I
N A S A C O N T R A C T O R
R E P O R T
*o
FLEXURAL VIBRATIONS OF THE PRESTRESSED TOROIDAL SHELL
1
by Artis A. Liepins
Prepared under Contract No . NASw-955 by DYNATECH CORPORATION Cambridge, Mass.
for
. . > . .
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. SEPTEMBER 1965
https://ntrs.nasa.gov/search.jsp?R=19650023048 2018-05-18T22:32:46+00:00Z
FLEXURAL VIBRATIONS OF THE PRESTRESSED TOROIDAL SHELL
By Artis A. Liepins
Distribution of'this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.
Prepared under Contract No. NASw-955 by DYNATECH CORPORATION
Cambridge, Mass.
for
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION ___- For sale by the Clearinghouse for Federal Scientific and Technical information
Springfield, Virginia 22151 - Price $3.00
1
". " "
SUMMARY
Results of a numerical study of the natural frequencies and modes of vibration of the pressure prestressed toroidal shell are presented. The analysis is based upon a linearized theory of vibrations of prestressed shells. The frequencies and mode shapes are obtained by trial and error in the Holzer fashion. The effects of wall bending stiffness on the frequencies of shells under varying degrees of pres t ress are shown.
iii
I
CONTENTS
Section
SUMMARY
ILLUSTRATIONS
NOMENCLATURE
1 INTRODUCTION
2 FUNDAMENTAL EQUATIONS
3 REDUCTION TO SECOND ORDER DIFFERENTIAL EQUATIONS
4 NUMERICAL ANALYSIS
5 R E S U L T S
6 CONCLUSIONS
7 R E F E R E N C E S
APPENDIX A
APPENDIX B
Page
iii
vii
viii
1
2
5
10
15
19
20
21
28
V
r
ILLUSTRATIONS
Figure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Geometry and Notation
Axisymmetric Modes, Symmetric, K = 0.002
Axisymmetric Modes, Antisymmetric, K = 0.002
n = 1 Modes, Symmetric, K = 0.002
n = 1 Modes, Antisymmetric, K = 0.002
n = 2 Modes, Symmetric, K = 0.002
n = 2 Modes, Antisymmetric, K = 0.002
Effect of Bending Stiffness on Mode Shapes, n = 0, K = 0.002, E = 0.75
Axisymmetric Modes, Symmetric, K = 0.0001
Axisyrnmetric Modes, Symmetric, K = 0 , h/R = 0.01
Axisyrnmetric Modes, Antisymmetric, K = 0, h/R = 0.01
n = 2 Modes, Symmetric, K = 0, h/R = 0.01
n = 2 Modes, Antisymmetric, K = 0, h/R = 0.01
Axisymmetric Mode Shapes, Symmetric, K = 0, h/R = 0.01
Axisymmetric Mode Shapes, Antisymmetric, K = 0, h/R = 0.01
Axisymmetric Modes, Symmetric, K = 0, h/R = 0.001
Axisymmetric Modes, Antisymmetric, K = 0, h/R = 0.001
n = 2 Modes, Symmetric, K = 0, h/R = 0.001
n = 2 Modes, Antisymmetric, K = 0, h/R = 0.001
Axisymmetric Mode Shapes, K = 0 , h/R = 0,001, E = 0.15
vii
NOMENCLATURE
a b amplitudes of ring flexural vibrations my m
h
k
P
9,
r
U
V
W
thickness of shell
( 1 - v ) K
pressure
2
D Q p 2
(1 - € C O S f f ) Y €
meridional displacement (Figure 1)
circumferential displacement (Figure 1)
normal displacement (Figure 1)
defined functions
Eh /12 (1 - v ) 3 2
Young' s modulus
membrane strains
May Me,
Na' Ne' Nae
Q d Q,
stress couples
stress resultants
transverse shear stress resultants
R radius of the generating circle of torus (Figure 1)
%' membrane prestress forces
ff
E
meridional position angle (Figure 1)
ratio of the two radii of the torus (Figure 1)
viii
I
0
K
KcYt3
x
V
circumferential position angle (Figure 1)
pR/Eh, prestress parameter
bending strains
( p R /E E ) w , frequency parameter 2 2 2
Poisson' s ratio
material density
rotations of the normal to the shell
circular frequency
(h/W2/ 12
spacing between finite difference stations
Matrices
a, b, c 5 x 5 matrices
X 1 x 5 column matrix
A , B, C, D, E, F, G, H, P 4 x 4 matrices
A , B, 3 x 3 matrices
Z 1 x 4 column matrix
Z 1 x 3 column matrix
-
-
Indices
i
I
m
n
station
last station
Fourier index of ring flexural vibrations
Fourier index of shell vibrations
ix
Section 1 INTRODUCTION
A study of the free vibrations of the pressure prestressed toroidal mem- brane [l] shows that the frequencies can be grouped into four families: a family of flexural vibrations associated with relatively low frequencies, and three families of vibrations with high frequencies. The flexural vi- brations are found to depend strongly on prestress . The vibrations of the other three families are practically insensitive to prestress. Since both the effects of prestress and the effects of wal l bending stiffness enter the fundamental equations in a similar manner (through terms whose coeffi-
cients are small compared to the coefficients associated with the membrane terms), it can be expected that bending stiffness has a strong influence on
the frequencies of the flexural vibrations. This report presents the results of a numerical study of the effect of wall bending stiffness on the frequencies and mode shapes of the flexural vibrations.
Several iterative 'numerical methods suitable for the analysis of the free vibrations of shells of revolution have appeared recently in the literature
[l] [2] [3]. Kalnins [2] uses a multisegment direct numerical integration approach and Reference [l] uses finite differences to evaluate a certain determinant corresponding to a trial value of the frequency. Cohen [3] uses a method which i terates on the mode shape instead of the frequency. Al- though the methods of References [2] and [3] a r e applied to shells of revolu- tion without prestress, their extension to shells with prestress appears to be straightforward. None of these methods, however, possesses a sub- stantial advantage over the others. This study uses the numerical method of Reference [l] .
Section 2
FUNDAMENTAL EQUATIONS
The present analysis of the free vibrations of the pressure prestressed
toroidal shell is based upon a set of equations which result from the
addition of the bending terms of the Sanders linear shell theory [4] to the
prestressed membrane equations presented in Reference [l 3. These are (refer to Figure 1):
Equilibrium
a a Ea + ' 6 [ a ~ - 4ae) a a 4 f f ) + (Ea- Ee) sin a ] + 'Sa (- -
+ pRr + phRo r u = 0 2
a N e + - a (rNae) + Nae sin a - Q, COS CY - - - a e act 2~ a a r r a cos a [(I + -1 Mael
a E e a +
+ 2Ea e sin a + d e cos f f 1 + rsa (Eae + dae)
+ pRr + phRo rv = 0 2
a a Qe r N a - Ne cos a - - ( r Qa) - - + Se [ - + 4a s in a - E cos a ] a a a e a e e
a 4, + Ea) - pRr (Ea + E ) - p h R w r w = 0 + rS (- 2
a a a e
a - ( r Ma) + - aMae - Me sin a -RrQa = 0 a a a e
a M e + - a a e a a Mae) + sin a -RrQ = 0 e
2
Strain Displacement
REcy - - - a u + w aa
rRE, = - a + u sin a. - w cos cT a e (7)
2rREae = r - a u - v sin cy (8) a v act +G
a 4 e a e r R K e = - + 4cy sin ct
- r R d e = - a w + v cos cy a e (13)
Constitutive Relations
EhEa = Ncy- v Ne
EhEe = N e - v Na
Eh = ( 1 + v ) Nae
Eh3 12 K, = Ma- v Me
Eh3 - 12 K e = Me- v M,
Eh3 - 12 Kcye = ( 1 + v ) Mae
3
The above constitute twenty equations for the twenty unknowns Na,
u, v, and w. The s t ress resul tants Sa and S are known functions of
prestress . 6
The state of pres t ress is determined from a separate analysis of the
toroidal shell subjected to static internal pressure. An analysis based
on the linear membrane theory [5] gives
1 1 - 3 E cosa Sa = pR 1 - E cos a
1 Se = 2 pR
Analyses of the toroidal membrane under internal pressure based upon
nonlinear theories [6, 71 show that the linear meridional stress resultant,
S , is in e r r o r by a negligible amount, but that the linear circumferential
stress resultant, S can be in error by 18%. The significant difference,
however, between the linear and nonlinear stress distributions is confined
to a small area of the torus at a = f n/2. Furthermore, an analysis [8] of the toroidal shell under internal pressure based upon equations which
consider wall bending stiffness and nonlinear behavior shows that the s t r e s s
resultants do not differ substantially from those obtained from nonlinear
membrane theory. Hence, it appears that the state of prestress determined
according to the linear membrane theory is adequately accurate for the
present analysis of vibrations.
a e’
4
Section 3
REDUCTION TO SECOND ORDER DIFFERENTIAL EQUATIONS
The solution of the fundamental equations (1-20) is started by separating the variables. Set
r 1
sin n e
Next, define
S = ( s i n a ) / r
C = ( cosc r ) / r N = n / r
Then, use of equations (22) and (23) in equations (1-14) yields
I + 2SNae - N N B - CQ, - 2 ~ 1 ( l + C ) ( M a i - S M ) Na 8 ae
+ S, ( - N E e + 2SEae + C4,) + + 44jo)
+ P R ~ ~ + phR w v = 0 2 2
5
Mde + 2SMae - N M e - R Q e = 0
Ea = U' + w
Eo = NV + SU - CW
2Eae = V I - SV - NU
- R K a - dol
%Y = - w ' + u
de = NW - CV
$cue = Nu + V I + Sv
In equations (24-37) and subsequent expressions the subscript n has been
dropped. Prime indicates differentiation with respect to a.
At this point the problem has been reduced to the simultaneous solution of
the ordinary differential equations (24-37) and (15-20). Our goal is to
derive four simultaneous second order differential equations for u, v, C$
and 9,. First eliminate Q from equations (25) and (26) using equation (28).
Then, substitute the strain displacement relations (29-34), two of the rota-
tion expressions (36, 37), and the constitutive relations (15-20) into the
equilibrium equations (24-27). The resulting equations together with (35)
a' e
6
constitute five equations for u, v, w, $,, and 9,. They may be written in
matrix form as
where
The elements of the a , b, c matrices are given in Appendix A.
W e now manipulate equations (38) so that w can be eliminated using equation (38c) (equilibrium of forces in the normal direction). In order that no deri- vatives higher than the second derivative appear in the final equations,
eliminate V I ! from equation (38c) using equation (38b) (equilibrium of forces in the circumferential directions) as follows:
1 v" = - (bZ1 u t + c u + b22vt + c22v + a wl!+ b wt + c w a22 21
+ b24 6 CY + ~ 2 4 4,)
23 23 23
Then equation (38c) reads:
- - - - - b ut + c31 u + b32vt + c32v + a w l l + E w1 + c w + b344; 31 33 33 33
where the coefficients are given in the appendix. From equation (35)
(39)
Af ter eliminating the derivatives of w using (41), equation (40) can be
written as
Qw = T u t + T u + T v' + T4v + T54; + T6d, 1 2 3 - rq; - r sq, (42)
7
and the derivative of w can .be written as
The T I s and Q are defined in the appendix.
Now, multiply equations (38a, b, d, e) by Q and eliminate Qwl from them with equation (43). Then multiply the resulting equations by Q and elimi-
nate Qw with equation (42). The resulting four equations for u, v, 4cr, and
q,may be written in matrix form as
where
AZ" + BZf + C Z = 0
and the elements of A, B, and C a r e given in Appendix A.
The basic equations (44) simplify for the case of axisymmetric vibrations,
n = 0, v = 0. The equation of equilibrium of forces in the circumferential
direction is then satisfied identically, and equations (44) become
where
(44)
The elements of A, E, and are the remaining elements of A, B, and C if
the second row and column are deleted, and N is set equal to zero.
8
The fundamental equations have now been reduced to the solution of two sets of second order differential equations as follows: 1) four equations (44) for the general case, 2) three equations (45) for the axisymmetric
case. These differential equations will be integrated numerically by a method described in Reference [l]. For completeness, the description of the numerical method is reproduced in the next section.
At this point w e remark that the fundamental equations (1-20) could be more easily reduced to four second order differential equations for
u, v, w, and Ma in the manner of Budiansky and Radkowski [ 9 ] . However, application of the numerical method described in the next section resulted in a convergence of the finite differences which was too slow for practical computation.
The fundamental equations can also be reduced to four equations for v, w, Ncy, and Ma. First , derive a set of five equations for u, v, w, N and M
cy cy’ then solve the equation of equilibrium of forces in the meridional direc- tion for u, and finally eliminate u from the remaining four equations. The application of the present numerical analysis to these equations resulted in efficient and accurate results except in the approximate range
2 2 < x < n n
2 (1 + v) (1 - E) 2 (1 + v ) (1 + E ) 2 2
In this range the coefficient of u in the equation of equilibrium of forces in the meridional direction has a zero.
9
Section 4 NUMERICAL ANALYSIS
Since the geometry and prestress are symmetrical about CY= 0 and CY= r, w e need to consider only one-half of the torus corresponding to the range
0 5 CY 5 n . Let this range be subdivided by I + 1 equally spaced stations.
Then the spacing between stations is
A = r / I
and the position angle for the ith station is
a. = i A 1 i = 0 7 1 , 2 , . . . . . I
The derivatives of Z at the ith station are approximated by the central
difference formulas
Z." = 2 (Zi+ 1 - 2zi + zi -
1
A
With these formulas we obtain from equation (44) the set of difference
equations
DiZi+ + EiZi + F i Z i - = o (47)
i = 0 , 1 , 2 , . . . . . I where
2 D. = - A . + B 1 A 1 i
E. = - - A . + 2 ACi 4 1 A i
2 F. = - Ai - Bi 1 A
10
The difference equations (47) are augmented by conditions at i = 0 and i = I. These end conditions are obtained from considerations of conti- nuity of the displacement functions, u, v, and w. Now, the solution of the fundamental equations (1-20), and the reduced equations (44) are such that either v and w are even, u is odd or v and w are odd, and u is an
even function of cy with respect to CY = 0 and T . These two groups of solu- tions will be denoted as symmetric and antisymmetric modes respectively. Hence, the end conditions are
where G is the diagonal matrix
and plus and minus signs refer to symmetric and antisymmetric modes respectively.
The difference equations (47) together with the end conditions (49) make up a set of homogeneous algebraic equations. The eigenvalues of these equa- tions are obtained by trial and error, using a special Gaussian elimination technique devised by Potters [lo].
The equations for this procedure are obtained as follows. Let
zi - - - Pi zi+l
Substitute this expression into the difference equations (47), and by com- paring the result with (50) obtain the recurrence relation
i = l , 2, 3, . . . . . 1 - 1
11
Now, write the difference equations (47) at i = 0 and eliminate Z using
the first of the end conditions (49). Again, by comparing the result with
equation (50) obtain
-1
Po = Eo -' + F GI lD0 - 0 (52)
where plus and minus signs refer to symmetric and antisymmetric modes
respectively. Equations (52) together with the recurrence relation (51) provide all the PI s up to P Finally, write the difference equations
(47) a t i = I. Eliminate ZI + using the second of the end conditions (49),
and ZI - using equation (50). The result is
1 - 1 '
[EI - (FI 5 DI G) PI - ,] Z I = HZ I = 0
Since ZI f 0, we must require that the determinant
v = 1 EI - (F + D G) P I - l I - I I vanish. Equation (54) is effectively a frequency equation. A value of X
which gives v = 0 is a natural frequency of vibration.
The mode shape corresponding to a natural frequency can be calculated
once a X for which V = 0 is obtained. Set the amplitude of one of the dis-
placements at i = I equal to unity and calculate the remaining unknowns
a t i = I from three of the equations (53). Thus, for the symmetric modes
For antisymmetric modes
z = I
- -
(53)
where
The remaining Zt s can then be calculated from equations (50) in the reverse order.
The w displacement is obtained from equation (42). In finite difference form the equation reads
w i = - { Qi 1 2a 1 l T 1 , i ( ' l , i + l - z l , i - l ) + T 3 , i ( z 2 , i + l - z 2 , i - l )
+ T5,i ( '3, i+l- '3, i-1) - r ( z 4 , i + l - ' 4 , i - 1 ) I
+ T2, i '1, i + T4, i '2, i + T6 '3, i - '' 1 '4, i 1 i = l , 2, 3, . . . . . 1 - 1 (5 8)
For symmetrical mode shapes
1 1 wO = - QO [A ( T 1 , 0 Z 1 , 1 + T 5 , 0 ' 3 , l - '4,l) + T4, 0 '2,O'
and for antisymmetrical mode shapes
wo = WI = 0
The procedure may be summarized as follows:
1. Assume a value of X ; 2. Calculate the elements of the A , B, C, D, E, F, and P
matrices at all stations from equations (44), (48), (51) and (52);
3. Calculate the determinant V from equation (54);
4. Repeat steps 1-3 and plot V versus X. F rom th is plot determine a X for which V = 0. This X is a natural frequency;
frequency from equations (50) and (55-60). 5 . Calculate the mode shape corresponding to a natural
The equations in this procedure were programmed for the IBM 7094
computer.
A number of natural frequencies were calculated with I = 50 and 100 and
compared. The difference was found to be less than one percent. There-
fore, all results were obtained with I = 50. For this number of finite
difference spacings the IBM 7094 required approximately 1.5 seconds to
evaluate one determinant v of the general vibrations.
14
Section 5 RESULTS
Results of the present numerical analysis are contained in Figures 2-20. These results show the effect of bending stiffness on the vibration of shells
under 1) high pres t ress , K = 0.002, 2) low pres t ress , = 0.0001, and 3) no prestress , K = 0. All results are for Poisson’s ratio v = 0.3.
The modes of the flexural vibrations of the prestressed toroidal shell may be thought to consist of two types of vibrations: 1) the modes of a torus whose meridional curve (cross-section) is not allowed to distort (overall
ring vibrations), 2) the modes of a pressure prestressed c i rcular r ing with radius R (cross-sectional or ring flexural vibrations). The first type
is approximated by the vibrations of a thin ring of radius R / E without p re s t r e s s [ I l l :
Bending modes
symmetric (in plane bending)
2 n (n - 1) 2 2 2
2 ( n +1) 2 x = E
antisymmetric (out of plane bending)
2 n (n - 1 ) 2 2 2 X = E n
Extensional modes (symmetric)
2 A = n + 1
Torsional modes (antisymmetric) n
1 n 2 1+ v
L A = ” (1 + -)
The frequencies of flexural vibrations of a radius R are derived in Appendix B:
pressure prestressed r ing of
[ K + r (m2- 111
The effect of bending stiffness on the frequencies of a shell with high pre-
stress a r e shown in Figures 2-7. These results are for n = 0, 1, and 2,
K = 0.002 and h/R = 0.01. I t can be seen that the frequencies of the first
mode of each type of vibration shown are practically unaffected by bending.
However, the effect of bending increases with the mode number so that
the increase in frequency of the fourth mode is of the order of 10%. This
trend is forecast by the frequencies of the uncoupled ring flexural vibra-
tions given by equation (65) in which the r part increases with (m - 1).
The effect of bending is larger on shells with high E (fat toroids). This is
also to be expected because for high E the mode shapes consist mostly of
local deformation of the meridional curve [l]. For E = 0.75, n = 0 sym-
metric vibrations, the mode shapes of the first four modes of a shell with
and without bending are compared in Figure 8. This figure shows that the
effect of bending on these mode shapes is small. In summary, w e can say
that for shells with K = 0.002 and h / ~ = 0.01 the effect of bending on the
first four modes of the n = 0, 1, 2 vibrations is only moderate.
2
The effect of bending on the axisymmetric, symmetric, frequencies of a shell with K = 0.0001 and / R = 0.01 is shown in Figure 9. In this case,
bending has increased the frequencies by approximately a factor of two
when E = 0.75. For smaller E the increase is smaller . For a shell with
K = 0.0001 and h/R = 0.001 the frequencies increase by less than 5%.
h
Frequency curves for a shell without pres t ress , K = 0, n = 0 and 2 and
h / ~ = 0.01 are shown in Figures 10-13. The dashed curves in these fig-
ures represent the frequencies of the uncoupled modes. The shell frequen-
cy curves are the solid curves. These curves exhibit transition regions
16
which occur as sharp jumps for the axisymmetric, symmetric, frequency curve, and as more gradual changes for the axisymmetric, antisymmetric, and n = 2 frequency curves. In a qualitative sense these frequency curves exhibit the same features as those of the prestressed membrane when K = 0.002 [l] (or refer to the dashed curves of Figures 2-7 of this report).
The axisymmetric mode shapes for a shell without prestress , K = 0, and h / ~ = 0.01 are shown in Figures 14 and 15. The f i rs t column of mode shapes in these figures shows essentially the uncoupled mode shapes for E
below the transition region. In this region the overall ring mode shape prevails for the symmetric vibrations, but for the antisymmetric vibra-
tions an irregular shape appears. The third column again shows essen- tially the uncoupled mode shapes. These are for E above the transition
region and a r e one order of complexity higher than those below the transi- tion region. The fourth column shows mode shapes for E: = 0.75 which are predominantly local oscillations of the meridional curve. In a qualitative sense, the mode shapes of the shell without prestress and h / ~ = 0.01 exhibit the same features as those of the prestressed membrane when K = 0. 002 [I].
Frequency curves for a shell without prestress , K = 0, n = 0 and 2 and h / ~ = 0.001 a r e shown in Figures 16-19. The significant feature of these curves is their tendency to pair up with nearly the same frequencies. Examples of the mode shapes, shown in Figure 20, associated with fre-
quencies that pair up appear to be mirror images of each other. That is,
if the mode shape of the second symmetric mode is folded on top of the mode shape of the first symmetric mode, then the two mode shapes are nearly the same. The third and fourth symmetric, second and third anti- symmetric and fourth and fifth antisymmetric mode shapes shown in Fig- ure 20 are mirror images also. Another significant feature of the mode
shapes presented in Figure 20, is the fact that the motion takes place mainly near the crowns.
The paired up frequencies with nearly the same mode shapes can be ex- plained by the following. Let the radius of the torus go to infinity (E-0).
Then the torus approaches a cylindrical shell. At every natural frequency
of the cylindrical shell there are actually two identical frequencies with
identical mode shapes. These frequencies separate slightly and the mode
shapes become slightly different when imperfections (for example, when
the cylindrical shell is not exactly circular) are present. T h u s the pairing
up of the frequencies of the toroidal shell are due to the weak influence of
the circumferential curvature which has the equivalent effect of imperfections
on the frequencies of the cylindrical shell.
18
Section 6
CONCLUSIONS
The following conclusions are drawn from this study:
Bending stiffness has only a moderate effect on the natural
frequencies and mode shapes of relatively thick shells, ( / R = 0.01) under high prestress ( K = 0.002). h
When pres t ress is small ( K = 0,0001) bending stiffness increases the natural frequencies of relatively thick and fat toroidal shells ( E large, h / ~ = 0.01) by as much a s a factor of two.
In the absence of pres t ress , when membrane theory pre- dicts a continuous frequency spectrum with discontinuous mode shapes, the consideration of bending stiffness leads to discrete frequencies with continuous mode shapes.
Bending stiffness should be considered in the calculation of the natural frequencies and mode shapes associated
with the flexural vibrations of toroidal shells.
Section 7
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Liepins, A. A. , Free Vibrations of the Prestressed Toroidal
Membrane, AIAA Paper 65-110 (1965)
Kalnins, A. , I t Free Vibration of Rotationally Symmetric Shells, 1 1
J. Acoust. Soc. Am., Vol. 36, No. 7, pp. 1355-1365 (1964)
Cohen, G. A. , Computer Analysis of Asymmetric Free Vibrations
of Orthotropic Shells of Revolution, I t AIAA Paper 65-109 (1965)
Sanders, J. L. , Jr. , "An Improved First-Approximation Theory for Thin Shells, I t NASA T R R-24 (1959)
Timoshenko, S. , Theory of Plates and Shellstt (McGraw-Hill
Book Co. , Inc. , New York, 1940) Chapter X
Jordan, P. F., l t Stresses and Deformations of the Thin-Walled
Pressurized Torustt J. Aerospace Sci. 29, 213-225, (1962)
Sanders, J. L., Jr. , and A. A. Liepins, Toroidal Membrane Under
Internal Pressure, AIAA Journal, Vol. 1, No. 9, 2105-2110 (1963)
Rosettos, J. N. , and J. L. Sanders, Jr. , Toroidal Shells Under
Internal Pressure in the Transition Range, I t AIAA Paper 65-145 (1965)
Budiansky, B. , and P. P. Radkowski, Numerical Analysis of Unsymmetrical Bending of Shells of Revolution, I t AIAA Journal, 1
(8), 1833-1842 (1963)
Potters, M. L. , A Matrix Method for the Solution of a Second Order
Difference Equation in Two Variables, I t Math. Centrum Report MR 19
(1 955)
Love, A. E. H. , The Mathematical Theory of Elasticity, (Dover
Publications, New York, 1927).
I
A P P E N D I X A
Elements of Matrices
The non zero elements of the a, b, and c matrices are:
a = 1 + k ( 1 + 2 C ) 1
a 22 = ~ ( l - v ) + k . ( l + ~ C ) + ' 1 1 (1 -v ) (1+3C) 2
1 a23 4
11
8 - - - - ( 1 - u ) r ( l + 3 C ) N = - a32
- 1 2 "(1-v) r N a33 - 2
a 54 =
bl 1 1 = (1+2 k) S
b12 = N [ Z ( l - v ) - - ( l - v ) 1 1 r ( l + C ) ( 1 + 3 C ) ] = - b 2 1 8
bl 3 = l - ~ C + k ( l + ~ C ) + - ( l - v ) r ( l + C ) N 4 1 1 2
b22 = ~ S [ l - v + k - - ( l - v ) r ( 1 + 3 C ) ( 5 + 3 C ) J 1 1
4
b23 = ~ ( 1 - u ) r N S (2+3C) 1
bz4 = r N [ ~ C + - ( l - v ) ( 1 + 3 C ) ] = - r b
b3 1 = l - v C + - k C + ~ ( l - ~ ) r ( l + C ) N 1 1 2 2
bS2 - - 2 ( 1 - v ) r N S
I 4 52
- 1
1 b33
b34 = k ( 1 + z C ) + ~ ( l + v ) r N
2 = ~ ( 1 - v ) r N S
1 1 2
21
b35
b43
= - r
= 1
1 2 b53 = ? ( l + u ) N
b54 = s
C = ( 1 - u 2 2 ) E x + u C - S 2 - ; ( l - u ) N 2 - i k ( N 2 2 1 + S ) - - ( l - u ) r ( l + C ) 2 2 N 11 8
C = N S [ - 2 ( 3 - u ) 1 - k + g ( l - ~ ) r ( l + C ) ( l + 3 C ) ] 1
'13 = S ( l + C ) [ l + ? k - T ( l - ~ ) r N 1 1 2 ]
C - - - [ k C + T ( l - u ) r ( l + C ) N 1 1 2 ]
12
14 - 2
'15 = r
C 21 = - N S [ z ( 3 - ~ ) + k + - ( l - ~ ) r ( l + C ) 1 3 2 ] 8
c22 = ( 1 - U ) E 2 2 x - ; ( 1 - u ) ( C + S ) - N 2 - ; k ( N 2 2 2 2 + S + C ) - k c - r N C 2 2
+ - ( I 1 - .) r ( 1 + 3 c ) [S 2 (5+3C) - C (1+3C)] 8 C = N { - u + C + k ( l + C ) + r N 2 1 C + T ( ~ - Y ) ~ [ C ( ~ + ~ C ) - ~ S 2 (l+C)]}
C = r N S [C - T ( ~ - u ) ] 1
23
24 1
C 31
'32
= S [ v - C - k (1+?C)]
= - N { - u + C + k ( l + C ) + r N 2 1 C - 2 ( 1 - u ) r [ S 2 1 - 2 C ( 1 + 3 c ) ] ]
2 2 2 1 1 2 4 2 c33
= - ( ~ - ~ ) E x + ~ - ~ u C + C + T k C ( S + C ) + z k N + r N + ( l - v ) r N C
1 2 C 34 = ? S [ k + ( 3 - u ) r N ]
c35 = - r ~
22
r, -1
A44 = - r Q T l l
= Q S ( 1 + 2 k ) + Q T7 (TI ' + T2) + T1 Ts 2 1 Bll
. B12 = 2 Q 1 2 N[(l+w)--(l-~)r(l+C)(1+3C)]+QT~(T3'+~~)+~3~8 4 1
B13
B14
= Q T7 (T51 + T6) -I- Ts T8
= - (QST7 + T8)
B24 = - r (QST9 + Tlo)
B3 1 = Q (Tl' + T2) - Q' T1
B32 = Q (T3' + "4) - Q' T3
B33 = Q (T5' + T6) - Q' T5
B34 = - ~ ( Q S - Q')
cll = Q [ ( ~ - u ) E A + u C - S ~ - ; ( ~ - ~ ) N ~ - ~ ~ ( N + S ) - g ( l - v ) r ( l + C ) N ] 2 2 2 2 2 1 2 2
2
+ Q T7 T2' + T2 T8
c12 = Q N S [ - Z ( ~ - U ) - ~ + ~ ( ~ - U ) ~ ( ~ + C ) ( ~ + ~ C ) + Q T ~ T ~ ' + T ~ T ~ 2 1 1
1 2 1 2 '13 = - 2 Q [ k C + T ( l - v ) r ( l + C ) N ] + Q T 7 T 6 ' + T 6 T 8
C14 = r [Q 2 - QT7 (C - S2) - S T8]
2 c21
= - Q N S [ 5 ( 3 - ~ ) + k + - ( l - ~ ) r ( l + C ) 1 3 2 ] + Q T 9 T i +T2T10
c22 = Q2{ ( ~ - u ) E x - ~ ( ~ - u ) ( C + S ) - N ~ - ; ~ ( N + S + C . ) - k c - r N C
8
2 2 1 2 2 2 . 2 2 2
+ L ( h ) r (1+3c) [ s2 (5+3C) - C (1+3C)] + QT T ' + T4T10 8 1 9 4
'2 3 = r Q 2 N S [C - 1 (1 - u)] + QTgTgl + T6T10
,C24 = - [QTg (C - S2) + STlo]
Cgl = - Q2 + QT2' - Q1 T2
C32 = QT4' - Q' T4
c33 = Q ~ + Q T ~ ' - Q' T~
c34 = - r [Q (c - s2) - Q' SI
'41 - - - (1 - U ) (1+C) N Q + QTll T2'+ T2 T12 - 4
Cd2 = Q 2 NS [C+ v ( 1 + C ) + - ( l - v ) 1 (1+3C)] +QT11T4'+T4T12
1 2 2
4
c43 = Q 2 [ U C - S2 - (1 - u)N 2 ] + QTll T61+ T6T12
c44 = - Q2 - r [QTll (C - S2) + S T12]
where T1 = 1 - v C + T k C - - ( l -v)rN (1-C) + 2 N T [ l + ~ + - ( l - ~ ) r ( l + 3 C ) ( l - C ) ] 1 1 2 1 1 4 0 4
1 1 2 1 2 2 0 2 T2 = S [ u-C - k ( l + - C ) + - ( l - u ) r N ] + N S T [ - ( 3 - ~ ) + k
+ - ( b v ) r (1 + c ) - ( h ) r (2+3C)] 3 2 1 8
- - - (1-U) r N S - - STo [ 1 - u + k + 4 ( I - u ) ~ ( 1 + 3 C ) ( S + 3 C ) ] 1 1 1 T3 - 2 2
T4 = N { u - C - k ( l + C ) + r [ - N 2 1 C+T( l -v )S 2 1 - ~ ( l - u ) ( 1 + 3 C ) C ] }
-To{ (1-u 2 2 ) e X-- ( l -v ) (C+s 1 ) - N -Zk (N + S + C ) - k C 2 2 1 2 2 2 2
- r N 2 C 2 + - ( l -u ) r (1+3C) [S2(5+3C) -C(1+3C) ]} 8 1-
T5 = k (1 + ZC) + r N - r N T [ v C + 2 ( 1 - u ) (1 + 3 C ) ] 1 2 1 0
~ r N T 0 [ l - u + ( 1 - 3 v ) C ] } 2
T, = l - u C + k ( l + - C ) + ~ ( l - v ) r ( 1 + C ) N 1 1 2 2
26
r
T8 = Q S ( l + C ) [ 1 + 2 k - T ( l - v ) r N 1 1 2 ] - Q ' T 7
Tg = T ( 1 - V ) r N S 1 (2+3C)
= QN [ - v + C + k ( l + C ) + r N C + Z ( l - u ) r ( 1 + 3 C ) ( C - 2 S ) 2 1 2 T1O
- - ( l - v ) r (1 - 3C) S ] - Q'Tg 1 2 2
1 2 Tll = 7 ( 1 + ~ ) N
T1 2 = - (2N2SQ + Q1Tl1)
and prime indicates differentiation with respect to CY. In differentiating N, S, and C the following formulas are useful:
N' = - N S
S' = c - s C' = - s (1+ C)
2
- A P P E N D I X B
Flexural Vibration of the Prestressed Circular Ring
The vibrations of a circular ring of radius R, prestressed by pressure p,
may be determined from equations (15-21) and (24-37) by setting E = Y = v
= S = 0. This results in the following equations: e
Equilibrium
1 M ; - R Q cy = O
Strain Displacement
E = u ' + w CY
RKcy - 4;
dcy = - w ' + u
-
Constitutive Relations
Ncy = EhEa
E h3 Ma - 12 KCY
-
Pres t ress
S = pR ct
Substitution of (B3-B9) into B1 anc n
039)
3 B2 yields
28
I
which with
u = a s i n m c y m w = brn cos m cy
yields the following frequency equation
For K << 1 and m not large the lowest root of this frequency equation is approximately
AXiS OF ROTATION
W a I
R €
t - -
FIGURE I: GEOMETRY AND NOTATION
0.75
0.50
E
0.25
0 0.5 I .o x
I .5
FIG, 2: AXISYMMETRIC MODES, SYMMETRIC, K=0.002
0.75 I I
K = 0.002 - h/R = 0.01
0.50 II h / R = 0
4th MODE
I I I 1 0 0.5 I .O I .5 2 -0
x FIG. 3: AXISYMMETRIC MODES,ANTISYMMETRIC, K = 0.002
L
E
0.75
0.50
0.25
K = 0.002 - h / R = O . O l "- h / R = O
0 0.5 1.0 I .5 2 .o
x FIG.4: n = i MODES, SYMMETRIC, K =0.002
0-7 5
0.50
0.25
r -1
K = 0.002
0 0.5 I .o 1.5 2 .o
x FlG.5: n = I MODES, ANTISYMMETRIC, K=0.002
0.75
0.50
E
0.25
0.75
0.50
E
0.25
K * 0.002 - h / R = O . O l "- h/R .O -
-
I I I I I I I I 0 0.5 1.0 1.5 2 .o
x FIG.6: n.2 MODES, SYMMETRIC, K=0.002
K = 0.002
h / R = 0 - h / R = 0.01 "- -
-
0.5 I .o 1.5 2 .o
x FIG.7: n * 2 MODES, ANTISYMMETRIC, K = 0.002
33
AXIS,OF ROTATION
h ( h / R = O ) 0.174 X(h/R=O.OI)0.179
i
X(h/R=O) 0.012 A(h/R=O.OI) 0.013
FIRST MODE
h / R = 0 h / R = 0.01
- - "
0.21'3 0.225
0.267 0.287
SYMMETRIC
0.206 0.2 I6
0.246 0.265
SECOND MODE THIRD MODE
ANTISYMMETRIC
0.338 0.379
0.323 0.354
FOURTH MODE
FIG.8: EFFECT OF BENDING STIFFNESS ON MODE SHAPES,n=O,K=0.002, rs0.75
E
0.75
0.50
E
0.25
0.75
0.50
0.25
K = 0.0001 - h/R=O.Ol -- - h/R= 0 -
-
0 0.5 I .o x
FIG.9: AXISYMMETRIC MODES, SYMMETRIC, K=0.0001
0 0.25 0.50 0.75 1.00 I .25
x FIG.10: AXISYMMETRIC MODES, SYMMETRlC,K=O,h/R=0.01
35
0.75
0.50
E
0.25
0 0.25 0.50 0.75 I .oo I .25
x FIG, I I: AXISYMMET-RIC MODES, ANTISYMMETRlC,K=O,h/R=O.Ol
E
0.75
0.50
0.25
0 0.25 0.50 0.75 I .oo x
FIG. 12: n = 2 MODES, SYMMETRIC, K = 0, h/R =0.01
0.75
0.50
E
0.25
/ -" /I
I I' OVERALL R I N G OUT ' OF PLANE BENDING
-
0 0.25 0.50 0.75 1.00
x FIG. 13: n = 2 MODES, ANTISYMMETRIC, K =O,h/R=O.Ol
37
E = 0.01
x = 0.999
I-@- € = 0.01 x = 1.157
€ = 0.03 x = 1.076
€ - 0.06 x = 1.0615
~ ~~
I I I FIRST MODE
E = O.'OI5
x = 0.789 E = 0.15
x = 0 .346
SECOND MODE ,
@- -@- E = 0.025 E = 0.04 x = 1.002 x = 0.817
4- -e- € - 0 . 0 5 € = 0.075 x - 1.008 x - 0.868
FOURTH MODE
E = 0.085 i - 0.11 x = 1.027 x = 0.974
E = 0.75
x = 0.130
E = 0 ! 7 5 x = 0.138
!
1 € - 0.75 x c 0.199
1
FIG.14: AXISYMMETRIC MODE SHAPES, SYMMETRIC, K = O , h./R=0.01
38
AXIS OF ROTATION
E = 0.001 x = 0.497
'0 i E =0.015 x= 1.141
i '4 I
E = 0.03 x= 1.082
E =0.055 x= 1.183
I
FIRST
E = 0.005 E = 0.1 x= 0.417 x= 0.0145
I SECOND MODE I
f I $ = 0.03 E P 0.05 x = 1.000 x= 0.682
@flll?b MO@
E = 0.04 E = 0.075 x= 0.987 x= 0.779 I FOURTH MODE I
E = 0,075 E = 0.15 x= I. 037 I x= 0.753
E = 0.75 x = 0.00239
E = 0.75 x= 0.137
I
E * 0.75 x= 0.155
-4- 1
E = 0.75 x= 0.239
FIG. 15; AXISYMMETRIC MODE SHAPES, ANTISYMMETRIC,K=O, hlR=O.OI
39
0.75
0.50
E
0.25
E
0.25
x FIG.16: AXISYMMETRIC MODES, SYMMETRIC,K=O,h
0.75
0.50
0.25
0
FIG. 17: A X ISYMMETR
0.25
x IC MODES, ANT
50
/R=0.001
0.50
ISYMMETRIC,K=O,h/R=O.OOI
r
E
x
E
FIG.18: n= 2 MODES, SYMMETRIC, K = 0, h / R = 0.001
0.75
0.50
0.25
r I
- - f i - 5th MODE
0 0.25 LO. 50
x FIG. IS: n = 2 MODES, ANTISYMMETRIC, K = O , h /R=0.001
41
AXIS OF ROTATION
I x = 0.1029 FIRST MODE
AXIS .OF ROTATION
x = 6.1032 SECOND MODE
x = 0.'00932 FIRST MODE
x = 0. IO31 SECOND MODE
x = 0.2176 THIRD MODE
SYMMETRIC
Q- x = 0.1'098
THIRD MODE
x = 0.2202 FOURTH MODE
x = 0.'2195
FOURTH MODE
x = 0.2336 FIFTH MODE
ANTISYMMETRIC
FIG. 20: AXISYMMETRIC MODE SHAPES, K = 0 , h/R=O.OOl ,€=0.15
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