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Free and forced torsional motion of joined concentric circular shells
J. Sound Fib. (1970) 12 (1), 113-123
FREE AND FORCED TORSIONAL MOTION OF JOINED CONCENTRIC CIRCULAR SHELLS E. J. BRUNELLE
Watervliet Arsenal, Watervliet, New York, U.S.A. and Rensselaer Polytechnic Institute, Troy, New York, U.S.A. (Received 9 May 1969) This paper describes the free and forced motion of two, thin, concentric, circular cylindrical shells joined at their ends by rigid diaphragms with inertial properties. Mathematically the problem is described by two special Sturm-Liouville systems with common eigenvalues coupled only through the boundary conditions. The systems possess a modified orthogonality principle and give rise to a modified expansion principle for the associated non-homogeneous systems. These principles are derived in detail and several free motion problems are presented to illustrate the theory.
1. INTRODUCTION Much has been written concerning the free and forced motion of various types of shells subjected to a variety of boundary conditions. However there appears to be no counterpart work involving shells that are concentrically nested and joined to one another in various ways.t (Such shells are hereafter called joined shells.) In order to understand clearly the modified principles and procedures that arise in the analysis of joined shells we now investigate the simplest type of a joined shell executing its simplest motion, i.e., two joined circular cylindrical shells executing torsional motion subjected to various boundary conditions. Referring to Figure 1, the non-dimensionalized equation of motion for the ith shell [1,2] is given by (1) and the associated effective tangential shear resultant is given by (2) where vi = twist angle of the ith cylinder ( = at/at), ~t = circumferential displacement of the ith cylinder, at --- radius of the ith cylinder middle surface, xt --- ~t/L = non-dimensional co-ordinate of ith cylinder, L -- length of all cylinders,~: At = [pdG,] I/2 = inverse of shear wave speed in the ith infinite medium, ht = thickness of the ith shell, Pi = tangential force per unit area of the ith middle surface, pi -- mass density of the ith shell, Ft = pilaf Pi hi. a 2 ~,(xi, t)/ax 2 + )t2[Fl(xi, t) - 02 ~t(xi, t)/at z] = 0; Tl(x,, t) = ht at Gt L -1 [I + h2/12a 2] ~,(x,, t)/axt;
i -- 1,2, i -- 1,2.
(1) (2)
$ These types of shells may be of interest when considering thermal protection systems and/or meteorite shielding systems for space vehicles. :[:This is assumed for convenience; problems with different L~ are easily formulated and solved with some increases in algebraic manipulation. 8 113
114
E.J. BRUNELLE
The various boundary conditions associated with this set of equations are (3) damped end, (4) end joined by weightless rigid diaphragm, and (5) end joined by rigid disk with inertial properties: fl(ffl, t) = v2(ff2,t) = 0;
(3)
~l(Xl, t )
= ~2(,~ 2, t ) ~. 2rra~ 2~(.~,, t) + 2zra~ T2(~2, t) = Ot '
Ul(3~l, t) = ~52(~2,t) 27ra2 Tl(2l, t) + 2rra 2 T2(~2, t) ~: I~ 02 ~,(~t, t)/Ot 2 = 0}'
(4) (5)
where the minus sign is used [in (5)] if ~ is zero and the plus sign is used if ~l is unity, and where ;~ = either 0 or 1 ; i = 1, 2, Ise = angular moment of inertia of the disk at the end )7.
Figure 1. Two joined circular cylindricalshells.
2. THE ORTHOGONALITY PRINCIPLE The homogeneous counterparts of (1) and (2) and the boundary conditions corresponding to (5) are now used to derive the orthogonality principle associated with this type of joined shell. It will be seen that the orthogonality principles using (3) and (4) will be Sl~cial cases of the orthogonality principle to be derived in this section. Hence the system to be investigated is (6) and (7), with the boundary conditions (8) through (11) where vt(xi) = e - l ' ' ~(xt, t ), to -- frequency of oscillation, = P l hi a~(p2h2a32) -t, A = AE/A,, z'2 = Al to, v% = Io/2zrpl ht La~, O, = Id2~r m hi La~, Io = angular moment of inertia of disk at Ix = angular moment of inertia of disk at
x t = x 2 = O, x t = x 2 = 1.
v~t(xi) + h2 to2 vl(xt) = 0,
(6)
v~(x2) -4- ,~2 2 to2 v2(x2) = O,
(7)
vl(O) = v,(O),
(8)
Vl(1) = v2(1),
(9)
~Lt~)(2[/)~(0) + 60 Q2 Vl(0)] _{_v2(0) = 0,
(10)
/x,~2[v[(1) _ v~ Q2 vt(1)] + v~(1) = O.
(11)
TORSIONAL MOTION OF JOINED SHELLS
I 15
Notice that primes have been used to denote differentiationwith respect to the argument so that, for example, v[(O)= avl(O)/axl and v~(O)= av2(O)/ax2,etc. Since (6)and (7)are validfor the ruthand nth modes of vibration,the standard orthogonality derivation procedures yield (12) and (13): l
[v,.(x~)v~.(x~) - v~.(xO vh(x0] l + (~9~ - ~ )
O,
(12)
[v2.(x2) v~m(x2) - v2m(x2)v~.(x2)]~ + A2(.Q2m- .Q~) f v2.(x2) v2.(x2) dx2 = 0.
(13)
~In(X,) ~lm(Xl) dxl 0 1 0
Using (8) through (11) we find that t [v2.(x2) vlm(x2) - v2m(x2)vi.(x2)]~ = v,.(1)[*9, Q~ v,~(1) - Vim(l)] It)t2 -- v,m(1)[~l ~2 v~.(1) - v/.(1)] ItS2 + v,.(O)[v[m(O) + *90~2~ v~=(0)] ItM --
- v,m(0)[vi,(0) + *90~9~ v,,(0)] ITS2.
(14)
Multiplying (12) by ITS2, adding the result to (13) and using (14), yields (15): [Q~ -- Q~][*9, V,,(1) V,m(1 ) + .9o Vln(0) Vim(0) ÷ 1 1
÷ f Vln(X,)Dlm(X,)dx, + It-' f 0
v2.(x2) v2.(x2) dx2] = 0.
(15)
0
Therefore if ~2~ # £2~ the desired orthogonality principle is given by (16) for the elastic shell m odes:
.9! v,.(1) v,m(1) + *90 v,.(0) vim(0) + 1
+ f vi.(x,)v,m(x,)dx, + It-' f v2.(x2) v2m(x2)dx2 = 0. O
(16)
0
The rigid shellmode isfound by letting~o be zero in (6) and (7) so that
V;o(XO = V~o(X2)= 0,
(17)
where we have put the mode index (m or n) equal to zero. Applying (8) through (11) to (17) we find the result (18) where A is a constant:
(18)
Vlo(X,) = V2o(X2)--- A.
The orthogonality principle for mixed elasticand rigid shell modes is found by observing that the sum of the angular inertialmoments for a freelyvibratingjoined shellmust be zero. Therefore, .9oa2 ~,(0,t)/at2 + .9!a2 ~,(I,t)/at2 + I
+
f (a=
I
t)/at 2) dx, + It-'
0
f (a= t~2(X2, t)/at z) dx2 .---O.
(19)
0
But since t~l.(xi, t) = C °~.tvl.(xt), rio(X,) =/;20(X2) ~- A, and since - t a l c °'.' ~ 0, (19) becomes (20) which is the desired mixed orthogonality principle; note that formal replacement of either rn or n by zero in (16) yields the same result as (20): *90vt.(0) vl0(0) + *9, v,.(1) V,o(1) + 1
+
f ,,,.(x,),,,o(x,)
0
l
dx, + It-'
f ,,..(x~) v~o(x~)~
0
= o.
(20)
116
E.J. BRUNELLE
If the end x~ = 0 is joined by an inertia-less disk (or is clamped) then the orthogonality principle reduces to (21): !
!
#, v,.(1)Ulm(1 ) -[- f Uln(Xl)Vlm(Xl)dx, +/z-' f vz.(X.)u2ra(x2)dx2 0
= O.
(21)
0
Finally, if both ends are joined by inertia-less disks (or one end clamped) the orthogonality principle further reduces to (22): 1
!
f l)ln(Xl)131m(Xl)dX I + ~-1 f l;2n(X2)V2m(X2)dx2 = 0 0
(22)
O.
3. THE FORCED MOTION EXPANSION PRINCIPLE We now seek solutions of(l) and (5) in the form e,(xi, t) = ~ ¢.(t)v,.(x,);
i= 1,2,
(23)
n=O
where the v~.(x,) are the mode shapes of the homogeneous equations associated with (1). Inserting (23) in (1) yields ¢.(t) v;;(x,) + A2 F,(x,, t) - A2 ~ ~.(t) v,.(x,) = O; i = 1,2, n~l
(24)
n=O
where dots are now used to denote differentiations with respect to time. Multiplying (24) by v..(x~)dxl and integrating over the xl interval 0 to 1 yields (25): !
n=l
1
1
¢.(t) f vi.,v'i'.dxi+;V f Fl(xi,t)vimdxl-A~.~.(') f vimvi,dxl=O; 0
n=O
0
i = 1,2.
(25)
0
Inserting (26), which may be obtained by suitably integrating the homogeneous form of (1), into (25) yields (27) where we note that in the sequel w~ ~ 0: I
1
vi. Vimdxi = -~q co.
0
vi. vim dx~, i = 1,2,
1
n-O
(26)
0 !
[~.(t)+o,2.~.(t)]fv,.v,.dxi=fF,(xi, t)v,mdx,; 0
i = 1,2.
(27)
0
Multiplying the second relation of (27) by ~-~ and adding this to the first relation of (27) yields (28): [~.(t)+to2~:.(t)l n=O 1
[/
V,mV,.dxt +! z-' 1
= f Fl(xi, t) Vim dxi + tz-I f F2(x2, t) U2rad x 0
/
v2.,v2.dx2
O
2.
1 (28)
0
Adding (29) to both sides of (28) and using the orthogonality relations (16) and (20) yields (30), where Mm is given by (31): [~.(t) + to2 ~:.(t)][#, yr.(l)v,m(1) + #o v,.(O)v,m(O)], n=O
(29)
117
TORSIONAL M O T I O N OF JOINED SHELLS I
I
[~m(t) + w~ ~:m(t)]Mm = f Fl(xl, t) Vimdx, + t~-l f F2(x2, t) v2mdx2 + o
0
+ i [~.(t) + w~ ~:n(t)][01v,n(1)vl.(1) + 00 vi.(0)V,m(0)],
(30)
n-0
1
1
0, v~.(l) + oo v~.(0) + / ~ . dx, ÷ ~-* f d.~x2
M.
o
(31)
o
Using (23) we easily determine (32), and using both (23) and the homogeneous form of (1) yields (33). These relations, when inserted in the summation term on the right-hand side of (30), use being made of(l), yield equations for the ~:, as given by (34), where En and Mn are given by (35) and (36), respectively.
a2 @,(x,, t)/at 2 = ~ ~.(t) v,.,
(32)
n~0
-Ai"2~5;(x,, t) = ~ co~~.(t) v,.,
(33)
n= 1
~.(t) + w~.(t) = M~' S.(t); n = 0, 1,2 .....
(34)
8.(t) = 0l vl.(1) El(l, t) + 0o vln(0)FI(0, t) + 1
1
+ f F,(X,, t)v,n dx, + Ix-' f F2(x2, t) V2a dx2, o
(35)
o 1
i
M. = 0, d.(1) + #o d.(o) + f re. dx, + ~-' f d. dx2. o
(36)
o
However, in order for (23) to satisfy (5), it can be shown that F,(O,t)= F~(l,t)= 0 (se¢ Appendix) so that the final form for the 8.(t) is given by i
1
z.(t) = f r,
(35a)
o
From a physical standpoint the actual response is affected by a negligible amount because of the FI(0, t) = FI(1, t) = 0 condition, since Fl(xl, t) may take on the desired values in the interval 0 + ¢~ < x~ < 1 - c2 where ~ and ¢2 may be as small as desired. Finally, once the initial conditions have been specified on ~5~(x~,t), all integration constants may be determined, and hence the forced motion expansion principle has been established. 4. FREE VIBRATIONS This section presents free vibration results (i.e., frequency equations and mode shape ratios) for cylindrical shells joined by rigid diaphragm(s). Of particular interest are the interpretations of the frequency equations as/~ --~ 0 and as/~ --~ oo, which quickly permit one to map out regions on the frequency plot in which solutions exist. This procedure saves substantial amounts of computer time when completing the solution details. 4.1.
CASE A : C Y L I N D E R S C L A M P E D A T X t = 0 A N D J O I N E D W I T H A R I G I D INERTIA-LESS DIAPHRAGM AT X I ~
I
The equations of free vibrations, (37) and (38), subject to the boundary conditions (39) through (41) yield the mode shapes (42), the amplitude ratio (43), and the frequency relation (44): v't(xl) + ~2 Vl(Xl) = 0, (37) V[(X2)+ )t2~22V2(X2)= 0, (38)
E. J. BRUNELLE
118
v~(O) = v2(O) = O,
(39)
vl(1) = v2(1),
(40)
# a 2 v[(1) + v~(1) = 0,
(41)
Vl~(Xl) = Ai~sin~Q, Xl
and
v2~(x2) = A 2 . s i n A ~ x 2 ,
(43)
A l~/A2, = sin M2~/sin ~ ,
(44)
sin .On cos L.Q. +/z)t sin M2. cos.On -- 0.
0
2
4
6
Figure 2. Case A: ?t vs. -Onsolution regions.
t
2
4
(42)
5
Figure 3. Case A : A vs..Q~ curves (n = 1,2,3,4) for tz = 0.3, 1-0, a n d 5.0.
TORSIONAL MOTION OF JOINED SHELLS
119
Since/~ is not in the arguments of (44) it is convenient to think in terms o f a ~ vs. ~2. plot with/~ as a parameter. Solutions independent of tz are given by (45) and (46) which are obtained by demanding that sin I2. = sin M2~ = 0 and cos ~2. = cos M2. = 0, respectively: I2.=mr ~ = (2n - 1)2
and
and
?t=m/n;
m,n=l,2,3 .....
)t = (2m - 1)/(2n - 1);
m, n = 1, 2, 3
(45)
(46)
Furthermore, the limiting solutions as/z ~ 0 and as/z ~ oo are given by (47) and (48), respectively: I2.=nrr
•r r
and
I2.=mr/A
.
12.=(2n-1)~,
and
12.=(2n-1)~; z
n=1,2,3 .... ,
(47)
n=1.2,3
(48)
Finally, since only positive values of/~ are physically possible, just the positive solutions of (49) are sought: t~ = - t a n Q/)t tan M2. (49) Using (45) through (48). the solution regions for the first five modes on the ~ vs .O. plot are easily sketched as shown by the shaded portions of Figure 2. The solution details are then obtained by numerical procedures and Figure 3 shows the completed )~ vs. ~ . plot (for the first four modes) for three values of the parameter/.. 4.2. CASEB: FRF~-F~E CYLrNDm~SJOINEDAT BOTHENDSBY RIGID INERTIA-LESS DIAPHRAGMS
The equations of free vibration, (37) and (38), subject to the boundary conditions (50) through (53) yield the mode shapes (54), the amplitude ratio (55), and the frequency relation (56): vl(0) = v2(0), (50)
and
vl(1) = v2(1),
(51)
t,~ ~ vl(o) + v~(o) = o,
(52)
p~2 v~(1) + v~(1) = O,
(53)
vn~(xl) = Al~sing2~xn + Bn.cosf2.xi --p)L~ I n sin M2~Xz + Bn,. cos M2. x2J'
(54)
A l./Bl. = (cos M2. - cos Q.)/(sin 12. +/x)t sin M2.),
(55)
2/~?t[1 - cosg2.cos LO.] + [1 + (/zA)2]s i n ~ . sinM2. = 0.
(56)
V2a( X2) =
The solutions of (56) independent of p are given by (57) and (58) and the limiting solution of (56), as both p --* 0 and p --* 0%is given by (59): f2. = 2mr
and
O. -- (2n - 1)or and
)t = m/n;
[m = 0, 1, 2 . . . . I n = l , 2. . . . ,
A = (2m - 1)/(2n - 1);
f2. = nrr and
f2. = nzr/?t;
m, n = 1, 2 . . . . ,
n = 1, 2 . . . . .
(57) (58) (59)
Dividing (56) by ~t and taking the limit as ?t --. 0 yields lim/~ = f2n sin f2J[2(cos f2. - 1)]. ~-,o
(60)
120
E.J. BRUNELLE
Additionally, solving (56) for ff results in (61), from which only the positive values of/z are sought, F = -[1 - cos-(2, cos L.Q, ~ Icos A.Q, - cos g2,]]/(A sin.62, sin L.Q,,). (61) Furthermore (61) has a double root when cosL-Qk- COSY2k, SO that the frequencies at which the double roots occur are given by (62), and the value of the double roots are given by (63). Equation (63) is obtained by noting that when g2 = g2k, (61) yields/z = (I -- cos-Q~cos AI2k)/(--A sin-(2ksin M2k) and that substituting this result into (56) yields (63) directly: -Qk=27rk/(A± 1);
k = 0 , 1, 2 . . . . ,
(62)
/z = ±,~-~.
(63) 5
0'3
5
I
0"3
I
I
7
~.
8
9
[~'
0"3
I
0'3
I0
~n
I2
II
I'0 0'3
/~=5"C ~1
5 0"3
0
I
2
4
5
6
an Figure 4. Case B : ,~ vs. Q . curves (n = 1,2,3,4) f o r F = 0"3, 1'0 and 5'0.
W h e n / z = Z-~ and (62) is satisfied, two successive modes w i l l have the same frequency and
since g2~ = .(2,2, the orthogonality relation (16) is not valid. Thus these modes must be chosen such that they are orthogonal so that the forced expansion principle may be used correctly. The worst example of this behavior is when ?t = 1 (i.e. cylinders made of the same material) for then all values of/~ will produce an infinity of paired equal frequencies. Furthermore, in regions close to frequency pairs, the successive modes will have nearly equal frequencies which will make experimental results very difficult to interpret; thus theory will play a decisive part in interpreting some of the experimental data obtained for joined shell vibrations. Using (57) through (59), the solution regions for the first four modes on the A vs..(2, plot are easily sketched and Figure 4 shows the completed solution details for three values of the parameter F.
TORSIONAL MOTION OF JOINED SHELLS
121
Problems A (section 4.1) and B (section 4.2) adequately represent the free torsional motion of actual joined cylinders when (i) the inertial properties of the joining disk(s) is(are) negligible and (ii) the joining disk(s) is(are) so stiff that it(they) may be considered rigid. The present theory, as given in section 2, permits the relaxing of assumption (i) but relaxing the second assumption requires a straightforward but laborious modification of the equations of motion and hence of the orthogonality principle and the expansion principle. As an example of relaxing assumption (i), problem A is now partially re-worked, letting the diaphragm have non-zero angular inertial properties. 4.3.
CASE C." CYLINDERS CLAMPED A T X ! = 0 A N D JOINED W I T H A R I G I D INERTIAL D I A P H R A G M AT X l = 1
This problem is basically the same as that problem stated by (37) through (41) except that (41) is replaced by (64) : / ~ ' 2 [ V ; ( 1 ) - - ~1 322
vl(1)] + v~(1) = 0.
(64)
Therefore using (37) to (40) and (64) yields the mode shapes (65), the amplitude ratio (66), and the frequency relation (67):
vj.(xJ = Aj.sin32.xl
and
v2.(x2) = A2.sin,kO.x2,
(65)
AI./A2. = sin,LQ./sin 32.,
(66)
sin 32. cos ,LQ. +/z)~(cos 32. - #1 32. sin 32.) sin ,~. 32. = 0.
(67)
Solving (67) for ~, and since only positive values of/z are physically possible, just the positive solutions of (68) are sought: /z = sin 32. cos M2./(# 132. sin 3-2. - cos 3-2.)A sin M2..
(68)
Letting 32* be the solutions of (cos32. - #~ 32. sin32.) = 0, the solutions of (67) independent of/z are given by (69) and (70): 32.=mr 32. = 32* and
and
A=m/n;
m,n=l,2
A = (2m - 1) ~r/232";
.....
(69)
m, n = 1, 2 . . . . .
(70)
Furthermore, the limiting solutions as/z --~ 0 and as/z ~ ~ are given by (71) and (72), respectively: 32.=mr and 12.=(2n-1)Tr/2A; n = l , 2 , . . . , (71)
32.=nzr/A and $2.=32*; n = 1,2 . . . . .
(72)
Notice that only one limiting boundary is different than the limiting boundaries obtained for problem A, that is, (48) has been replaced by (72). Thus it is easy to visualize how the limiting solution boundaries shift with increasing t~ as is shown in Figure 5. The main feature to be noticed is the increasingly abrupt change of ,~ vs 32 (for increasing v~l) in the regions near 32 = mr. These abrupt changes appear to be similar to those observed by Mindlin for certain plate vibration problems governed by transcendental equations [3]. Finally, although the .Q* values may be found via the computer, it is useful to have approximate solutions for the 32* for both large and small values of#~. These approximate solutions are now presented for the roots 32* of (73): cos 32.- #1 32.sin 32. = O,
(73)
122
E . J . BRUNELLE
w h i c h m a y be r e - w r i t t e n as cot £2. = 0'l g2n.
(74)
F o r large ~1 we m a y a s s u m e £2* = n~- + ~ for n = 0, 1, 2 . . . . . T h u s cot £2* = cot (nrr + ~) ~ ~a n d hence (74) b e c o m e s
~-t ~ #~(n~ + ~).
(75)
, ~ - ( n r r / 2 ) + [(nrr/2) 2 + ( O l ) - l ] 1/2.
(76)
S o l v i n g (75) f o r E yields
~= 0"10
~=0"01 4 X
~ x45,
2
6
q=l-o
2
0
4
~ : I0"0
4
2
6
6
£Z. Figure 5. Case C: 3, vs .Q, solution regions for u% = 0.01, 0.10, 1.0 and 10. T h u s , for large values o f v~l, £2* is given b y
g-2* "~ [ (~l)-z/2;
n =0
= ~nrr[1 + ( n z r ) - 2 ( ~ t ) - l ] ;
(77) n = 1, 2,
I t s h o u l d be n o t e d t h a t this a p p r o x i m a t i o n i m p r o v e s as n increases. F o r s m a l l v~ we m a y a s s u m e .(2* = (2n - 1)(~r/2) - ~ for n = 1, 2 . . . . . T h u s cot.Q* ~ ~ a n d hence (74) b e c o m e s
, ~ #1[(2n - l)(~r/2) + "l.
(78)
, ~ ~l(rr/2)(2n -- 1)/(1 -- z~l).
(79)
S o l v i n g (78) for • yields
Thus, for s m a l l values o f # ~ , £2* is given by -(2* ~ (2n - 1)(~r/2)[1 --~L(1 - - 9 ~ ) - 1 ] ;
n = 1, 2 . . . . .
It s h o u l d be n o t e d t h a t this a p p r o x i m a t i o n gets w o r s e as n increases.
(80)
TORSIONALMOTIONOF JOINEDSHELLS
123
REFERENCES 1. H. GARNET, M. A. GOLDBERGand V. L. SALERNO1961 J. appl. Mech. 28, 4, 571. Torsional vibrations of shells of revolution. 2. E. J. BRUNELLE1969 Pro& llth Midwestern Mechanics Conf. Torsional vibrations of thin shells of revolution. 3. R. D. MINDLIN,A. SCHACKNOWand H. DERESIEWICZ1956 J. appl. Mech. 23, 3, 430. Flexural vibrations of rectangular plates. APPENDIX We now demonstrate the conditions under which the assumed forced solution (23) satisfies the second expression in (5). When (5) is written in expanded form for ~ = 1 it becomes /zA2 0t~(1, t)/Oxt + 0t)2(1, t)/Ox 2 + ~9!/~A2)i2 02 ~t(1, t)/Ot 2 = 0.
(A1)
Inserting (23) in (A1) yields #A2 ~ ~n(t)v~n(1)+ ~ ~,(t)v~n(1)+#~l~A2A202~l(1,t)Ot2~O, n=0
(A2)
n=0
but from (11). and using Q. = ?q con. we have /z~2 v~,(1) + v~,(1) =/z)~2 A2#~ v~,(1)0 2, so that (A2) becomes /zA2A12#l {~9 (~:n(t)v'"(1)°J2)+ 02v'n(l't)/Ot2} 7=0" Now using (6), (A3) becomes
{"n=~o
/~A2#l --
~n(t) v~n(1) +
)
Ai202 t~ln(1, t)/Ot 2 ~=0,
(A3)
(A4)
and noting that 01(1. t) ------~
n=0
~:n(t)v~.(1)
and using (1), (A4) becomes /~A2#1 [A2Fi(1, t)] ~ O.
(A5)
Therefore (A5) and hence (5) is satisfied when Fl(1, t) = 0, and in a like fashion wc can show that F~(0, t) must be zero for (5) to be satisfied when ~ = 0.
FREE AND FORCED TORSIONAL MOTION OF JOINED CONCENTRIC CIRCULAR SHELLS E. J. BRUNELLE
Watervliet Arsenal, Watervliet, New York, U.S.A. and Rensselaer Polytechnic Institute, Troy, New York, U.S.A. (Received 9 May 1969) This paper describes the free and forced motion of two, thin, concentric, circular cylindrical shells joined at their ends by rigid diaphragms with inertial properties. Mathematically the problem is described by two special Sturm-Liouville systems with common eigenvalues coupled only through the boundary conditions. The systems possess a modified orthogonality principle and give rise to a modified expansion principle for the associated non-homogeneous systems. These principles are derived in detail and several free motion problems are presented to illustrate the theory.
1. INTRODUCTION Much has been written concerning the free and forced motion of various types of shells subjected to a variety of boundary conditions. However there appears to be no counterpart work involving shells that are concentrically nested and joined to one another in various ways.t (Such shells are hereafter called joined shells.) In order to understand clearly the modified principles and procedures that arise in the analysis of joined shells we now investigate the simplest type of a joined shell executing its simplest motion, i.e., two joined circular cylindrical shells executing torsional motion subjected to various boundary conditions. Referring to Figure 1, the non-dimensionalized equation of motion for the ith shell [1,2] is given by (1) and the associated effective tangential shear resultant is given by (2) where vi = twist angle of the ith cylinder ( = at/at), ~t = circumferential displacement of the ith cylinder, at --- radius of the ith cylinder middle surface, xt --- ~t/L = non-dimensional co-ordinate of ith cylinder, L -- length of all cylinders,~: At = [pdG,] I/2 = inverse of shear wave speed in the ith infinite medium, ht = thickness of the ith shell, Pi = tangential force per unit area of the ith middle surface, pi -- mass density of the ith shell, Ft = pilaf Pi hi. a 2 ~,(xi, t)/ax 2 + )t2[Fl(xi, t) - 02 ~t(xi, t)/at z] = 0; Tl(x,, t) = ht at Gt L -1 [I + h2/12a 2] ~,(x,, t)/axt;
i -- 1,2, i -- 1,2.
(1) (2)
$ These types of shells may be of interest when considering thermal protection systems and/or meteorite shielding systems for space vehicles. :[:This is assumed for convenience; problems with different L~ are easily formulated and solved with some increases in algebraic manipulation. 8 113
114
E.J. BRUNELLE
The various boundary conditions associated with this set of equations are (3) damped end, (4) end joined by weightless rigid diaphragm, and (5) end joined by rigid disk with inertial properties: fl(ffl, t) = v2(ff2,t) = 0;
(3)
~l(Xl, t )
= ~2(,~ 2, t ) ~. 2rra~ 2~(.~,, t) + 2zra~ T2(~2, t) = Ot '
Ul(3~l, t) = ~52(~2,t) 27ra2 Tl(2l, t) + 2rra 2 T2(~2, t) ~: I~ 02 ~,(~t, t)/Ot 2 = 0}'
(4) (5)
where the minus sign is used [in (5)] if ~ is zero and the plus sign is used if ~l is unity, and where ;~ = either 0 or 1 ; i = 1, 2, Ise = angular moment of inertia of the disk at the end )7.
Figure 1. Two joined circular cylindricalshells.
2. THE ORTHOGONALITY PRINCIPLE The homogeneous counterparts of (1) and (2) and the boundary conditions corresponding to (5) are now used to derive the orthogonality principle associated with this type of joined shell. It will be seen that the orthogonality principles using (3) and (4) will be Sl~cial cases of the orthogonality principle to be derived in this section. Hence the system to be investigated is (6) and (7), with the boundary conditions (8) through (11) where vt(xi) = e - l ' ' ~(xt, t ), to -- frequency of oscillation, = P l hi a~(p2h2a32) -t, A = AE/A,, z'2 = Al to, v% = Io/2zrpl ht La~, O, = Id2~r m hi La~, Io = angular moment of inertia of disk at Ix = angular moment of inertia of disk at
x t = x 2 = O, x t = x 2 = 1.
v~t(xi) + h2 to2 vl(xt) = 0,
(6)
v~(x2) -4- ,~2 2 to2 v2(x2) = O,
(7)
vl(O) = v,(O),
(8)
Vl(1) = v2(1),
(9)
~Lt~)(2[/)~(0) + 60 Q2 Vl(0)] _{_v2(0) = 0,
(10)
/x,~2[v[(1) _ v~ Q2 vt(1)] + v~(1) = O.
(11)
TORSIONAL MOTION OF JOINED SHELLS
I 15
Notice that primes have been used to denote differentiationwith respect to the argument so that, for example, v[(O)= avl(O)/axl and v~(O)= av2(O)/ax2,etc. Since (6)and (7)are validfor the ruthand nth modes of vibration,the standard orthogonality derivation procedures yield (12) and (13): l
[v,.(x~)v~.(x~) - v~.(xO vh(x0] l + (~9~ - ~ )
O,
(12)
[v2.(x2) v~m(x2) - v2m(x2)v~.(x2)]~ + A2(.Q2m- .Q~) f v2.(x2) v2.(x2) dx2 = 0.
(13)
~In(X,) ~lm(Xl) dxl 0 1 0
Using (8) through (11) we find that t [v2.(x2) vlm(x2) - v2m(x2)vi.(x2)]~ = v,.(1)[*9, Q~ v,~(1) - Vim(l)] It)t2 -- v,m(1)[~l ~2 v~.(1) - v/.(1)] ItS2 + v,.(O)[v[m(O) + *90~2~ v~=(0)] ItM --
- v,m(0)[vi,(0) + *90~9~ v,,(0)] ITS2.
(14)
Multiplying (12) by ITS2, adding the result to (13) and using (14), yields (15): [Q~ -- Q~][*9, V,,(1) V,m(1 ) + .9o Vln(0) Vim(0) ÷ 1 1
÷ f Vln(X,)Dlm(X,)dx, + It-' f 0
v2.(x2) v2.(x2) dx2] = 0.
(15)
0
Therefore if ~2~ # £2~ the desired orthogonality principle is given by (16) for the elastic shell m odes:
.9! v,.(1) v,m(1) + *90 v,.(0) vim(0) + 1
+ f vi.(x,)v,m(x,)dx, + It-' f v2.(x2) v2m(x2)dx2 = 0. O
(16)
0
The rigid shellmode isfound by letting~o be zero in (6) and (7) so that
V;o(XO = V~o(X2)= 0,
(17)
where we have put the mode index (m or n) equal to zero. Applying (8) through (11) to (17) we find the result (18) where A is a constant:
(18)
Vlo(X,) = V2o(X2)--- A.
The orthogonality principle for mixed elasticand rigid shell modes is found by observing that the sum of the angular inertialmoments for a freelyvibratingjoined shellmust be zero. Therefore, .9oa2 ~,(0,t)/at2 + .9!a2 ~,(I,t)/at2 + I
+
f (a=
I
t)/at 2) dx, + It-'
0
f (a= t~2(X2, t)/at z) dx2 .---O.
(19)
0
But since t~l.(xi, t) = C °~.tvl.(xt), rio(X,) =/;20(X2) ~- A, and since - t a l c °'.' ~ 0, (19) becomes (20) which is the desired mixed orthogonality principle; note that formal replacement of either rn or n by zero in (16) yields the same result as (20): *90vt.(0) vl0(0) + *9, v,.(1) V,o(1) + 1
+
f ,,,.(x,),,,o(x,)
0
l
dx, + It-'
f ,,..(x~) v~o(x~)~
0
= o.
(20)
116
E.J. BRUNELLE
If the end x~ = 0 is joined by an inertia-less disk (or is clamped) then the orthogonality principle reduces to (21): !
!
#, v,.(1)Ulm(1 ) -[- f Uln(Xl)Vlm(Xl)dx, +/z-' f vz.(X.)u2ra(x2)dx2 0
= O.
(21)
0
Finally, if both ends are joined by inertia-less disks (or one end clamped) the orthogonality principle further reduces to (22): 1
!
f l)ln(Xl)131m(Xl)dX I + ~-1 f l;2n(X2)V2m(X2)dx2 = 0 0
(22)
O.
3. THE FORCED MOTION EXPANSION PRINCIPLE We now seek solutions of(l) and (5) in the form e,(xi, t) = ~ ¢.(t)v,.(x,);
i= 1,2,
(23)
n=O
where the v~.(x,) are the mode shapes of the homogeneous equations associated with (1). Inserting (23) in (1) yields ¢.(t) v;;(x,) + A2 F,(x,, t) - A2 ~ ~.(t) v,.(x,) = O; i = 1,2, n~l
(24)
n=O
where dots are now used to denote differentiations with respect to time. Multiplying (24) by v..(x~)dxl and integrating over the xl interval 0 to 1 yields (25): !
n=l
1
1
¢.(t) f vi.,v'i'.dxi+;V f Fl(xi,t)vimdxl-A~.~.(') f vimvi,dxl=O; 0
n=O
0
i = 1,2.
(25)
0
Inserting (26), which may be obtained by suitably integrating the homogeneous form of (1), into (25) yields (27) where we note that in the sequel w~ ~ 0: I
1
vi. Vimdxi = -~q co.
0
vi. vim dx~, i = 1,2,
1
n-O
(26)
0 !
[~.(t)+o,2.~.(t)]fv,.v,.dxi=fF,(xi, t)v,mdx,; 0
i = 1,2.
(27)
0
Multiplying the second relation of (27) by ~-~ and adding this to the first relation of (27) yields (28): [~.(t)+to2~:.(t)l n=O 1
[/
V,mV,.dxt +! z-' 1
= f Fl(xi, t) Vim dxi + tz-I f F2(x2, t) U2rad x 0
/
v2.,v2.dx2
O
2.
1 (28)
0
Adding (29) to both sides of (28) and using the orthogonality relations (16) and (20) yields (30), where Mm is given by (31): [~.(t) + to2 ~:.(t)][#, yr.(l)v,m(1) + #o v,.(O)v,m(O)], n=O
(29)
117
TORSIONAL M O T I O N OF JOINED SHELLS I
I
[~m(t) + w~ ~:m(t)]Mm = f Fl(xl, t) Vimdx, + t~-l f F2(x2, t) v2mdx2 + o
0
+ i [~.(t) + w~ ~:n(t)][01v,n(1)vl.(1) + 00 vi.(0)V,m(0)],
(30)
n-0
1
1
0, v~.(l) + oo v~.(0) + / ~ . dx, ÷ ~-* f d.~x2
M.
o
(31)
o
Using (23) we easily determine (32), and using both (23) and the homogeneous form of (1) yields (33). These relations, when inserted in the summation term on the right-hand side of (30), use being made of(l), yield equations for the ~:, as given by (34), where En and Mn are given by (35) and (36), respectively.
a2 @,(x,, t)/at 2 = ~ ~.(t) v,.,
(32)
n~0
-Ai"2~5;(x,, t) = ~ co~~.(t) v,.,
(33)
n= 1
~.(t) + w~.(t) = M~' S.(t); n = 0, 1,2 .....
(34)
8.(t) = 0l vl.(1) El(l, t) + 0o vln(0)FI(0, t) + 1
1
+ f F,(X,, t)v,n dx, + Ix-' f F2(x2, t) V2a dx2, o
(35)
o 1
i
M. = 0, d.(1) + #o d.(o) + f re. dx, + ~-' f d. dx2. o
(36)
o
However, in order for (23) to satisfy (5), it can be shown that F,(O,t)= F~(l,t)= 0 (se¢ Appendix) so that the final form for the 8.(t) is given by i
1
z.(t) = f r,
(35a)
o
From a physical standpoint the actual response is affected by a negligible amount because of the FI(0, t) = FI(1, t) = 0 condition, since Fl(xl, t) may take on the desired values in the interval 0 + ¢~ < x~ < 1 - c2 where ~ and ¢2 may be as small as desired. Finally, once the initial conditions have been specified on ~5~(x~,t), all integration constants may be determined, and hence the forced motion expansion principle has been established. 4. FREE VIBRATIONS This section presents free vibration results (i.e., frequency equations and mode shape ratios) for cylindrical shells joined by rigid diaphragm(s). Of particular interest are the interpretations of the frequency equations as/~ --~ 0 and as/~ --~ oo, which quickly permit one to map out regions on the frequency plot in which solutions exist. This procedure saves substantial amounts of computer time when completing the solution details. 4.1.
CASE A : C Y L I N D E R S C L A M P E D A T X t = 0 A N D J O I N E D W I T H A R I G I D INERTIA-LESS DIAPHRAGM AT X I ~
I
The equations of free vibrations, (37) and (38), subject to the boundary conditions (39) through (41) yield the mode shapes (42), the amplitude ratio (43), and the frequency relation (44): v't(xl) + ~2 Vl(Xl) = 0, (37) V[(X2)+ )t2~22V2(X2)= 0, (38)
E. J. BRUNELLE
118
v~(O) = v2(O) = O,
(39)
vl(1) = v2(1),
(40)
# a 2 v[(1) + v~(1) = 0,
(41)
Vl~(Xl) = Ai~sin~Q, Xl
and
v2~(x2) = A 2 . s i n A ~ x 2 ,
(43)
A l~/A2, = sin M2~/sin ~ ,
(44)
sin .On cos L.Q. +/z)t sin M2. cos.On -- 0.
0
2
4
6
Figure 2. Case A: ?t vs. -Onsolution regions.
t
2
4
(42)
5
Figure 3. Case A : A vs..Q~ curves (n = 1,2,3,4) for tz = 0.3, 1-0, a n d 5.0.
TORSIONAL MOTION OF JOINED SHELLS
119
Since/~ is not in the arguments of (44) it is convenient to think in terms o f a ~ vs. ~2. plot with/~ as a parameter. Solutions independent of tz are given by (45) and (46) which are obtained by demanding that sin I2. = sin M2~ = 0 and cos ~2. = cos M2. = 0, respectively: I2.=mr ~ = (2n - 1)2
and
and
?t=m/n;
m,n=l,2,3 .....
)t = (2m - 1)/(2n - 1);
m, n = 1, 2, 3
(45)
(46)
Furthermore, the limiting solutions as/z ~ 0 and as/z ~ oo are given by (47) and (48), respectively: I2.=nrr
•r r
and
I2.=mr/A
.
12.=(2n-1)~,
and
12.=(2n-1)~; z
n=1,2,3 .... ,
(47)
n=1.2,3
(48)
Finally, since only positive values of/~ are physically possible, just the positive solutions of (49) are sought: t~ = - t a n Q/)t tan M2. (49) Using (45) through (48). the solution regions for the first five modes on the ~ vs .O. plot are easily sketched as shown by the shaded portions of Figure 2. The solution details are then obtained by numerical procedures and Figure 3 shows the completed )~ vs. ~ . plot (for the first four modes) for three values of the parameter/.. 4.2. CASEB: FRF~-F~E CYLrNDm~SJOINEDAT BOTHENDSBY RIGID INERTIA-LESS DIAPHRAGMS
The equations of free vibration, (37) and (38), subject to the boundary conditions (50) through (53) yield the mode shapes (54), the amplitude ratio (55), and the frequency relation (56): vl(0) = v2(0), (50)
and
vl(1) = v2(1),
(51)
t,~ ~ vl(o) + v~(o) = o,
(52)
p~2 v~(1) + v~(1) = O,
(53)
vn~(xl) = Al~sing2~xn + Bn.cosf2.xi --p)L~ I n sin M2~Xz + Bn,. cos M2. x2J'
(54)
A l./Bl. = (cos M2. - cos Q.)/(sin 12. +/x)t sin M2.),
(55)
2/~?t[1 - cosg2.cos LO.] + [1 + (/zA)2]s i n ~ . sinM2. = 0.
(56)
V2a( X2) =
The solutions of (56) independent of p are given by (57) and (58) and the limiting solution of (56), as both p --* 0 and p --* 0%is given by (59): f2. = 2mr
and
O. -- (2n - 1)or and
)t = m/n;
[m = 0, 1, 2 . . . . I n = l , 2. . . . ,
A = (2m - 1)/(2n - 1);
f2. = nrr and
f2. = nzr/?t;
m, n = 1, 2 . . . . ,
n = 1, 2 . . . . .
(57) (58) (59)
Dividing (56) by ~t and taking the limit as ?t --. 0 yields lim/~ = f2n sin f2J[2(cos f2. - 1)]. ~-,o
(60)
120
E.J. BRUNELLE
Additionally, solving (56) for ff results in (61), from which only the positive values of/z are sought, F = -[1 - cos-(2, cos L.Q, ~ Icos A.Q, - cos g2,]]/(A sin.62, sin L.Q,,). (61) Furthermore (61) has a double root when cosL-Qk- COSY2k, SO that the frequencies at which the double roots occur are given by (62), and the value of the double roots are given by (63). Equation (63) is obtained by noting that when g2 = g2k, (61) yields/z = (I -- cos-Q~cos AI2k)/(--A sin-(2ksin M2k) and that substituting this result into (56) yields (63) directly: -Qk=27rk/(A± 1);
k = 0 , 1, 2 . . . . ,
(62)
/z = ±,~-~.
(63) 5
0'3
5
I
0"3
I
I
7
~.
8
9
[~'
0"3
I
0'3
I0
~n
I2
II
I'0 0'3
/~=5"C ~1
5 0"3
0
I
2
4
5
6
an Figure 4. Case B : ,~ vs. Q . curves (n = 1,2,3,4) f o r F = 0"3, 1'0 and 5'0.
W h e n / z = Z-~ and (62) is satisfied, two successive modes w i l l have the same frequency and
since g2~ = .(2,2, the orthogonality relation (16) is not valid. Thus these modes must be chosen such that they are orthogonal so that the forced expansion principle may be used correctly. The worst example of this behavior is when ?t = 1 (i.e. cylinders made of the same material) for then all values of/~ will produce an infinity of paired equal frequencies. Furthermore, in regions close to frequency pairs, the successive modes will have nearly equal frequencies which will make experimental results very difficult to interpret; thus theory will play a decisive part in interpreting some of the experimental data obtained for joined shell vibrations. Using (57) through (59), the solution regions for the first four modes on the A vs..(2, plot are easily sketched and Figure 4 shows the completed solution details for three values of the parameter F.
TORSIONAL MOTION OF JOINED SHELLS
121
Problems A (section 4.1) and B (section 4.2) adequately represent the free torsional motion of actual joined cylinders when (i) the inertial properties of the joining disk(s) is(are) negligible and (ii) the joining disk(s) is(are) so stiff that it(they) may be considered rigid. The present theory, as given in section 2, permits the relaxing of assumption (i) but relaxing the second assumption requires a straightforward but laborious modification of the equations of motion and hence of the orthogonality principle and the expansion principle. As an example of relaxing assumption (i), problem A is now partially re-worked, letting the diaphragm have non-zero angular inertial properties. 4.3.
CASE C." CYLINDERS CLAMPED A T X ! = 0 A N D JOINED W I T H A R I G I D INERTIAL D I A P H R A G M AT X l = 1
This problem is basically the same as that problem stated by (37) through (41) except that (41) is replaced by (64) : / ~ ' 2 [ V ; ( 1 ) - - ~1 322
vl(1)] + v~(1) = 0.
(64)
Therefore using (37) to (40) and (64) yields the mode shapes (65), the amplitude ratio (66), and the frequency relation (67):
vj.(xJ = Aj.sin32.xl
and
v2.(x2) = A2.sin,kO.x2,
(65)
AI./A2. = sin,LQ./sin 32.,
(66)
sin 32. cos ,LQ. +/z)~(cos 32. - #1 32. sin 32.) sin ,~. 32. = 0.
(67)
Solving (67) for ~, and since only positive values of/z are physically possible, just the positive solutions of (68) are sought: /z = sin 32. cos M2./(# 132. sin 3-2. - cos 3-2.)A sin M2..
(68)
Letting 32* be the solutions of (cos32. - #~ 32. sin32.) = 0, the solutions of (67) independent of/z are given by (69) and (70): 32.=mr 32. = 32* and
and
A=m/n;
m,n=l,2
A = (2m - 1) ~r/232";
.....
(69)
m, n = 1, 2 . . . . .
(70)
Furthermore, the limiting solutions as/z --~ 0 and as/z ~ ~ are given by (71) and (72), respectively: 32.=mr and 12.=(2n-1)Tr/2A; n = l , 2 , . . . , (71)
32.=nzr/A and $2.=32*; n = 1,2 . . . . .
(72)
Notice that only one limiting boundary is different than the limiting boundaries obtained for problem A, that is, (48) has been replaced by (72). Thus it is easy to visualize how the limiting solution boundaries shift with increasing t~ as is shown in Figure 5. The main feature to be noticed is the increasingly abrupt change of ,~ vs 32 (for increasing v~l) in the regions near 32 = mr. These abrupt changes appear to be similar to those observed by Mindlin for certain plate vibration problems governed by transcendental equations [3]. Finally, although the .Q* values may be found via the computer, it is useful to have approximate solutions for the 32* for both large and small values of#~. These approximate solutions are now presented for the roots 32* of (73): cos 32.- #1 32.sin 32. = O,
(73)
122
E . J . BRUNELLE
w h i c h m a y be r e - w r i t t e n as cot £2. = 0'l g2n.
(74)
F o r large ~1 we m a y a s s u m e £2* = n~- + ~ for n = 0, 1, 2 . . . . . T h u s cot £2* = cot (nrr + ~) ~ ~a n d hence (74) b e c o m e s
~-t ~ #~(n~ + ~).
(75)
, ~ - ( n r r / 2 ) + [(nrr/2) 2 + ( O l ) - l ] 1/2.
(76)
S o l v i n g (75) f o r E yields
~= 0"10
~=0"01 4 X
~ x45,
2
6
q=l-o
2
0
4
~ : I0"0
4
2
6
6
£Z. Figure 5. Case C: 3, vs .Q, solution regions for u% = 0.01, 0.10, 1.0 and 10. T h u s , for large values o f v~l, £2* is given b y
g-2* "~ [ (~l)-z/2;
n =0
= ~nrr[1 + ( n z r ) - 2 ( ~ t ) - l ] ;
(77) n = 1, 2,
I t s h o u l d be n o t e d t h a t this a p p r o x i m a t i o n i m p r o v e s as n increases. F o r s m a l l v~ we m a y a s s u m e .(2* = (2n - 1)(~r/2) - ~ for n = 1, 2 . . . . . T h u s cot.Q* ~ ~ a n d hence (74) b e c o m e s
, ~ #1[(2n - l)(~r/2) + "l.
(78)
, ~ ~l(rr/2)(2n -- 1)/(1 -- z~l).
(79)
S o l v i n g (78) for • yields
Thus, for s m a l l values o f # ~ , £2* is given by -(2* ~ (2n - 1)(~r/2)[1 --~L(1 - - 9 ~ ) - 1 ] ;
n = 1, 2 . . . . .
It s h o u l d be n o t e d t h a t this a p p r o x i m a t i o n gets w o r s e as n increases.
(80)
TORSIONALMOTIONOF JOINEDSHELLS
123
REFERENCES 1. H. GARNET, M. A. GOLDBERGand V. L. SALERNO1961 J. appl. Mech. 28, 4, 571. Torsional vibrations of shells of revolution. 2. E. J. BRUNELLE1969 Pro& llth Midwestern Mechanics Conf. Torsional vibrations of thin shells of revolution. 3. R. D. MINDLIN,A. SCHACKNOWand H. DERESIEWICZ1956 J. appl. Mech. 23, 3, 430. Flexural vibrations of rectangular plates. APPENDIX We now demonstrate the conditions under which the assumed forced solution (23) satisfies the second expression in (5). When (5) is written in expanded form for ~ = 1 it becomes /zA2 0t~(1, t)/Oxt + 0t)2(1, t)/Ox 2 + ~9!/~A2)i2 02 ~t(1, t)/Ot 2 = 0.
(A1)
Inserting (23) in (A1) yields #A2 ~ ~n(t)v~n(1)+ ~ ~,(t)v~n(1)+#~l~A2A202~l(1,t)Ot2~O, n=0
(A2)
n=0
but from (11). and using Q. = ?q con. we have /z~2 v~,(1) + v~,(1) =/z)~2 A2#~ v~,(1)0 2, so that (A2) becomes /zA2A12#l {~9 (~:n(t)v'"(1)°J2)+ 02v'n(l't)/Ot2} 7=0" Now using (6), (A3) becomes
{"n=~o
/~A2#l --
~n(t) v~n(1) +
)
Ai202 t~ln(1, t)/Ot 2 ~=0,
(A3)
(A4)
and noting that 01(1. t) ------~
n=0
~:n(t)v~.(1)
and using (1), (A4) becomes /~A2#1 [A2Fi(1, t)] ~ O.
(A5)
Therefore (A5) and hence (5) is satisfied when Fl(1, t) = 0, and in a like fashion wc can show that F~(0, t) must be zero for (5) to be satisfied when ~ = 0.