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Dynamic response of large space structures
Acta Astronautica Vol. 9, No. 6-7, pp. 455--471, 1982 Printed in Great Britain.
0094-5765/82/060455-17503.00/0 Pergamon Press Ltd.
DYNAMIC RESPONSE OF LARGE SPACE STRUCTURES PAOLO SANTINI,CLAUDIO BOTTIGLIERIand MARIO MARCHETTI University of Rome, Istituto di Tecnologia Aerospaziale, Via Eudossiana N.16, 00184 Rome, Italy (Received
21 December 1981)
Abstract--In the first part the case of a one-dimensional string of bays is firstly considered, and the equations for the most general case of mass and stiffness are derived. A procedure very similar to numerical integration is discussed. When there are many bays, the procedure becomes too heavy, and a "finite elements" approach is very useful. In the second part, the case of two dimensional structures, consisting of several adjacent strings is treated, and solved by method very similar to those applicable to partial differential equations; in particular, finite elements and separation of variables. Numerical results illustrate the procedures, for sake of comparison; graphs and Appendices complete the work.
1. ONE-DIMENSIONAL STRUCTURES
INTRODUCTION
The problem of large structures for space is receiving more and more careful attention from the scientists of all the world. This is obviously due to the increasing size of the spacecrafts, present and future, and, also, to the very little knowledge about this kind of structures. In particular, the dynamic behaviour is one of the most important information the designer needs, even starting from the calculation of natural modes and frequencies. This is a very old problem, and accurate and quite reliable programs exist for its solutions: however, when we have to analyze a structure with a tremendous number of degrees of freedom, the size of the system and the relevant computing time make the use of general purpose programs practically impossible. For this reason, since long[l-3], and on account of the "modularity" of large space structures (consisting of several elements all almost equal one to another), the concept of "periodic" structures has been introduced. However, available solutions are practically confined to the case where all the elements are equal-not "almost" equal. Therefore, it is worth while to see whether also the more general approach to not completely uniform elements could give a satisfactory response. Another important point is the fact that, when the elements are really too-many, also "exact" solutions may lead to some difficulties from a numerical standpoint. The use of "finite elements" technique is very helpful in several of such cases; it is based on the idea of treating a discrete problem as a continuous problem. And this approach is even more fruitful when we have to solve two-dimensional problems; i.e. when the elements are distributed on a surface; in this case the resulting matrix-difference equation of the problem is very similar to a partial differential equation; so the relevant techniques, with the due changes, can be applied: finite elements-as said-, separation of variables (when applicable) singularities (not included in this paper) may give a substantial support to the analysis of large space structures. This attempt is illustrated in the following pages.
?Please refer to the list of symbols at the end of the paper.
1. G e n e r a l c o n s i d e r a t i o n s
(a) Let us consider a structural element, Fig. l(a), with inherent mass and flexibility (to be defined more clearly later) and a "string" of such elements, not necessarily all equal, Fig. l(b). Anyway we assume that some strong similarities exist between any two of the elements; e.g., they may have the same general shape, and the same kind of degrees of freedom, and consist of the same members; although lengths, cross sections, mass, etc., may be different. We shall denote the ith elements as the ith bay, B " ) ( i = 1, .. N ) . t (b) For necessary generality, we must also assume that two consecutive bays are connected through an "interface", such that the degrees of freedom of the elements joining upon it are the same as those of the junction, Fig. 2; the interface may have mass and flexibility. (c) At this point, we must specify the unknowns of our problem. The ith bay is connected at its "left" end to the ( i - l)th interface, J~-l, Fig. 3, so we must also introduce a 0th "interface" at the "left" end of the whole structure, and an Nth "interface" at the "right" end. It is also convenient to define separately the internal degrees of freedom of B "), and those pertaining to the adjacent interfaces. More exactly: we denote by X~ the vector of the degrees of freedom common to B (°, J~, B "+1) Fig. 4, and by x ") the remaining degrees of freedom--if any--pertaining to B "), Fig. 4. (d) As a consequence, the ridigity matrix k ") of B ") will be of the type sketched in Fig. 5. A similar arrangement holds also for the mass-matrix, m "), with the same subscripts: it should be pointed out, however that off-diagonal matrices are denoted, e.g. by ^b(i) Ac, + m~)c. Also note that the matrices Xnc,"")k~-b are not necessarily square, since two consecutive interfaces may have a different number of degrees of freedom, and that, of course: _
k A( ai )--_
bfi)
^hA.r,
• etc.
(1)
For J, on account of its definition, we have a single matrix Kj, and a corresponding mass matrix Mj, both centered on Xi. 455
456
P. SANTINIet al.
k,.,:
~ v
-¢
(N)
,-,-Li, ,
) _k~i~
-
Cll
k(i~ c,
,b kcc
B(t)
Fig. 1. One-dimensionalstructure.
Fig. 5. Rigidity matrix for the ith bay.
y
X°
x~ ) X 1
Xi_l
X (i)
Xi
XN- 1 X('N) X N
l
i
Fig. 2. Definitionof bays and interfaces.
22
i
] I
-k~
I Jo
4 i4
J i
Fig. 6. General rigidty matrix.
Fig. 3. Numbering of structural elements.
our system. For B ") we have, Fig. 6:
~
~
/ i
.
i
.
.
.
.
(1) ~
.
--
b ( i ) "V
+ k~'~x (', - k~,~X, +/~".
(3)
,v- + m ~ k ~ " ) + M~g~ + m ~ ' ) ~ "+~ m (~) CB.'Xi-t + #t~BC __,+1) /~, - k ~ x ") + K~Xi t i + l = -- k ~ X i - i
(3)
Similarly, for J~:
//
b(i+l)~(i+l) b(i+l)g - - ~ ' B A "~ - - a ' B C ~IH+ 1 + F / .
For sinusoidal behaviour, under free vibration conditions:
Fig. 4. Internal and external degrees of freedom.
By assembling the matrices of the elements and of the interfaces, we obtain, for the entire structure, the rigidity matrix of Fig. 6, where the element painted in black has the value: K, = Ks., + k ~ + k ~ ' .
(2)
In eqn (2), the last term is missing for i = N, the last but one is missing for i = 0. (e) Let us now write the equations of equilibrium for
[ k ~ - to2ma~lx ('~- (k~)a + toZm~a)Xi_l - (k~c + to2m ~)c)X~ = 0; (i = 1. . . . N) - ( k ~ + to2m ~)X~-~ - ( / ~ k + o2m~)x
") + ( K i -
~o2M,)X,
(4)
- (k"~" + to2O+,)~.+.mnA p~ --~.,ncll'"+l) + ~02m~l))Xi+l = 0; (i = 0 . . . . N) The "boundary" conditions are obtained by deleting in the second of eqn (4) terms with Xi-t, x (i) for i = 0, and
Dynamic response of large space structures terms with x " + ' , X~+I for i = N. Letting: L(i) KAA--to
2
(i) mAA
For a free-free structures the end conditions are simply
(5)
we may write (4) in the more compact form: [ l"~x"~
t")v
I")v-a.(i=l,
-- l ABZ'Xi-I -- t AC/~i -- u, V I (i) ~(i)..I. I V - - tCA., . ~ a-~iz'l i -I(i+l)V
--I(i)
-'~i+1 = 0; (i = 0 . . . . .
,Bc
N)
. •., (i+l)
l(i+l)~ tB A
ICBI,~i_I
-
(6)
N).
1.2 A numerical integration scheme [or multibays structures
(a) For the special case where all the bays and interfaces are equal, a well known [1, 2] matrix difference equation can be obtained and solved (see Appendix 1). For the general case of quasi-periodic, or non periodic structures, it is convenient to make a further distinction. The matrix w) • AA, eqn (5) can be considered as the sum of three further matrices: IAA.S,[(i)IAA.O,(i) 1")AA,C.t The first of them corresponds to the contribution of the members having one end on J~_~ (on the left of B "~) and the other end inside B"): similarly for 'AA.D. ~"~ • obviously, l ~ . c corresponds to the elements who have no connections but inside B "). It is clear therefore that: S
(i) -- i(i) -- tAA.S
U
v
x(i)
I(i) --IAB~i--I~
.)
t.)v..-a
.~o)_t.)
-- IAA,DX
-- tAC,,"xi,
•i
- 1 -~...,
[ -- u,...,
~,r a~
~r
~
1"¢ - - I
(7)
are the reactions, exerted from left and right respectively from the interfaces to the bay, Fig. 7. We can of course define similar things for the interface: • -- I(i) V i(i) sr Oi -- tCC./Xi -- tCB.,'xi-I FI -- l(i) V /(i+l) I.,q--tBB.rXi--tBA X
'So=0
-- 1(i) -- IAA
k ~ + t o 2 m ~ : 1 ~ etc.
l
457
I(i) ( i ) . • __ I l~r -- tcAX , I - - t , . . . , 1"¢ I(i+l)v • " -- N --IBC ,.'li+l, [--~, .....
(i+1)
(8)
I.DN = 0
(10)
(b) On the basis of the foregoing approach, a numerical solution scheme can be developed, as follows: (i) Assume to is given, so that all the matrices appearing in the foregoing development can be computed; assume also that we have started from the left end side, and we are now on the ith interface, where we know S~, X~; physically, the reaction on the left of ~, and the values of its degrees of freedom. From the second of eqns (9) we immediately have D~; therefore, by associating the second of eqns (8), written for i, and the first of eqns (6) written for i + I, we have: I(i+l)~(i+l)± ,BA .~ l(i+l)w(i+l)
T
I (i) V tBC"Xi+I I(i+l)V
-- I (i) V -- "BB/Xi_ I(i+l)v
!'~ L~i
(11)
From eqns (11), by a routine inversion scheme, we compute x (~+', X~+~, and from the first of (8), S;+~; so we are now in a position for the next step. (ii) For practical calculations, we give successive tentative values to to; furthermore we have So = 0, and set Xo = I (the unit matrix of order NE, the number of degrees of freedom of the interface), obtaining the final value DN for the reaction right of JN). Now, if to* denotes the true frequency, and X* the true value of Xo, we would have: D * = DNX~ = O.
(12)
Note that DN is a square matrix, whereas D* is a vector. However, from (12) we have the condition for to:
N-1. det(DN) = 0 By the word "reaction" we denote the real force as projected on the degrees of freedom of the single elements. Now eqns (6) read very simply: l(i) ¢(i).a_ , AA,C.~ r
S
(i)
+ d (i)
LjiX~ + S~ + Di = 0;
= 0;
i = I, N i = 0 . . . . N.
///al
/
~f /
Js i
(9)
from which X* is computed; subsequently, by repeating the "integration procedure", we obtain the mode. To illustrate the procedure, we can follow Fig. 8. At the beginning, we know (or arbitrarily give) Xo, and compute the terms described as I; so we can solve the set of simultaneous equation II; compute the terms III, and so on. A further illustration can be gained through a particular case. Let us consider a type of elements, for which no internal degrees of freedom are present, so that the matrices ^BA,~'")k~,)B, k ~ (and the corrisponding lmatrices) are absent; now (7) are missing and the first of eqns (9) as well. Thus, by suitable combination of eqns (8) and of the second of eqns (9), we obtaining the following "Runge Kutta" scheme: f __ (i+1) --I (i) (i+1) --1 [3(/+1 - {[lac ] [lab + Lji]}Xi + [IBc ] Si C -- fl(i+l)[l(i+l)]-l/l(i) -L T ~ /(i+I)l~.¢r I O i + l - l , c c t,ac J ~,~sT,-~JIJ-,c8 ~.'*i (14) ) |
L
~Ji-I Fig. 7. Reactions exerted on a bay.
tltalian: C = Centro: S = sinistra (left); D = destra (right).
(13)
+SL(i+l)tl(i+~),-I~, , ~ s t nc I 1oi;
!•= 0~, . . . . .N.
to be integrated with the initial condition So = 0, Xo = I. 1.3. "Finite elements" technique (a) The above described formulation gives exact solu-
etal.
458
P. SANTINI Xo
\II/ computing X
~:~lvlng slmultanllous,
equations
Fig. 8. Numerical integration scheme.
tions. In other words, in the foregoing pages all what has been done can be thought of as an indirect way of considering the sparseness of the matrix for the complete structure, and an attempt the reduce it to a simple element approach. In practical analyis, and when there are many bays (and interfaces) this procedure may prove to be very heavy. For this purpose it may be convenient to try an approach where approximate formulations can give satisfactory results, such as is the case of the finite element techniques. This is a well known procedure of reducing continuous to discrete problems; in the following, we shall use it to reduce a problem with many degrees of freedom to another one with a lesser, or much lesser, number of degrees of freedom. (b) The strain-energy corresponding to the matrix of Fig. 6 can be written:
Now, in the spirit of the finite element technique, we give the approximate law:
[
x°)=xo(l-N)+XNN;
(17)
X~ = X o ( 1 - ~ ) + XN ~-; i = 0 ..... N
corresponding to a linear variation of x (°, X. between the end values defined by Xo, Xo, xN, XN, obtaining, for the two basic quadratic forms: 1
~: = ~
[XoTXorXNTxNr ] I I too PNo ~0N
| "
-- --/~
~(i)b(i)
',/~"T"A,~" ~l,
~(I) j_
T
1 i N
III
0i N
gON, T
L.. P o n , T
~(i)L-(i) V ~=d .~ T r* B A , C X i ~ l - - ~ . l ~ t - l . T ~ Ii li N --~ ~(i)lt(i) V L i d ~ T r~ A C ' ° ' t • 1 i
b(t) V" BC/Xi
(18) 1
•
I yoo
yoo, r
yo~
I F l,~'oT
- - H I ~
YNO,T
(19)
i N
-
II -
-- YON.T -- gON, T
N
or_ 1 / ~ .~(o_o) .:(i)± ~o
li N
- Go~
goo
- GoN, T --
+ 2 ~o)~(o ~ "~ T H t B A l t i
DNN.T
3- = ~ [XoTYCoTX,.,~XNTI Goo
J-l. I i N
RNN
[XoT1 ,oNr / o,::j j::j poN1
(15)
The physical meaning of each of the terms under (15) is evident. By the same token we obtain for the kinetic energy: N
II IV
I RoN,T PNO,T
X, r K , Xi
--~
i=1 ..... N
± ~. ~ 1T ~ #x t-
~o) x7 I.Ttlt BC/~M
(16)
li
+ Z ~(o~(o xk "~ T n t A C ~ t i • I i
By leaving the x°'s, X/s unconstrained, and through the application of the principle of minimum potential energy, we obtain eqns (3) and (3').
The expression of the submatrices constituting the elements of the rigidity matrix R defined by (18), and of the mass matrix G defined by (19), are given in Appendix 2. Finally, by considering several adjacent elements, Fig. 9(a), and using the well known assembly technique, Fig. 9(b), one can obtain mass and stiffness in finite element approach for the entire structure.
459
Dynamic response of large space structures
C-
1
®
:3
B~1~)
~
Jh=
B (h+U
O-
P
®
O
I'l÷!
h-1
O
Fig. 9. Finite element approach. For the generic gridpoint, h, Fig. 9(b), we have the matrix equation: to(h)
i i~lV
2~(hn.J_{R~h+1)
- (.o L I I V l .
--
(o2G~h+')}][Xn]
Xh
+
t Xh+1 J '°'" [l'* II! - -
where I, II, III, IV refer to the four submatrices described in formula (19). As end conditions, for a free-free structures, we must delete the terms underlined, for h = 0, and the terms overlined for h = P, the number of subintervals. Here too, however, a numerical integration scheme can be applied. As a matter of fact, assume that the underlined terms at h are known and Xh, xh are known too. Then, by means of the uplined terms we may proceed to h + I, and so on: at the beginning the underlined terns are effectively known, since they are zero. For
(b) The structural connections between the various elements are rather more complicated here. We must now account for interfaces Ju in the direction of increasing i, (we shall refer to it as to the axis of i0, and interfaces J22 on the axis of/2. Of course, we must also account for a coupled interface J~2. (c) Although no special difficulties exist to describe a general formulation, in order to avoid problems with subscripts, we shall refer in the future to a simplified scheme where rigidity and mass connection have the simple aspect of Fig. 11. This is done with the aim of illustrating some of the possible procedures to solve two-dimensional large structures, without being involved in complicated algebra. (d) For such a simple structure we have the equations generalizing (8), Section 1: CC,!i2)Xil,i 2 -- I(~i])Xii-i , , i2 t rOo), , i l , i 2 ----- I(il' I(ii,i2) ~ r I(il+l,t2) ~ r J~'l,il,i 2 -- , B B , , .~Xil.i 2 - - I BC, I (i ,i
S2.i,..i2 = l ~ . ~
D=.i,.i2 = lg'~2Xi,.~2 ,
[ XhJ we take [I0 Oi]where I, i are the unti
of dimension equal to those of X, x respectively. The value of ~o is found from the condition that the sum of the overlined terms is zero as h = P.
ZXil+l,i 2
(20)
Xi,.i2 - l(~i.~ Xit.i2-i
~(~i2+,)~, /Xil,i2+l
-- IBC, 2
so the general equation of equilibrium reads: i, -- 0 . . . . . N, i2 = 0,...,N2'
(S, + D, + S2+ D2)iti24- Lili2Xili2= 0
2. T W O D I M E N S I O N A L S T R U C T U R E S
(21)
2.1 General Considerations (a) By adjoining (and structurally connecting) several one-dimensional structures as those of Fig. 1, one obtains a two-dimensional large space structure, Fig. I0. A single bay is now defined by the two subscripts i,; (i, = 1. . . . . N0 and/2; (i2 = 1. . . . N2).
, ~ /
~ ii
Fig. 10. Two dimensional structure.
t2÷l
'i2
i2-~ Fig. 11. Simplified mass and rigidity connections.
ii,l,i2
P. SANTINIet al.
460 The boundary conditions are written simply: Sial2=0 i2=0 . . . . . N2 D," N1 i2 = 0 i2 = 0 . . . . . N2 S21t o'= 0 i l = 0 . . . . N D2".i,'.N2 = 0 il = 0 . . . . . N,.
(22)
The corresponding terms should be deleted in eqns (21). 2.2 Separation of variables (a) As already said, most of the techniques valid for partial differential eqns can be applied to two-dimensional problems of large structures. As an example consider eqns (20)--(21) when all matrices are independent of il and of 6, splitting also the L and 1 matrices into the respective mass and stiffness components, so as to write:
The pertinent boundary conditions to eqn (26) state that quantities containing kcc, kca, mcc, mac are missing for i2 = 0 and quantities containing kaa, kac, maa, mac are missing for i2 = N> From such a set of matrix algebraic equations, we can extract frequencies and modes. 2.3 Finite elements technique (i) A more powerful approach is obtained through the use of finite elements technique, as an extension of the approach described in Section 1.4. Let us consider again the very simple scheme of Fig. 11; now we have the potential energy: 1 Nl
~_ b(il,i2)~ ~r ~, BB ]-Otil,i2 NI N2
{[kcc., + kaa.i + K] - to2[(mcc.l + maa.,
- E ,, ~, ,2 X , , _ , . , 2 . r k ~ X , i . ~
+ M)]}Xi,~- [kca.1 + ~o2mca.,]Xil l.i2 -- [kac. I + o)2macA]Xi~+,, i~+ {[kcc2 + kaa.2]
1 NI
(23)
o
- [kac.2 + to2mac.2]Xi,, i2+, = O.
0 N2
b(ili2) - ~. ,, ~, ~ ..~,,.,2_~.r^ac2X,,., 2
- o22(mcc2 + maa.2l}Xi,12 -- [kca.2 + 022mca.2]Xi,.,2 ,
I
(28)
A similar expression holds for the kinetic energy. Now we set:
Now let us introduce the matrix W~l of the modes obtained when only the first three terms of eqn (23) are taken into consideration. The matrix W~, is consistent with the equation:
il
+XN,
i2 '
{[kcc., + kaa.i + K] - f~2[mcc.i + maa.l
+ M]} IVi, - [kca.1 + II2mca.l] Wil-,
/*42
°l¢'= ~ ~.o ,l ~.o ~ X, li2r(Kili2 + k~'2'
NI]N2
(24)
-- [ k B c , I + f l 2 m a C , l] W/l+l = 0
where ~ is the diagonal matrix containing all the frequencies of the one dimensional structure located along the axis of il, so that:
i, 6 + XNIN2 NI N2"
Thus, by performing the operations indicated under (27) we obtain the final result:
NI
~=~Z~Z
i, W/1.T[(mcc.1 + mao., + M)W/, + mac.i W/I l 1
+mcB., W/,+,] = I.
(25)
where: 1
Thus, by introducing (24) into (23), multiplying by W~lr and taking into account (25) we obtain the matrix difference equation:
Z T = ~ [Xoo,T "~XNIO,T ; XON2,T ; X N I,N2,T],
"
A 6 ~ - B6~ i - C6~+t = 0
(26)
Roooo; Roo N,o; RoooN2; Roo NtN2 RoolN,o.r; RN,o N,oi RN,o ON2; RN,o N,N2
. RoolNiN2.'r; RNIO.NIN2,T; RON2.NIN2,T
where
-] --
RNIN2.NIN2J
(29) and a similar expression is obtained for the mass matrix. The expressions of the submatrices appearing in eqn (28) are left to the reader. Here, too, successive procedures of matrix assembling will provide the final result.
N1
A = (fl 2 - ~2I) + ~.. il W~,,r[kcc,2 + kaB.2 l
- ~o2(mcc.~ + maaa)l IV/, NI
(27) 1 NI
C = Y~ ,, ~l.r(ka<2 + ~O%aca) ~ , I
3. NUMERIC.M.EXAMPLES 3.1 Simple structure (1) Most of the techniques referred to in the foregoing pages were tested and compared to each other with
461
Dynamic response of large space structures reference to the simple example of Fig. AI (Appendix 1). This was done because mass and rigidity matrices are here particularly simple. All the relevant programs can be easily adapted to the most complicated cases by simply changing such matrices, which appear as a subroutine in the programs themselves. (2) The results of the application of the finite elements technique to a 60 bays structure with Ko = 0, a = 45 °, (see Appendix 1) is shown in Figs. 12-17. Here the whole structure is divided into a number of elements, each consisting of a certain number of bays: both numbers are written in the first row of Table 1; note that that the product is constant and equal to 60. Comparison is done with the exact solution, eqn (A15): obviously, the number of available frequencies is increasing with the decreasing of the number of bays entering in each element; for each of them the matrices of eqns (18), (19): must be constructed. Table 1. Frequencies of simple structures (O Bays, [] Elements) Exact
®xm ®xr ®x® ®×N 0.0371 0.0755 0.113 0.148 0.186 0.224 1.4073
0.0375 0.0774 0.122 0.162 0.220 0.243 1.4037
0.0380 0.0816 0.132 0.172 --1.4029
0.0398 ----1.379
O Bays, [] Elements. By considering the first six modes, we note the variation in accuracy with the number of elements in agreement with the increase in frequency with the decreasing of the number of elements (first number on the top row of Table 1). Figures 12-15 show the comparison of the modes for the two cases 4 x 15, 6 x 10; the difference is small for the first mode and becomes significant for the other three modes. Note that the exact solution in the first mode is
practically coincident with the 6 x 10 solution. Figure 16 gives the 5th and the 6th mode for the 6 x 10 case (for 4 x 15 there are no such modes). Now we go to consider the last row of Table 1, there we have a frequency whose accuracy shows a trend opposite to that of the other frequencies. This is a local mode, symmetric with respect to the axis A - A of Fig. A1, whereas the others were antisymmetric. It is interesting to note that there are other frequencies very near to the one discussed here, whose modes present a decreasing number of nodes with increasing frequency. In other words, starting from low frequencies, we have firstly a group of modes with an increasing number of nodes: then we have a "black-out" range, which is narrower and narrower as the number of elements increases: then we have the high frequencies group, corresponding to local modes, with a decreasing number of nodes. The highest frequency mode has a constant, antisymmetric movement. This trend is shown in Fig. 15, which refer to an antisymmetric movement, whereas Fig. 17 refers to a symmetric movement. It is the authors' opinion that this is a characteristic behaviour of the "large structures". Large structure are those where we have two classes of modes: a low frequency group, similar to that of "non-great" structure, and a high frequency group, especially sensitive to local movements of the individual elements. (3) Some results referring to the variable-length bays are shown in Fig. 18. Here we have a total number of 10 bays, and the case 5 x 2 is considered. The structure is the same of Fig. AI, but the lengths of the bays of each element are varying with the following law: di= 1~+ k(l -/3)(1 - 4iv(1 - iv)), where:
1
k=
I +~(I - v2)" N
The constant k is choose in such a way that Xi di= N, I
[]
®
[]
[]
®
[]
®
[]
[]
r I
r
r
i I
~)
I ~=,0.0375
/
E
I / I I I I
I
I I
I I
I I
I 1 I
i
I --
Fig. 12. Comparison of modes for simple structure. 1° mode.
--~
4x15 6~10
®
P. SANTINIet al.
462
[] ®
[] II
(~
I
®
I
1 I
S
816
jr-.,-
):0.0774
-~
______
4,45 6,,10
Fig. 13.2 o Mode.
[] (D
/
,I~
[4-] ®
,~
,[~
[] ®
,
//'~%'%.
/ "Z
%'%%
~
4,15 6x10
Fig. 14. 3° Mode.
I
i
®
'
®
I
/
, I
162
I I I I i
I_ _ ~ Fig. 15.4o Mode.
~
4,,15 6.10
Dynamic response of large space structures
463
[]
Ai /!~I
/ I X . - -'>,,
~,
X/
/
6=10
....
",,. !/
__/._\
~.~
0.243
I t
ELEMENTS 5°MODE
6 ° MODE
Fig. 16. 5% 6° Mode.
[]
I[]
(B
' L [
;[] ®
[]
j[]
®
I
i[] ®
)
I
--X-
.__~
V//1.4037
/,/
',
/
",, / ,z
',
! ~
4xt5 6~I0
Fig.
17. High frequency mode.
i
,,.
t.5.
tO.
.5 .... IJ
I
.....
Fig.
lg. Frequencies of variable
(¢ngth bays structure,
(I ~ ii
!! ~I
t60
L3
464
P. SANTIN1et al.
i.e. in such a way that the lengths of the structures be independent of i. (4) Two dimensional structures (Fig. 19) were also studied by means again of the finite elements technique. Since the inversion program for the general two-dimensional case (corresponding to the numerical integration of (21) is not yet ready, the case of one single element 4 gridpoints) was only considered. The frequencies are collected in Table 2 for several aspect ratio of the rectangular grid. The results are being analyzed, in order to see whether they agree with other results. Anyway it is evident that the well known "Pitagoric rule"; ~o2= ~o~2+to22, valid for two dimensional membranes, applies here too.
(4) The modes of the 5-bays structures are shown in Figs. 22(a-c), where also the conventions for the sign of horizontal displacements u., u", and of vertical displacements v', v", are shown. 3.3 Three-dimensional bay The third example refers to a typical three-dimensional bay, very similar to those really employed in large space structure: a further step will be made by considering stiff-jointed trusses instead of pin-jointed trusses. Some of the results provided by the program do not agree with NASTRAN solution, and investigations are being conducted to understand the reason for this. Probably such reason lies in some instability associated with the
Table 2. Two dimensional structure frequencies Qo
1
20 40 80 100
0.0245 0.0245 0.0245 0.0245
2
3
0.114 0.0565 0.0305 0.0245
4
0.1167 0.0625 0.039 0.0345
5
1.9155 1.949 1.967 1.9705
3.2 Pin-jointed trusses (1) Further results refer to the structure represented in the Fig. 20, i.e. a pin-jointed truss: here the element is practically the same of Fig. AI, with the only exception that also horizontal displacements are now allowed. Therefore, we have now a 4 × 4 stiffness and a 4 x 4 mass matrix. (3) The results of Fig. 20(a, b), refer to the three-bays structure; essentially the aim was to see the influence of the variation of mass and stiffness on the truss members. Figure 20(a) provides the value of the first 16 frequencies (zero-frequencies are not represented) vs it, the ratio of the concentrated mass on each of the nodes to the mass of vertical members; for the other members, the mass per unit length is the same as for vertical bars. Figure 20(b) refers to unit values of the rigidity R of the bars. Fig. 20(a) to R = 0,5. Figures 21(a), 21(b) provides the same results for a variable length structures.
1.9285 1.959 1.973 1.9795
6 I.%15 1.973 1.978 1.980
1.9715 1.983 1.989 1.9905
4. CONCLUSIONS There are several procedures that can be used in order to solve dynamic problems of large space structures, instead of using the big general purposes programs which are currently employed. For periodic structure, a constant coefficients matrix difference equation can be established: numerical difficulties associated with such a procedure lie in the need of obtaining all the roots and all the modes of the characteristic algebraic equation. For the reason, a kind of "numerical integration scheme" can be developed; numerical difficulties are here associated
1
• ~ays f--.
ooOo.-.. Qo_( oob..
P=lOO
numerical integration scheme discussed in Section 1. Figure 24 describes te first mode for the 3 bays structure (numbering of the various degrees of freedom in shown in Fig. 23), for symmetric modes with respect to the vertical axis. Here all the members have unit rigidity, a unit mass is attached at every node; the bay length is l, the angle 213 is 600.
I
P.lO0 bays
7
-L Fig. 19. Two dimensional grid structure.
Dynamic response of large space structures
465
2.0 v
v
iS.
tO.
05. o.) 0-
lb
O
21o
1.5
Fig. 20a. Pin-jointed truss-frequencies (R = 0.5).
2°t 1.5
i
1.0.
0.5.
0
o.'5
0
fo
oJ
720
""
Fig. 20b. Pin-jointed truss-frequencies (R = 1).
\
2.0
1.5.
tO.
05 o)
0
6
6s
lb
Fig. 21a. Variable length bays structure (R = 0.5).
2~o
w
466
P. S^NTINI et al.
\
2£
t
1..5. tO. 0.5.
\ to
0
r
lb
0.5
Fig. 2lb. Variable length bays structure (R = 1).
o3=0,3209]
~~ - - ~
u:-u"
V,'- V"
Fig. 22a. Five-bays pin jointed truss-l ° mode.
UI_U
Iw
v,'-v"
////
//
\\
\\ /
/
/
\
Fig. 22b. Five bays pin-jointed truss-2° mode.
\
\
\
Dynamic response of large space structures
[~o=
] U~-U"
V Fig. 22c. Five-bay pin-jointed truss-3 ° mode.
2
3 Fig. 23. Three dimensional bay.
467
468
P. SANTINIet al.
4
U"
/
=0.4592
..\~=V.m.-
/
\ \ \ v,~,x,
Fig. 24. Three dimensional structurel-I ° mode.
with some instability which depends, in turn, from the fact that, for frequencies different from the eigenfrequencies of the system, the response is amplified in either directions. At this point there is more confidence in finite elements approach, which, at least for not too high frequencies, yields a satisfactory method of solution; furthermore, it can be easily extended to non uniform bays structures, and to two dimensional structures as well. Acknowledgements--The Authors are greatly indebted to Mr. Alberico Blasi for his careful preparation of the drawings of the present report.
NOMENCLATURE
Section 1. B<° /. the bay D reaction from right F external force on a bay
G I J K Kj L M Mj N R S .9" X d f k 1 m s x w
mass matrix in finite elements unit matrix interface between bays stiffness matrix interface stiffness matrix combination of mass and stiffness mass matrix interface mass matrix number of bays stiffness matrix in finite elements reaction from left kinetic energy potential energy vector of degrees of freedom of the bay reaction on an interface from right external load stiffness matrix combination of stiffness and mass-matrix for a bay: eqn (5), P.I. mass matrix reaction on an interface from the left vector of degrees of freedom of a bay frequency
Dynamic response of large space structures
469
Note that, in principle, the coefficients of the end interfaces may be different from the others.
Subscripts i order of interface A, B, C position of structural connections T transpose of a matrix
(2) Let: XO) = X0 Ai,
Superscripts i order of the bay
X,. = Xo,~~
(AI.3)
Section 2. A, B, C coeffÉcients of eqn 16 D reaction on a bay K rigidity matrix M mass-matrix N number of bays R reduced stiffness matrix S reaction on a bay W modal matrix potential energy
where the vectors Xo, Xo, and the parameter A must be determined from the equations:
Subscripts and superscripts 1, 2 axis direction
where:
IAAXo -- (IAC + IAB]'~)Xo = O,
(L - Icd)t - IBca)Xo -
(lea + IBAa )X0 =
0.
(AI.4)
Hence, the equations determining 0, 3(o reads (see also [1], [2]): Det[P - O,~ - O ; a ] ---0, ( P - Q,~ - Qrl,OXo = 0,
P = L - ICAI-AIAlac- laAI-A~IAB, O = IBAI-AIAIAC+ IOC.
Appendix 1 Same symbols of Section 1--Furthermore D matrix (A.I.8) I number of imaginary roots N, (number of roots)/2 P matrices, eqn (AI.6) R number of real roots Ko axial stiffness z eqns (AI.I1)(AI.12) X eigenvalue # eqns (AI.15) (AI.16) a Fig. A1 (angle of diagonal bars)
(AI.5)
(AI.6)
As well known, if Nj is the number of degrees of freedom of the interface, the 2Nj roots of the first of (A1.5) can be classified into NI pairs of reciprocal values. (3) As a practical procedure we give tentative values to to, appearing in eqns (4), P.I., and determine the corresponding values of ~ from the first of eqns (A1.5). In general, there will be R real roots ,~k and I = N j - R imaginary roots A~,+j,~ with the corresponding conjugate (A~,-jA~). By denoting the pertinent eigenvectors by Ak, A'h +-jA~ respectively, the general solution of our matrix difference system (AI.1) can be written: R
Xi = ~ k Ak( Ck'~ki + dk,~k-i)
Appendix 2 ~ } eqns (A2.1)
+~
I C'h[Ah cos (i~p~)- A~ sin (i~ph)]
/
t.
-C'~[Ahsin(ieh)+ A'hcos(i~h)]},
(AI.7)
(~oh = tan -~ ,~;,] X
i) --
-1
- IAA(IABXi_ + IAcXi),
where the constants Ck, dk, C'h, C~ for a total number of 2N1, must be determined through the 2Nt scalar conditions corresponding to eqn (AI.2). This is a very simple, although timeconsuming exercise, and is left to the reader. Anyway we have the equation: REFERENCES
I. P. H. Denke, G. R. Eide and J. Pickard, Matrix differences equation analysis of vibrating periodic structures. AIAA J. 13, 160-166 (1975). 2. L. Meirovitch and C. Engels Remi, Response of periodic structures by the Z-transform method. AIAA J. 15, 167-174 (1977). 3. I. U. Ojalvo and M. Newman, Vibration modes of large structures by an automatic matrix-reduction method. AIAA J. 8, 1234-1239 (1970). APPENDIX I
(I) Let us write eqns (6) when all the coefficients are independent of i:
{
D(to)C = 0
(A1.8)
where the vector C is that of the unknowns ck, dk, C~, C~. By changing to, the value of the determinant of D is computed, until it reaches the zero value; at this point eqns (A1.8) provides the constants appearing in (AI.7) and the problem is completely solved. (4) Consider the example of Fig. AI. Here we have a periodic structure; each bay consists of two diagonal bars of unit axial stiffness, and no mass: the verticals have axial stiffness Ko and no mass. Horizontal displacements of the nodes are prevented: a unit mass is applied at every node, including the first and last of them. Equation (A1.5) are written Now:
= 0; i = 1. . . . . N LXI - laax "+t) (AI.1) i=l ..... N-1.
IAAxO)- IABXi_t-- IAcXi
- lcsXi_l
- ICA xO) +
--[ecXi+l'~-O;
The end conditions read:
LoXo - lna.oxm - lsc.oX1 = 0 LNXN - ICa.NXN-1 -- ICA,Nxe~) = 0
(,, or
(A1.2)
(Ko + 2 cos2 a - to2)= +- ( (,~ + l ) cos2 a + Ko).
(Al.10)
470
P. SANTINI
et al.
Here we have two pairs of reciprocal roots. The first pair is given by the + sign in eqn (AI.10):
RON
=
N
=2-
o'fri
i
o) 2
~+
O'i'ri- 1
COS Ot
=2cos h
(AI.11)
corresponding to modes antisymmetric w.r. to A A , Fig. (AI.1). The second pair corresponds to symmetric modes:
~o-~-2(1+co-~)
= 2cos ~2
~A112)
For each of the two cases, we have the general solution X~ = uiXo, with: ui = sin[zdN - i)1 ---sin[zk i1; k = 1, 2,
k•)A +
k~)c . (A2.4)
L [L:i-''- ~,_,] L:¢'J /J
o: U,oNNj=
Similar expression also for mass matrices. To give an idea of the application of the finite-elements method, let us come back to the simple structure of Fig. AI.I. Practically, the vertical bars are the interfaces, and the diagonal bars are the bays; the horizontal bars are unessential. Since all the displacements belong to the interface, the vector x (°, and the corresponding matrices, are missing. Furthermore: kAA,C = ( I -- J)go; kAA,D = kAA,S = J COS2 a~;
(A1.13)
kac = J cos 2 a,
where the + sign correspond to symmetric motions about the vertical axis B B of the structure, and the - sign to antisymmetric modes. So, depending on symmetry of motions, we have four possiblities. Application of the end condition provides the equation for the frequencies:
(A2.5)
where: l=[1001];J=[
01 -10].
(A2.6)
m
(A2.7)
As far as masses are considered
~k sin ( N z D + [sin[(N - l)z~] -+sin zk] = 0; k = 1,2 (Al.14)
--
where: ~1 = 1 - 2 c o s zl,
(Al.15)
#2 = - 1 - 2 cos z2.
(A1.16)
v.
We assume that each of the verticals has a mass 1 attached at its upper end and mass 1 attached at its lower end. We have now for all the bays: Roo = RNN = A I - fl.J, RNO = RoN = a I - M,
(A2.8)
where:
Bi to!
f,^ = t,,.+v2 ~ o - o ) 2), ( l + v X6 2 + v ) .+ T2 C OS 3=K°(l+v~
2+v)
2 a,
2L~_cos2a;v=llN,
(A2.9)
1 1- v2 -
r
1 - v2
2
= T ( K o - ~ )+--T-cos a, 8 =-~-~
K°-~
~-~ c°s2 a'
(A2.1O)
Fig. A.I. Simple periodic structure. Let us consider the case of two gridpoints only; in this case, the frequencies are obtained from: APPENDIX2
P , ~ o - RONXN = O, - RoNXo + RNNXN = 0.
Let: i i o', = 1 - ~ ; ~ , = ~
(A2.1)
t
From eqns (17), (16), we obtain. roo 1 _ l ~N/ ro':2/-1 k.) I ro~ -~-Ti+,l~,~,l ~ '
L rNN_J
(A2.2)
L~TJ
~l=__,
(A2. i 1)
t
I
t
I
i
r~, r~,~]
~,,I N+ll~,l~ml K, R~J L L¢?J 2--r, 1 / " ' - ' " / k ~ tL~',-,r,j
L2":;J
(A2.3)
!
!
I
®
Fig. A.2. Vibration modes of simple structure via finite elements.
Dynamic response of large space structures Table AI.
Due to the symmetry of the problem, we have two different classes of solutions, Xo = ---XN, and the corresponding equation reads: ;t_+~;/3+_8 {det[//__. ,; A __~ ] } = 0.
(A2.12)
By means of simple algebra, we obtain the following four roots LOI2 = 0 ,
4 cos 2 a toz2=2K°4 l + v ' 12v2 COS2 ot t°32 (l + v)(l +2v)'
(A2.13)
to42= 2Ko. I 4(1 - vZ)cos2 a (1 + v)(l +2v)" They correspond to conditions of symmetry with respect to the two axes AA, BB, Fig. Al, acccording to Table Al, and to the schematic representation of Figs. A2.a, b, c, d
471
Mode
Axis AA
Axis B
Fig. A2
I 2 3 4
ANTIS SYM ANTIS SYM
SYM SYM ANTIS ANTIS
a b c d
The frequency to~ is common to all free-free structures; co3 is very similar to the first non zero frequency of a free4ree rod and is very sensibly proportional to the inverse of its length: ~o2 and ~4 are directly related to the rigidity of the bars and very little influenced by the number of the bays; in other words, they are essentially depending on the single element. It can be inferred that similar circumstance are common to all kinds of large structures. It should also be pointed out that the each of the said four frequencies is the first of the four groups corresponding to the four possible conditions of symmetry, described by the roots of eqn (AI.14). Comparison of th~ values sub (0) with the results of (Al.14) yields the following results in Table A2.
Table A2.
N=5 Approx.
Mode
Value of
Exact
2 3 4
to22- 2/(o to3 to42- 2go
1.366 0.366 1.345
1.291 0.378 1.143
N=20 Exact Approx.
N = 100 Exact Approx.
1.410 0.106 1.409
1.414 0,0220 1.414
1.380 0.114 1.314
1.407 0.0241 1.393
0094-5765/82/060455-17503.00/0 Pergamon Press Ltd.
DYNAMIC RESPONSE OF LARGE SPACE STRUCTURES PAOLO SANTINI,CLAUDIO BOTTIGLIERIand MARIO MARCHETTI University of Rome, Istituto di Tecnologia Aerospaziale, Via Eudossiana N.16, 00184 Rome, Italy (Received
21 December 1981)
Abstract--In the first part the case of a one-dimensional string of bays is firstly considered, and the equations for the most general case of mass and stiffness are derived. A procedure very similar to numerical integration is discussed. When there are many bays, the procedure becomes too heavy, and a "finite elements" approach is very useful. In the second part, the case of two dimensional structures, consisting of several adjacent strings is treated, and solved by method very similar to those applicable to partial differential equations; in particular, finite elements and separation of variables. Numerical results illustrate the procedures, for sake of comparison; graphs and Appendices complete the work.
1. ONE-DIMENSIONAL STRUCTURES
INTRODUCTION
The problem of large structures for space is receiving more and more careful attention from the scientists of all the world. This is obviously due to the increasing size of the spacecrafts, present and future, and, also, to the very little knowledge about this kind of structures. In particular, the dynamic behaviour is one of the most important information the designer needs, even starting from the calculation of natural modes and frequencies. This is a very old problem, and accurate and quite reliable programs exist for its solutions: however, when we have to analyze a structure with a tremendous number of degrees of freedom, the size of the system and the relevant computing time make the use of general purpose programs practically impossible. For this reason, since long[l-3], and on account of the "modularity" of large space structures (consisting of several elements all almost equal one to another), the concept of "periodic" structures has been introduced. However, available solutions are practically confined to the case where all the elements are equal-not "almost" equal. Therefore, it is worth while to see whether also the more general approach to not completely uniform elements could give a satisfactory response. Another important point is the fact that, when the elements are really too-many, also "exact" solutions may lead to some difficulties from a numerical standpoint. The use of "finite elements" technique is very helpful in several of such cases; it is based on the idea of treating a discrete problem as a continuous problem. And this approach is even more fruitful when we have to solve two-dimensional problems; i.e. when the elements are distributed on a surface; in this case the resulting matrix-difference equation of the problem is very similar to a partial differential equation; so the relevant techniques, with the due changes, can be applied: finite elements-as said-, separation of variables (when applicable) singularities (not included in this paper) may give a substantial support to the analysis of large space structures. This attempt is illustrated in the following pages.
?Please refer to the list of symbols at the end of the paper.
1. G e n e r a l c o n s i d e r a t i o n s
(a) Let us consider a structural element, Fig. l(a), with inherent mass and flexibility (to be defined more clearly later) and a "string" of such elements, not necessarily all equal, Fig. l(b). Anyway we assume that some strong similarities exist between any two of the elements; e.g., they may have the same general shape, and the same kind of degrees of freedom, and consist of the same members; although lengths, cross sections, mass, etc., may be different. We shall denote the ith elements as the ith bay, B " ) ( i = 1, .. N ) . t (b) For necessary generality, we must also assume that two consecutive bays are connected through an "interface", such that the degrees of freedom of the elements joining upon it are the same as those of the junction, Fig. 2; the interface may have mass and flexibility. (c) At this point, we must specify the unknowns of our problem. The ith bay is connected at its "left" end to the ( i - l)th interface, J~-l, Fig. 3, so we must also introduce a 0th "interface" at the "left" end of the whole structure, and an Nth "interface" at the "right" end. It is also convenient to define separately the internal degrees of freedom of B "), and those pertaining to the adjacent interfaces. More exactly: we denote by X~ the vector of the degrees of freedom common to B (°, J~, B "+1) Fig. 4, and by x ") the remaining degrees of freedom--if any--pertaining to B "), Fig. 4. (d) As a consequence, the ridigity matrix k ") of B ") will be of the type sketched in Fig. 5. A similar arrangement holds also for the mass-matrix, m "), with the same subscripts: it should be pointed out, however that off-diagonal matrices are denoted, e.g. by ^b(i) Ac, + m~)c. Also note that the matrices Xnc,"")k~-b are not necessarily square, since two consecutive interfaces may have a different number of degrees of freedom, and that, of course: _
k A( ai )--_
bfi)
^hA.r,
• etc.
(1)
For J, on account of its definition, we have a single matrix Kj, and a corresponding mass matrix Mj, both centered on Xi. 455
456
P. SANTINIet al.
k,.,:
~ v
-¢
(N)
,-,-Li, ,
) _k~i~
-
Cll
k(i~ c,
,b kcc
B(t)
Fig. 1. One-dimensionalstructure.
Fig. 5. Rigidity matrix for the ith bay.
y
X°
x~ ) X 1
Xi_l
X (i)
Xi
XN- 1 X('N) X N
l
i
Fig. 2. Definitionof bays and interfaces.
22
i
] I
-k~
I Jo
4 i4
J i
Fig. 6. General rigidty matrix.
Fig. 3. Numbering of structural elements.
our system. For B ") we have, Fig. 6:
~
~
/ i
.
i
.
.
.
.
(1) ~
.
--
b ( i ) "V
+ k~'~x (', - k~,~X, +/~".
(3)
,v- + m ~ k ~ " ) + M~g~ + m ~ ' ) ~ "+~ m (~) CB.'Xi-t + #t~BC __,+1) /~, - k ~ x ") + K~Xi t i + l = -- k ~ X i - i
(3)
Similarly, for J~:
//
b(i+l)~(i+l) b(i+l)g - - ~ ' B A "~ - - a ' B C ~IH+ 1 + F / .
For sinusoidal behaviour, under free vibration conditions:
Fig. 4. Internal and external degrees of freedom.
By assembling the matrices of the elements and of the interfaces, we obtain, for the entire structure, the rigidity matrix of Fig. 6, where the element painted in black has the value: K, = Ks., + k ~ + k ~ ' .
(2)
In eqn (2), the last term is missing for i = N, the last but one is missing for i = 0. (e) Let us now write the equations of equilibrium for
[ k ~ - to2ma~lx ('~- (k~)a + toZm~a)Xi_l - (k~c + to2m ~)c)X~ = 0; (i = 1. . . . N) - ( k ~ + to2m ~)X~-~ - ( / ~ k + o2m~)x
") + ( K i -
~o2M,)X,
(4)
- (k"~" + to2O+,)~.+.mnA p~ --~.,ncll'"+l) + ~02m~l))Xi+l = 0; (i = 0 . . . . N) The "boundary" conditions are obtained by deleting in the second of eqn (4) terms with Xi-t, x (i) for i = 0, and
Dynamic response of large space structures terms with x " + ' , X~+I for i = N. Letting: L(i) KAA--to
2
(i) mAA
For a free-free structures the end conditions are simply
(5)
we may write (4) in the more compact form: [ l"~x"~
t")v
I")v-a.(i=l,
-- l ABZ'Xi-I -- t AC/~i -- u, V I (i) ~(i)..I. I V - - tCA., . ~ a-~iz'l i -I(i+l)V
--I(i)
-'~i+1 = 0; (i = 0 . . . . .
,Bc
N)
. •., (i+l)
l(i+l)~ tB A
ICBI,~i_I
-
(6)
N).
1.2 A numerical integration scheme [or multibays structures
(a) For the special case where all the bays and interfaces are equal, a well known [1, 2] matrix difference equation can be obtained and solved (see Appendix 1). For the general case of quasi-periodic, or non periodic structures, it is convenient to make a further distinction. The matrix w) • AA, eqn (5) can be considered as the sum of three further matrices: IAA.S,[(i)IAA.O,(i) 1")AA,C.t The first of them corresponds to the contribution of the members having one end on J~_~ (on the left of B "~) and the other end inside B"): similarly for 'AA.D. ~"~ • obviously, l ~ . c corresponds to the elements who have no connections but inside B "). It is clear therefore that: S
(i) -- i(i) -- tAA.S
U
v
x(i)
I(i) --IAB~i--I~
.)
t.)v..-a
.~o)_t.)
-- IAA,DX
-- tAC,,"xi,
•i
- 1 -~...,
[ -- u,...,
~,r a~
~r
~
1"¢ - - I
(7)
are the reactions, exerted from left and right respectively from the interfaces to the bay, Fig. 7. We can of course define similar things for the interface: • -- I(i) V i(i) sr Oi -- tCC./Xi -- tCB.,'xi-I FI -- l(i) V /(i+l) I.,q--tBB.rXi--tBA X
'So=0
-- 1(i) -- IAA
k ~ + t o 2 m ~ : 1 ~ etc.
l
457
I(i) ( i ) . • __ I l~r -- tcAX , I - - t , . . . , 1"¢ I(i+l)v • " -- N --IBC ,.'li+l, [--~, .....
(i+1)
(8)
I.DN = 0
(10)
(b) On the basis of the foregoing approach, a numerical solution scheme can be developed, as follows: (i) Assume to is given, so that all the matrices appearing in the foregoing development can be computed; assume also that we have started from the left end side, and we are now on the ith interface, where we know S~, X~; physically, the reaction on the left of ~, and the values of its degrees of freedom. From the second of eqns (9) we immediately have D~; therefore, by associating the second of eqns (8), written for i, and the first of eqns (6) written for i + I, we have: I(i+l)~(i+l)± ,BA .~ l(i+l)w(i+l)
T
I (i) V tBC"Xi+I I(i+l)V
-- I (i) V -- "BB/Xi_ I(i+l)v
!'~ L~i
(11)
From eqns (11), by a routine inversion scheme, we compute x (~+', X~+~, and from the first of (8), S;+~; so we are now in a position for the next step. (ii) For practical calculations, we give successive tentative values to to; furthermore we have So = 0, and set Xo = I (the unit matrix of order NE, the number of degrees of freedom of the interface), obtaining the final value DN for the reaction right of JN). Now, if to* denotes the true frequency, and X* the true value of Xo, we would have: D * = DNX~ = O.
(12)
Note that DN is a square matrix, whereas D* is a vector. However, from (12) we have the condition for to:
N-1. det(DN) = 0 By the word "reaction" we denote the real force as projected on the degrees of freedom of the single elements. Now eqns (6) read very simply: l(i) ¢(i).a_ , AA,C.~ r
S
(i)
+ d (i)
LjiX~ + S~ + Di = 0;
= 0;
i = I, N i = 0 . . . . N.
///al
/
~f /
Js i
(9)
from which X* is computed; subsequently, by repeating the "integration procedure", we obtain the mode. To illustrate the procedure, we can follow Fig. 8. At the beginning, we know (or arbitrarily give) Xo, and compute the terms described as I; so we can solve the set of simultaneous equation II; compute the terms III, and so on. A further illustration can be gained through a particular case. Let us consider a type of elements, for which no internal degrees of freedom are present, so that the matrices ^BA,~'")k~,)B, k ~ (and the corrisponding lmatrices) are absent; now (7) are missing and the first of eqns (9) as well. Thus, by suitable combination of eqns (8) and of the second of eqns (9), we obtaining the following "Runge Kutta" scheme: f __ (i+1) --I (i) (i+1) --1 [3(/+1 - {[lac ] [lab + Lji]}Xi + [IBc ] Si C -- fl(i+l)[l(i+l)]-l/l(i) -L T ~ /(i+I)l~.¢r I O i + l - l , c c t,ac J ~,~sT,-~JIJ-,c8 ~.'*i (14) ) |
L
~Ji-I Fig. 7. Reactions exerted on a bay.
tltalian: C = Centro: S = sinistra (left); D = destra (right).
(13)
+SL(i+l)tl(i+~),-I~, , ~ s t nc I 1oi;
!•= 0~, . . . . .N.
to be integrated with the initial condition So = 0, Xo = I. 1.3. "Finite elements" technique (a) The above described formulation gives exact solu-
etal.
458
P. SANTINI Xo
\II/ computing X
~:~lvlng slmultanllous,
equations
Fig. 8. Numerical integration scheme.
tions. In other words, in the foregoing pages all what has been done can be thought of as an indirect way of considering the sparseness of the matrix for the complete structure, and an attempt the reduce it to a simple element approach. In practical analyis, and when there are many bays (and interfaces) this procedure may prove to be very heavy. For this purpose it may be convenient to try an approach where approximate formulations can give satisfactory results, such as is the case of the finite element techniques. This is a well known procedure of reducing continuous to discrete problems; in the following, we shall use it to reduce a problem with many degrees of freedom to another one with a lesser, or much lesser, number of degrees of freedom. (b) The strain-energy corresponding to the matrix of Fig. 6 can be written:
Now, in the spirit of the finite element technique, we give the approximate law:
[
x°)=xo(l-N)+XNN;
(17)
X~ = X o ( 1 - ~ ) + XN ~-; i = 0 ..... N
corresponding to a linear variation of x (°, X. between the end values defined by Xo, Xo, xN, XN, obtaining, for the two basic quadratic forms: 1
~: = ~
[XoTXorXNTxNr ] I I too PNo ~0N
| "
-- --/~
~(i)b(i)
',/~"T"A,~" ~l,
~(I) j_
T
1 i N
III
0i N
gON, T
L.. P o n , T
~(i)L-(i) V ~=d .~ T r* B A , C X i ~ l - - ~ . l ~ t - l . T ~ Ii li N --~ ~(i)lt(i) V L i d ~ T r~ A C ' ° ' t • 1 i
b(t) V" BC/Xi
(18) 1
•
I yoo
yoo, r
yo~
I F l,~'oT
- - H I ~
YNO,T
(19)
i N
-
II -
-- YON.T -- gON, T
N
or_ 1 / ~ .~(o_o) .:(i)± ~o
li N
- Go~
goo
- GoN, T --
+ 2 ~o)~(o ~ "~ T H t B A l t i
DNN.T
3- = ~ [XoTYCoTX,.,~XNTI Goo
J-l. I i N
RNN
[XoT1 ,oNr / o,::j j::j poN1
(15)
The physical meaning of each of the terms under (15) is evident. By the same token we obtain for the kinetic energy: N
II IV
I RoN,T PNO,T
X, r K , Xi
--~
i=1 ..... N
± ~. ~ 1T ~ #x t-
~o) x7 I.Ttlt BC/~M
(16)
li
+ Z ~(o~(o xk "~ T n t A C ~ t i • I i
By leaving the x°'s, X/s unconstrained, and through the application of the principle of minimum potential energy, we obtain eqns (3) and (3').
The expression of the submatrices constituting the elements of the rigidity matrix R defined by (18), and of the mass matrix G defined by (19), are given in Appendix 2. Finally, by considering several adjacent elements, Fig. 9(a), and using the well known assembly technique, Fig. 9(b), one can obtain mass and stiffness in finite element approach for the entire structure.
459
Dynamic response of large space structures
C-
1
®
:3
B~1~)
~
Jh=
B (h+U
O-
P
®
O
I'l÷!
h-1
O
Fig. 9. Finite element approach. For the generic gridpoint, h, Fig. 9(b), we have the matrix equation: to(h)
i i~lV
2~(hn.J_{R~h+1)
- (.o L I I V l .
--
(o2G~h+')}][Xn]
Xh
+
t Xh+1 J '°'" [l'* II! - -
where I, II, III, IV refer to the four submatrices described in formula (19). As end conditions, for a free-free structures, we must delete the terms underlined, for h = 0, and the terms overlined for h = P, the number of subintervals. Here too, however, a numerical integration scheme can be applied. As a matter of fact, assume that the underlined terms at h are known and Xh, xh are known too. Then, by means of the uplined terms we may proceed to h + I, and so on: at the beginning the underlined terns are effectively known, since they are zero. For
(b) The structural connections between the various elements are rather more complicated here. We must now account for interfaces Ju in the direction of increasing i, (we shall refer to it as to the axis of i0, and interfaces J22 on the axis of/2. Of course, we must also account for a coupled interface J~2. (c) Although no special difficulties exist to describe a general formulation, in order to avoid problems with subscripts, we shall refer in the future to a simplified scheme where rigidity and mass connection have the simple aspect of Fig. 11. This is done with the aim of illustrating some of the possible procedures to solve two-dimensional large structures, without being involved in complicated algebra. (d) For such a simple structure we have the equations generalizing (8), Section 1: CC,!i2)Xil,i 2 -- I(~i])Xii-i , , i2 t rOo), , i l , i 2 ----- I(il' I(ii,i2) ~ r I(il+l,t2) ~ r J~'l,il,i 2 -- , B B , , .~Xil.i 2 - - I BC, I (i ,i
S2.i,..i2 = l ~ . ~
D=.i,.i2 = lg'~2Xi,.~2 ,
[ XhJ we take [I0 Oi]where I, i are the unti
of dimension equal to those of X, x respectively. The value of ~o is found from the condition that the sum of the overlined terms is zero as h = P.
ZXil+l,i 2
(20)
Xi,.i2 - l(~i.~ Xit.i2-i
~(~i2+,)~, /Xil,i2+l
-- IBC, 2
so the general equation of equilibrium reads: i, -- 0 . . . . . N, i2 = 0,...,N2'
(S, + D, + S2+ D2)iti24- Lili2Xili2= 0
2. T W O D I M E N S I O N A L S T R U C T U R E S
(21)
2.1 General Considerations (a) By adjoining (and structurally connecting) several one-dimensional structures as those of Fig. 1, one obtains a two-dimensional large space structure, Fig. I0. A single bay is now defined by the two subscripts i,; (i, = 1. . . . . N0 and/2; (i2 = 1. . . . N2).
, ~ /
~ ii
Fig. 10. Two dimensional structure.
t2÷l
'i2
i2-~ Fig. 11. Simplified mass and rigidity connections.
ii,l,i2
P. SANTINIet al.
460 The boundary conditions are written simply: Sial2=0 i2=0 . . . . . N2 D," N1 i2 = 0 i2 = 0 . . . . . N2 S21t o'= 0 i l = 0 . . . . N D2".i,'.N2 = 0 il = 0 . . . . . N,.
(22)
The corresponding terms should be deleted in eqns (21). 2.2 Separation of variables (a) As already said, most of the techniques valid for partial differential eqns can be applied to two-dimensional problems of large structures. As an example consider eqns (20)--(21) when all matrices are independent of il and of 6, splitting also the L and 1 matrices into the respective mass and stiffness components, so as to write:
The pertinent boundary conditions to eqn (26) state that quantities containing kcc, kca, mcc, mac are missing for i2 = 0 and quantities containing kaa, kac, maa, mac are missing for i2 = N> From such a set of matrix algebraic equations, we can extract frequencies and modes. 2.3 Finite elements technique (i) A more powerful approach is obtained through the use of finite elements technique, as an extension of the approach described in Section 1.4. Let us consider again the very simple scheme of Fig. 11; now we have the potential energy: 1 Nl
~_ b(il,i2)~ ~r ~, BB ]-Otil,i2 NI N2
{[kcc., + kaa.i + K] - to2[(mcc.l + maa.,
- E ,, ~, ,2 X , , _ , . , 2 . r k ~ X , i . ~
+ M)]}Xi,~- [kca.1 + ~o2mca.,]Xil l.i2 -- [kac. I + o)2macA]Xi~+,, i~+ {[kcc2 + kaa.2]
1 NI
(23)
o
- [kac.2 + to2mac.2]Xi,, i2+, = O.
0 N2
b(ili2) - ~. ,, ~, ~ ..~,,.,2_~.r^ac2X,,., 2
- o22(mcc2 + maa.2l}Xi,12 -- [kca.2 + 022mca.2]Xi,.,2 ,
I
(28)
A similar expression holds for the kinetic energy. Now we set:
Now let us introduce the matrix W~l of the modes obtained when only the first three terms of eqn (23) are taken into consideration. The matrix W~, is consistent with the equation:
il
+XN,
i2 '
{[kcc., + kaa.i + K] - f~2[mcc.i + maa.l
+ M]} IVi, - [kca.1 + II2mca.l] Wil-,
/*42
°l¢'= ~ ~.o ,l ~.o ~ X, li2r(Kili2 + k~'2'
NI]N2
(24)
-- [ k B c , I + f l 2 m a C , l] W/l+l = 0
where ~ is the diagonal matrix containing all the frequencies of the one dimensional structure located along the axis of il, so that:
i, 6 + XNIN2 NI N2"
Thus, by performing the operations indicated under (27) we obtain the final result:
NI
~=~Z~Z
i, W/1.T[(mcc.1 + mao., + M)W/, + mac.i W/I l 1
+mcB., W/,+,] = I.
(25)
where: 1
Thus, by introducing (24) into (23), multiplying by W~lr and taking into account (25) we obtain the matrix difference equation:
Z T = ~ [Xoo,T "~XNIO,T ; XON2,T ; X N I,N2,T],
"
A 6 ~ - B6~ i - C6~+t = 0
(26)
Roooo; Roo N,o; RoooN2; Roo NtN2 RoolN,o.r; RN,o N,oi RN,o ON2; RN,o N,N2
. RoolNiN2.'r; RNIO.NIN2,T; RON2.NIN2,T
where
-] --
RNIN2.NIN2J
(29) and a similar expression is obtained for the mass matrix. The expressions of the submatrices appearing in eqn (28) are left to the reader. Here, too, successive procedures of matrix assembling will provide the final result.
N1
A = (fl 2 - ~2I) + ~.. il W~,,r[kcc,2 + kaB.2 l
- ~o2(mcc.~ + maaa)l IV/, NI
(27) 1 NI
C = Y~ ,, ~l.r(ka<2 + ~O%aca) ~ , I
3. NUMERIC.M.EXAMPLES 3.1 Simple structure (1) Most of the techniques referred to in the foregoing pages were tested and compared to each other with
461
Dynamic response of large space structures reference to the simple example of Fig. AI (Appendix 1). This was done because mass and rigidity matrices are here particularly simple. All the relevant programs can be easily adapted to the most complicated cases by simply changing such matrices, which appear as a subroutine in the programs themselves. (2) The results of the application of the finite elements technique to a 60 bays structure with Ko = 0, a = 45 °, (see Appendix 1) is shown in Figs. 12-17. Here the whole structure is divided into a number of elements, each consisting of a certain number of bays: both numbers are written in the first row of Table 1; note that that the product is constant and equal to 60. Comparison is done with the exact solution, eqn (A15): obviously, the number of available frequencies is increasing with the decreasing of the number of bays entering in each element; for each of them the matrices of eqns (18), (19): must be constructed. Table 1. Frequencies of simple structures (O Bays, [] Elements) Exact
®xm ®xr ®x® ®×N 0.0371 0.0755 0.113 0.148 0.186 0.224 1.4073
0.0375 0.0774 0.122 0.162 0.220 0.243 1.4037
0.0380 0.0816 0.132 0.172 --1.4029
0.0398 ----1.379
O Bays, [] Elements. By considering the first six modes, we note the variation in accuracy with the number of elements in agreement with the increase in frequency with the decreasing of the number of elements (first number on the top row of Table 1). Figures 12-15 show the comparison of the modes for the two cases 4 x 15, 6 x 10; the difference is small for the first mode and becomes significant for the other three modes. Note that the exact solution in the first mode is
practically coincident with the 6 x 10 solution. Figure 16 gives the 5th and the 6th mode for the 6 x 10 case (for 4 x 15 there are no such modes). Now we go to consider the last row of Table 1, there we have a frequency whose accuracy shows a trend opposite to that of the other frequencies. This is a local mode, symmetric with respect to the axis A - A of Fig. A1, whereas the others were antisymmetric. It is interesting to note that there are other frequencies very near to the one discussed here, whose modes present a decreasing number of nodes with increasing frequency. In other words, starting from low frequencies, we have firstly a group of modes with an increasing number of nodes: then we have a "black-out" range, which is narrower and narrower as the number of elements increases: then we have the high frequencies group, corresponding to local modes, with a decreasing number of nodes. The highest frequency mode has a constant, antisymmetric movement. This trend is shown in Fig. 15, which refer to an antisymmetric movement, whereas Fig. 17 refers to a symmetric movement. It is the authors' opinion that this is a characteristic behaviour of the "large structures". Large structure are those where we have two classes of modes: a low frequency group, similar to that of "non-great" structure, and a high frequency group, especially sensitive to local movements of the individual elements. (3) Some results referring to the variable-length bays are shown in Fig. 18. Here we have a total number of 10 bays, and the case 5 x 2 is considered. The structure is the same of Fig. AI, but the lengths of the bays of each element are varying with the following law: di= 1~+ k(l -/3)(1 - 4iv(1 - iv)), where:
1
k=
I +~(I - v2)" N
The constant k is choose in such a way that Xi di= N, I
[]
®
[]
[]
®
[]
®
[]
[]
r I
r
r
i I
~)
I ~=,0.0375
/
E
I / I I I I
I
I I
I I
I I
I 1 I
i
I --
Fig. 12. Comparison of modes for simple structure. 1° mode.
--~
4x15 6~10
®
P. SANTINIet al.
462
[] ®
[] II
(~
I
®
I
1 I
S
816
jr-.,-
):0.0774
-~
______
4,45 6,,10
Fig. 13.2 o Mode.
[] (D
/
,I~
[4-] ®
,~
,[~
[] ®
,
//'~%'%.
/ "Z
%'%%
~
4,15 6x10
Fig. 14. 3° Mode.
I
i
®
'
®
I
/
, I
162
I I I I i
I_ _ ~ Fig. 15.4o Mode.
~
4,,15 6.10
Dynamic response of large space structures
463
[]
Ai /!~I
/ I X . - -'>,,
~,
X/
/
6=10
....
",,. !/
__/._\
~.~
0.243
I t
ELEMENTS 5°MODE
6 ° MODE
Fig. 16. 5% 6° Mode.
[]
I[]
(B
' L [
;[] ®
[]
j[]
®
I
i[] ®
)
I
--X-
.__~
V//1.4037
/,/
',
/
",, / ,z
',
! ~
4xt5 6~I0
Fig.
17. High frequency mode.
i
,,.
t.5.
tO.
.5 .... IJ
I
.....
Fig.
lg. Frequencies of variable
(¢ngth bays structure,
(I ~ ii
!! ~I
t60
L3
464
P. SANTIN1et al.
i.e. in such a way that the lengths of the structures be independent of i. (4) Two dimensional structures (Fig. 19) were also studied by means again of the finite elements technique. Since the inversion program for the general two-dimensional case (corresponding to the numerical integration of (21) is not yet ready, the case of one single element 4 gridpoints) was only considered. The frequencies are collected in Table 2 for several aspect ratio of the rectangular grid. The results are being analyzed, in order to see whether they agree with other results. Anyway it is evident that the well known "Pitagoric rule"; ~o2= ~o~2+to22, valid for two dimensional membranes, applies here too.
(4) The modes of the 5-bays structures are shown in Figs. 22(a-c), where also the conventions for the sign of horizontal displacements u., u", and of vertical displacements v', v", are shown. 3.3 Three-dimensional bay The third example refers to a typical three-dimensional bay, very similar to those really employed in large space structure: a further step will be made by considering stiff-jointed trusses instead of pin-jointed trusses. Some of the results provided by the program do not agree with NASTRAN solution, and investigations are being conducted to understand the reason for this. Probably such reason lies in some instability associated with the
Table 2. Two dimensional structure frequencies Qo
1
20 40 80 100
0.0245 0.0245 0.0245 0.0245
2
3
0.114 0.0565 0.0305 0.0245
4
0.1167 0.0625 0.039 0.0345
5
1.9155 1.949 1.967 1.9705
3.2 Pin-jointed trusses (1) Further results refer to the structure represented in the Fig. 20, i.e. a pin-jointed truss: here the element is practically the same of Fig. AI, with the only exception that also horizontal displacements are now allowed. Therefore, we have now a 4 × 4 stiffness and a 4 x 4 mass matrix. (3) The results of Fig. 20(a, b), refer to the three-bays structure; essentially the aim was to see the influence of the variation of mass and stiffness on the truss members. Figure 20(a) provides the value of the first 16 frequencies (zero-frequencies are not represented) vs it, the ratio of the concentrated mass on each of the nodes to the mass of vertical members; for the other members, the mass per unit length is the same as for vertical bars. Figure 20(b) refers to unit values of the rigidity R of the bars. Fig. 20(a) to R = 0,5. Figures 21(a), 21(b) provides the same results for a variable length structures.
1.9285 1.959 1.973 1.9795
6 I.%15 1.973 1.978 1.980
1.9715 1.983 1.989 1.9905
4. CONCLUSIONS There are several procedures that can be used in order to solve dynamic problems of large space structures, instead of using the big general purposes programs which are currently employed. For periodic structure, a constant coefficients matrix difference equation can be established: numerical difficulties associated with such a procedure lie in the need of obtaining all the roots and all the modes of the characteristic algebraic equation. For the reason, a kind of "numerical integration scheme" can be developed; numerical difficulties are here associated
1
• ~ays f--.
ooOo.-.. Qo_( oob..
P=lOO
numerical integration scheme discussed in Section 1. Figure 24 describes te first mode for the 3 bays structure (numbering of the various degrees of freedom in shown in Fig. 23), for symmetric modes with respect to the vertical axis. Here all the members have unit rigidity, a unit mass is attached at every node; the bay length is l, the angle 213 is 600.
I
P.lO0 bays
7
-L Fig. 19. Two dimensional grid structure.
Dynamic response of large space structures
465
2.0 v
v
iS.
tO.
05. o.) 0-
lb
O
21o
1.5
Fig. 20a. Pin-jointed truss-frequencies (R = 0.5).
2°t 1.5
i
1.0.
0.5.
0
o.'5
0
fo
oJ
720
""
Fig. 20b. Pin-jointed truss-frequencies (R = 1).
\
2.0
1.5.
tO.
05 o)
0
6
6s
lb
Fig. 21a. Variable length bays structure (R = 0.5).
2~o
w
466
P. S^NTINI et al.
\
2£
t
1..5. tO. 0.5.
\ to
0
r
lb
0.5
Fig. 2lb. Variable length bays structure (R = 1).
o3=0,3209]
~~ - - ~
u:-u"
V,'- V"
Fig. 22a. Five-bays pin jointed truss-l ° mode.
UI_U
Iw
v,'-v"
////
//
\\
\\ /
/
/
\
Fig. 22b. Five bays pin-jointed truss-2° mode.
\
\
\
Dynamic response of large space structures
[~o=
] U~-U"
V Fig. 22c. Five-bay pin-jointed truss-3 ° mode.
2
3 Fig. 23. Three dimensional bay.
467
468
P. SANTINIet al.
4
U"
/
=0.4592
..\~=V.m.-
/
\ \ \ v,~,x,
Fig. 24. Three dimensional structurel-I ° mode.
with some instability which depends, in turn, from the fact that, for frequencies different from the eigenfrequencies of the system, the response is amplified in either directions. At this point there is more confidence in finite elements approach, which, at least for not too high frequencies, yields a satisfactory method of solution; furthermore, it can be easily extended to non uniform bays structures, and to two dimensional structures as well. Acknowledgements--The Authors are greatly indebted to Mr. Alberico Blasi for his careful preparation of the drawings of the present report.
NOMENCLATURE
Section 1. B<° /. the bay D reaction from right F external force on a bay
G I J K Kj L M Mj N R S .9" X d f k 1 m s x w
mass matrix in finite elements unit matrix interface between bays stiffness matrix interface stiffness matrix combination of mass and stiffness mass matrix interface mass matrix number of bays stiffness matrix in finite elements reaction from left kinetic energy potential energy vector of degrees of freedom of the bay reaction on an interface from right external load stiffness matrix combination of stiffness and mass-matrix for a bay: eqn (5), P.I. mass matrix reaction on an interface from the left vector of degrees of freedom of a bay frequency
Dynamic response of large space structures
469
Note that, in principle, the coefficients of the end interfaces may be different from the others.
Subscripts i order of interface A, B, C position of structural connections T transpose of a matrix
(2) Let: XO) = X0 Ai,
Superscripts i order of the bay
X,. = Xo,~~
(AI.3)
Section 2. A, B, C coeffÉcients of eqn 16 D reaction on a bay K rigidity matrix M mass-matrix N number of bays R reduced stiffness matrix S reaction on a bay W modal matrix potential energy
where the vectors Xo, Xo, and the parameter A must be determined from the equations:
Subscripts and superscripts 1, 2 axis direction
where:
IAAXo -- (IAC + IAB]'~)Xo = O,
(L - Icd)t - IBca)Xo -
(lea + IBAa )X0 =
0.
(AI.4)
Hence, the equations determining 0, 3(o reads (see also [1], [2]): Det[P - O,~ - O ; a ] ---0, ( P - Q,~ - Qrl,OXo = 0,
P = L - ICAI-AIAlac- laAI-A~IAB, O = IBAI-AIAIAC+ IOC.
Appendix 1 Same symbols of Section 1--Furthermore D matrix (A.I.8) I number of imaginary roots N, (number of roots)/2 P matrices, eqn (AI.6) R number of real roots Ko axial stiffness z eqns (AI.I1)(AI.12) X eigenvalue # eqns (AI.15) (AI.16) a Fig. A1 (angle of diagonal bars)
(AI.5)
(AI.6)
As well known, if Nj is the number of degrees of freedom of the interface, the 2Nj roots of the first of (A1.5) can be classified into NI pairs of reciprocal values. (3) As a practical procedure we give tentative values to to, appearing in eqns (4), P.I., and determine the corresponding values of ~ from the first of eqns (A1.5). In general, there will be R real roots ,~k and I = N j - R imaginary roots A~,+j,~ with the corresponding conjugate (A~,-jA~). By denoting the pertinent eigenvectors by Ak, A'h +-jA~ respectively, the general solution of our matrix difference system (AI.1) can be written: R
Xi = ~ k Ak( Ck'~ki + dk,~k-i)
Appendix 2 ~ } eqns (A2.1)
+~
I C'h[Ah cos (i~p~)- A~ sin (i~ph)]
/
t.
-C'~[Ahsin(ieh)+ A'hcos(i~h)]},
(AI.7)
(~oh = tan -~ ,~;,] X
i) --
-1
- IAA(IABXi_ + IAcXi),
where the constants Ck, dk, C'h, C~ for a total number of 2N1, must be determined through the 2Nt scalar conditions corresponding to eqn (AI.2). This is a very simple, although timeconsuming exercise, and is left to the reader. Anyway we have the equation: REFERENCES
I. P. H. Denke, G. R. Eide and J. Pickard, Matrix differences equation analysis of vibrating periodic structures. AIAA J. 13, 160-166 (1975). 2. L. Meirovitch and C. Engels Remi, Response of periodic structures by the Z-transform method. AIAA J. 15, 167-174 (1977). 3. I. U. Ojalvo and M. Newman, Vibration modes of large structures by an automatic matrix-reduction method. AIAA J. 8, 1234-1239 (1970). APPENDIX I
(I) Let us write eqns (6) when all the coefficients are independent of i:
{
D(to)C = 0
(A1.8)
where the vector C is that of the unknowns ck, dk, C~, C~. By changing to, the value of the determinant of D is computed, until it reaches the zero value; at this point eqns (A1.8) provides the constants appearing in (AI.7) and the problem is completely solved. (4) Consider the example of Fig. AI. Here we have a periodic structure; each bay consists of two diagonal bars of unit axial stiffness, and no mass: the verticals have axial stiffness Ko and no mass. Horizontal displacements of the nodes are prevented: a unit mass is applied at every node, including the first and last of them. Equation (A1.5) are written Now:
= 0; i = 1. . . . . N LXI - laax "+t) (AI.1) i=l ..... N-1.
IAAxO)- IABXi_t-- IAcXi
- lcsXi_l
- ICA xO) +
--[ecXi+l'~-O;
The end conditions read:
LoXo - lna.oxm - lsc.oX1 = 0 LNXN - ICa.NXN-1 -- ICA,Nxe~) = 0
(,, or
(A1.2)
(Ko + 2 cos2 a - to2)= +- ( (,~ + l ) cos2 a + Ko).
(Al.10)
470
P. SANTINI
et al.
Here we have two pairs of reciprocal roots. The first pair is given by the + sign in eqn (AI.10):
RON
=
N
=2-
o'fri
i
o) 2
~+
O'i'ri- 1
COS Ot
=2cos h
(AI.11)
corresponding to modes antisymmetric w.r. to A A , Fig. (AI.1). The second pair corresponds to symmetric modes:
~o-~-2(1+co-~)
= 2cos ~2
~A112)
For each of the two cases, we have the general solution X~ = uiXo, with: ui = sin[zdN - i)1 ---sin[zk i1; k = 1, 2,
k•)A +
k~)c . (A2.4)
L [L:i-''- ~,_,] L:¢'J /J
o: U,oNNj=
Similar expression also for mass matrices. To give an idea of the application of the finite-elements method, let us come back to the simple structure of Fig. AI.I. Practically, the vertical bars are the interfaces, and the diagonal bars are the bays; the horizontal bars are unessential. Since all the displacements belong to the interface, the vector x (°, and the corresponding matrices, are missing. Furthermore: kAA,C = ( I -- J)go; kAA,D = kAA,S = J COS2 a~;
(A1.13)
kac = J cos 2 a,
where the + sign correspond to symmetric motions about the vertical axis B B of the structure, and the - sign to antisymmetric modes. So, depending on symmetry of motions, we have four possiblities. Application of the end condition provides the equation for the frequencies:
(A2.5)
where: l=[1001];J=[
01 -10].
(A2.6)
m
(A2.7)
As far as masses are considered
~k sin ( N z D + [sin[(N - l)z~] -+sin zk] = 0; k = 1,2 (Al.14)
--
where: ~1 = 1 - 2 c o s zl,
(Al.15)
#2 = - 1 - 2 cos z2.
(A1.16)
v.
We assume that each of the verticals has a mass 1 attached at its upper end and mass 1 attached at its lower end. We have now for all the bays: Roo = RNN = A I - fl.J, RNO = RoN = a I - M,
(A2.8)
where:
Bi to!
f,^ = t,,.+v2 ~ o - o ) 2), ( l + v X6 2 + v ) .+ T2 C OS 3=K°(l+v~
2+v)
2 a,
2L~_cos2a;v=llN,
(A2.9)
1 1- v2 -
r
1 - v2
2
= T ( K o - ~ )+--T-cos a, 8 =-~-~
K°-~
~-~ c°s2 a'
(A2.1O)
Fig. A.I. Simple periodic structure. Let us consider the case of two gridpoints only; in this case, the frequencies are obtained from: APPENDIX2
P , ~ o - RONXN = O, - RoNXo + RNNXN = 0.
Let: i i o', = 1 - ~ ; ~ , = ~
(A2.1)
t
From eqns (17), (16), we obtain. roo 1 _ l ~N/ ro':2/-1 k.) I ro~ -~-Ti+,l~,~,l ~ '
L rNN_J
(A2.2)
L~TJ
~l=__,
(A2. i 1)
t
I
t
I
i
r~, r~,~]
~,,I N+ll~,l~ml K, R~J L L¢?J 2--r, 1 / " ' - ' " / k ~ tL~',-,r,j
L2":;J
(A2.3)
!
!
I
®
Fig. A.2. Vibration modes of simple structure via finite elements.
Dynamic response of large space structures Table AI.
Due to the symmetry of the problem, we have two different classes of solutions, Xo = ---XN, and the corresponding equation reads: ;t_+~;/3+_8 {det[//__. ,; A __~ ] } = 0.
(A2.12)
By means of simple algebra, we obtain the following four roots LOI2 = 0 ,
4 cos 2 a toz2=2K°4 l + v ' 12v2 COS2 ot t°32 (l + v)(l +2v)'
(A2.13)
to42= 2Ko. I 4(1 - vZ)cos2 a (1 + v)(l +2v)" They correspond to conditions of symmetry with respect to the two axes AA, BB, Fig. Al, acccording to Table Al, and to the schematic representation of Figs. A2.a, b, c, d
471
Mode
Axis AA
Axis B
Fig. A2
I 2 3 4
ANTIS SYM ANTIS SYM
SYM SYM ANTIS ANTIS
a b c d
The frequency to~ is common to all free-free structures; co3 is very similar to the first non zero frequency of a free4ree rod and is very sensibly proportional to the inverse of its length: ~o2 and ~4 are directly related to the rigidity of the bars and very little influenced by the number of the bays; in other words, they are essentially depending on the single element. It can be inferred that similar circumstance are common to all kinds of large structures. It should also be pointed out that the each of the said four frequencies is the first of the four groups corresponding to the four possible conditions of symmetry, described by the roots of eqn (AI.14). Comparison of th~ values sub (0) with the results of (Al.14) yields the following results in Table A2.
Table A2.
N=5 Approx.
Mode
Value of
Exact
2 3 4
to22- 2/(o to3 to42- 2go
1.366 0.366 1.345
1.291 0.378 1.143
N=20 Exact Approx.
N = 100 Exact Approx.
1.410 0.106 1.409
1.414 0,0220 1.414
1.380 0.114 1.314
1.407 0.0241 1.393