Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

Journal

of Sound

COUPLED

and Vibrurion (1987) 117(l),

LONGITUDINAL

A ROTATING C.

A.

SHAFT

PAPADOPOULOS

Machine Design Laboratory,

(Received

81-93

AND BENDING WITH AND

A.

VIBRATIONS

AN OPEN D.

University

OF

CRACK

DIMAROG0NAS.t

f Patras, Patras, Greece

19 June 1986, and in revised form 24 October 1986)

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6 x 6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3 x 3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.

1. INTRODUCTION

It is well known that the presence of a transverse crack in a structural member introduces local flexibility, which for a beam can be described by a local flexibility matrix, the dimension of which depends on the number of the degrees of freedom considered, the maximum being 6 x 6. Such a matrix was first introduced for beams of rectangular cross-section with transverse surface cracks by Dimarogonas and Paipetis [l] for 5 degrees of freedom, with torsion neglected. Some of the elements of this matrix, identified as direct compliances, were computed previously by several authors. Thus Okamura [2] and Liebowitz et al. [3,4] have computed the diagonal element corresponding to tension, Rice and Levy [5] have computed the four elements corresponding to tension, bending and their coupling terms, Dimarogonas and Massouros [6] have computed the diagonal element related to shear parallel to the crack edge and Dimarogonas and Paipetis [l] have given a complete 5 x 5 matrix. Petroski [7] investigated the forced response of cracked beams, evaluating the crack rotational stiffness by correlation with the three point test result available for cracked beams. A full 6 x 6 flexibility matrix for a transverse surface crack on a shaft was introduced by Papadopoulos and Dimarogonas [8]. The elements of this matrix were computed by t Now at the Department of Mechanical Engineering, Washington University, St. Louis, Missouri 63130, U.S.A. 81 0022460X/87/160081 + 13 $03.00/O @ 1987 Academic Press Limited

82

C. A. PAPADOPOULOS

AND A. D. DIMAROGONAS

way of the analysis presented in reference [l]. This analysis is based on available expressions of the Stress Intensity Factor (SIF) and the associated expressions of the Strain Energy Density Function (SEDF). The diagonal element of the flexibility matrix corresponding to pure bending has been the subject of several previous investigations, [9-201. In general, of the full 6 x 6 stiffness matrix available to model a crack, only the diagonal terms have been utilized for dynamic analysis of cracked rotors. Use of the off-diagonal terms will couple the equations of motion with interesting consequences on coupled vibrations. This is the subject of the present investigation.

2. CRACK MODEL

Consider a shaft with given stiffness properties, radius R = D/2, and a transverse crack of depth (Y(see Figure l(a) and (b)). The shaft is loaded with axial force P,, shear forces P2 and P3, bending moments P4 and Ps and torsional moment P6. The dimension of the local flexibility matrix depends on the number of degrees of freedom, here 6 x 6.

.

.

.

.

(a)

(b) Figure 1. (a) A cracked shaft element in general loading; (b) the crack section of the shaft.

ROTATING

SHAFT

WITH

Paris’ equation [21] gives the additional the i direction, as

AN OPEN

CRACK

displacement

83

ui due to a crack of depth

(Y,in

(2.11 where J(a) is the Strain load. The SEDF is [l]

Energy

Density

Function

(SEDF)

and Pi the corresponding

(2.2) where E’ = E or E / ( 1 - Y’) for plane stress and plane strain respectively, of elasticity, m = 1+ V, v is the Poisson ratio (Y = 0.3 for steel) and Stress Intensity Factors (SIF) for the i = Z, ZZ, ZZZ modes and for j = 1, index. The local flexibility due to the crack per unit width is, by definition

E is the modulus K, are the crack 2, . . . ,6, the load [l],

au+ j-;

J(a) da],

Cc=z-apia4 or, after integration

along the width 2b of the crack,

“=aP; a*

The values of SIF in of unit thickness with permissible to integrate is variable and that the that this approximation reference [213,

J(a) da dx

(2.3)

1.

(2.4)

equation (2.2) are well known from the literature [21] for a strip a transverse crack. Since the energy density is a scalar, it is along the tip of the crack it being assumed that the crack depth stress intensity factor is given for the elementary strip. It is known yields acceptable results to engineering accuracy [l]. From K,, = a,&F,(a/h),

WI= P,/(nR*),

K,d= a,&F,(a/h),

a, = (4P4/vR4)x, u5= (4P,/rR4)(R2-x2)“*,

K,, = q&F&/h),

K,* = K,, = K,, = 0, u3 = kP,/( rR*),

K,,3 = ud=F,,(ffIh),

weI1 = 2P,x/rR4,

K,M= ~d=F,,(~lh), K,, , = J&,12

=

K,,2

=

&,4

=

=

w2 =

~,J?rcuF,,,(~lh),

K ,116= u,,,,J?rcYF,,,(alh), K ,111- K,,,, =

&,s

0,

kP,/( rR*),

u6,,, = 2P,( R* - x2)“*/ rR4, K,,,4

=

&,,s

=

(2.5)

0,

where F,(~/h)=(tanh/h)“2[0~752+2~02(a/h)+0~37(1-sinA)3]/cosh, F,(a/h)

= (tan A/A)“*[0*923+0*199(1

-sin

A)4]/cos

A,

F,,(a/h)=[1~122-0~561(c~/h)+0~085((u/h)~+0~18(a/h)~]/(l-~u/h)”~, F,,,(a/h)

= (tan A/A)“‘.

Here k = 6( 1 + v)/ (7 + 6 v) is a shape coefficient

A = m/(2h).

for circular

cross section.

84

C. A. PAPADOPOIJLOS

AND

A. D. DIMAROGONAS

Combining relations (2.2), (2.4) and (2.5) yields the dimensionless terms of the compliance matrix: c,, = ?rERc,,/(l-v2)=4 &= aER*~,~l(l-

v*) = 16

IYIb 0 0

y(l -Z’)F;(@

d

6

II0

0

v*) = 32

?I‘$= ?rER*c,,J( 1- v’) = 8 E45=~ER3~45/(1-v2)=64

I?33 =

J(1 -n’)“*F,(@F,(@

Ub 0 0

& = PER’c,, / ( 1 - v*) = 64 & = 7~ER~c~/(l-

IYb 0 0

Ub 0 0

~ERc~~l(l-

XF;( K) dx dj,

lilb 0 0

v2) = 4

&l

-

Ub 0 0

dff dy, df dy,

f*JF:( 6) dff djj, zjF:( i) dff dy, -Z*F,(@F,(@

dff dJ,

pF;,( 6) dff dj,

jF:,,( Ii) d2 dy,

Fe2= rER2ce2/

Eh3=

(1 - v’) = 8

rrER*c,,/(l-

Eh6= PER~cJ(I

v’) = 8 - v’) = 16

1*1* 0 0

/“lb 0 0

zj#,(

Q) df dj,

[A, + mA,] dZ dJ,

where A, = f’jjFf,( E), A2 = (1 - f*)yFf,,( 6) and P = x/R, 7 = y/R, The dimensionless compliance matrix is then Cl1 0

E=

0 CT**

0

E,4

E,5

0

0

0

0

C26 3336

0

0

c33

0

0

c41

0

0

r44

c45

0

F54

F,,

0

0

0

&j

C51

0

0

i;6*

4,

0

6 = y/h,

(2.6) 6= b/R.

.

elements of this matrix were computed and are plotted in Figure 2(a)-(d). Acceptable values of the compliance matrix elements are obtained up to a crack depth a/D = 0.8. This results from the accuracy of functions F, , F2, F,, and F,,, of equation (2.5). This is not a serious limitation since the area of practical interest is confined to crack depths o/D < 0.5. To calculate the global stiffness matrix of the cracked shaft one must add the local flexibility due to the crack to the total flexibility of the untracked shaft and then invert the global flexibility matrix: i.e., KG = [C,,, + Clot]-‘, where C,,, = diag [L/AE, L/4kAG, L/4kAG, L/4EI, L/4EI, L/G&], because the off-diagonal terms of C,,, are zero.

The

ROTATING

SHAFT

WITH

AN OPEN

CRACK

86

C. A. PAPADOPOULOS

AND

A. D. DIMAROGONAS

For the local flexibility matrix due to the crack equations (2.6) and (2.7) yield

GlR 0 0

Got=+-0

0 c22R

0

0

0 G3R 0

E41 Gl -

0

EL4

0 z.52

0 43

G5

0 0

0 0

0 c26 c36

c44IR

?45/R

0

G4IR

G/R

0

0

0

-

’

EMI R-

where cU (i,j=1,2 ,..., 6) are the dimensionless compliance coefficients and Fo= vER'/(l- v*). It must be pointed out that the above analysis applies to a crack which remains open during the shaft rotation. This assumption is not always valid. This is the case, for example, if the static deflection of the shaft is large, compared with the vibration amplitude. In this case, the crack opens and closes periodically with the shaft rotation. If the static deflection of the shaft is small, as in most light shafts, the crack remains open if the unbalance is in the direction of the crack, as usually happens. If the unbalance is in the opposite direction, the crack remains closed and does not affect the shaft behavior, except in the case of violent transients. The utility of the present analysis is maintained, however, even in the case of a closing crack because most of the features of rotor behavior under an open crack are qualitatively maintained in the case of a rotor with a closing crack. 3. UNDAMPED

VIBRATIONS

Consider a de Lava1 rotor (see Figure 3) of length L and radius R,with a transverse surface crack of relative depth cw/D (D = 2R). The mass of the disc is m and the stiffness of the untracked shaft EZ.

Figure 3. The model of a cracked de Lava1 rotor.

The rotor revolves at constant angular velocity w. With the mass assumed to be lumped at the center of the disc, three degrees of freedom can be used (Figure 3). If (x, y, z) is the stationary co-ordinate system and (5, 7, u) is the rotating one then [22] (3.1)

ROTATING

SHAFT

If L, = L/2 is the crack location,

WITH

AN

the equations

OPEN

of motion

are [23]

= &,W2+g cos ot,

-w2)~+w:4q+&4

&2wrj+(&

87

CRACK

~+2w~+w:55+(0:4-~2)r]+~:,~=~2~2-gsinwt,

ii+&+“:‘$7)+&4

=o,

(3.2)

where 0; = kg/m (i, j = 1, 4, 5), E, and are the e2 unbalance amplitudes and g is the acceleration of gravity. For example for R = 0.01 m, L = 1 m, E = 2.1 x IO” N/m2, the stiffness matrices for the shaft without a crack and for the shaft with a crack depth cr/D = 0.5 are, respectively, in N/m, 77 660

0

0

0O-O

77 0660

6.47 x 10’

1 [

-9 171

61 313

-1.64x

lo5 .

9

K=

[

The

system of equations

3.1.

FREE

Consider

-1.64 x 1O-5 -9 171

KC=

(3.2) is linear.

Three solutions

-38 802 73 408

6.24 x 10’ -38 802

1

can be distinguished.

VIBRATION

the homogeneous

system of equations ih,r

5=&e

,

7

=

7.

(3.2). A solution

eiAo',

u

=

u.

is sought in the form

e%',

(3.3)

where &,, T,, and u0 are constants and A0 is the angular velocity of the free vibrations in the moving co-ordinate system. If A is the angular velocity of these vibrations in the stationary co-ordinate system, then A =/A,+w. Substitution

of equations (45

-

(3.3) in the homogeneous W*-A;)&+(&&-

(w&+2iwA&+((w&-

(3.4) system yields

2iwA,)~,+(w:,)u0=0, w2-A&,+(~:,)u,,=0,

(~:5)~~+(~:4)~70+(~:1-~2)u~=0.

(3.5)

The condition for the existence of non-trivial solution of equations (3.5) is that the determinant of the coefficients be equal to zero. The evaluation of this determinant leads to the characteristic equation A:+g,A:+g,A;+g,=O,

(3.6)

where g,= -(w:,+w:,+w:j+2aJ2), g2=l-a-

w2)(w~-~2)+(w:~+w:5+2w2)“:,-((w:5+w~~+w~~)],

g3 = [ 0;,(w~-w2)+w~&J:,

-w2)+w&f1

-(w~5-w2)(o~4-~2)o~1 -~oJ~~w~~w~~].

A Newton-Raphson procedure has been used to compute the roots of equation (3.6) for a cracked shaft of a/D = O-5 as functions of o. In Figure 4 these results are plotted in a form used extensively by Tondl and Dimentberg [23,24]. On this graph, o is plotted along the horizontal axis and A along the vertical axis. To each w correspond six values of A (A = A,,+ w) which are the roots of equation (3.7). The points of intersection of the associated curves with the A axis correspond to the frequency of the natural vibrations

88

C. A. PAPADOPOULOS

AND

Figure 4. Eigenvalues

A. D. DIMAROGONAS

versus the rotational

speed.

of a non-rotating shaft. The points of intersection of the curves with the line A = o correspond to the values of critical speeds of forward precession, and the points of intersection with the line A = -o correspond to values of the critical speeds of reverse precession. It has already been shown [25] that between o1 and w2 there is instability. In this case there exists one more critical speed corresponding to the longitudinal eigenvalue of the shaft near which instability also appears. The first instability, which exists in the interval w55< w < 44, is called by Tondl [23] the instability of the second kind; the second instability, which lies in the vicinity of w,, , is called the instability of the first kind. 3.2.

THE

The

EFFECT

OF ECCENTRICITY

OF THE

CENTRE

OF GRAVITY

system of equations (3.2) for g = 0 gives &2w*+(&-

w2)5+ “&VI + w:,u = E,W2,

~+2w~+“L5+(0~4-W2)~++0:,U This system is non-homogeneous

[I[ 70 uo 50 =

= &*02,

ii+w:,~+o:4~+w:,u=o.

(3.7)

and the solution is

b&w*) WI5 w45 2

44

(d4- “142 07

4,

011 w:, 2

I[

&*W2 1 w* 0

(3.8)

In Figure 5 to, no and u. are plotted against w. Maximum values are observed near wU and wS5 which are the two main eigenvalues for the main directions of the crack section. The existence of u. is caused by the coupling terms of the stiffness matrix. If the crack depth is zero then for i # j+wz = 0 and u. = 0. Figures 5(a)-(c) are plotted for a typical rotor of m = 3 kg, L = 1000mm crack depth CY/D = O-5at L,= L/2 and eccentricity E, = O-001 m and ~~0-004 m, and yield w=,~= 136 s-’ and w4.,= 161 s-l. If no influence of the coupling is taken into account then w55= 143 s-’ and w+, = 156 SC’.

ROTATING

SHAFT

AN OPEN

WITH 1

‘-O-

0.9 0.6 -

89

CRACK

Ib)

0.6 -

FO?0 “E06-

7 0.7 s x 0.6 -

g 0.5 -

IO.5 -

j 0,4-

3 0.4 -

‘ 0.3 -4p o-2 01 0 0

5:::I ~0

1 60

90

o-1 0 0

I I I 120 150 160 210 240 210 300

’ 30

’ 60

90

120 150 160 210 240 270 300

l3otatbnd speed, w (rod/s) 5

(c)

Figure 5. The effect of eccentricity: amplitude of vibration in the three main directions versus the rotational speed. Crack depth (I/O = 0.5; eccentricity E, = 0.001, e2 = 0.004. (a) Vertical vibration; (b) horizontal vibration; (c) axial vibration.

3.3.

THE

EFFECT

OF THE

DISC

Let the shaft be perfectly for the rotating co-ordinatate

WEIGHT

balanced, so that e1 = e2 = 0. The system system gives

of equations

(3.2)

&-2w7j + (w:, -w2)~+w:~~+o:,u=gcoswc, w*)q + W&U = -g sin wt,

ij+2w&+&+(w~-

ii+w:,~+w:4~+o:1u=o.

(3.9)

Let 5=50e

With the above

representation

-ior

,

77 = v.

em’“‘,

of the harmonic

(45 -2w2)50+

u = ~0 e-‘“‘.

functions,

equations

(3.9) become

(44 +2i02)~o+w:1uo=g,

(&5 -2iw2)to+(w&-

20*)r],+0:,U0=

-g,

o:5~o+w:,~o+(w:,-02)uo=o. Solution of this system yields the response. to = 5, + i&, then I~ol=&:+G

(3.10)

and

(3.11)

If the complex q5=tan-’

solution

(t2/5,).

for k. is of the form

(3.12)

In Figure 6 the influence on the amplitude of vibration and of the disc weight is illustrated. The first peak corresponds to the well known [25] sub-critical resonance at about half the first critical speed: w, = w55w44/[2(w~+

0J:5)]“2.

(3.13)

90

C.

A.

PAPADOPOULOS

AND

A.

D.

DIMAROGONAS 10

0.1

0.00

-O.&

-

x

P E s $ Q

E -0.6

0,06-

i L $

B 3

-0.4

oo+

;

I .z

.g

) 70.2

0.02 -

00

201

40I

60

100 I

60

120 I

3000 1

4OQO I

5000 I

ai-

6O:O

Rotational speed. w (red/s) Figure 6. The effect of the disc weight:

amplitude

of vibration

versus the rotational

speed,

for two scales.

This value is not affected by coupling. For w55= 135 s-l and wM = 161 s-l from the above relation w, = 78 SC’. This is verified by the results in Figure 6. On the right of the diagram one can see a second resonance which is caused by the coupling effect and corresponds to the first longitudinal eigenvalue of the shaft (wl, = 4562 SC*).

4. DAMPED

VIBRATIONS

If damping is present, the equations of motion for the rotating co-ordinate system are [231 4’+2s,(&o77)-2w7j+(w:,

-0*)~+&+&4

= qw2+g

cos wt,

~+2Sz(7j+o~)+2w~+w~S~+(~~4-~2)~+0241u=~202-gsinwt, ii+2s$i+“:&+w:,~

+o:,u

=o,

(4.1)

where of = k,/m (i, j = 1,4,5), 2Si = c/m (i = 1, 2, 3), cl = C&i&$, c2 = JG, and c, = 4’Jm, in which 5 is the percentage of critical damping. In particular 26, = 25uS5, 2S2 = 2&&,, and 2& = 25~~~. Upon introducing the dimensionless parameters <= t/L, fj = q/L, ti = u/L, E, = Ei/ L (i = 1,2), g = g/(Lx.o’) and r = wt, and defining 0; = CM~/W’ the dimensionless differential equations become ~+25.n,,(~-ii)-2rSr+(n:,-1)~+~:,ii+~:,a=E,+gcoswt, jin+2~~44(jj’+F)+2~+~n:5~+(~~-l)ij+n:,a=E2-gsinwt, ~“+25.n,,a’+~n:sT+n:,ii+n:,a=o,

(4.2)

where the primes denote differentiation with respect to T. When time domain analysis of the equations of motion is desired, it is preferable to express the equations as a set of first-order differential equations. Let c=x,,

7j=x2, ii=x3,

.T$=x_$,jj’=x5,ii’=x,.

(4.3)

The equations of motion in normal form then become ir=[A]x+B(t),

(4.4)

ROTATING

SHAFT

91

AN OPEN CRACK

WITH

where

0

[Al =

0

0

0 0 -(G-l)

0 0

0 0 -(G--25&)

4, -n:, -n:,

-(Of,-1) -.n:,

-(&+2&U -fi,S B=[O

0

0

1 0 0 (-2&S,) -2 0

(Ei+g cos ot)

0 1 0 2

0 0 1 0

3

0 (-2!%4) 0 1 (-25G,)

(&--g sin wt)

OIT.

As computed by using the Runge-Kutta numerical integration method for a de Lava1 rotor of length L = 1 m, radius R = 1 cm, mass of disc m = 3 kg and with a crack at the center of depth (Y/D = 0.5, the rotor orbits in the x, y plane are shown in Figure 7(a) for the disc weight effect only. In Figure 7(b) one can observe the x, y, z space orbit of the center of gravity of the disc, where z is the direction of the axis of shaft. It is evident that motion exists in this direction because of the coupling. Similar results are obtained for the eccentricity effect. -06mm

-O$,,,,,,

r-i-7 :

!

!

:

::

.?

:

I

:

: :

+0.6mn

t0.6mm

:b)

Figure 7. The disc weight effect of the damped

system in (a) the x-y plane and (b) in x-y-z

space. a/D

= 0.5.

5. DISCUSSION

From the foregoing discussion, it is apparent that substantial coupling of longitudinal and bending vibration of a rotating shaft exists due to an open transverse surface crack, evidenced by the coupling terms in the stiffness matrix associated with the existence of a surface crack on a linear elastic structural element. The assumption of an open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions.

92

C. A. PAPADOPOULOSAND

A. D. DIMAROGONAS

due to the presence of the crack was represented by way of a 6 x 6 The local flexibility matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Shear was not considered and three degrees of freedom were used, bending in the two main directions and extension, leading to a 3 x 3 stiffness matrix with coupling terms. For short shafts the shear terms could be included with a Timoshenko beam formulation. It is clearly demonstrated that an instability region exists, as is usual for rotors with dissimilar moments of inertia undergoing natural vibration. In addition the variation of the eigenfrequencies due to the crack is noticeable, but small for small crack depths. A much more pronounced manifestation of the existence of the crack appears in the vibration spectrum (see Figure 6), where both the longitudinal and lateral vibration frequencies coexist on the same spectrum. This is due only to the surface crack and can be used for an unambiguous identification of the existence of the crack. The coupling of the longitudinal and lateral motion was also demonstrated by way of a dynamic response analysis of the damped system. In fact, for moderate crack depths, the coupling of the motion can be clearly observed. For machinery vibration monitoring, the on-line analysis of the longitudinal vibration signal can yield valuable information for the development of surface cracks on rotors and shafts because the longitudinal vibration of such members is usually associated with a low level of noise; therefore the resolution of the crack detection is relatively high. Cracks with depth ratios of the order of a/d = 0.1 can be easily detected.

REFERENCES

1. A. D. DIMAROGONASand S. A. PAIPETIS1983 Analytical Methods in Rotor Dynamics. London: Applied Science Publishers. 2. H. OKAMURA 1969 Engineering Fracture Mechanics 1, 547 A cracked column under compression. 3. H. LIEBOWJTZ,H. VANDERVELDT and D. W. HARRJS 1967 International Journal ofSolids and Structures 3, 489-500. Carrying capacity of notched columns. 4. H. LIEBOWTIZand W. D. CLAUS 1968 Engineering Fracture Mechanics 1,379-383. Failure of notched columns. 5. J. R. RICE and N. LEVY 1972 Journal of Applied Mechanics 3, 185. The part-through surface crack in elastic plate. 6. A. D. DIMAROGONAS and G. MASSOUROS1980 Engineering Fracture Mechanics 15,439-444.. Torsional vibration of a shaft with a circumferential crack. 7. H. J. PETROSKJ1984 Journal of Applied Mechanics 51, 329-334. The permanent deformation of a cracked cantilever struck transversely at its tip. 8. C. A. PAPADOFOULOS and A. D. DIMAROGONAS 1987 Zngenieur Archive 57, 495-505. Coupling of bending and torsional vibration of a cracked Timoshenko shaft. 9. A. J. BUSH 1976 Experimental Mechanics 16, 249-257. Experimentally determined stressintensity factors for single-edge-crack round bars loaded in bending. 10. J. A. HENRY 1978 The Design Application of Industrial Drives, Z.E.E. Conference and fiblication 170, 37-44. Monitoring rotating machinery to detect the growth of rotor cracks. 11. I. W. MAYES and W. G. R. DAVIES 1980 Institution of Mechanical Engineering Conference, Paper No C254/80, Vibration in Rotating Machinery. A method of calculating the vibrational behaviour of coupled rotating shafts containing a transferse crack. 12. T. PAFELIAS 1974 Technical Information Series, General Electric, No DF-74LS-79. Dynamic behaviour of a cracked rotor. 13. T. A. HENRY and B. E. OKAH-AVAE 1976 institution of Mechanical Engineers Conference Publication, Vibrations in Rotating Machinery, Paper No C162/76. Vibration in cracked shafts. 14. I. W. MAYES and W. G. R. DAVIES 1976 Institute Mechanical Engineers Conference Pubfication, Vibration in Rotating Machinery, Paper No C168/76. The vibrational behaviour of a rotating shaft system containing a transverse crack.

ROTATING

SHAFT WITH AN OPEN CRACK

93

15.R.GASCH 1976Institution of Mechanical Engineers Conference Publication, Vibration in Rotating Machinery, Paper No C 178/76. Dynamic behaviour of a simple rotor with a cross-sectional crack. 16. H. ZIEBARTH, H. SCHERDTFEGER and E. E. MUHLE 1978 VDI-Berichte 320, 37-43. Auswirkung von Querissen auf das Schwingungsverhalten von Rotoren. (In German.) 17. B. GRABOWSKI 1979 ASME Design Engineering Technology Conference, St. Louis, Paper No 79-DET-67. The vibrational behaviour of a turbine rotor containing a transverse crack. 18. 1.W. MAYES and W. G. R. DAVIES 1983 ASME Design and Production Engineering Technical Conference, Dearborn, Michigan Paper No 83-DET-84. Analysis of the response of a multi-rotorbearing system containing a transverse crack in a rotor. 19. W. G. R. DAVIES and I. W. MAYES 1983 ASME Design and Production Engineering Technical Conference, Dearborn, Michigan Paper No 83-DET-82. The vibrational behaviour of a multishaft, multi-bearing system in the presence of a propagating transverse crack. 20. T. INAGAKI, H. KANKI and K. SHIRAKI 1981ASME Design Engineering Technical Conference, Hartford, Connecticut, Paper No 81-DET-45. Transverse vibrations of a general cracked-rotor bearing system. 21. H. TADA, P. C. PARIS and G. R. IRWIN 1973 The Stress Analysis of Cracks Handbook. Hellertown, Pensylvania: Del Research Corp. 22. A. F. SOUZA and V. K. GARG 1984Advanced Dynamics, Englewood Cliffs, New Jersey: Prentice Hall, Inc. 23. A. TONDL 1965 Some Problems of Rotor Dynamics. Prague, Publishing Hause of the Crechoslovac Academy of Sciences. 1961 Flexural Vibrations of Rotating Shafts. London: Butterworths. 24. F. M. DIMENTBERG 25. A. D. DIMAROGONAS and C. A. PAPADO~OULOS 1983 Journal ofSound and Vibration 91 583-593. Vibration of cracked shafts in bending.