracks of this

type occur in the case of two-sided bending of a smooth beam . It is assum ed that the

191

0022-460X/91 ;200191 + 1 03.00 0

(: 1991 Academ ic Press Limited


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192

W. M. OSTACHOWICZ AND M . KRAW CZUK

/

Crack 1

i--

f

Crack

2

/

4

X

A

A

2

/.

/_I _/

/

L 2

?F

I7

Figure 1. Schematic diagram of a beam with two cracks.

t

Farce

Figure 2. Crack model due to first stress intensity factor.

a

da

Figure 3. Double-sided crack dimensions).

decrease in the elastic deformation energy of the crack w ith regard to a plane state of

stress can be expressed in the form [ l o ]

A

s

A

Ec i

K 2 dA ,

(2.1)

where

K

is the stress intensity factor,

E

is Youngs modulus and

A

is the area of the crack.

The stress intensity factor is calculated as follows

[ I O ]

K = K ,= o f (a /H ) ,

(2.2)

where (r denotes the stress in the crack,

a

is the crack depth,

H

is half the height of the

beam, and, according to Hellan [l l] and Haisty and S pringer [12], the functionf(a/H)

the form

f ( a /H ) = 1.003 - 1.349(a/H ) + 5.896(a/H )2.

(2.3)


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EFFECT OF CRACKS ON BEAM FREQUENCIES

193

Assuming that

A U= M2/2k,,,

0 = 3M/2BH,

dA = 2B da,

(2.4 2.6)

where M is the bending moment, after simple transformation (with T=a/H) one can

arrive a t the relationship for the equivalent stiffness at the crack location:

kZD=BHEj9~y2(0.5335-0.9293/+3.500y2-3.181y3+5.793y4).

(2.7)

Relation (2.7) can be presented in a simplified form as

kzD= BH2E/9ry2fD(Y) = BH2KD,

(2.8)

where

Kn=E/9Wtf,(Y),

f~(y)=O~5335-O~929y+3~500~2-3~181y3+5~793~4.

The flexibility at the crack location becomes

0 =

EJ/kznL, (2.9)

where J is the geometrical moment of inertia and L is the length of the beam. Substitution

of equation (2.7) into equation (2.9) yields

0,,=6zy%(1/)(HiL),

(2.10)

or

t3,,=(H/L)OD.

(2.11)

wher e BD = 67~7 rD( 7). Diagram s of the stiffness

KD and flexibility 0, functions are

presented in Figures 4 and 5.

Relative depth o f c rack 7

F igure 4. Relation between flexibility and relative depth of a crack 7. ~~,

QJ;

on

2.2. OPEN SINGLE-SIDED CRACK

The open single-sided crack is illustrated in Figure 6. Under service conditions cracks

of this type occur under fluctuating loads. The equivalent stiffness at the single-sided crack

location is calculated in a similar way as that for a double-sided crack, by adopting the

correction function given by Anifantis and Dim arogon as (61, in the form

f(a/H)= 1.13- 1.374(a/H)+5.749(ajH)2-4.464(a/H)3.

(2.12)


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194

W. M. OSTACHOWICZ AND M. KRAWCZUK

a

\

?8-

z

\

GJ

\

\

\

\

4-

\

\

\

\

--__

--___

O-I 0~150~20~250~30~350~40~450~50~550~60~650~70~75

Relative depth of crack, 7

Figure 5. Relation between stiffness and relative depth of a crack 7. ---~-, K,;

. K,,

Figure 6. Single-sided crack dimensions)

Figure 7. Clamped beam with two cracks.

Assuming, as previously, K in the form (2.2) and also

AU =M i 2k Z J,

cs = 6M /BH= .

dA=Bda,

(2.13-2.15)

after some simple transformations (with r = a /H ) one obtains

kZ J=B H E /72~~ 2(0~6384-1~035y+3~7201y2-5~17731 /3

+7.553y4-7.332y5+2.4909y6).

(2.16)


5/11


Relation (2.16) can be written in the form

195

(2.17)

where

The flexibility at the single-sided crack location is calculated as

~s.T=W%(F)(HIL) or

QsJ= H IL)8J,

(2.18)

where

&=67ry2fJ(/).

Diagram s of the stiffness

KJ

and flexibility QJ functions are show n in Figures 4 and 5.

3. CALCULATION OF NATURAL VIBRATION FREQUENCIES OF A BEAM WITH

TWO CRACKS

One can take the natural vibration frequencies equation of the beam in the well know n

form

EJ Z4y(x, t), + pF ; y(x, t)iiit= 0,

(3.1)

where

p

is the material density,

F

is the cross-sectional area of the beam , y(x, t) is the

deflection of the beam and

J

is the geometrical mom ent of inertia of the beam cross-

section. By introducing elasticity elements in crack locations one obtains a system of three

beam s. T he equivalent stiffness of the elasticity element is calculated as in section

2.

The model of the problem is shown in Figure 7. The boundary conditions, in terms of

the non-dim ensional beam length

(=x/L,

can be expressed as follows: y,(O) =O , zero

displacement of the beam at the restraint point; y;(O) =O , zero angle of rotation of the

beam at the restraint point; y,(e,) =y2(eI), compatibility of the displacement of the beam

at the location of the first crack; y;(el) -y;(e,)= OlyG (e,), total change of the rotation

angle of the beam at the location of the first crack; y y(e,) = yg(e,), com patibility of the

bending mom ents at the location of the first crack; y(e,) = y;(e,), compatibility of the

shearing forces at the location of the first crack; y2(e2) = y3(e2), compatibility of the dis-

placements of the beam at the location of the second crack; y;(e2) -y (el) = e2y (e2), total

change of the beam rotation angle at the location of the second crack; y (e2) =y4(e2),

compatibility of the bending mom ents at the location of the second crack; y?(e2) =

yy(e2), com patibility of the shea ring forces at the location of the second crack;

y ;( 1) = 0, zero bending mom ent at the end of the beam; y Y(1) = 0, zero shearing force at

the end of the beam. Here eI and e2 are the distances between the end of the cantilever

beam and the crack locations.

The solution of equation (3.1) is sought in the form

y( 4, t) =y( 5) sin ut.

(3.2)

Substituting this solution into equation (3.1), after simple algebraic transformation, one

has

y( 5 ) - B 4_V ( ) = 0,

(3.3)


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96

W. M. OSTACHOWICZ AND M . KRAWCZ UK

=

3

cc cc

6

-0


7/11


197

where p 4 = 02pF/L4EJ. Taking the function y( 5) in the form of a sum of three functions,

vi(5)=A1cosh(P5)+&sinh(P5)+Gcos(P + G ~0 s (P5 ) + D3 s i n (P5 1 ,

SE(e2 ,

11,

(3.4)

and taking into account the boundary conditions one obtains the characteristic equation

(see Figure 8) which is to be solved to determine the characteristic roots. The roots are

used for the calculation of the natural vibration frequencies,

mi= (PiIL)Jm>

i=1,2 )...) n,

where W; is the ith natural vibration frequency of the beam and pi is the ith

root.

I.8

I

1.7 c__c------

_----

,,. . . . . . ...

. . . . . .

--.I.-

I.6 -

_..

. . ...

. ..

.H

.-

,_..-.--

. ..

....

.H

l-5 -

.

.

B 1,4- .

.

I.3 -

I.2 I-

I.1 -

3.5)

characteristic

I.0 1

0.1

0.2

/

I

0.3

0.4 0.5 0.6

0.7

O-0

Pos i t ion o f c rack 110 .2 L 2 / L

Figure 9. Effect of the second crack upon the fundamental frequency of the beam with two double-

sided cracks. The first crack: location

L /L=O.I

size

a /H=0.3

coefficient p, = 1.7602. The second crack:

-.

aZ/H=O I; - -- a2/H=0.3; .. .. a2/H=0.5; m.m aZ/H=0.7.

16

,

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Pos i t ion o f c rock no .2

L z / L



L /L=O.I

size

a /H=O.S

coefficient PI= 1.5837. The second crack:

-.

aJH=O.l

; -,

aZ/H=0.3;

..

a2/H=0.5; -

--. -.

a2/H=0 7.

The method of calculating the vibration frequencies of a beam w ith two cracks supported

in a different way is similar. It is only necessary to change the boundary conditions at the

beginning and the end of the beam .


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198

W .

M. OSTACHOWICZ AND M. KRAWCZUK

P I.4 :

-------.~~.~.=~~~.___.

_.

_._.-._a

1.3 ....

. . . . . . . . . .

.k

_ (

_. -

12-

1.1 -

IO

0.1

0.2

0.3

0.4

05

0.6

0.7

O-8

POSItiOn Of crack no.2, L2/~


sided cracks. The first crack: location L,/L=O.l, size

a,/H=0.7,

coefficient PI = 1.3906. The second crack:

-. aJH=O.l;

,

aJH=O.3;

,

a2,fH=0,5; ~.

- ,

a,/H=0.7.

4. NUMERICAL RESULTS

In order to carry out the num erical calculations the authors wrote a computer program

called BETA . The program was written in Fortran 77 language for the IBM PC /AT

microcompu ter. By means of this program it became possible to analyze the effect of the

positions and the magnitudes of two cracks upon the vibration frequencies of a cantilever

beam of the following dim ensions: length

L =

1 m, depth

H= 0.1

m, width B= 0.1 m. It

was assumed that the beam was made of steel with the parameters density p = 7860 kg/

m3 and Y oungs modulus E= 2.1 x 10 N/m2.

1.8

I.7

I.6

I.5

P

I.4

I.3

I.2

III

I

I

__-----

. . . . . .

- __--

--

,,......

_.

,,.+/

./.-.

-________

,Yf

.~.

.

I

,..

./

0

I

.

,.

/

I

/

/

0.;

I

/

I i

- /

I

/

I

I

I

I I

I-O 1

0, I

0.2

t 1 I

J

03 0.4 0.5

O-6 O-7 0.0

PositIon of crack no.2

L z / L

Figure 12. Effect of the second crack upon the fundamental frequency of the beam with two single-


L,/L=O.l,

size

al/H=0.3,

coefficient PI = 1.7665. The second crack:

-, u,jH=O.l: -,

aJH=0.3; , aJH=0.5; ~

-, a2jH =0,7.

In Figures 9- 11 is shown the effect of the second crack upon the fundamental frequency

of the cantilever beam with two double-sided cracks. The first is located at L,IL = 0.1. Its

size is defined by the ratio al/H. p appears to denote the characteristic root connected

with the fundam ental frequency of a cracked beam (see equation (3.5)). For the untracked


9/11


99

beam pP= 1.8751 (for the fundam ental frequency). j?, denotes the root (3.5) of a cracked

beam w ith one crack. The thin discontinuous lines show two localizations of the crack for

the same fundam ental frequency.

P

1.5 - ____-_-----,y , , . . . .. . . . .. . . . . . . . - ,_.-

-____

. . . . .

j-__-_

._..

i_ T

I.4 _.....

I

I .

.A

I

I

13 -

. Y

.I

I.2 -

i

I

0

I.1 1

1

I.0

I

0.1 o-2

t

0.3 0 4 0.5

0.6

0.7 0.8

Poshon of crack no.2.

Lz L


sided cracks. The first crack: location L,/L=O.l, size a,/H=0.5, coefficient PI = 1.5258. The second crack:

----.

aJH=O.l; -, a2/H=0.3; 1aJH=0.5; - -, a,, H=0,7.

1.8

1

0.3

0.4 O-5

0.6

0.7 08

Position of crack 130.2.

Lz L



L,/L=O.I,

size

a,/H-0.7.

coefficient j?, = 1.1856. The second crack:

--.

a2/H=0.1; - ~., 2/H=0.3;. .,a2/H=0.5; -. ~. -, a2/H=0.7.

In Figures 12-14 is show n the effect of the second crack upon the fundam ental frequency

of this same beam with two single-sided cracks. The first crack is located as previously.

Its size is defined by the ratio al/H. For the untracked beam the coefficient p, = 1.1875 1.

The calculated results are also presented in Tables 1 and 2. j3 is presented as a function

of the localization and size of the cracks. Here, as previously, L, and

L 2

denote distances

between the end of the beam and cracks, and

a l / H

and

az /H

dimensionless heights of the

cracks.

5. CONCLUSIONS

The results of the calculations indicate the following relationships between the position,

the magn itude of the crack a nd the first frequencies of the cantilever beam natural

vibrations.


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200

W. M. OSTACHOWICZ AND M. KRAWCZUK

TABLE

1

Sing l e - si ded crack : roo t s o f t he cha rac t er i s t i c equa t i on ( Ji r s t na t u ra l v i b ra t i on

f requency)

a z l H

L,/L=O.ll L,/L=O.2

L,/L=O.l, a,/H=0.3, /?, = 1.7665

0.1 I.7580 1.7604

0.3 1.6896 1.7063

0.5 1.4897 1.5448

0.7 1.1813 I.2561

L,/L=O.l, a,/H=0,5, p,= 1.5258

0.1 1.5207 1.5226

0.3 1.4852 1.4958

0.5 1.3702 1.4031

0.7 1.1376 1.1992

L,/L=O.l,

a,/H=0.7, p, =

1.1856

0.1 1.1840 1.1845

0.3 1.1738 1.1770

0.5 1.1362 1.1462

0.7 1.0214 1.0564

LJL=O4

L,/L=O.6

L,/L=O.S

1.7642 1.7661 1.7664

1.7414 1.7601 1.7659

1.6581 1.7402 I.7621

1.4500 1.6582 1.7563

1 5246 1.5250 1.5258

1.5134 1.5227 1.5250

1.4801 1.5113 1.5231

1.3452 1.4701 1.5210

1.1850 1.1856 1.1856

1.1823 1.1842 1.1847

1.1694 1.1831 1.1836

1.1252 1.1814 1.1824

TABLE

2

D oub l e - si ded c rack : roo t s o f t he cha rac t er i s t i c equa t i on ( f i r s t na t u ra l v i b ra t i on

f requency)

az/H

L /L=O.ll

L,/L=O.2 L,/L=O.4 L,/L=O.6 L,,L=O.8

L,/L=O.l, a,/H=0.3, p, = 1.7602

0.1 1.7496 1.7506

0.3 1.6762 I.6964

0.5 1.5354 1.5831

0.7 1.3689 1.4356

L,/L=O.l, a,/H=0.5, p,=l.5837

0.1 I.5774 1.5792

0.3 1.5318 1.5452

0.5 1.4426 1.4731

0.7 1.3062 1.3623

L,/L=O.l,

a,/H=0.7, PI =

1.3906

0.1 1.3869 1.3879

0.3 1.3712 1.3721

0.5 1.3061 1.3243

0.7 1.2212 1.2601

1.7564 1.7593 I.7601

1.7337

1.7512 1.7598

1.6772 1.7321

1.7562

1.5871

1.6812 1.7558

1.5817 1.5829

1.5835

1.5684

1.5791 1.5826

1.5342 1.5708 1 5794

1.4762

1.5491 I.5864

1.3884

1.3901 1.3905

1.3812 1.3896 1 3900

1.3643

1.3831 1.3892

1.3303

1.3742 1.3874

1. The positions of the cracks in relation to each other affect significantly the changes

in the frequencies of the natural vibrations in the case of an equal relative depth of the

cracks. Any decrease in the natural frequency is largest if the cracks are near to each

other; when the distance between the cracks increases the frequencies of the beam natural

vibrations also tend to the natural vibration frequencies of a system with a single crack.

2. In the case of two cracks of different depths, the larger crack has the most significant

effect on the natural vibration frequencies (see Tables 1 and 2 and Figures 9 and 10). This

is evident for the first natural vibration of a cantilever beam . F or other modes of vibration

this is not so clear, because the influence of a crack located at a node is negligible.


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E:FFECl OF CRACKS ON BEAM FREQUENCIES

LO1

3. Double-sided cracks affect the vibration frequencies to a smaller de gree than single

cracks with the same relative depth of crack and the same position [13]. How ever. the

differences are not very large.

ACKNOWLEDGMENTS

This research was carried out under Grant No. RPB P-02.8-3.7.2.2 and supported by the

Polish Academy of Sciences. The financial support of Polish Academy of Sciences is

gratefully acknowledged.

REFERENCES

1 B 0 DIRR and B. K. SCHMAIXORST 1987 Proceedings ofthe ASME Conference I 1h Biennial

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shaft by vibration measurem ent.

2. G. GOUNARIS and A. D. DIMAROCONAS 1988 Computers and Structures 28(3), 309~ 313.

A

finite element of a cracked prismatic beam for structural analysis.

3. C. A. PAPA~OPOULOS and A. D. DIMAROGONAS 1987 Journal of Sound and Vibration 117,

81L93.

Coup led longitudinal and bending vibrations of a rotating shaft with an open crack.

4. I. W. MAYES and W. G. R. DAVIES 1984 Transactions of the Am erican Society

q

Mechanical

Engineers, Jo urnalof Vibration, Acoustics, Stress, and Reliability in Design 106, I36 145. Analysis

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and stability of cracked thick rotating blades.

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l-291. Stability of columns with a single crack subjected to follower and vertical loads.

7. J. WAVER 1989 Proceedings

qf

the 16th International Conference on Dynamics of Machines.

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8. C. A.

PAPADOPOULOS and A. D. DIMAROGONAS 1988

Transaction s of the America n Soc,iety

o

Mecha nical Engineers, Journa l of Vibration, Aco ustics, Stress, and Reliability in Design 110.

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10. S. N.

ATLURI 1986

Computational Methods in the Mechanics of Fracture. Amsterdam : North-

Holland.

I I K. HELLAN 1984 Introduction to Fracture Mechanics. New York: McGraw-Hill.

12. B. S. HAISTY and W. T. SPRING ER 1988 Trunsaction s of the Ame rican Society, of Mechu nical

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Documents

Effect of Crack on Natural Frequency1