A unified approach to free vibration of locally damaged beams having various homogeneous boundary conditions

Journal of Sound and Vibration (1991) 147(3), 47.5-488

A UNIFIED DAMAGED

APPROACH BEAMS

TO

FREE

HAVING

BOUNDARY

VIBRATION

VARIOUS

OF

LOCALLY

HOMOGENEOUS

CONDITIONS A.

JOSHI

Aeronautical Engineering Department, Indian Institute of Technology, Bombay-76, India AND

B. S.

MADHUSUDHAN

Structures Division, National Aeronautical Laboratory, Kodihalli, Bangalore-17, (Received

India

15 March 1990, and in revised form 23 July 1990)

The problem of free vibration of uniform beams containing a local material damage has been studied with the aim of arriving at accurate closed form analytical expressions for the natural frequency for various homogeneous boundary conditions. Elementary beam theory is used, with the local material damage modelled as an effective reduction in Young’s modulus, and the exact solution of the transcendental equations for the frequency parameter is obtained for various symmetric and unsymmetric boundary conditions in terms of the identified damage parameters. The numerical results for the natural frequency show that it is possible to arrive at a simple polynomial type of representation which is the same for various boundary conditions and mode numbers. The results also show that the nature of changes in frequency with respect to the damage location is of the same type as the local curvature in the undamaged beams. Finally, the study demonstrates that the analytical expressions derived for representing the damage dependence can also be used, with slight modification, in situations where there is local stiffening. 1. INTRODUCTION Beams are one of the most commonly used structural elements in numerous engineering applications and thus experience a wide variety of static and dynamic loads. Even though in the design one assumes the material property to be invariant throughout its designed life, there are many situations in practice in which these properties could change because of continuous wear and tear suffered during operation. This is particularly so in cases in which the component is designed to take loads beyond the material yield limit. The growth of microcracks under cyclic loading is another case in which a reduction in local bending stiffness could result. However, even if the area under damage is too small to endanger overall structural integrity and reliability, its influence on the dynamic characteristics could be important. This is particularly relevant from the dynamic response point of view, as a shift in natural frequency or changes in the mode shape could either increase or decrease the response and this makes the problem of vibration of locally damaged beams quite important. The earliest attempt to tackle stiffness discontinuities in beams dates back to 1949, when Thompson [l] analyzed the influence of a small slot on the free vibration characteristics of uniform cantilever beams. It may be noted that a slot, while modelling a change in the stiffness pattern, also involves a reduction in the inertia and therefore does not truly represent local material damage. In 1983 Sato [2] revisited the same problem with 475 00?2-460X/91/120475+14$03.00/0

0 1991 Academic

Press Limited

476

A. JOSH1

AND

B. S. MADHUSUDHAN

a new approach which he called the Transfer Matrix Method, and combined it with the finite element method. Dimarogonas and Papadopoulos [3] in their study have analyzed the influence of an edge crack of a specified size on the vibration of a circular shaft. However, instead of modelling the crack as a local loss of stiffness, they have related the crack to an overall reduction in the bending stiffness of the whole shaft. Only Cawley and Adams [4] and recently Yuen [5] have modelled the local changes in the Young’s modulus as an independent local effect using the finite element procedure. Whereas Cawley and Adams [4] used a kind of stiffness matrix sensitivity approach for estimating the effect of the damage strength, Yuen [5] used the Timoshenko beam formulation and presented frequency results only for a cantilever beam and has also restricted himself to damage of a fixed size. It would be extremely useful at the design stage if one could estimate, reasonably accurately, the changes in the natural frequency due to damage for a variety of boundary conditions using simple algebraic expressions. The present study has been aimed at exploring the nature of variations of the natural frequency with respect to the size of the damage and of its extent and location for cases of classical boundary conditions, with a view to arriving at such a unified algebraic expression. For this purpose, a uniform slender beam, the motion of which can be described by using elementary beam theory, has been chosen for analysis. 2. FORMULATION AND SOLUTION In Figure 1 is shown the geometry of the damaged beam, in three separate segments, and the co-ordinate systems to be used in each of the three beam segments. The governing differential equations for the three segments can be written in dimensionless form as (a list of nomenclature is given in the Appendix) (a”w,/aZf) - Aqwi= 0,

(1) where subscript i refers to the ith beam segment and hi is the dimensionless frequency parameter in the ith segments. It may be noted that as the modulus values in the first and the third segment are the same, the frequency parameter definitions in these segments are same and that the frequency parameter in the second segment can be defined in terms of either of the other two frequency parameters. It may also be noted that as the frequency parameter in each of the three segments is normalized with respect to the total beam length, the dimensionless co-ordinate R in the first segment varies from 0 to z$, that in the second segment from 0 to (fs - ff) and that in the third segment varies from 0 to (1 - &).

Figure

1. Geometry

of a locally

damaged

uniform

slender

beam and segmental

co-ordinate

systems.

VIBRATION

OF

LOCALLY

DAMAGED

The general solution of equation (1) for displacement written as Wi

=

Ai cash h~i + Bi sinh A& + Ci

COS

477

BEAMS

in all the three segments can be A&

+

Di

sin h&i,

(2)

where Ai, Bi, Ci and Oi are four arbitrary constants which describe the exact solutions in each of the three beam segments, and hi and Xi are as explained above. The above general solution of equation (2) contains a total of 12 unknown constants which are to be determined by using 12 conditions on the displacement. These conditions are as follows: four boundary conditions at the two end points and eight continuity conditions on displacement, slope, shear force and bending moment at the two junction points of the three segments. In dimensionless form these 12 conditions are $w,/d$

(q)

Wl (Zf) = wz (O), (0) (1 -

= d2w,/ds;

wz (2s) = a*w,/aff:

(q) = a2w3/a$

W/W Q j,

(Xf) = W/G a3w,/af:

aw,/aff,

W3 (Oh

(O)/( 1 - CT),

(O),

(iyj =a3w2/a$

(zqj =aw,/aff,

a3w2/a$

(0) (1 -cY),

(O),

(fs)=a3~3/a37:

(0)/l

-a),

Therefore, by application of conditions described by equations (3 j to the general solution (2), one can obtain a 12 x 12 characteristic determinant for various cases of homogeneous boundary conditions. 3. RESULTS AND DISCUSSION 3.1. INFLUENCE OF THE DAMAGE In Tables l-4 are presented numerical results which bring out the influence of the damage parameters (Yand d on the normalized frequency parameter, ;i, for four different TABLE

Variation

_

1

of the normalized frequency parameter, h, with the damage parameter, symmetric boundary conditions; d = O-0, d = 0.2, .ZO= O-5

CK,for

Boundary conditions

(Y

c-c

0.0

1 .oooo

1 .oooo

1 .oooo

1 .oooo

1 .OOOO

0.1

0.9928

0.9981

0.9928

0.9951

0.9953

P-P

1,

AZ

G

AS

i,

0.2

0.9846

0.9958

0.9849

0.989

0.3

0.9751

0.9928

0.9763

0.9817

0.9847

0.4

0.9638

0.9888

0.9669

0.9723

0.9786

1

0.9902

0.5

0.9505

0.9834

0.9565

0.9600

0.9715

0.6

0.9343

0.9754

0.9449

0,9434

0.9627

0.7

0.9141

0.9626

0.9317

0.9200

0.9506

0.0

1 *oooo

1 .oooo

1 .oooo

1 .oooo

1.0000

0.1

0.9895

0.9986

0.9921

0.9959

0.9948

0.2

0.9771

0.9967

0.9834

OX@6

0.9893

0.3

0.9623

0.9948

0.9738

0.9841

0.9834

0.4

0.9441

0.9919

0.9632

0.9757

0.9769

0.5

0.9213

0.9880

0.9512

0.9646

0.6

0.8915

0.9822

0.9377

0.949

0.7

0.8509

0.9727

0.9222

0.9264

0.9696 1

0.9610 0.9500

478

A. JOSH1 AND

B. S. MADHUSUDHAN TABLE

2

Variation of the normalized frequency parameter, ;i, with the damage length parameter, (r, for symmetric boundary conditions; d = 0.0, X0 = O-5, LY= 0.7 Boundary conditions d c-c

P-P

1,

L

A3

14

is

0.05 0.10 0.15 0.20 0.25 0.30

1~0000 0.9647 0.9409 0.9248 0.9141 0.9075 0.9038

1~0000 0.9993 0.9945 0.9826 0.9626 0.9371 0.9102

1~0000 0.9585 0.9397 0.9329 0.9321 0.9312 0.9266

1~0000 0.9976 0.9824 0.9520 0.9200 0.8991 0.8904

1~0000 0.9657 0.9578 0.9572 0.9506 0.9290 0.8989

o*oo 0.05 0.10 0.15 0.20 0.25 0.30

1~0000 0.9489 0.9089 0.8769 0.8509 0.8293 0.8115

1~0000 0.9995 0.9962 0.9878 0.9727 0.9514 0.9255

1~0000 0.9572 0.9354 0.9254 0.9222 0.9219 0.9210

1~0000 0.9981 0.9858 0.9582 0.9264 0.8996 0.8837

1~0000 0.9641 0.9541 0.9531 0.9500 0.9354 0.9087

0.00

TABLE

3

Variation of the normalized frequency parameter, i, with the damage parameter, unsymmetric boundary conditions; d = 0.0, d = 0.2, X0 = 0.5 Boundary conditions (Y

XI

1,

x,

L

CY,for

L

C-P

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1~0000 0.9930 0.9848 0.9751 0.9636 0.9494 0.9318 0.9089

l*OOpO 0.9975 0.9945 0.9908 0.9863 0.9804 0.9724 0.9604

1~0000 0.9931 0.9854 0.9770 0.9675 0.9568 0.9444 0.9295

1*0000 0.9952 0.9895 0.9825 0.9738 0.9624 0,9472 0.9263

1~0000 0.9950 0.9895 0.9834 0.9765 0.9684 0.9586 0.9456

C-F

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1~0000 0.9973 0.9940 0.9899 0.9844 0.9770 0.9662 0.9495

1~0000 0.9898 0.9780 0.8642 0.9478 0.9279 0.9032 0.8715

1~0000 0.9980 0.9954 0.9922 0.9880 0.9822 0.9737 0.9601

1~0000 0.9929 0.9851 0.9767 0.9675 0.9573 0.9460 0.9331

1*0000 0.9951 0.9891 0.9818 0.9724 0.9602 0.9437 0.9204

cases of boundary conditions and the first five modes of vibration. The boundary conditions considered in the present study are clamped-clamped (C-C) and pinned-pinned (P-P), representing symmetric configurations, and clamped-pinned (C-P) and clamped-free (C-F), representing unsymmetric configurations. The damage parameter, LY,is a measure of the local reduction in the modulus and is defined as a ratio of the reduction in modulus to the undamaged modulus. It can be seen from Tables 1 and 3 that as (Y increases from

479

VIBRATIONOF LOCALLYDAMAGEDBEAMS TABLE 4

Variation of the normalized frequency parameter, ;i, with the damage length parameter, d, for unsymmetric boundary conditions; d = O-0, X0 = 0.5, a = 0.7

Boundary conditions C-P

0.00 0.05 0.10 0.15 0.20 0.25 0.30

1.oooo 0.9662 0.9416 0.9232 0.9089 0.8974 0.8877

1.oooo 0.9933 0.9861 0.9757 0.9604 0.9404 0.9176

1.oooo 0.9636 0~9452 0.9355 0.9295 0.9243 0.9182

1.oooo 0.9924 0.9789 0.9540 0.9255 0.9043 0.8922

1.oooo 0.9696 0.9593 0.9534 0.9456 0.9291 0.9037

C-F

0.00 0.05 0.10 0.15 0.20 0.25 0.30

1.oooo 0.9869 0.9742 0.9618 0.9495 0.9372 0.9248

1~0000 0.9504 0.9153 0.8899 0.8715 0.8580 0.8481

1.oooo 0.9992 0.9941 0.9824 0.9601 0.9324 0.9027

1.oooo

1.oooo

0.9591 0.9407 0.9341 0.9331 0.9324 0.9274

0.9976 0.9825 0.9521 0.9204 0.8998 0.8913

-

zero to O-7, the normalized frequency parameter, 1, decreases for all the cases of boundary conditions and mode numbers. The results for natural frequency show that for 6= 0.2, 20= O-5 and (Y= 0.7, the reductions in the value of 1 in the first mode of the C-C, P-P, C-P and C-F cases are 8*5%, 15%, 10% and 5%, respectively. It may be noted here that the frequency parameter, I\, is proportional to & and therefore the percentage reduction in o will be about twice this value. It can be seen that the changes for the same span-wise location (ZO= 0.5) and damage length are different for different boundary conditions as well as in different modes, and this indicates that, as expected, local curvature effects are quite important. Similarly, it can be seen from Tables 2 and 4 that as the damage length parameter, a, increases from 0 to 0.3, the frequency parameter, h, decreases monotonically. In Figures 2 and 3 are shown the effects of damage on the first deformation mode shape for all end conditions, for symmetric damage location. It is interesting to note that while for symmetric boundary conditions (C-C and P-P) the peak location remains the same and only the curvature pattern changes, for unsymmetric end conditions the location of the peak amplitude also changes. 3.2. NATURE OF THE DAMAGE DEPENDENCE It is logical that when either cx or d is zero, the effect of the damage must disappear, irrespective of its location, and this is indeed so, as can be seen from the results presented in Tables 1-4. This clearly shows that the influences of (Y and d must appear as a polynomial product, and, therefore, one can investigate the existence of a simple, yet sufficiently general, expression for the normalized frequency parameter, h,, as follows: ii, = 1 - (Y=-Pf(qJ.

Here, m and n are exponents which are dependent on the boundary conditions and the mode number and f(3,) is a function which depends on the location of the damage. In Table 5 are presented the complete range of values of exponents m and n which are estimated by a least-squares technique. The term f0 in Table 5 is the damage influence

480

A. JOSH1

AND

B. S. MADHUSUDHAN

06

o-4

0.2

0.0

02

0.4

0%

I.0

06

Figure 2. Variation of the norm_alized displacement, w, for the first mode for the cases of (a) C-C and (b) P-P end conditions; Z,, = 0.5 and d = 0.2. -, (Y = 0.0; - - -, (Y = 0.7.

0.6 -

Figure 3. Variation of the normalized displacement, w, for the first a =o.o; - - -, (I =0.7. C-F end conditions; Z0= 0.5 and d = 0.2. -,

mode

for the cases of (a) C-P and (b)

VIBRATION

OF

LOCALLY TABLE

DAMAGED

481

BEAMS

5

Exponents of the damage parameter, (Y, and the damage length parameter, Li;, and the averaged multiplier, fO, at f0=0*5, for dejning the algebraic function for the normalized frequency parameter, h Coefficient boundary conditions

Mode

1

Mode 2

Mode 3

Mode 4

Mode

5

1.4309

1.9094

C-P P-P C-F

1.5036 1.5917 1.8583

1.7081 1.9103 1.4799

1.2355 1.3063 1.2664 1.9025

1.7018 1.6571 1.7758 1.2318

I.3900 1.3943 1.2991

n/C-C C-P P-P C-F

0.4270 0.5969 0.6562 0.9783

2.4516 1.5856 2.6455 0.5238

0.2688 0.4117 0.2063 2.4635

1.5120 1.2499 1.7600 0.2797

1.0558 0.9731 0.9288 1.5066

_&/C-C C-P P-P C-F

0.2804 0.4093 0.7665 0.5024

3.8369 1.0282 3.9399 0.5047

0.1638 0.2197 0.1666 4.1651

1.5368 0*9557 2.2033 0.1637

0.5726 04467 0.4393 1.5151

m/C-C

1.7002

function for the case in which the damage is located at mid-span. It can be seen from Table 5 that while values of the exponent m for all cases lie in the interval {1.0 < m < 2-O}, values of the exponent n have a greater dispersion. However, one can see that for symmetric boundary conditions cases, represented by C-C and P-P, both m and n are more or less close to each other, and that for the case C-P the values always lie in between the values for the two symmetric cases. The case of unsymmetric end condition C-F, is quite different from the other three cases because of the presence of a free end, and this is clearly brought out in values of m and n for this case. It may be mentioned here that while the unified expression in equation (4) contains explicit expressions for both (Y and a, the dependence on damage location f, is not explicit. In fact, Table 5 gives averaged values of f0 which are values of the function f(%,) for the case when f0=0*5. Tables 6 and 7 completely describe the function f(&) for various values of &, for all the cases of boundary conditions and modes. In Figures 4 and 5 are presented typical changes in the first mode when the damage location is made unsymmetric at Z,,= 0.1, and it can be seen that even the symmetric modes become unsymmetric and the peak amplitude location shifts towards the damage by about 5%. Furthermore, it can be seen from Tables 6 and 7 that the variation of the function f(%) appears to be similar to the pattern of the curvature variation and this hypothesis is confirmed in Figure 6. Therefore one can reasonably accurately express the function f (x0) as f(3,)

It can be seen the normalized normalized to the normalized representation

= f~{a2w(n,)/an2}/{~2w(0~5)/d~~},

(5)

that in order to quantify the effect of the damage location, one needs only modal curvature variation of the undamaged structure which has been its value at f0 = 0.5. Thus, one can conclude that algebraic expressions for frequency parameter, i, given by equations (4) and (5) provide a complete of the effect of the local material damage.

482

A. JOSH1 AND

R. S. MADHUSUDHAN TABLE

6

Variation of the damage location function, f(.$,), with the damage location parameter, x0, for symmetric boundary conditions Boundary conditions

20

Mode

1

Mode

2

Mode

3

Mode

4

Mode

c-c

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0.2345 0.1431 0.0858 0.0745 0.1080 0.1676 0.2275 0.2696

6.8154 5.4871 5.9060 7.1321 7.7452 7.1730 5.7936 4.4039

0.1875 0.1451 0.1303 0.1334 0.1413 0.1485 0.1552 0.1609

1.4410 0.9642 1.1357 1.5456 1.5080 1.2152 1.1545 1.4808

0.5217 0.4490 0.6753 0.6053 0.4499 0.5747 0.7373 0.5963

P-P

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

0.1385 0.2445 0.3632 0.4773 0.5762 0.6549 0.7117 o-7451

10.2781 13.8670 15.4032 14.9703 13.0850 10.3060 7.2617 4.7899

0.1736 0.1771 0.1590 0.1347 0.1178 0.1180 0.1358 0.1594

1.9845 2.1189 2.3238 2.3366 2.1285 1.9013 1.9557 2.2246

0.4046 0.5676 0.5556 0.4202 0.3607 0.5095 0.5790 0.4606

TABLE

5

7

Variation of the damage location function, f(%,), with damage location parameter, &,, for unsymmetric boundary conditions Boundary conditions

.f,

Mode

1

Mode

2

Mode

3

Mode

4

Mode

5

C-P

0.10 0.20 0.30 0.40 0.60 0.70 0.80 0.90

0.3677 0.1349 0.083 1 0.2234 0.5192 0.5170 0.3847 0.1671

1.4914 1.1705 1.8548 1.7014 0.9085 1.7722 2.4259 1.8878

0.2393 0.1808 0.1762 0.1793 0.2114 0.1824 0.2090 0.2325

0.9774 0.6790 0.9909 0.7587 0.8842 0.8343 1.0395 0.7938

0.4692 0.5700 0.4236 0.6228 0.6023 0.4330 0.5858 0.4661

C-F

0.10 0.20 0.30 0.40 0.60 0.70 0.80 0.90

1.8483 1.4867 1.1232 0.7766 0.2408 0.0946 0.0253 0.0037

0.2722 0.1060 0.1646 0.3612 0.5239 0.4033 0.1926 0.0398

6.9093 5.9223 7.6990 5.7560 7.1379 11.3043 11.4601 4.9352

0.1906 0.1326 0.1436 0.1563 0.1602 0.1731 0.2093 0.1864

1.4271 1.1252 1.4961 1.1439 1.1211 1.5500 1.6785 14485

VIBRATION

OF

00

LOCALLY

02

DAMAGED

04

O-6

BEAMS

483

08

.F,(x/a)

Figure 4. Variation P-P end conditions;

of the normalized displacement, w, for the first mode for the cases of (a) C-C 6=0.2. -, o =O.O and P0=0.5; - - -, (Y=0.7 and ~,=0.1.

I&

, -

,

(

/

,

,

(

and (b)

,

(b)

0.8 -

0.6 -

0.0

0.2

04

06

10

?,?,(x/u)

Figure 5. Variation of the normalized displacement, w, for the first mode for the cases of (a) C-P and (b) C-F end conditions; d = 0.2. -, a = 0.0 and Z0 = 0.5; - - -, a = 0.7 and X0 = 0.1.

484

A. JOSH1

AND

B. S. MADHUSUDHAN

0.8

06

0.4

0.0

01

I 0.3

I 0.5

I 0.7

0.9

x0, (&Jo)

Figure 6. Variation of the damage location function, f(&,) us. the damage location parameter, x0, for (a) @ C-P, @ C-F, and (b) @ C-C, a P-P end conditions. d = 0.2 and a = 0.7.

3.3. ERROR ESTIMATES AND EXTENSION TO LOCAL STIFFENING It was mentioned earlier that the exponents M and n and the functional amplitude f0 have been arrived at by carrying out a least-squares fit over the complete sample. It is also clear from Tables l-4 that the selected range of the damage parameter, (Y,is up to 0.7 and the selected range for the damage length parameter, 4, is up to 0.3. It is felt that these limits are more than adequate for representing any practical damage which is to be classified as localized. Therefore, one can say that equations (4) and (5) are valid in the range defined by (0.0 < LY< 0.7) and (0.0 < d < 0.3) and are expected to give reasonably accurate estimates for the frequency of the damaged beam. However, it is necessary to assess the nature and extent of the error band which is inherent in these two expressions. For this purpose three different cases of cy and d have been considered for error analysis, covering all the modes and all the boundary conditions, and the results are presented in Tables 8-10. It is clear from these results that, except for a few isolated cases in which the errors in frequency prediction are around 3%, in all other cases the error is 1% or less. Therefore, it can be concluded that the algebraic expressions (4) and (5) not only represent a reasonably accurate solution to the problem of vibration of locally damaged beams, but also describe a unified solution for the frequency parameter for various boundary conditions and modes. An interesting offshoot of the present study is the situation in which, instead of a local damage, there is a local stiffening of the cross-section, which could be the result of a repair carried out on a local damage. Logically, an expression similar to the one presented in equation (4) should be valid for this case also, and to that purpose an attempt has been made to interpret equation (4) for the case when there is local stiffening. It has been found after investigation that it is possible to arrive at a similar expression for 1 for the case of local stiffening, namely i, = 1 +P”d”f(n,),

(6)

VIBRATION

OF LOCALLY TABLE

DAMAGED

485

BEAMS

8

Estimates of the error in the algebraic expression for the normalized frequency parameter, h; a = 0.7, d = 0.2, f0 = 0.5 B.C./mode

x,

10

c-c/

1 2 3 4 5

0.9141 0.9626 0.9317 0.9200 0.9506

0.9153 0.9624 0.9316 0.9265 0.9362

-0.0012 0.0002 0~0001 -0.0065 0.0144

-0-1359 0.0156 0.0106 -0.7088 1.5109

2 3 4 5

0.9089 0.9604 0.9295 0.9263 0.9456

0.9084 0.9564 0.9289 0.9289 0.9433

0.0005 0.0040 0.0006 -0.0026 0.0023

0.0555 0.4135 0.0618 -0.2818 0.2472

5

0.8509 0.9727 0.9222 0.9264 0.9500

0.8489 0.9718 0.9239 0.9312 0.9380

0.0020 0.0009 -0.0017 -0.0048 0.0120

0.2361 0.0939 -0.1858 -0.5143 1.2622

2 3 4 5

0.9495 0.8715 0.9601 0.9331 0.9204

0.9464 0.8719 0.9599 0.9327 0.9269

0.003 1 -0.0004 0.0002 0.0004 -0.0065

0.3295 -0.0417 0.0196 0.0383 -0.7045

C-P/l

P-P/l 2 3 4 C-F/l

TABLE

A;i

% Difference

9

Estimates of the error in the algebraic expression for the normalized frequency parameter, X; (Y = 0.3, d = 0.2, X0= O-5 B.C./mode C-C/l

A,

1, 0.9748 0.9926

A#i

% Difference

4 5

0.9751 0.9928 0.9763 0.9817 0.9847

2 3 4 5

0.975 1 0.9908 0.9770 0.9825 0.9834

0.9826

3 4 5

0.9623 0.9948 0.9738 0.9841 0.9834

0.9608 0.9944 0.9740 0.9847 0.9794

0.0015 0.0004 -0*0002 -0.0006 0.0040

0.1587 0.0393 -0.0185 -0.0622 0.4088

2 3 4 5

0.9899 0.9642 0.9922 0.9767 0.9818

0.9889 0.9634 0.9920 0.9763 0.9827

0~0010 0.0008 0.0002 0.0004 -0.0009

0.1017 0.0797 0.0199 0.0394 -0*0903

2 3

C-P/l

P-P/l 2

C-F/l

0.9804

0.0043

0.0292 0.0249 0.0319 -0.0942 0.4404

0.9744

0.0007 0~0010 0.0005 o*oOOO 0.0008

0.0741 0.1058 0.0510 -0.0041 0.0824

0.9760 0.9826

O-9898 0.9765

0.9825

0*0003 0*0002 0.0003 -o*ooo9

A. JOSH1

486

AND

B. S. MADHUSUDHAN

TABLE

Estimates

B.C./mode C-C/l

x,,

AA

x0

% Difference

2 3 4 5

0.9038 0.9102 0.9266 0.8904 0.8989

0.8993 0.8985 0.9237 0.8644 0.9022

0.0045 0.0117 0.0029 0.0260 -0.0033

0.4935 1.2817 0.3103 2.9254 -0.3635

2 3 4 5

0.8877 0.9176 0.9182 0.8922 0.9037

0.8833 0.9171 0.9160 0.8820 0.9158

0.0044 0.0005 0.0022 0.0102 -0.0121

0.4943 0.0515 0.2382 1.1438 - 1.3409

2 3 4 5

0.8115 0.9255 0.9210 0.8837 0.9087

0.8028 0.9175 0.9173 0.8595 0.9097

0.0087 0.0080 0.0037 0.0242 -0*0010

1.0684 0.8612 0.4045 2.7405 -0.1057

2 3 4 5

0.9248 0.8481 0.9027 0.9274 0.8913

0.9203 0.8415 0.8912 0.9247 0.8653

0.0045 0.0066 0.0115 0.0027 0.0260

0.4907 0.7731 1.2791 0.2948 2.9150

C-P/l

P-P/l

C-F/l

where

10

of the error in the algebraic expression for the normalized frequenc_y parameter, h; (Y = 0.7, 2 = 0.3, X0 = 0.5

p is a modified

stiffening

parameter

and is defined

as

p = E/E’. are presented in Table The results of such investigations conditions and modes for a particular stiffening, and it can be predict the effect of local stiffening for values of p up to 0.5, about 3%. Thus it can be seen that, within reasonable error predict the effect of local stiffening by using expression (6).

(7) 11, for all the boundary seen that it is possible to with a maximum error of bounds, it is possible to

4. CONCLUSIONS

In this study the problem of free vibration of locally damaged beams has been investigated, with the damage being taken as an effective reduction in local Young’s modulus. Elementary beam theory has been employed to arrive at an exact solution for various cases of homogeneous boundary conditions and for a range of damage parameters. The results show that it is possible to express the entire range of frequency results by a single algebraic expression in which only the exponents and the location function depend on the boundary conditions and mode number, and these have been tabulated for four cases of boundary conditions. This simple expression is found not only to provide a unified framework for analysis of the free vibration of locally damaged beam, but also to be quite accurate in the range of interest. Finally, the study has also established that, with slight re-interpretation, the same algebraic expression can be used to predict reasonably accurately, the effect of local stiffening.

VIBRATION

OF LOCALLY TABLE

Errors

DAMAGED

487

BEAMS

11

in the prediction of the e$ect of focal sti$ening on i, as obtained by using the damage representation; p = O-5, d = 0.2, &, = O-5

B.C./mode c-c/

i,

Kl

A/i

% Difference

1

1.0409

2

1.0086

1.0523 1.0198

-0.0114 -0.0112

3 4 5

1.0497 1.0234 1.0363

1.0451 1.0414 1.0399

0.0046 -0.0180 -0.0036

-1.1056 0.4349 -1.7633 -0.3516

2 3 4 5

1.0384 1.0130 1.0454 1.0251 1.0365

1.0552 1.0245 1.0458 1.0407 1.0355

-0.0168 -0.0115 -0.0004 -0.0156 0~0010

-1.6210 -1.1377 -0.0379 -1.5224 0.0973

2 3 4 5

1.0551 1.0126 1.0516 1.0198 1.0395

1.0885 1.0148 I.0497 1.0379 1.0400

-0.0334 -0.0022 0.0019 -0.0181 -0.0005

-3.1609 -0.2208 0.1818 -1.7721 -0.0519

2 3 4 5

1.0124 1.0566 1.0092 1.0490 1.0233

1.0287 1.0779 1.0211 1.0444 1.0413

-0.0163 -0.0213 -0.0119 0.0046 -0.0180

- 1.6098 -2.0139 -1.1826 0.435 1 -1.7554

C-P/l

P-P/l

C-F/l

- 1.0960

REFERENCES 1. W. T. THOMSON

1949 Journal ofApplied Mechanics 16, 203-207. Vibration of slender beams with discontinuities. 2. H. SATO 1983 Journal ofSound and Vibration 89, 59-64. Free vibration of beams with abrupt changes

in cross-section.

and C. A. PAPADO~OULOS 1983 Journal of Sound and Vibration 91, 3. A. D. DIMAROGONAS 583-593. Vibration of cracked shaft in bending. 4. P. CAWLEY and R. D. ADAMS 1979 Journal of Strain Analysis 14(2), 49-57. The location of defects in structures from measurements of natural frequency. 5. M. M. F. YUEN 1985 Journal of Sound and Vibration 103, 301-310. A numerical study of the eigen parameter of a damaged cantilever.

APPENDIX: :

A,, 4, C, Q

NOMENCLATURE

beam span cross-sectional area arbitrary constant in equation (2) clamped boundary condition = d/a, dimensionless damage length undamaged Young’s modulus value damaged/stiffened modulus value damage location function value off(&) at R, = 0.5 free boundary condition area moment of inertia exponent of 0 exponent of d pinned boundary condition

parameter

488 Wi 5 .fi X0 % -3 % ; a/ax A hi A

A -0

L 0

A. JOSH1

AND

B. S. MADHUSUDHAN

transverse z-displacement length co-ordinate in ith beam segment = x,/a, dimensionless value of x, dimensionless location of the damage = Z. - d/2,first junction point location = go+ d/2, second junction point location = 1 - ZS,,beam right end location = {E - E'}/ E,damage parameter = E'/E,modified stiffening parameter partial differential operator = pAw*a"/ EI,frequency parameter = pAo2a4/E,Z, segmental frequency parameter - A/A,, normalized frequency parameter value of A for a = 0 algebraic expression of equations (4) and (6) radian frequency of vibration