Natural frequencies of rotating prestressed cylinders

Journal of Sound and Vibration (I 973) 31(4), 469--482

NATURAL

FREQUENCIES

PRESTRESSED

OF

ROTATING

CYLINDERS

J. PADOVAN

Department of 3lechanical Engineerhrg, Unirersityof Akron, Akron, Ohio 44325, U.S.A.

(Receired 26 J~Iarch 1973, and ht rerisedform 18 July 1973)

A vibration analysis is described, involving the use of complex Fourier series, by means of which the complete frequency eigenvalue spectrum and associated mode shapes for rotating axially, radially and torsionally prestressed cylindrical shells can be obtained. Due to its generality, the solution procedure can also handle the effects of constructional and material anisotropy together with arbitrary boundary conditions. From the solution, it is found that torsional prestress, rotation effects and constructional and/or material anisotropy each independently cause bifurcations in the frequency spectrum. In fact, it is found that the combined effects of prestress, rotation and anisotropy can cause an eightfold bifurcation in the frequency eigenvalue spectrum.

1. INTRODUCTION In numerous engineering applications (advanced gas turbines), certain rotor systems are essentially cylindrical shells rotating about their longitudinal axes. Dynamic analysis of such systems is necessary in order to influence design. Although numerous papers have been published on the free vibrations of cylindrical shells [1-5], few of these include the effects of rotation together with prestress. The effects of centrifugal and Coriolis forces have been discussed by Macke [6] and Armstrong and Christie [7] using ring theory, and again by DiTaranto and Lessen [8] including membrane effects of the infinitely long cylindrical shell. Srinivasan and Lauterbach [9] included the additional effects of torque to the membrane prestresses of a rotating infinitely long cylindrical shell. More recently, Penzes and Kraus [I0] studied the frequency eigenvalue spectrum of rotating prestressed cylindrical shells with arbitrary boundary conditions. Although the investigations of references [4], [9], and in particular [10], are of fairly wide scope, through the use ofincomplete eigenfunction bases, half of the frequency eigenvalue branches were entirely missed for orthotropic cylinders. The importance of these missed branches stems from the fact that certain of their frequency eigenvalues are of the order of the lowest shell frequency. The purpose of this paper is to present a vibration analysis which obtains the complete frequency eigenvalue spectrum of cylindrical shells rotating about their longitudinal axes and subject to axial, radial and torque prestress. In particular the analysis presented herein will develop a complete eigenfunction basis from which all the frequency eigenvalues and associated physical mode shapes are obtained for infinite cylinders. Furthermore, as will be seen later, due to the generality of the solution developed herein, the effects of arbitrary boundary conditions applied to finite cylinders can be handled as well. 469

470

j. PADOVAN 2. EQUATIONS

Figure I shows the shell geometry used in the analysis. To extend the scope of the analysis, the cylindrical shell is considered as either orthogonally [10] or spirally stiffened [11] and under the following loadings: (i) normal pressure, p, and "moderate" axial load, F, (ii) centrifugal and Coriolis forces, (iii) torque, T. The displacement equilibrium equations of motion used are based on Fliigge [5, 10, 12] type shell theory and include the buckling terms due to (i)-(ii) and the effects of (iii); they are (the basic nomenclature is given in Appendix V) aZ u L , ( u , v, Iv) = p R ~ ~ t ~ ,

(1)

)

[ a 2v 2 0 aw L , ( u , v, IV) = p R 2 [ - ~ - + - ~ - - 0 2o ,

[a'w

av_~ )

L3(U , v, w) = - p R z k at2 -- 2 0 ~

Iv ,

(2)

(3)

where for simplicity the differential operators LI, L2, L3 are defined in Appendix I.

Figure 1. Shell geometry. 3. SOLUTION Due to the appearance of the torque, T, as well as the anisotropic stiffnesses, E~3 and E23, the typical solution characterized by the usual Fourier decomposition procedure cannot satisfy equations (1)-(3). Furthermore, although appropriate independent variable transformations can be used to reduce equations (1)-(3) to one of the canonical forms, the associated change in the domain of definition distorts the boundary surfaces of the cylinder, thereby making the boundary value problem intractable. To avoid these difficulties, the procedure discussed by Padovan [5, 13-16] will be used to obtain the complete eigenfunction basis and thereby the entire eigenvalue spectrum.

471

ROTATING PRESTRESSED CYLINDERS

Independent of the appearance of T,I Ea3. E23 and 12, due to the periodic nature of the coordinate space, it follows that the shell displacements, strains and stresses are themselves periodic in the 0 variable depicted in Figure 1. Hence u, v and w can be formally represented by the following complex Fourier series [5. 13, 14]: ~

(u(x,O,t), v(x.O.t), w ( x , O , t ) ) =

(u,.(x,t), v , . ( x , t ) , w ( x , t ) ) e ]~e,

(4)

wherej = v / ( - l ) and

,f

2ff

(u,., v,.. win) = ~

(5)

(u, v, w) e-j"~ dO.

0

such that e J"~ m = 0, _+1, +2 . . . . . form a complete basis space. Applying the transform depicted by equations (5) to equations (1)-(3) yields the following complex partial differential equations: 02 U m LI,.(u,.. vm. w,.) = p R 2 - ~ - .

L2m(u"'v"'w")=PRZ\

a t.'~

0w,.

[a ~Wm

Or,.

(6)

)

at 2 +212"-'~t - 122t,...

La,.(u,,. v,,,, w,.) = p R ' ~ at z -- 2 0 ~

)

-- 12' w,. .

(7)

(8)

where L.,.. L2,. and L3,. are defined in Appendix II. Due to the appearance of r, E~3 and Ez3, unlike the situation for orthotropic torque free cylinders, the real and imaginary parts of urn, v,., w,. are coupled. Without a loss in generality, a further reduction of equations (6)-(8) can be obtained for an infinite cylinder by considering

(u,.,v,.,w,.)= ~

~" \~mn,

"~ r a n , '" ~.Jt.~IL~x .-ran; -,

(9)

tl.i~

where

1j

L



dx,

(IO)

-L

such that R L is the wavelength of the lowest frequency eigenvalue obtainable by the characterization depicted by equation (9). Since the choice of L is arbitrary, the entire frequency spectrum of equations (1)-(3) can be obtained. Note that, upon letting L approach o% equation (9) reduces to the usual bilateral Fourier inversion integral and equation (10) to i" 9 denotes torque prestress (3 = T/2nR2).

472

j. PADOVAN

the usual infinite Fourier integral transform. Applying equation (10) to equations (6)-(8) yields [ - N 2 al - tuNa2 - tn 2 a3 + N 2 F + 2 m N r + p R n t 2] u . . + [ - N 2 a4 - tuNa5 - m 2 a6] Vmn -k" d z um. + [jNa7 + j m a a - j m 2 N a 9 - j m N 2 a l o - j N a a 1 1 - j m 3 a 1 2 + j p N ] w,~. = p R 2

dt 2

(i1) [-N 2 bl - m 2b2 - mNb3] Urn. + [--N 2 b4 - m 2b5 - mNb6 + N 2 F + m 2 p R + 2mNz] v,.. + + [ - j N 3 b7 - j m 3 ba - j m 2 Nb9 - j m N 2 blo + j N b l l + ntbl2 - j 2 N z =pR

-jmpR] w.. =

2/d2 [ ~ v.. + 2f2 dW..d-tQ Z v . . . ' )

(12)

[ - i N 3 cl - - j m a c2 - j m N 2 c3 - j m 2 No4 + j N c 5 + j m c 6 + j N p R ] Urn. + [ - j N 3 c7 - - j m z Ncs --- j m N z c9 + j N c m + j m c l l - j 2 N ~ - j m p R ] vm. + [N ~ c12 + nt 4 c13 + m 2 N 2 C14 + +

Dt 3

Nc~s + m N

=-pR2~

dt 2

3

c16 - - m 2 Car - - mNc~a + c~9 - N 2 F -

do..

)

m 2 p R - 2mNz] w.. =

2f2--~--- 122,v.. ,

(13)

where N = nn/L. As before, due to the inherent nature of equations (1 I)-(13), the real and imaginary parts of u . . , v',.. and w.. are coupled. The frequency eigenvalues of equations (11)-(13) can now be obtained in the usual manner by assuming (u,.., v,.., w,.~)= (Urn., I'm., W,..) eJ'~,,.L

(14)

Inserting equation (14) into equations (11)-(13) yields the following complex polynomial matrix: [O92A o "Jr O9A 1 "Jr A2] ~ = 0,

(15)

where for convenience, Ao, AI and A2 are given in Appendix III and the transpose of tj is cr={u..,

(16)

I'm., W . . } .

Since Ao is non-singular, the pencil of equation (15) is regular. Hence, equation (15) can be equivalenced to the following complex linear eigenvalue problem in o9,.. [16]:

Ao

A,

o9m.+

,4,

L r

J

"

Although the pencil of equation (17) is non-Hermitian, oo.. can be obtained by the usual complex eigenvalue procedure.s [16, 17]. In contrast to corresponding results of previous studies [9], the characteristic polynomial of equation (15) is complex. For the present paper this result is directly due to the effects of torque and constructional or material anisotropy. Hence for given m and tt, since equation (15) represents a complex eigenvalue problem, the o9.. no longer occur in conjugate pairs. Instead, given a stable cylinder configuration (r < z buckling), for fixed values of z and 12, equation (15) will yield a total of six distinct frequency eigenvalues, o9... This leads to the

ROTATING PRESTRESSED CYLINDERS

473

fact that in a manner similar to that of rotation effects, anisotropy and torque each independently induce bifurcations in the eigenvalue spectrum of infinite cylinders. In particular, from the form of the pencil of equation (15) it follows that an infinite orthotropic cylindrical shell'l" under the combined influence of torque and rotation has 12 separate frequency eigenvalue branches and associated mode shapes. By setting z and 12 to zero the characteristic polynomial of equation (15) becomes purely real. Hence, it follows that the 12 frequency eigenvalue branches noted earlier are bifurcations about the three branches of the orthotropic static case with no torque prestress. For the case ofanisotropic composite construction or spiral stiffening, it follows from the form of equation (l 5) that 24 independent frequency eigenvalue branches are possible, depending upon the choice of E~3, Eza, 9 and f2. In a manner similar to that of the previous case, these branches are bifurcations about the six branches of the stati6 anisotropic case with no torque prestress [5]. From equations (4), (9) and (14) it is evident that u, v and w take the form

o8) such that L

2n

(tt'n'Vmn'W"n)=eff (u'v'w)e-J''no§ -L

(19)

o

Hence, the complete eigenfunction basis takes the form

(u, v, w> = ( V,.., V,.., W...> e J('~

(20)

where m = 0, +1, +_2. . . . . n = 0, +I, +_2. . . . . Due to the form of equations (1)-(3) it follows that for m > 0, n > 0 and nt < 0, n < 0, since the pencil of equation (15) is conjugated, the mode shapes associated with such m and n values are given by

(u,v,w> = I eJ~'e+(""/L'x+'~"")+ .,n

+ ( U . . , V,.., W . . )

I e -'l("~

--m, --n

(21)

Furthermore, from the inherent nature of equation (I 5), equation (21) may be reduced to u = 2[Re(Urn.) cos (mO + (mr~L)x + to,.. t) - Im(U..) sin (mO + (nn/L) x + o9.,. t)], v = 2[Re(I'm.) cos (toO + (nn/L) x + o9,.. t) - Ira(I'm.) sin (too + (mr~L) x + to,.. t)], w = 2[Re(W,..) cos (too + (mr~L) x + ogre. t) - Im(W,..) sin (m0 + (mr~L) x + w,.. t )], (22) where m, n = 0, 1, 2 . . . . . shape takes the form

In a similar fashion, for m > 0, n < 0 and m < 0, n > 0, the mode

(u,o,w)=(Umn,

Zmn, Wm.)

I e Jt-me+tnnlL)x§ . . . . ,,q_ --m. n

+ ( U . . , I'm., W,..)

[ e jt'e-("'/L) . . . . . . ", ..

1"For Kirchhoff-type shell theory.

--n

(23)

J. PADOVAN

simply u = 2[Re(U_,.~) cos (-too + (nrc/L) x + to_,.~ t) -- Im(U_"~) sin (-too + On/L) x + to_,.~ t), v = 2[Re(V_"~)cos (-mO + (nn/L)x + co_"~ t) - Im(V_m,)sin (-mO + (nn/L)x + to_"~ t), w = 2[Re(W_,.~) cos (-mO + (nrr/L)x + co_,.~ t) - Im(W_,.~) sin (-mO + (nn/L)x + co_,.~t), (24) lere m, i1 = 0, 1, 2 . . . . . 4. DISCUSSION With the use of complex Fourier series, expressions have been developed from which the tire frequency eigenvalue spectrum can be obtained for infinite rotating cylindrical shells composite construction or with orthogonal or spiral stiffening and subject to axial, radial d torsional prestress. Due to the generality of the technique, the complete eigenfunction sis can be developed for finite cylinders with arbitrary boundary conditions. To implement .~ procedure for the finite case, instead of expanding u,,, v,. and w" in complex Fourier 9ies in x, let

(us(x, t), v"(x, t), w"(x, t)) = ~r ea'x+J~

(25)

ch that ~r = {Us, V", W"}.

(26)

gerting equation (25) into equations (6)-(8) yields the following complex polynomial ltrix: [Bo 2,.9 + Bl 2,.3 + B2 2~ + B3 2,. + B4] ~ = 0,

(27)

~ere B0. . . . B4 are complex three-three matrices given in Appendix IV. Although the ncil of equation (27) is irregular/f since B4 is nonsingular, the latent vectors, ~,., correonding to distinct latent roots, 2,., are linearly independent. Evaluating the determinant 'the pencil of equation (27) yields a complex polynomial of the form

eo2S +jet27m+e226 +jea2~ +e42~ +jes2~ +e62~ + j e 7 2 , . + e s = O ,

(28)

acre eo. . . . el are real functions of E~j, f2, m, N, p, F, p and z. Since the odd coefficients 'equation (28) are complex, 2,. take the form 2,. = _+Re(;.,.) + j Im0.,.).

(29)

t terms of equations (25) and (29), u, v and w are given by

<.,v,w>--

m---~

1-1

(30)

ence, the complete eigenfunction basis is given by 8

(u,., v,., w,.) = Y. Crie~'~x+J~,.e+'~"~, 1-1

here nt = 0, +1, +2 . . . . . Bo is singular; see Appendix IV.

(31)

ROTATING

PRESTRESSED

475

CYLINDERS

Ir,20C

,, ~,

.~

////..

(r.n~

\ \ \- _J / I

I

2

4.

I 6

I 8

I I0

m

Figure 2. Effects of variations in m on the lowest frequency eigcnvalue branch. T= 2 x 104 in lb (2.31 x 102 m-kg); E = 3 x 107 lb]in2 ( 2.115 x 101~kg/m2); 12 = 3600 rev/min; v = 0"25; RL = 48 in (109-9 cm); n = 1; R = 10 in (25-4 cm); H = 0.03 in (0-76 turn); p = 0.28 lb/in 3 (0.775 x 10' kg/m3). F r o m this point, by using eigenfunctions of the form of equation (31), the frequency eigenvalues of equations (1)-(3) can be obtained in the manner of Forsberg [3], Penzes and Kraus [10], or Padovan [5]. From the form of the pencil of equation (27), it follows that the eigenfunctions associated with m and - m give rise to conjugate eigenvalues, (o,,, and latent vectors, ~,,. Hence the ph~'sical mode shape associated with a given O~mvalue takes the form ( Ilm' I)m' Wm> = 2 {~L e ~''x+j(""~'''~ + eL e~"x+J'-"~

(32)

1.1

where the overbar denotes complex conjugation. Note that, due to the appearance of conjugates, equation (32) is purely real as would be expected. This is in contrast to the peculiar complex modes described in reference [10]. To illustrate the substantial effects of z( = T]2rcR2) and f2 on the frequency eigenvatue spectrum, Figures 2-8 present various aspects of the eigenvalue spectrum of an infinite isotropic cylinder. In particular, Figures 2 and 3 illustrate all the possible bifurcations of the lowest branch for given values of T, ~ and N (= nn/L). Although the lowest branches I

I

300

N I

~ , 200

100

I

I

I

I

I

2

a

6 m

0

lO

Figure 3. Effects of variations in m on the lowest frequency eigenvalue branch. Legend as for Figure 2 except n - 2.

476

J. PADOVAN I

!

80C

GOC N "1-

40(::

I

cr,-~)\ C~-E2)

2OO

I 2

/7

I 4

Figure 4. Effects of variations in N on the lowest frequency eigenvalue branch. Legend as for Figure 2 except m = 6 and n is variable.

depicted retain the usual non-monotone [1] characteristic as the m Fourier harmonic is varied, the effects of T and f2 are extremely pronounced for the lowest frequency eigenvalues of the branches illustrated. Such effects are much less pronounced for the higher-order branches. Thus although bifurcation does occur, the differences between certain of the higher-order branches are negligible. As noted earlier, one of the primary effects of Tand s on the frequency eigenvalue spectrum of infinite isotropic or orthotropic cylinders is the fourfold bifurcation of the static torque

~(Hz)

3

0

"f~(rev/minxlO-3} ~

T(IbxlO-4)

-36~2

Figure 5. Effects of variations in Tand -Q on lowest frequency eigenvalue branch, m = 3; n = 1 ; R L = 48 i n (109.9 cm); R = I0 in (25"4 cm); H = 0"03 in (0.76 m m ) ; E = 3 Ib/in 3 (0-775 x 104 kg/m3); v = 0.25.

x l0 T lb/in 2 (2.115 x 10 t~ k g / m ' ) ; p = 0 " 2 8

477

R O T A T I N G PRESTRESSED CYLINDERS

-=E-:~-.

1147 ~

91 9 -

(Hz) 36

-2

,/2(~

-36~2 Figure 6. Effects of variations in Tand/2 on lowest frequency eigenvalue branch. Legend as for Figure 5 except m = 4. free branches. Interestingly, as illustrated in Figures 2 and 3, sign reversals between I2 and T give rise to a doubling of branches. Figure 4 illustrates the effects of variations in N on the lowest branch for given values of T, f2 and m . A s might b e anticipated on physical grounds, the effects of T and 12 are less pronounced as N increases. This is particularly true for the higher-order branches. The effects of variations in T and 12 are presented in Figures 5-8 for given N and m values. Each of the four quadrants of these figures illustrates the important topological redistributions due to variations in T and I2 for each of the bifurcations of the lowest branch of the static torque free case. In particular, the quadrants associated with (T, I2) and ( - T , - f 2 ) represent one set of bifurcations, while the quadrants associated with (-T, t2) and (T,-I2)

715-" ~

~594

l ,[2(rev/minx10-3)e

~_ ~ ~

-' 2

r(~xlO-4I

Figure 7. Effects of variations in Tand/2 on lowest frequency eigenvalue branch. Legend as for Figure 5 except m = 5.

478

J. P A D O V A N

94.7-

-86I ~

~(Hz)

J ~ ( r e v / m i n x lTO(6~2 -I 3b )x~l O - ' l - 3

)

Figure 8. Effects of variations in Tand g2 on lowest frequency eigenvalue branch. Legend as for Figure 5 except m = 6. represent the other set. Interestingly two bifurcations occur in the lowest frequency branch for both the T = 0, 12-r 0 or T ~ - 0 , f2---0 cases. Hence, backward and forward traveling waves can be associated with static cylinders subjeoted to t o r q u e p r e s t r e s s as well as to rotating cylinders solely under the influence o f Coriolis forces. With the introduction ofmaterial or structural'f anisotropy, although an eightfold increase in the frequency branches occurs, the trends remain somewhat the same as those just depicted for the isotropic case.

REFERENCES I. R. N. ARNOLDand G. B. WARBURTON1953 Proceedings of the Institute ofAfeehanical Enghteers 167, 62-74. The flexural vibrations of thin cylinders. 2. A. E. ARMENAKASand G. HERRMANN1963 American Institttte of Aeronautics a/td.4stronatttics Jottrnal 1, 10(O106. Vibrations of infinitely long cylindrical shells under initial stress. 3. K. FORSBERt3 1964 American hzstitute of ,4eronautics and Astronautics Jottrnal 2, 2150-2157. Influence of boundary conditions on the modal characteristics of thin cylindrical shells. 94. L. R. KOVALand E. T. CRAr~CH 1962 Proceedings ofthe 4th U.S. National Congress of Applied Mechanics 1, 107-117. On the free vibrations of thin cylindrical shells subjected to an initial static torque. 5. J. PADOVAN 1971 bzternational Journal of Solids and Strttctures 7, 1449-1466. Frequency and buckling eigenvalues of anisotropic cylinders subjected to nonuniform lateral prestress. 6. H. J. MACr:E 1966 Journal ofPowerfor btdastry 88, 179-187. Traveling-wave vibration of gas turbine engine shells. 7. E. K. ARMSTRONGand P. 1. CHRISTIE1966 AppliedAlechanics Convention, Cambridge, England, Paper No. 10. Vibrations in cylindrical shafts. 8. R. A. DITARA~,rrO and M. LESSEN 1964 Journal of Applied Mechanics 31, 700--701. Coriolis acceleration effect on the vibration of a rotating thin walled circular cylinder. 9. A. V. SRINIVASANand G. F. LAO'rERnACH1971 JournalofEnghzeerhtgfor Industry93, 1229-1231. Traveling waves in rotating cylindrical shells. 10. L. E. PrszES and H. KRAUS 1972 American htstitute of Aeronatttics and Astronautics Journal 10, 1309-1313. Free vibration of prestressed cylindrical shells having arbitrary homogeneous boundary conditions. 1"Spiral stiffening.

ROTATINGPRESTRESSEDCYLINDERS

479

I 1. TSAI-CHEN SOONG 1969 American b~stitute o f Aeronautics and Astronautics Journal 7, 65-72. Buckling of cylindrical shells with eccentric spiral-type stiffeners. 12. W. FLOGOE 1967 Stresses h: Shells. Springer-Verlag, 4th printing. 13. J. PADOVANand J. L~TIr~GI 1973 Journal o f the Acoustical Society o f America (in press). Natural frequency of monoclinie circular plates. 14. J. PADOVAN 1972 American lnstitttte o f Aeronautics attd Astronatttics Journal 10, 1364-1366. The solution of stresses and strains for laminated cylinders. 15. J. PADOVAN 1972 American btstitute o f Aeronautics and Astronautics Jottrnal 10, 60-64. Temperature distributions in anisotropic shells of revolution. 16. J. PADOVANand J. LESTINGI 1972 American btstitttte o f Aeronatttics and Astronautics Jottrna110, 1 2 3 9 - 1 2 4 1 . Mechanical behavior of fiber reinforced cylindrical shells. 17. K. K. GUPrA 1973 btternationalJournalfor Numerical Methods in Eng#teerhrg 5, 395-418. Free vibration analysis of spinning structural systems.

APPENDIX I a2 u

02 u

a2u

"' ~--~+ ~ b - ~ + - 3 ~ -

02 v

a3 w

a3 w

+01o ~

02 v

+a, a T + . , ~

02 v

aw

+.o-~-+a,~

03 w

02 u

aw

a3 w

+a. - ~ +.~ ao--Z-~a ~+ a 2 II

a2 u

- - F a x 2 - 2 r Ox00 - p R ~-~ +

+ a,i--~-- +012 - ~ -

aw + P T ; = L,(,,,,,,,~,),

a2u

b, ~

02 u

a 2 it

(M)

a2 v

a2 v

a2 v

a3w

a3 w

a3 w

+ b2-~r + b3 3 - ~ + b. ~-~ + b,-~- + b, a--~ + b, ~ - +b. -~- + b, a--x-~ + 03 w

aw

+ b.,o ~

aw

oz v

+ bn-~x + b,2-~ --Fox 2 -

02 v pR-ff~ -2T

02 v

aw

0x0~0 - 2~ ax

-

aw --pR

03 u

(A2)

-ffff = L2(u, v, w),

03 u

03 u

03 u

~,T~ + ~ 2 ~ + c ~ ~ av

au

au

03 v

03v

+ c" ao-~ax + c~ ~ + ~ , ~ + ~ , ~ - + ~ a--~-; +r av

a 4 Iv

a~w

a:* w

0-* w

a3 v

a--~-~ + a'* w

+ C,o ~ + c,, ~ + c,2 g~- + ~,3gffa- + c,, a0-azgZax~+ ~,~ a0-ffW;ax+ ~,o a - T gff + azw

a2w

02 w

au

av

+ ~" g ~ - + ~'%-;~ + ~'~' + e~-x~ + PR ~ - 2 ~ x a 2w + 2"r ~

= L3(u, v, w),

av

azw

- pR~# + p R - ~ - + (A3)

480

J. PADOVAN

al = R E t t k

b, = R E I 3 k + El3kb

a2 = 2 R k E t 3

b2 = RE23(k + kv) - E2a kb

a3 = RE33(k + kp)

b 3 = .R(EI2 + E 3 3 ) k

a,, = REI3(k + kv)

b, = RE33(k + kp) + 2E33 kb

a s = R k ( E t 2 + E33)

b5 = R k E 2 2

a6 = RkE23

b6 = 2(RkE23 + E23 kb)

a7

b7 = - R k t , Eta - Et3 kb

=

RE12 k

as = RE23(k + kv) a 9 = RE33

bs = RE22 kv

kv

E22 kb

-

b9 = R k v E23 - - 3E23 kb

alo = - R E 1 3 kp

blo = - R E 3 3 kv - kb(2E33 + E12)

at i = - R E I t kv

bll = R E 2 3 k

(A4)

al2 = RE23kv,

bt2 = R E 1 2 ( k

+

kv)

cl = --Elx kt,

clt = RE22k

c2 = E23kb

ct2 = EH kb

c3 = - E l 3 kb

c13 = E22 kb

c,L =/?33 kb

ct,, = 2(Era kb + 2/?33 kt,)

C5 = R E I 2

k

cm

= 4E23

E22kb,

(AS)

kb

c6 = RE23(k +/%)

c~6 = 4E13/%

c7 = '--2El3kb

ca = -2E23 kb

Ct7 = RE22kv + E22kb cls = RE2a kv + E23 kb

C9 = -3Eaa kb -- El2 kb

ct9 = R E , 2 ( k + ku),

ClO -----RE23

-

(A6)

k

k = h/R,

kv = h3/(12R3),

kb = h3/(12R2).

(A7)

APPENDIX II a 2 um aura 0 2 v,~ arm aw,, a, ~ + jma2 -~x - m2 a3 u= + a, ~ + j'mas ~ - m 2 a6 v= + a7 ~ + jmaa w= awm - - m 2 a9 ~

0 2 Wm

0 3 W,.

" ~ ax +jmato ax e + att ~

- j m 3 a12 w m - F

a 2 um aura + Ox2 - 2jmz

+ p R m 2 Um + p OWm = Ll~(Um, Vm, W,,,),

(A8)

ox

0 2 u,~ aura 0 2 vm arm aav,b, ~ - m 2 b2 u,,, + j , nba ~ + b4 ~ - m 2 b5 v,,, + jmb6 "~x + b7 ~ - j m aw,.

0 2 w=

- ,,t 2 b9 -~x +.l'mbm ~ aum

aW m

aw~,

3 bs w= -

0 2 v=

+ b~t -~x +.#nb12 w,. - F ~ x 2 + m2pRv,. -

-- 2j'mr "~x -- 2z -~x - j m p R w m = L2=(u,., vm, Win),

(A9)

ROTATING PRESTRESSED CYLINDERS ~3/,/at

ct 0x 3

.fitla C2 lim + j m c 3 02 Um

Ox 2 - - m

Ov m

2

0 2 w,.

02 Wrn

Oum Ollm 03 vm c 4 "~X + C S - ~ X + j m c 6 um + c7 - O-x 3 - -

02 v,,,

-- m c x , - - ~ -

+ F ~

2

Or,,,,

aw,.

- - j m 3 cts ~ x

481

0 t w,.

0 3 w,.

.

+ jmc,6 -~-

aw,.

-- m2 ct7 w.. + j m c t s ~

OUm + pR--~x - 2T OOv"; - j m p R v m -

+ ct9 wm+

m 2 p R w . + j 2 m ~ Ow,. Ox

(A10)

= L3,,,(u,,,, v=, w,,,).

A P P E N D I X III

Ao = p R 2

lio l ~176 I

(All)

,

0

A t = - - 2 j p R 2 ~'~

A2=

[

0

1

0

0

,

(A12)

dl3]

dlt

dl2

d21

(,/22 d23 / ,

dax

da2

(A13)

da3J

dxl = - N 2 al - tuNa2 - m z aa + N 2 F + 2 m N ~ + p R m 2

dx2 = - N 2 a4 - tuNas -- m z a6 d13 = j N a ~ + j m a s - j m 2 Nao - j m N z alo - j N 3 all - j m 3 al2 + j p N d2t = - N 2 bl - m 2 b2 - m N b a d.2 = - N 2 b4 - m 2 bs - m N b 6 + N z F + m 2 p R + 2 m N ~ + p R ~ 2 d23 = - j N 3 b7 - j m 3 bs - j m 2 Nb9 - j m N

dat = - j N a cl - j m 3 c2 - j m N

- flnpR

2 c3 - j m 2 Nc,, + j N c s + j m c 6 + j N p R

da2 = - j N a c7 - j m 2 N c s - j m N

da3 = N* ct2 + m 4 c13 +

2 blo + j N b l l + jmb12 - j 2 N ~

2 c9 + j N c t o + j m c t t - j 2 N x - j m p R

m 2 N 2 c14 + n # N e t s + m N a c16 - m 2 cvl - m N c l 8 + cl9 -- N 2 F -

- m 2 p R - 2 r a N , - p R 2 .(22.

(AI4)

482

J. PADOVAN A P P E N D I X IV

T h e coefficients o f Bo . . . . . B~ a r e defined as matrices whose elements are given B~ = [Bt,,]; i = 0, 1 , . . . , 4; r, s = 1, 2, 3, where Boil = Bol2 = a o l s = Bo2l = Bo22 = 8023 = Boat = Boa2 = O, o33 = C12,

B i l l = BII2 = Bl2l = BI22 = 0, Bll3~all,

B132=c7,

B12a=bT,

BI33 = j m c l 6 ,

BI31=Cl,

B211 = a l - F ,

B223 = j m b i o ,

B2jz=a~,

B~al = j m c a ,

B2ls=jmaio,

B2a2 = j m c g ,

B22t=bl,

B2aa = - m 2 Cl, + F,

B222=b,-F, BaH=jma2--2jmr,

B323 = - m 2 b9 + bit - 2r,

Bal2=jma~,

Baal = - m 2 c4 + c5 + p R ,

g313 = a7 -- DI2a9 + p ,

Baaz = - m 2 ca + clo - 21",

Ba2l = j m b a ,

Baaa = j m a Cls + j m c i a + j 2 t m ~

Ba22 = j m b 6 - 2jmr, B4II = - - m 2 a3 + p R m 2 + p R z 092, B412 = -rrt2 a6, B*13 = jntas -- J m 3 a l 2,

B421 ------nil b2, B422 = - m Z b5 + m 2 p R + pR2 m 2 + p R 2 0 2, B423 = - j m 3 bs + j m b l 2 - j m p R

- 2 j p R 2 I2ca,,,

8431 = - - j n l 3 C 2 + j m c 6

B432 = j m c t 1 -- j m p R -- 2 j p R 2 12r B433 = m 4 c l z -- m z ci7 + ci9 - m Z p R - p R z m 2 - p R 2 0 2.

APPENDIX V NOMENCLATURE Ell material stiffnesses Ira( ) imaginary part of ( ) Re( ) real part o f ( ) t time shell displacements in axial, circumferential and radial directions P density I2 rotational speed O) natural freq ency