Delamination in composite rotors

Composites Part A 28A (1991) 523-521 0 1997 Published by Elsevier Science Limited

Printed in Great Britain. All rights reserved 1359-835X/97/$17.00

PII: S1359-835X(96)00152-2

ELSEVIER

Delamination

Hikan

in composite

rotors

L. Wettergren

Department of Mechanical Engineering, Linktiping University, S-58183 (Received 2 September 1996; revised 15 November 1996)

Linkd;ping, Sweden

For a composite rotor with a delamination, the effects on the eigen-frequencies due to the delamination increase with increasing thickness of the composite shaft. The position of the delamination and its length are also important. For small delaminations the effects on the eigen-frequencies increase almost linearly with increasing length of the delamination. However, compared to other imperfections, the effect of the delamination can probably be neglected and will be difficult to detect using, for instance, diagnostic techniques for condition monitoring, as employed in other similar studies. 0 1997 Published by Elsevier Science Limited (Keywords: delamination; composite rotors; diagnostic techniques)

INTRODUCTION dynamics, as in many other areas, diagnostic techniques for condition monitoring are receiving growing attention. To be able to use such techniques the different types of phenomena that arise have to be understood. For composite rotors some new aspects have to be considered, such as the influence of imperfections due to different manufacturing processes and material damping. Wettergren’ studied the behaviour of a rotating composite shaft due to the influence of imperfections. His studies included differences due to divergences in the radius, thickness and filament winding angle in certain parts of the composite shaft. Wettergren2 also studied material damping in composite rotors and found that there exists an optimal filament winding angle interval to obtain the lowest possible dissipation energy for three different design criteria: lateral eigen-frequency, maximum torque due to static stress and maximum torque due to buckling. Wettergren3 showed that the combined effects of fluid film bearings and material damping have to be taken into account in analyses of composite shafts carried by fluid film bearings. The highest allowable rotational frequency for a rotor is decreased by internal damping. Another failure mechanism is the interlaminar separation known as delamination, which can effect the loadcarrying capacity of components made from laminated composites. The separation may be either local or covering a wide area. It can occur during cure or in subsequent life. Chang4 showed that delamination significantly affects the buckling load and response of cylindrical composite shells subjected to external pressure loadings. In rotor

Saravanos and Hopkins’ analytically predicted the natural frequencies, modes and mode damping in composite beams with delamination cracks. They also conducted experiments on composite beams with a single delamination, and measured natural frequencies and modal damping. The correlations between predicted and measured data showed good agreement for the case of natural frequencies but not such good agreement for the case of modal damping. It was also found that natural frequencies were fairly insensitive to interfacial friction. Rapid damping changes may, however, provide reliable indications of delamination damage and growth. In this study attention has been focused on the differences in the behaviour of the eigen-frequencies of a cylindrical shaft due to delamination in a region of varying size and position. A comparison is also made of the differences due to delamination and other types of imperfections.

FE FORMULATION In the analyses a specially developed FE program with a 10 degrees-of-freedom thin-walled finite element for laminate composite beams, developed by Wu and Sun6, has been used. Their element, here called the WS element, was further developed and extended by Wettergren7. The element allows transverse shear and warping and, moreover, the mass is not distributed on the centre-line of the beam. Let (x, y, z) be a fixed Cartesian coordinate system (see Figure la) with an axis parallel to the axis of the beam. The length of the element is h. The plane normal to the longitudinal axis z cuts the middle surface of the beam

523

Delamination

in composite

rotors: H. L. Wettergren

given by:

B z

X

h

aw Ezo =&

(7)

au 24 Ed=--+as a

(8)

the shearing strains are given by:

(a>

(9) (10)

(b) Figure 1

Geometrical

the changes of curvature of the shell are given by:

relations

&12

cross-section along the curved line called the contour (see Figure lb). A local contour coordinate system (n,s, z) is placed on the middle surface, where n and s are the directions normal and tangential to the contour, respectively. To derive the finite element for a laminated composite thin-walled beam, three assumptions were made:

1) The

U(S,z) = U(z) sin O(s) - V(z) cos O(s) - @(z)T,(s) (1) w(s,2) = U(z) cos O(s) + V(z) sin O(S)+ Q(z)r,(s)

(2)

where U, V are the displacements of the pole P in the x and Y directions, respectively, and @ is the rotation angle around the pole axis. The other variables are defined in Figure 1. The Saint-Venant assumption of deformation of a twisted shaft is valid. This assumption states that the

deformation of a twisted shaft consists of rotations of cross-sections of the shaft, as in the case of a circular shaft and of warping of the cross-sections, which are the same for all cross-sections. This leads to the axial displacement w on the contour being expressible as: W(S> z) = W(z) + G)x(s)

+ MY(S) + Ww(s)

(3)

where W is the average displacement of the beam in the z direction. In the classical theory of thin-walled beams, <=--U’, q=-V, tI=Q’ and w is the warping function.

E = EdIs

cxs

Ez = Ezo -

d-x,

(5)

(11)

( aS

(12)

a>

and the twist of the shell is given by:

a au

21

KS2 = Z ( 8s

a)

(13)

au2

f%s= dzds

(14)

The variable a(s) is the radius of curvature of the contour, defined as positive when the centre of the curvature lies on the negative n axis (see Figure lb). The thickness of the wall of the laminated beams is built up of an arbitrary number of bonded orthotropic layers having different thicknesses and fibre orientations. The stresses and strains with respect to the structural principal axes n, s, z (Figure lb) in each layer are related by:

[i;;

i:

ZJ

(:$}

(15)

where C& are the reduced stiffnesses for the unidirectional fibre composite relative to the (n, s, z) coordinates. The strain energy for each element is given by: U= ;

WTIK1{d

(16)

where [K] is the element stiffness matrix, expressed as:

PITNPld~) Wldz WI= JohIHIT(Jus

(17)

and (4) =

1+ C/a

&

xs=%

assumption in classical shell 3) The Kirchhof-Love theory remains valid for thin-walled composite beams.

This means that the strains of the middle surface are given by:

=

a au-- v

contour does not deform in its own plane. This

assumption leads to the displacements of a point on the middle surface U,w in the x and Y directions, respectively, being expressible as:

2)

xz

[Wl

El 71 Ql

w2 G 72 *2

Ul

Vl fh

ui

u2 v2 a2 G

vi vi

@l

@iIT

(18)

The kinetic energy for each element is given by: where the unit strains of the middle surface are

524

T = 4 W=WI{4

(1%

Delamination

in composite

rotors: H. L. Wettergren

where [M] is the element mass matrix, expressed as: M=

J

)I*,’ (J 0S,C]T&, ds [H*] dz >

(20)

Using the above described FE formulation, the imperfections presented in the next two sections may be studied. -0

30

DELAMINATION

60

90

120 150 180

Imperfectionangle,cp[“I

For some reason there is a delamination in a freely supported composite shaft. Delamination may, for instance, be introduced by impact, manufacturing defects or voids. In this paper a delamination is treated as if there is no shear stress in the plane where the delamination is situated, i.e. rrV and 7,; are equal to zero. The geometry of the delamination is shown in Figure 2. The delamination, with length &t, is symmetrically situated in the axial direction but the position in the radial direction may vary. The part of the periphery positioned symmetrically around the y-axis, given by cp, is, however, the same in all calculations. The position in the radial direction is given by t&t. In this first study the position of the delamination is on the middle of the shaft with a length L/8. The ratios of the difference between the two eigen-frequencies in the x and y directions and the eigen-frequency for the shaft without delamination are shown. These values have been found for different ratios between the radius and the thickness. From the results in Figure 3 it is noted that the

Figure 4 Plot of results for three different delamination positions but same thicknesses

3”

1 3” - 3”

0

30

60

90

120 150 180

Imperfection angle, cp [“I Figure 5 Plot of results for varying delamination length but same thicknesses and delamination position

‘00

‘:

300

3”

0209

1 3

D;larnination

- 3”

100 0LLA 0

0,25

0,50

0,75

LOO

Imperfection length, L&l& I

Delakination

Figure 6 Plot of results for varying delamination length but same thicknesses and delamination position

.Figure 2

Geometry of delamination

400)

R/t = 5

1

Figure 4.

‘00 *

-

200 fg

- 3”

delamination has the largest influence for the smallest radius-to-thickness ratio. When the interlaminar separation does not take place symmetrically in the middle of the thickness of the shell, i.e. t/& is not equal to two, the effect due to the delamination will decrease, as shown in

100 0 -0

30

60

90

120 150 180

Imperfection angle, cp [“I Figure 3 Plot of results for three different thicknesses but same delamination position

To investigate how the size of the delamination affects the behaviour of the rotor, the same shaft as in Figure 2 will be used. The position of the delamination is still in the middle of the shaft, but the width varies. The results are shown in Figures 5 and 6. In Figure 5 the variation of the difference ]wY- w,]/ w. has been plotted for four different lengths of delamination. The maximum values of the variation are used to plot Figure 6. It is noted in Figure 6 that for small imperfection lengths the effect increases almost linearly with the imperfection length.

525

Delamination

OTHER

in composite

rotors: H. L. Wettergren

IMPERFECTIONS

To be able to estimate the significance of the effects due

to delamination, the results from another study by Wettergren’ have been summarized. Besides delamination there can be other imperfections in a composite shaft. Yurgartis’ showed that continuous, non-woven fibres in high-performance composites have small angular misalignments about their mean direction. According to Yurgartis, fibre misalignment has been suspected to have a significant influence on composite properties such as longitudinal tensile modulus, longitudinal compression, strength and delamination fracture toughness. Due to dynamic forces, imperfections in thin-walled shafts will lead to changes in geometry. To see what kind of changes in geometry these imperfections cause and to see how the eigen-frequency would be influenced, Wettergren studied a thin circular cylindrical beam with radius R. The cylindrical beam with the lay-up [(45”, -45’)&i6 had a different filament winding angle over a certain part of the periphery, given by cp, positioned symmetrically around the y-axis. The cylindrical beam was studied both when the imperfection was over the whole axial direction and also for a local imperfection stiffened by the surroundings. Here the cylindrical beam stiffened by the surroundings is called a ‘cylindrical beam with local imperfection’ and the cylindrical beam with an imperfection over the whole axial direction is called a ‘cylindrical beam with global imperfection’. The hatched area in Figure 7 has a filament winding angle different from the other parts. The width of the imperfection area is defined by the imperfection angle cp, described in Figure 2. The variation of the difference ]wy - wXl/wOhas been calculated. This has been done for a cylindrical beam with a global imperfection and is shown in Figure 8. In this figure the filament winding angle changes from cyto (a - Ao) with cp = loo”, where Ao is 1, 2 or 3”. From Figure 8 it is noted that the rotor has different eigenfrequencies in the x and y directions. In the interval w, < R < We the rotor is unstable. Figure 8 shows that this interval is smallest when the filament winding angle is as small as possible. The stiffness of the shaft will also be highest in this region.

0 0

15 30 45 60 15 90 Filament winding angle, a [“I

Figure 8 Eigen-frequency ratio versus filament winding angle a. The thickness to radius ratio (t/R) is 0.04, the rotational speed R is 20 000 rev min-’ and the density p is 1382kg m-3

0

15 30 45 60 75 90 Filament winding angle, a [“I

Figure 9 Eigen-frequency ratio versus filament winding angle a. The filament winding angle changes from a to a - 3” in the region ‘p = 100”. The thickness to radius ratio (t/R) is 0.04 and the rotational speed R is 20 000 rev min-’

The variation of the difference Iwy- w,I/wO for different filament winding angles when one-third of the length has delaminated has been plotted in Figure 9. The result is also compared with a global imperfection. Figure 9 shows that the influence of the imperfection in the region where the stiffness of the shaft is highest (10’ < Q < 157, is only half as great as for the shaft with a local imperfection. The optimum filament winding angle for shafts optimized for torsional load is 45”; unfortunately the maximum influence on the eigenfrequencies appears when the filament winding angle a! is 45-60”.

CONCLUSIONS

-c R

Figure 7 Diagram of three composite cylindrical beams with 45” filament winding angle. The central cylindrical beam has a different filament winding angle. The thickness to radius ratio (t/R) is 0.04 and the radius to length ratio (R/I) is 0.05

526

The effects on the eigen-frequencies due to delamination in a composite rotor increase with increasing thickness of the composite shaft. The position of the delamination and its length are also important. For small delaminations the effects on the eigen-frequencies increase almost linearly with increasing length of the delamination. However, compared to other imperfections, the effect of the delamination can probably be neglected and will be difficult to detect using, for instance, diagnostic techniques for condition monitoring, as employed in other similar studies.

Delamination

REFERENCES Wettergren, H.L., Influence of imperfections of a rotating composite shaft. J. Sound Vibration, in press. Wettergren, H.L., Material damping in composite rotors. J. Compos. Mater., submitted. Wettergren, H.L., Dynamic instability of composite rotors. In Proceedings of Fourth International Conference on Rotor Dynamics, Chicago, Illinois, Sept. 1994. IFToMM, pp. 2877292.

Chang, F.K. and Kutlu, Z., Delamination effects on composite shells. J. Eng. Muter., 1990, 111(3), 336-340. Saravanos, D.A. and Hopkins, D.A., Effects of delaminations on the damped dynamic characteristics of composite laminates. Analysis and experiments. J. Sound l’ibration. 1996, 192(5), 9717993.

Wu, X.X. and Sun, C.T., Vibration of thin-walled composite structures. In Vibration and Behavior of’ Composite Structures. Vol. 14. The American Society of Mechanical Engineers, New York, 1989, pp. 47 -52. Wettergren, H.L., Rotordynamic analysis with special reference to composite rotors and internal damping. Ph.D. thesis, Linkoping University, 1996. Yurgartis, SW.. Measurement of small angle fibre misalignments in continuous fibre composites. Compos. Sci. Technol.. 1987, 30. 279- 293.

NOMENCLATURE a

PI # 61 E F Fij [HI, w*1 h ;;I L Ldel

bl [Ml

radius of curvature of contour part of strain-displacement matrix strain-displacement matrix element damping matrix displacement field elastic modulus load stress parameter in stress space shape function element length structure stiffness matrix element stiffness matrix beam length delamination length element mass matrix structure mass matrix

in composite

rotors: H. L. Wettergren

mass force per unit length component in n, s, z coordinate system component in [S] reduced stiffness for unidirectional fibre nodal displacements radius coordinate in n, s. z system coordinate in n, s! z system stiffness matrix length of contour component in n, s, z coordinate system thickness of wall, time position of delamination displacement in x direction, strain energy displacement in y and z directions displacement components in (n, s, z) direction Cartesian coordinates component in n, s. z coordinate system fibre orientation angle dissipation energy strains thickness coordinate Y -I!’ curvatures element in {E} Poisson’s ratio 7,

N-

L

mass density normal stress shear stress phase angle element in {E> -0’ angular rotational frequency of shaft angular frequency Eigen-frequency for shaft without deformations sectorial coordinate or warping function

527