Composites Engineering, Vol. 3, Nos l-8, pp. 633-643. 1993. Printed in Great Britain.

096-9526193 16.CQt NJ 0 1993 Pergamon Press Ltd

CRITICAL SPEED ANALYSIS OF LAMINATED COMPOSITE, HOLLOW DRIVE SHAFTS

CHUN-DO KIM and CHARLES W. BERT School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman,

OK 73019, U.S.A.

Abstract-A theoretical analysis is presented for determining the critical speeds of a rotating circular cylindrical hollow shaft with layers of arbitrarily laminated composite materials by means of the thin- and thick-shell theories. The theory used is the dynamic analog of the Sanders best first approximation shell theory. By means of tracers, the analysis can be reduced to that of various simpler shell theories, i.e. Love’s first approximation, Loo’s, Morley’s and Donnell’s shell theories and, also, to that of the more precise, but complicated Fhigge’s shell theory. The theories include the combined effects of torsion and rotation. The rotational effects contain the centrifugal and Coriolis forces. As an example of the application of the theory, a closed form solution is presented for a simply supported drive shaft. In order to validate the analysis, numerical results are compared with existing results for various special cases. Also, the effects of the various shell theories, thickness shear flexibility and bending-twisting coupling on the critical speed are investigated.

INTRODUCTION

Cylinders constructed of laminated anisotropic materials are being used as structural components with increasing frequency in the aerospace industry. Because of their high modulus, high strength and low density, a significant weight saving can be realized. Also, by appropriate design of the layup configuration, a number of desirable structural as well as thermal characteristics can be obtained. Furthermore, the use of composite cylindrical tubes would permit the use of longer shafts for a specified critical speed than is possible with conventional aluminium alloy or steel shafts.

In contrast to the extensive body of analytical and experimental information on the dynamics of disk-type rotors, engineering data on cylindrical-shell or drum-type rotors are quite sparse. Bryan (1890) is often credited with the first analysis of this problem. However, in fact, he considered only a rotating thin ring. Brzoska (1953) used Love’s first approximation shell theory to analyze the critical speeds of a thin-walled drum-type rotor constructed of isotropic material. His work was followed by that of Grybos (1961), who formulated the same problem in terms of isotropic elasticity theory, but obtained a solution only for the special case of a thin-walled cylinder.

The effect of Coriolis forces was added by DiTaranto and Lessen (1964), who considered an infinitely long, thin-walled, isotropic, circular cylindrical shell. Additional work on thin-walled, isotropic cylindrical shells rotating about their axes was due to Srinivasan and Lauterbach (1971), who considered traveling wave type solutions, Zohar and Aboudi (1973), who employed an exponential matrix expansion of the so-called “fundamental matrix”, and Wang and Chen (1974).

Shells constructed of orthotropic material were considered by Penzes and Kraus (1972), Padovan (1973) who used the orthotropic version of Fhigge’s thin shell theory and Chamis and Kiraly (1976), who made a NASTRAN finite element analysis.

The critical speeds of rotating anisotropic cylindrical shafts have been investigated by Zinberg and Symonds (1970) (whose experiments have proven the advantages over aluminum alloy shafts) and dos Reis et al. (1987), who evaluated the shaft of Zinberg and Symonds (1970) by the finite element method. Recently, a simplified theory for calculating the first critical speed of a flexible drive shaft constructed of composite material was developed (Bert, 1992) and hence the effect of bending-twisting coupling on the critical speed was investigated.

633

634 CHUN-DO KIM and C. W. BERT

In the present study, a theoretical analysis is presented for determining the critical speeds of a rotating circular cylindrical hollow shaft with layers of arbitrarily laminated composite materials by means of the various thin- and thick-shell theories.

BASIC EQUATIONS

Hypotheses

The following assumptions are made concerning the description of the shell motions:

(0

(ii)

(iii)

(iv) 09

04 (vii) (viii)

The displacement components vary linearly across the thickness of the shell except for the radial component, W, which is constant for fixed x, 8 and t. Displacements are small compared to the shell thickness so that the strain- displacement relations may be taken as linear. The layer material is linearly elastic and macroscopically homogeneous and may be monoclinic, orthotropic, transversely isotropic or isotropic, as desired. The shell is circular cylindrical with a uniform wall thickness. The shell is rotating at a constant angular velocity. Body forces and body moments are negligible. Dissipative effects, such as damping, are negligible. Acoustical fluid interactions are neglected, i.e. the shell is assumed to be vibrating in a vacuum.

Equations of rnotion

The coordinate system and displacement notation used here are shown in Fig. 1.

(1) Thin-shell theory. The equations of motion in terms of stress resultants in the absence of body forces and body moments can be written as:

NM + N6,y - WRM,y - 2Tu,, = PU,,,

&,x + Nz,~ + @,/R&f,,, + (CI/R)M,,~ - 2T@,, + W./R) = P(u,rt - &-J*~J + 2aW,,) (1) Ml,, + 2k& + ii&,, - N2/R - 2T& - V,,/R) = p(W,,, - a*W - x31,),

where Ni r in surface stress resultants, Mi 9 stress couples or moment resultants, T = shear load, 0 = rotating speed of shaft about x-axis, R = mid-surface radius, ci = thin-shell theory tracers, u, u and w = displacements in the axial, circumferential and normal direction, t = time, and ( ), = partial differentiation with respect to the subscript following the comma. Here, it has been assumed that the mass distribution is symmetric about the middle surface. The normal inertia is:

l

h/2

P’ pdz (2) -h/2

where h is the thickness of cylindrical wall.

Fig. 1. Coordinate system

Critical speed analysis of hollow drive shafts

Table 1. Thin-shell theory tracers and their values

635

Theory Cl c2 c3 c4 C5

Fliigge 1 1 1 0 1 Sanders 1 1 3/2 l/2 l/2 Love’s first approximation 1 1 1 0 0 Loo 1 0 1 0 0 Morley 1 0 0 0 0 Donnell 0 0 0 0 0

The shell theory used is a generalized first approximation one that can be reduced to the theories of Flugge (1973), Sanders (1959), Love’s first approximation (1927), Loo (1957), Morley (1959) and Donnell (1933) by use of the tracers listed in Table 1.

The kinematics of deformation can be expressed as follows:

Ej = &P + ZKi (i = 1,2,6) (3)

where mid-surface strains can be written as:

El O=u 9X’ E; = u,~ + w/R, &;=U +u ,Y ,x (4)

and curvatures can be written as:

Kl = -w,,, Kz = -w,yy + (cJR)u,~ (5)

~~ = -2w,, + (q/R)u,, - (@R)u,~

where the radial displacement, w, is positive in the outward direction. The laminate constitutive relations are given as follows:

(6)

Here the respective in plane, bending-stretching, and bending or twisting stiffnesses are:

s

h/2

tAijs Bij, Oij) = -h,2 Qij(ls 2, z2) b (i,j = 1,2,6). (7)

Substitution of eqns (2-7) into eqn (1) yields a set of operator equations in terms of displacement components;

[CiIq (24, u, Wy-‘0 (8)

where “T” denotes the matrix transposition. The elements of the 3 x 3 matrix of differential operator [d=] are:

cI1 = ArId: + (2Ar,j - [(cd + c,)/R]B,, - 2T]d,d,

+ b466 - [(cd + +)/RI& + (Cq+/R 2)&)d; - pd:. &, = [AI6 + (c,/R)&,]d: + (A,, + A,, + WW3,, + [(C3 - Cd/R146

- (CdR2)&)dxdy + (A26 + b2 - diRlB26 - (C2cdR2)&jld;

d:,, = -B,,d; - [3& - (c&?)D16]d:dy - [&2 + 2&6 - (2c,/R)&]d,d;

- [&, - @q/R)&@; + (An/RN, + [(A26/R) - (c,~,,/R2)]dy (9) 221 = [A,, + WW,,ld,2 + (A12 + A66 + (c,/W,2 + KC, - C5)/R1&

- (C,C,/R2)&,]d,dy + (&j + [(Cl - $)/RI&j - (c,c,/R2)&&$ 222 = b466 + ww66 + (C3/W2Da61d: + tu26

+ [(cl + c2 + 2c,)/R]& + [(c1c3 + c2c3)/R2]Dz6 - 2T)d,d,

+ (A22 + [(cl + c,)/R]B,, + (c,c2/R2)D2,Jd; - pdf + pa2

636 CHUN-DO KIM and C. W. BERT

223 = - [B,6 + (cJW~,,ld: - L&z + 24 + WRP,, + (WW,,1dtd,

- ]3B2, + ](2c, + c,)/W,,ld,d; - P22 + WW,,ld: + [A,,/R + c3(B2JR2) - 2T/R]d, + [(A22/R) + c,(B2,/R2)]d, - 2pSZd,

C,, = -B,,d; - [3B16 - (cs/R)DIa]d,2d,, - [B,, + 2B,, - (2c,/R)D,,]d,d;

- &a - WW,,ld; + (An/W, + [@26/R) - csUWR2)1dy (9) (continued)

J&2 = -h + WW,,ld~ - P12 + 2B66 + (c,/RP,, + (WWd~dy

- ]3B,, + UC, + dW%&W; - P22 + WW&; + [-426/R

+ (c,B~~/R~) - 2T/R]d, + [A22/R + (c2B22/R2)]d,, - 2pQd,

C,, = Dll d; + 4D,,d;d,, + @I,, + 4D,,)d:d, + 4D,,d,d; + D,, d; - (2B,,/R)d,Z

- [4(B,,/R) - 2T]d,d,, - 2(B2,/R)d, + A2,/R2 - Pa2 + pd:

where d, = ( ),,, d,, = ( ),u, d: = ( ),tt, and so on.

(2) Thick-shell theory. The equations of motion in terms of stress resultants in the absence of body forces and body moments can be written as:

N1.x + N6.y - (c2/2R)M6,y - 2Ty, = PM,,, + (Z/R)Yl,tt

N6,, + N2,y + WR)Q4 + (c2/2R)M6,x - WY, + w,,/R)

= p(u,,, - CJ2u + 2Ow,,) + (Z/R)(Y,,,, - Cl’Y’,) (10)

Q5.x + Q.t,y - WR - 27-(~,~ - v,,/R) = p(w,tt - Q2w - 2Qv,t) - 2(Z/R)CN2, t

ML, + M5.y - Qs = ml,,, + WR)u,tt

M6,x + M2,y - Q4 = ZY2,tt + (Z/R)y,, + (Z/R)(2&, - Q’z.J - RQ2Y2)

where Ni, Mi, T, &I, R, u, U, w, t and ( ), are the same symbols as in the thin-shell theory but Ci are thick-shell theory tracers, Y, and Y2 are bending slopes in the xz- and yz-planes, respectively. Here, it has been assumed that the mass distribution is symmetric about the middle surface. The normal and rotatory inertias are:

s

h/2

(P9 Z) = (1, Z2)P dz. (11) -h/2

The shell theory used is a generalized first approximation one that can be reduced to the shear deformation versions of the theories of Fliigge (1973), Sanders (1959), Love’s first approximation (1927), Loo (1957), Morley (1959) and Donnell (1933) by use of the tracers listed in Table 2.

The kinematics of deformation can be expressed as follows:

Ei = &i” + ZKi (i = 1,2,6)

e4 = ‘I”, + w,~ - (c,/R)u, Es = Yy, + w,,

Table 2. Thick-shell theory tracers and their values

Theory Cl C2 c1

(12)

Fliigge 1 0 2 Sanders 1 1 1 Love and Loo 1 0 0 Morley and Donnell 0 0 0

Critical speed analysis of hollow drive shafts

where mid-surface strains can be written as:

637

El O=u

3X’ E2 ’ = u,~ + w/R,

and curvature changes can be written as:

es0 = u,~ + v,~ (13)

Kl = Yl,x* K2 = y2.y 3 K6 = YI,~ + y2,x + (C2u.x - C3U,,9)42R). (14)

The coefficients ci appearing in eqns (lo), (12) and (14) are thick-shell tracers which are different from those of the thin-shell theory and take on the values listed in Table 2. Note that in contrast to the case of the thin-shell theory, Loo’s approximation offers no simplification over Love’s first approximation theory and Donnell’s theory coincides with Morley’s.

The laminate constitutive relations are the same as eqn (6), and also:

fQi1 = [si~I(Ejl (i, j = 4, 5). (15)

Here, respective stiffnesses are the same as for eqn (7) and thickness-shear stiffnesses are defined as:

Sij = s

h/2

Ki Kj Qij dZ (i,j = 4, 5). (16) -h/2

Here, Qij denotes the plane-stress reduced stiffness defined as:

(ail = [QijItEjl* (17)

Here, Ki and Kj are the thickness-shear motion correction factors which are taken as K2 = 7c2/12.

Substitution of eqns (11-17) into eqn (10) yields a set of the operator equations in terms of displacement components.

[Ju% u, w, y’, , y’,Y = 0 (18)

where T, also, denotes the matrix transposition. The elements of the 5 x 5 matrix of differential operator [.C] are:

&I = A,d + I”,6 - (c,/R)B,,ld,d, + [&j - (CZ/R&j

+ (C2/2R)2D66]d; - pdf

$1, = IA,, + (@R)B,,ld: + [A,, + -46, - (C2/2R)2&ld,dy 2

+ IA26 - (‘4/2R)&,ld,

$13 = (Au/W, + [(A26/R) - @2/2R2)&ldy

2 h = Bdb + I246 - (c,/2R)D,,]d,d, + [B,, - (C2/2R)Da6]d,2 - (Z/R)d;

gc,, = hid: + 142 + B66 - @,/2R)&l‘%d, + I826 - WWD,,ld,2 &r = [A,6+ (c,/2R)B,,ld: + ]A12 + A,, - (cJ2R)2D6,j1d,dY

+ [A 26 - (cd2WU d;

$2, = [& + @2/R&, + (C2/2R)2&jld: + [u,, + (c,/R)B,,ld,d,

+ A22d; - (cJR)~S,, + pa2 - pd:

$23 = [&JR) + c,(&JR) + (C2/2R2)&Idx + [(A22 + ClS&R]d, - 2&W

224 = [B,b + (c2/2R)D16]df + LB,, + B66 + (C,/2R)D,,]d,d, + B,,d,2 + (c,/R)S,,

g25 = [B6, + (c2/2R)D661dt + [2B26 + @dWWW, + B22d; + (q/R)&, + (Z/R)(L-12 - d;)

C,, = - (A,,/R)d, - ([A26 - (c,~WB,,l~Rldy

(19)

638 OWN-DO KIM and C. W. BERT

C,, = - [(cl S,,/R) + (A,,/R) + (c2/2R2)Z& - 2(T/R)]d, - [(cr S,, + A,,)/R]d,,

C,, = S,,d; + (2&s - 2T)d,d,, + S,,d; - Azz/R2 + p&J2 - pd;

C,, = 6, - B,,/R)d, + (S,, - B,,/R)d,

& = (S,, - B,,/R)d, + (S.,, - B,,/R)d, + 2(Z/R)Qd,

J&r = Brrd,Z + [2& - (cJ2R)D,JdXdY + [B,, - (c3/2R)&Jd; - (Z/R)d:

$42 = [h + WWhild: + i&2 + B66 + (‘2/2R)&ild,dy

&, = @%2/R - S,,)dx + (B,,/R - S,,)d,

&44 = D,,d,Z + 2D,,d,d,, + D66dy2 - Ss5 - Id:

245 = D,,d; + (012 + D,,)dxd, + D26d; - $5

25, = B,,d; + LB12 + B66 - (C,~2R)D,,]d,d, + IB26 - @~/2R)&ild;

252 = [& + @,/2R)D,,ld: + [2Bz6 + (c,/2R&ild,d,

(19) (continued)

+ B,,d; + &JR + (Z/R)(Q2 - d:)

J&3 = (R26/R - S,,)d, + (B,,/R - S,,)d, + 2(Z/R)Qd,

&4 = Dd,2 + (012 + &i)dxdy + %d; - s45

c& = D66d; + 2Dz6dxdy + Dz2d; - S.,, + Z(Q2 - d:)

where d,, d,, , dz , df , etc. designate the same symbols as in eqn (9).

Boundary conditions

The method of analysis is general and is used with any desired boundary conditions at the ends, x = 0, L. For comparison with other investigators’ results, simply supported conditions are chosen. Accordingly, one can have the boundary conditions at the ends as shown in Table 3.

Table 3. Boundary conditions

Theory Simply supported without axial Simply supported with axial constraint (freely supported) constraint

Thin-shell u=w=M,=N,=O u=v=w=M,=O Thick-shell v=w=Mx=N,=Yl=O u=v=w=M,=y,=o

METHOD OF ANALYSIS

Since the object of the present analysis is to predict the critical speeds, the solutions of the equations of motion, eqn (8) or (18), and the boundary conditions are assumed to be harmonic functions in a so-called semi-inverse method. The procedure follows closely that suggested by Fhigge (1973) and demonstrated by Forsberg (1964) for thin isotropic shells and Vanderpool and Bert (1981) for materially monoclinic, thick-wall, circular cylindrical shells. For brevity of analysis, the thick-shell theory is considered. The same method can be applied for the thin-shell theory.

Assumed displacements

Because of the materially anisotropic nature of the shell in the present analysis, care must be exercised in selecting the displacement functions. The primary difficulty

Critical speed analysis of hollow drive shafts 639

encountered here is readily seen in the various combinations of odd and even derivatives in the equations of motion. Noticing that in eqn (18) each equation either contains an even or odd number of differentiations of each displacement component, but not both even and odd, one conveniently assumes:

(u, u, w, Y, , Y2) = (U, V, W, X, Y)eahre’(oy-or) (20)

where (Y = mn/L, j3 = n/R, I = axial wave parameter, U, V, W, X, and Y = amplitudes of u, U, w, Y, , and Y2. Here, m and n denote the number of axial half-waves and circum- ferential waves, respectively.

Substituting eqn (20) into eqn (18) and carrying out the differentiations, one obtains the following system of equations of motion.

. - . u 0

V 0

W = 0

X 0

Y 0 - where cij are functions of the shell stiffnesses, m, n, Q, o and 1.

Cl1 Cl2 Cl3 Cl4 Cl5

c21 c22 c23 c24 c25

c31 c32 c33 c34 c35

c41 c42 c43 c44 c45

_ c51 c52 c53 c54 c55

(21)

Eigenvalue problem solution method

For a given longitudinal wave number m, circumferential wave number n, rotating speed 0, and angular frequency o, one can have five homogeneous equations in the five unknown amplitudes. This forms an eigenvalue problem with respect to the axial wave parameter, rl. In order to solve for the eigenvalues, one follows the procedure outlined in Gourlay and Watson (1973) wherein one rewrites eqr

I/- V

W

RX

RY

l(

o- 0

=

Id

0

0

0

IDI

(21) in matrix form as:

(22)

Here [D] is a so-called I matrix defined by:

[D] = [A”]A2 + [A’]A + [A21 (23) The components of [A’], [A’] and [A21 can be deduced directly from eqn (21). The characteristic roots, A, in eqn (23), are the same as those of the standard eigenvalue problem:

WI - WI = PI where [Zj is the 10 x 10 matrix defined as:

(24)

WI [II Ia = [B2] [I?‘] [ 1 (25)

and [B2] = - [A”]-‘[A2],

[B’] = - [Ay[k].

Having found the characteristic values for A, one can see that the complete solution of eqn (21), for a given m, n, Q and o is:

(u, u, w, Y’,, Y2) = E (U,, V,, W,, X,, Yk)euxkxe’~~y-w’). k=l

(27)

640 CHUN-DO KIM and C. W. BERT

For each value of k (i.e. each eigenvalue, A) one can calculate a set of amplitude (eigen- vector) ratios. For the present case, all eigenvectors were normalized by Wk. Since these values are known for each eigenvalue, let a starred quantity designate an appropriate eigenvector ratio. For example:

x,* = x,/w,. Then eqn (27) can be rewritten in appropriate form and the resulting equations substituted into the boundary conditions. Thus, one can see that there is a set of 10 homogeneous equations in the 10 unknown W,, for a given m, n, Q and w. The coefficient matrix for the W, must be singular in order to have possible solutions to the boundary conditions. An iterative process is employed on w, for given m, n and CJ, in order to find the set of eigenvalues and eigenvectors for which the boundary conditions are satisfied. Practically, though, one merely observes a change of sign in the determinant of the coefficient matrix for the boundary conditions. Hence, one may calculate resonant frequencies for any shell geometry and for any boundary conditions to any degree of accuracy.

NUMERICAL EXAMPLES AND COMPARATIVE RESULTS

Comparisons with previous investigations In order to check the validity of the analytical approach as well as the algebra and

programming, comparison with the results obtained by other investigators for special cases is presented. Bray and Egle (1970) compared experimental resonant frequencies with those obtained by analysis using the Fliigge thin-shell theory, for a homogeneous, isotropic, thin-cylindrical shell. In applying the present analysis to the same shell, two sets of frequencies were obtained. The first set was the result of using the elastic properties from a plane-stress assumption for a thin shell. The second set was the result of using the same properties for a thick-shell. It can be seen in Table 4 that the comparison between the previous analysis and the present analysis shows very good agreement.

As a second example for use in testing the analysis and programming, consideration is given to calculation of the first critical speed of a rotating circular cylindrical drive shaft with layers of arbitrarily laminated composite material. This thin-walled composite drive shaft was studied by Zinberg and Symonds (1970) and dos Reis et al. (1987) and is made of boron-epoxy with a specific weight pg of 0.071 lb ine3. The individual layers are assumed to be orthotropic, with the following elastic material properties:

E, = 30.6 x 106psi

E2 = 3.5 x 106psi

G12 = 1 .O x lo6 psi

V 12 = 0.36.

Table 4. Comparisons among theoretical resonant frequencies (Hz), for a homogeneous, isotropic, thin cylindrical shell with freely supported ends [for first axial mode (m = l)]. Case: (a) experimental results from Bray and Egle (1970); (b) Fliigge theory from Bray and Egle (1970), as listed numerically in Bert ef al. (1969); (c) present analysis by using Fliigge

thin-shell theory; (d) present analysis by using Fhigge thick-shell theory

n a b c d

1 2 3 4 5 6 7 8 9

10

- -

1001 715 534 410 393 426 495 587

3270.5 3270.5 3270.5 1862.0 1862.0 1862.0 1101.8 1101.8 1101.8 705.7 705.7 705.7 497.5 497.5 497.5 400.1 400.1 400.1 380.7 380.7 380.7 416.7 416.8 416.7 488.6 488.7 488.6 583.9 583.9 583.8

Material properties: E = 29.5 x lo6 psi, v = 0.285, specific weight = 0.2836 lb ine3 Shell dimensions: L = 11.74it-1, R = 5.836in, h = 0.02in.

Critical speed analysis of hollow drive shafts 641

The layup consists of a total of 10 layers, each with a thickness of 0.0052 in, for a total shell thickness equal to 0.052 in. The winding angles for each layer, from the inner to the outer surface of the cylinder, are given by [90”, 45”, -45”, OS’, 90’1. The length of the shaft L is 97.25 in, and the mid-surface diameter D is 4.948 in.

The first critical speed of the shaft was predicted by Zinberg and Symonds (1970) based on beam theory using a longitudinal equivalent Young’s modulus E, under the assumption that the shaft was uniform with a thin-walled circular cross section and with pinned supports at each end. Under these assumptions, Zinberg and Symonds predicted a first critical speed of 5840rpm. This prediction, however, does not include the effects of bearings or end fittings. They also performed a test on this shaft to experimentally determine its critical speeds. This test was performed by rotating the shaft with a variable speed motor with the deflections of the shaft being recorded at different speeds. Extrapolating the results of this test, Zinberg and Symonds predicted a critical speed of about 6000 rpm, while excessive vibrations occurred at 5 100 rpm.

However, the critical speed predicted by dos Reis et al. was 4942 rpm which was obtained by using 20 equal finite elements to model the static stiffness of the shaft. In this prediction, the bearing damping coefficients were neglected and the bearing stiffness coefficients were given values of large quantities to represent the rigid supports.

The critical speed predicted by the present approach, using Fhigge thin-shell theory, is found to be 5883 rpm, which is 0.74% higher than the critical speed predicted by Zinberg and Symonds using the classical beam theory with a longitudinal equivalent Young’s modulus and 1.95% lower than the experimentally obtained critical speed. Taking into account that no information on the actual effects of bearings, end fittings or mounting conditions was available, this prediction is certainly within engineering accuracy. In comparison with the critical speed predicted by dos Reis et al., however, the present critical speed is 19.0% higher.

Discussion

One of the most informative ways of presenting numerical data on rotor dynamics is the Campbell diagram or ‘whirling map’ originated by Campbell (1924) in connection with turbine-disk flexural vibration. This diagram is a plot of various natural whirling fre- quencies of lateral vibration vs rotational speed. A Campbell diagram for this thin-walled composite drive shaft, as predicted by the present analysis using Fhigge thin shell theory is shown in Fig. 2. As the rotational speed (Q) is increased, the whirling frequency ‘splits’ into two distinct branches. The deviation between two branches associated with a given zero-speed mode is caused by the gyroscopically tilting moments induced in the masses which are free to tilt. The branch having the higher frequencies is associated with forward

E IO’ 5 C

10’ 10’ 102 102 10’ 10’

Rototmai speed (rev mm-‘) Rototmai speed (rev mm-‘)

Fig. 2. Campbell diagram for thin-walled composite drive shaft. Fig. 2. Campbell diagram for thin-walled composite drive shaft.

642 CHUN-DO KIM and C. W. BERT

Table 5. Critical speed (rpm) predicted by various shell theories

Theory Thin-shell theory Thick-shell theory

Fliigge 5883 5885 Sanders 5872 5874 Love’s first approximation 5892 5894 Loo 5856 5894 Morley 5878 6401 Donnell 6399 6401

whirling; i.e. the shaft is whirling in the same direction as the rotation. Likewise, the branch having the lower frequencies is associated with backward whirling; i.e. the direc- tion of whirling is opposite to the direction of rotation.

In order to validate the various shell theories, the first critical speed was predicted for the same drive shaft, the results of which are shown in Table 5.

Although Bert and Reddy (1982) showed that relatively simple Donnell theory generally provides acceptible accuracy if the radius-thickness ratio of the shell exceeds 20 for composite cylindrical shells, they also noted that Donnell’s shell theory is not appro- priate for long shells. Hence, for this drive shaft, one can see that Donnell’s shell theory does not produce appropriate results. Here, as far as thick-shell theories are concerned, stiffnesses Qd4 and Qss have been assumed as the same as Qe6 (= GJ since they were not given by Zinberg and Symonds (1970).

In order to check the effects of the quantities of Q.+, and Qss on the critical speeds, several cases have been investigated, i.e. fixing one of Qd4 and QSS, varying the other quantity. The results are shown in Table 6. Hence, one can see that the effects of the transverse shear moduli on the critical speeds are negligible for this very thin-walled drive shaft.

The simplest laminate having the greatest bending-twisting coupling would be an off- axis parallel-ply laminate having all of the layers oriented at an acute angle 8. For a drive shaft with such a laminate having 10 layers, the critical speed was calculated for various values of 8, by use of the Sanders thin-shell theory. The results are listed in Table 7. From Table 7, it is clear that an off-axis parallel-ply laminate is less desirable than one having all of the fibers oriented axially (0”). For 0” < 0 < 90”, this is due to the combination of shear-normal coupling and low transverse extensional stiffnesses, while for 0 = 90”, it is due to the latter effect only. Also shown in Table 7 are the results of a simplified anisotropic-beam critical speed analysis by Bert (1992).

Table 6. Effect of transverse shear stiffness on critical speed (rpm) by

using Sanders’ shell theory

Qd4 and Qss Critical speed

G,, 5874 (5874) G,,/2.0 5874 (5874) G,,/3.0 5874 (5874) G,,/4.0 5874 (5874) GJ5.0 5874 (5874)

Fix QSS = G12(QW = G,,/2.0).

Table 7. Critical speed (rpm) for a thin-walled, off-axis, boron-epoxy drive shaft

Lamination angle 0, deg. 0 15 30 45 60 75 90

Present 5714 4432 3048 2529 2425 2512 2582 Bert (1992) 7141 4955 3722 3011 2569 2433 2415

Material properties: E, = 30.6 msi, Ez = 3.5 msi, G,, = 1 .O msi, viz = 0.36, specific weight = 0.071 lb/K3 Shaft dimension: L = 97.25 in, D = 4.948 in, ply thickness = 0.0052 in, total layers = 10.

Critical speed analysis of hollow drive shafts

CONCLUSIONS

643

The present approach provides a powerful tool for determining the critical speeds of a rotating circular cylindrical hollow shaft with a wide variety of layers of laminated com- posite materials which may be monoclinic, orthotropic, transversely isotropic or isotropic, as desired.

Comparison of the present results with those obtained with the classical beam theory approach using an equivalent longitudinal Young’s modulus and by experiments shows that the present approach gives results with engineering accuracy. Among the various shell theories, Donnell’s theory is not effective for a long shaft. Also, for this very thin-walled drive shaft, the effects of the transverse shear moduli on the critical speeds are negligible.

REFERENCES

Bert, C. W. (1992). The effect of bending-twisting coupling on the critical speed of a drive shaft. 6th Japan-U.S. Conf. on Composite Materials, 22-24 June, Orlando, FL, U.S.A.

Bert, C. W. and Reddy, V. S. (1982). Cylindrical shells of bimodulus composite materials. J. Engng Mech. Div. Proc. AXE 10, 675-688.

Bert, C. W., Baker, J. L. and Egle, D. M. (1969). Free vibrations of multilayer anisotropic cylindrical shells. J. Compos. Mater. 3, 480-499.

Bray, F. M. and Egle, D. M. (1970). An experimental investigation of the free vibration of thin cylindrical shell with discrete longitudinal stiffening. J. Sound Vibr. 12, 153-164.

Bryan, G. H. (1890). On the beats in the vibrations of a revolving cylinder or bell. Proc. Cambridge Phi/. Sot. 7,101-111.

Brzoska, Z. (1953). Critical speeds of short drums. Tech. Lotn (in Polish) 8, 151-156. Campbell, W. (1924). The protection of steam-turbine disk wheels from axial vibtation. Trans. Am. Sot. Mech.

Engnrs 49, 31-160. Chamis, C. C. and Kiraly, L. J. (1976). Rim-spoke composite flywheels-stress and vibration analysis. NASA

TN D-8339. DiTaranto, R. A. and Lessen, M. (1964). Coriolis acceleration effect on the vibration of a rotating thin-walled

circular cylinder. J. Appl. Mech. 31, 700-701. Donnell, L. H. (1933). Stability of thin-walled tubes under torsion. NACA Report 479. Fliigge, W. (1973). Stresses in Shells, 2nd edn. Springer, NY. Forsberg, K. (1964). Influence of boundary conditions on the modal characteristics of thin cylindrical shells.

AIAA J. 2, 2150-57. Gourlay, A. R. and Watson, G. A. (1973). ComputationalMethods of Matrix Eigenproblems, Chap. 14. Wiley,

NY. Grybos, R. (1961). The problem of stability of a rotating elastic solid. Archiwum Mechaniki Stosowanej 13,

761-774. Loo, T. T. (1957). An extension of Donnell’s equation for a circular cylindrical shell. J. aeronaut. Sci. 24,

390-391. Love, A. E. H. (1927). A Treatise on theMathematical Theory of Elasticity, 4th edn (reprinted in 1944). Dover,

NY. Morley, L. S. D. (1959). An improvement on Donnell’s approximation for thin-walled circular cylinders. Q. J.

Mech. appl. Math. 12, 89-99. Padovan, J. (1973). Natural frequencies of rotating prestressed cylinders. J. Sound Vibr. 31, 469-482. Penzes, L. E. and Kraus, H. (1972). Free vibration of prestressed cylindrical shells having arbitrary

homogeneous boundary conditions. AIAA J. 10, 1309-1313. dos Reis, H. L. M., Goldman, R. B. and Verstrate, P. H. (1987). Thin-walled laminated composite cylindrical

tubes; part III-critical speed analysis. J. Compos. Technol. Res. 9, 58-62. Sanders, J. L. (1959). An improved first-approximation theory for thin shells. NASA Reporr R-24. Srinivasan, A. V. and Lauterbach, G. F. (1971). Traveling waves in rotating cylindrical shells. J. Engng Ind.

Trans. ASME 93, 1229-1231. Vanderpool, M. E. and Bert, C. W. (1981). Vibration of a materially monoclinic, thick-wall circular cylindrical

shell. AIAA J. 19, 634-641. Wang, S. S. and Chen, Y. (1974). Effects of rotation on vibrations of circular cylindrical shells. J. Acoust. Sot.

Am. 55, 1340-1342. Zinberg, H. and Symonds, M. F. (1970). The development of an advanced composite tail rotor drive shaft.

Presented at the 26th Annual National Forum of the American Helicopter Society, June, Washington, DC. Zohar, A. and Aboudi, J. (1973). The free vibrations of a thin circular finite rotating cylinder. Int. J. Mech. Sci.

15, 269-278.