Vibration of cracked annular plates

Mechanics Vol. 49, No. 3, pp. 371-319. 1994 Engineering Fracaerure

Pergamon

Elsevierscience Ltd Printed in Great Britain.

0013-7944(!34)00161-8

0013-7944/94 s7.00 + 0.00

VIBRATION

OF CRACKED

N. K. ANIFANTIS,t

ANNULAR

PLATES

R. L. ACTIS# and A. D. DIMAROGONAS#

tMachine Design Laboratory, Department of Mech. Engineering, University of Patras, Greece SDepartment of Mech. Engineering, Washington University, St. Louis, MO 63130, U.S.A. Almtrac-A surface peripheral crack of an annular plate is modelled as a local rotational flexibility for vibration analysis. Computed values of the local stiffness are very close to those computed by using strain energy arguments and data for beam-like structures. From the solution of the appropriate eigenvalue problem, the decrement of the fundamental natural frequency of the plate, due to the presence of the crack, is computed. Results are used for crack identification in annular plates.

NOMENCLATURE 4 E G J K K, K,(Q) Nd M, R R, R, w, a % r rC 5 r. t, a,0 6 : ; P 7 rp w

plate flexural rigidity, Di = Efyr(l - v2) Young’s modulus shear modulus strain energy release surface function nondimensional stiffness of crack, K = ZK,(u)t,/D, stress intensity factor local rotational stiffness of crack radial bending moment radial normal force outer radius of plate, Fig. 1 inner radius of plate, Fig. 1 radius of thickness change, Fig. 1 nondimensional deflection of plate crack depth flexibilities to simulate crack behavior nondimensional coordinate, r = a/R nondimensional crack position = R,/R = R,,IR

half-thickness of plate, Fig. 1 polar coordinates transverse displacement of inner plate boundary crack ratio, r) =a/2t, nondimensional frequency parameter, 1 = u~(pR’/Et:)“~ Poisson’s ratio thickness ratio parameter, r = (t/t2)1’2 mass density of material time rotation of inner plate boundary circular frequency.

1. INTRODUCTION THE PRESENCEof

intrinsic flaws or cracks in machine elements is a source of local flexibility which in turn influences the dynamic behavior of the system. By measuring the response of a cracked body to white noise broad band excitations, the shift in the vibration spectrum and the presence of coupled modes can be correlated with the crack depth and its location. This yields an interesting methodology for an on-line, real-time, early warning system to identify critical cracks in various structures. The local flexibility of cracked structures can be determined by using the fracture parameters which were originally introduced to predict fracture instability. Thus, on the basis of the energy release rate formulation and by application of Castigliano’s theorem, the Paris integral yields the displacement of elastic bodies at any desired point. This of course, requires a knowledge of the 0 Crown copyright (1994). 371

312

N. K. ANIFANTIS ef al.

energy release rate values or the corresponding stress intensity factors along the ligament region ahead of the crack surfaces. In this way, the local flexibility may be considered as a single spring relating only one force to the deflection in the same direction [11.Vibrations of cracked plates have been investigated modelling the crack as a continuous local flexibility [2]. The local flexibility method for the dynamic analysis of cracked structures is restricted only to simple structural elements, because it requires knowledge of expressions giving the stress intensity factor as a function of the crack depth [3-51. For more complicated structures, the local flexibility of the cracked section may be computed by numerical or experimental techniques [6-S]. The change of vibration behavior of plates and shells, as a tool for crack identification, is well-known [2,9-l 11.Improving understanding of how cracks affect vibration behavior of annular plates, which is the subject of this paper, will be used for a non-invasive crack detection technique in applications such as heavy rotating drums and road simulation flywheels. The latter problem, encountered in an industrial facility during high speed testing of aircraft tyres, motivated this investigation.

2. CRACK COMPLIANCE COMPUTATION An annular plate, as shown in Fig. 1, has a peripheral surface crack of constant depth a, at the inner boundary r = ri. The plate is considered as a linear elastic structure with stepwise varying thickness, as shown in Fig. 1. Further, it is assumed that the crack remains always open, and that the usual linear elastic fracture mechanics conditions are valid. In the present analysis, since bending vibrations are considered, the rotational crack compliance is assumed to be the dominant one in the local flexibility matrix neglecting the other terms. In this way, the crack is treated as a local distributed rotational stiffness of constant K,(a). The exact relationship between the crack depth and the local stiffness is difficult to be determined by the strain energy approach, because, stress intensity factor expressions for this complex geometry are not available. For the computation of the local crack compliance, a finite element method was used. It is assumed that the outer plate boundary is fixed. Then under imposed assumptions, we consider two statically equivalent boundary states in the inner plate boundary. The first state represents the real situation of the fixed boundary with a crack at the inner boundary of the plate. The second state represents a constrained boundary with radial and transverse elastic supports. Under a particular bending loading, the inner plate boundary will be displaced due to elastic

2R Fig. 1. Geometry of an annular plate with varying thickness.

Vibration of cracked annular plates

313

W

r

-

Fig. 2. Finite element mesh for a cracked annular plate. Singular elements have been shaded.

supports. Let the transverse deflection be 6, and the rotation cp,, at r = r,, where the load applies. Then, the equivalent rotational cIl, and coupling c12 flexibilities are given by the relations Cl1

=

~12 =

wrp, Ml4

9

(1)

where M, is the bending moment at r = ri. The equivalent local rotational stiffness K,(a), when a crack of depth a, is present at r = ri, is by definition

M,(a) %(a) ’

K,(u) = -

where M,(u) is the bending moment and q,(u), the rotation of the inner plate boundary for a given crack depth. From compatibility and equilibrium conditions, 6, - 6, = cpo -

~,(a)

=

M,(aYc,z

MAah,

(3)

where 6,, cp,,are the load point vertical displacement and rotation, respectively, for the simply supported case. Combining eqs (2) and (3), the local stiffness is found to be

Equation (4) is used to compute K,(u) from two different boundary states at the boundary r = r,. All the deflections appearing in this equation, are referred to the same loading conditions, and may be computed by the use of the finite element method. The load point vertical displacement of plate boundary was computed for various crack lengths. To improve the accuracy, quarter-point singular elements were used in the crack tip [12-141. To solve this problem, a mesh of 20 quadratic axisymmetric elements and 83 nodes was used, as shown in Fig. 2. A remeshing technique is used in order to change the density of the mesh and the location of the singular elements as the crack advances. Results of the computations are shown in Table 1. The vertical and rotational displacements for crack free plate were computed with the same loading condition. Two sets of linear springs were specified along the inner boundary. One set in the transverse direction, very stiff, to simulate a roller support and one set of springs in the radial direction to counteract against the free rotation of the border. Results are shown in Table 2, for various values of c,, . The values of 6, and cpOmay be found from Tables 1 and 2.

314

N. K. ANIFANTIS et al.

Table 1. Load point vertical displacement for a given crack length

Table 2. Load point displacements for given radial spring constant

Crack ratio a

l/c,, (kp/mm’)

Vertical displacement 6, (nun) 3.6179 3.7389 3.8202 3.9319 4.0843 4.2697 4.4628 4.7650 5.1410 5.7319

2t, 0.0

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4, (rad x lO-5) 0.01 5.68 25.5 46.1 77.0 128.6 148.9 156.6 193.4 203.0 208.7 215.2 231.6

2 ::S’ 1:s x 10’ 2.8 x 5.5 x 1.4 x 2.0 x 2.8 x 5.5 x 8.3 x 1.1 x 1.4 x 2.8 x

In Fig. 3, shown by solid line is the nondimensional the relation

10’ 10’ lOr 102 IO2 IO2 lo2 10’ 10’ lti

1787 1745 1593 1436 1200 807 664 525 313 240 196 146 13

5.7379 5.6896 5.5118 5.3289 5.0546 4.5949 4.4272 4.2647 4.0183 3.9370 3.8862 3.8252 3.6779

local rotational stiffness K, defined by

K=K.(a)$. 2

Values for K were computed from eq. (4) by the use of data from Tables 1 and 2. Alternatively, the local stiffness may be computed approximately, by using fracture mechanics, and strain energy arguments. Known results for the stress intensity of cracked rectangular beams may be used. The rotation of a strip with an edge crack due to bending moment M may be written [2] @,(a) = ~ a

‘Jda aM,(c) s o ’

where the strain energy release surface function J, for the opening mode of fracture and a strip of width R,dp, with a crack of depth a is J=

s

2nK: 7 dp, o E

where E’ = E is the Young’s modulus for plane stress and E’ = E/(1 - v’) for plane strain, and K, is the stress intensity factor for the opening mode of fracture. Table 3. Nondimensional

r, 0.5 0.6 0.7 0.8

1 28.1916

frequency parameter A,, for annular plate fixed inside and outside

1.5

2

2.5

3

26.9745 21.7971 18.2226 17.2985

24.6345 17.1320 13.6467 12.2198

21.9539 13.9561 10.6467 9.5329

19.4428 11.7241 8.7771 6.9574

Table 4. Nondimensional frequency parameter d, for annular plate hxed inside and free outside

r, 0.5 00s 0.8

1

1.5

2

2.5

3

3.4188

3.4127 3.3902 3.4062

3.3923 3.2410 3.3861

3.3817 3.0312 3.3728

3.3904 2.8028 3.3668

3.1254

2.7495

2.4016

2.1039

375

Vibration of cracked annular plates

The local stiffness is defined by the relation [2]

z

1

K,(a)=aM;

s

(8)

‘Jda

o

*

From eqs (6)-(8), the local stiffness may be computed by appropriate use of K, for the cracked strip. (1) From ref. [l 11,by suitable rearrangement of terms, integration and substituting in eqs (7) and (8), the dimensionless stiffness constant is K=-,

1 3F, (tl)

where F,(q) = 1.862~’ - 3.95~~ + 16.375~~ - 37.226~~ + 75.81~~ - 126.9~’ + 172.5#‘+ 143.97~‘+ 66.56~‘~ (10) rl = a/2tz. (2) From ref. [15], after some mathematical compliance is K=-,

(11)

manipulation

using eqs (7) and (8), the crack

1

(12)

3F,(rl)

where F,(q) = 1.98~~ - 3.28~~ + 14.43~~ - 31.26$+

63.56~~ - 103.36~’ + 147.52$-

127.69~~ + 61.5~“.

(13)

(3) Finally, from ref. [16J (14) where

F3(q) = (1.193 - 1.98~ + 4.478~~ + 4.443~~ + 1.739~3/(1 - q2).

(15)

All three alternative expressions can be used for crack length up to 0.8 of the beam height. In Fig. 3, the computed results from eqs (9), (12) and (14), are shown by dots for comparison with eq. (5). It can be seen, that the two methods agree very well, proving that the localised nature of the crack effect can be used in analysis using fracture mechanics results instead of finite element modelling. The boundary conditions at the outer plate boundary and the thickness ratio < do not affect the crack compliance. This is due to the localised effect of the crack. 3. NATURAL FREQUENCY

OF THE PLATE

The transcendental equations which predict the eigenvalues representing the frequencies of the transverse natural modes of a non-uniform thickness annular plate with different boundary conditions can be obtained as an extension of the work of Vogel and Skinner [17]. The theory used in this analysis assumes small deflections and neglects the effects of rotary inertia and the additional deflection caused by shear forces. Then, the differential equations governing the displacement of a point along the central plane of the plate may be written as

(16) Replacing for the radial coordinate Wi(a,8)=

the expression a = Rr,and assuming a solution of the form

W,(r)sinn@-fI,)sinw(r

-to),

i= 1,2

(17)

N. K. ANIFANTIS et al.

316

eq. (16) becomes

i (-$+i$ -s)2W,-/jfW,=(),

= 1,2

(18)

with j?; = 3(1 - v2)po2R4/Etf,

i=l,2.

(19)

The radial bending moment and the radial normal force are, respectively,

(3 - v)d r3

1

Wi .

The general solution of eq. (18) may be written as Wi(r)

=

4,J,(/V)

+

42

Y,(Bir)

+

434(Bir)

+

44JG(Bir)9

i =

1,2,

(21)

where J, and Y. are Bessel functions, and Z. and K. are modified Bessel functions. The coefficients A,, i = 1,2, j = 1,4, determine the mode shape and are computed from the boundary conditions. At the position r = ro, the compatibility conditions are

Wi(r,) = w2(rO) W; (r,) = Wr,) Wi (r,) =

M2bJ

Ml

M2kJ

(r,)

=

If there is a crack at position r = r,, then the compatibility

(22)

and equilibrium conditions are

W,(r,‘) = W,(r;) Wi(r:)

- Wl(r;)

= M,i(r,+)/K.

(23)

For a plate with geometry as shown in Fig. 1, the boundary conditions at the inner border r = ri, are W, (ri) = 0 W; (ri) = 0.

(24)

Similarly, at the outer boundary, r = 1, if it is fixed W,(l) = 0 w;(l) = 0

(25)

and if it is free W,(l)

=

0

&2(l) = 0.

(26)

The frequency equation is determined by substituting eq. (21) into the compatibility conditions (22), (23) and boundary conditions (24), (25). Then, considering the recursion relationships of Bessel functions, the following equation is obtained A(&K,r,,r,,O=o, where 1 is the nondimensional

(27)

frequency parameter Iz

=

(~l/b8)2 3(1 - v2)

(28)

Vibrationof crackedannular plates

317

w

Fig. 3. Nondimensionallocal stiffnessvs crack ratio. (0, After ref. [2]; 0, eq. (8), J from ref. [IS]; 0, eq. (8), J from ref. [16];-, FE solution).

4 =

BIre

(29)

is a nondimensional parameter of the plate geometry. For the free outer boundary of the plate, eqs (26) must be used to construct eqs (27).

4. NUMERICAL

RESULTS

Equation (27) represents a 8 x 8 determinant, that includes the geometry, and the physical properties of the cracked plate. As k + 00 the problem degenerates to that of an untracked and, as K-+0, to plate simply supported at the location of the crack. The nondimensional frequency parameter a, was calculated. Only the fundamental frequency was determined because this is the most crack sensitive. For this case v is taken equal to unity. The Bessel functions were replaced by their approximate polynomial equivalents. Results are affected very little by the values of Poisson’s ratio. Hence the results presented in this work for Y equal to 0.3 should be reasonably accurate for most engineering materials. Due to the large number of the parameters involved, results are given here only for some important cases. For simplification it is assumed that the crack is in the inner boundary region, which is the most dangerous one. Table 3 gives the parameter rZ,for various values of the geometry parameters r, and fi, where 1, is the fundamental frequency parameter of an untracked plate, eq. (28). In this case, the plate is fixed at the inside and outside boundaries. Table 4 gives the variation of the same parameter for a plate fixed at the inside and free at the outside boundary. The frequency ratio A/&, for both cases is shown in Figs 4 and 5, respectively. The frequency ratio is given as a function of the local crack stiffness and plate geometry. In general n/n,, < 1, because the cracked structure is more flexible than the untracked one. The decrement off, is a function of the crack compliance and has a lower bound for K-*0. This limiting value corresponds to a plate pinned at the crack position which is at the inner boundary.

N. K. ANIFANTIS et al.

378

.8

.8

F

/’

,/’

--

10'

K

loo

Fig. 4. Frequency ratio vs local stiffness for annular plate tixed inside and outside.

5. CONCLUSIONS For complex annular plates, the crack compliance can be computed by combining finite element analysis and energy arguments. This method gives accurate results for crack ratios from 0.2 to 0.8. The frequency ratio may be computed from the appropriate eigenvalue problem, for structures with complex geometry. Results show maximum decrement of fundamental frequency ratio about 1

I

.9

.8

.7

.8 10’ K Fig. 5. Frequency ratio vs local stiffness for annular plate fked inside and free outside.

Vibration of cracked annular plates

319

0.4. This frequency drop is sensitive to the crack depth. This measurable frequency decrement provides a methodology to identify cracks in annular plates. The crack in this analysis was presumed open. This is not a severe limitation because in engineering practice, cracks are associated usually with static loads which keep the crack open. If the static loads act in the direction of closing the crack, the method cannot be used.

REFERENCES [l] A. D. Dimarogonas and S. A. Paipetis, Analytical Meihoak in Rotor Dynamics. Elsevier Applied Science, Barking (1983). [2] T. G. Chondros and A. D. Dimarogonas, Identification of cracks in circular plates welded at the contour. ASME paper 79-DET-106, St. Louis (1979). [3] C. Papadopoulos and A. D. Dimarogonas, Coupled longitudinal and bending vibration of a rotating shaft with an open crack. J. Sound Vibr. 117, 81-93 (1987). [4] N. K. Anifantis and A. D. Dimarogonas, Identification of peripheral cracks in cylindrical shells. ASME paper No. 83-WA/DE-14, Boston, MA (13-18 November 1983). [5] J. Y. Yao and A. D. Dimarogonas, Vibration of a circular ring with a transverse crack. ASME paper No. 88-WA/DSG-42, Chicago, IL (27 November-2 December 1988). [6] G. Gounaris and A. D. Dimarogonas, A finite element of a cracked prismatic beam for structural analysis. Comput. Structures 28, 309-313 (1988). [I G. Gounaris, N. K. Anifantis and A. D. Dimarogonas, Dynamics of cracked hollow beams, Engng Fracture Mech. 39, 931-940 (1991). [8] R. Rice and N. Levy, The part through surface crack in an elastic plate. J. appl. Mech. BE, 185-194 (1972). [9] P. C. Gauley and R. D. Adams, Defect location in structures by a vibration technique. ASME paper No. 79-DET-46, St. Louis, MO (lo-12 Septermber 1979). [lo] H. J. Petroski, On the cracked bell. J. Sound Vibr. 96, 485-493 (1984). [ll] N. Papaeconomou and A. D. Dimarogonas, Vibration of cracked beams. Compur. iUech. 5, 88-94 (1989). 1121 E. Akin, The generation of elements with singularities. Int. J. numer. Meth. Engng 10, 1249-1259 (1976). [13] M. Tracey and T. S. Cook, Analysis of power type singularities using finite elements. Int. J. numer. Meth. Engng 12, 1225-1233 (1977). [14] W. Kelley and C. T. Sun, A singular finite element for computing time dependent stress intensity factors. Engng Fracfure Mech. 1% 13-22 (1979). [15] H. Okamura, H. Liu and C. Chu, A cracked column under compression. Engng Fracture Mech. 1, 547-564 (1969). [16] Standard test method for Jti, a measure of fracture toughness. ASTM E813-87. [17] B. G. Vogel and D. W. Skinner, Natural frequencies of transversely vibrating uniform annular plates. J. Appl. Mech. 32,926-931

(1965). (Received 19 September 1993)

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