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Natural frequencies and normal modes of a four plate structure
Journal of Sound and IZibration (1973) 28(2), 259-275
NATURAL FREQUENCIES AND NORMAL MODES OF A FOUR PLATE STRUCTURE A. L. ABRAHAMSONt British Aircraft CorporationLimited, lYeybridge, Surrey KT13 0SF, England
(Received 18 October 1972, and in revisedform 7 February 1973) A simple model of a four plate structure is assumed and joint continuity conditions enable solution of the differential equations in terms of a single transcendental equation, involving non-dimensional parameters, for each of the four symmetry cases. The bounds within which solutions must fall are defined in terms of the natural frequencies of the composite plates with fully-fixed or simple supports at joints. Solutions are given for the case of plates with zero aspect ratio.
1. INTRODUCTION It is of great importance to the structural designer to have at his disposal a rapid means of estimating natural frequencies and normal modes of built-up structures [1, 2]. Most previous work, aimed at the production of data sheets, has considered elementary structures. For example, Warburt0n [3] treated the vibration of thin plates with various combinations of edge conditions. Recently [4, 5], design data sheets for obtaining the natural frequencies of skin-stringer 'arrays and singly curved plates, or sandwich panels, have been compiled. This paper is concerned with a built-up, four plate structure (Figure 1), and attempts to develop a design procedure for it. Z
T w~l t
I
I1'
/J-
"b
-
"
Y
-'
Figure 1. Model of a four plate structure. The following nomenclature and simplifications are applied: (1) plate in x-y plane at z = d]2 - p l a t e in x - y plane at z = -d/2; these two plates will be known as Type-A plates; (2) plate in z-y plane at x = a/2 - plate in z-y plane at x = -a/2; these two plates will be known as Type-B plates; t Now with Wyle Laboratories, 3200 Magruder Boulevard, Hampton, Virginia 23366, U.S.A. 259
260
A.L. ABRAHAMSON
(3) all plates have simple supports at y = • (4) only rotation (that is no translation) is allowed for plate joints; (5) the angles of plate joints are assumed to remain equal to rc[2 radians throughout the vibratory motion; (6) deflections are assumed to be small; thus simple bending theory is applicable. Although the assumptions appear restrictive the model has a wide range of applications from skin-rib-spar structures, to fairings, pods, and rectangular ducts. 2. PARTITIONING OF THE PROBLEM The classical differential equation of motion for the transverse displacement W o f a plate is pt V' W + - ~ i f ' = 0, (1) where p is the mass density of the plate material, t is the thickness of the plate and D is the flexural rigidity of the plate. Also Et a D = 12(1 - v2) ' (2) where E is the Young's modulus of the plate material and v is its Poisson's ratio. Dots indicate differentiation with respect to time and V4 = Vz V2 where V2 is the Laplacian operator. In Cartesian coordinates V4 is given by V4
04
04
+ ~
+
2 ~
.
(3)
The solution of equation (1) for the rectangular plates A (Figure l) with simple supports at y = • and identical boundary conditions at x --- a]2 and x = -a]2 may be separated into two orthogonal displacement functions with opposite symmetries about the line x = 0. The same type of conditions also apply to plates B. Thus there are four independent sets of displacement functions for the four plate structure shown in Figure I : (a) symmetric displacement function on Type A and B plates WA,(x,y) = - W A . ( x , y ) =sW~,
(4)
W.,(z,y) = - W..(z,y) = s W . ;
(5)
(b) symmetric displacement function on Type A plates, antisymmetric on Type B plates
Wa,(x,y) = Wa2(x,y) -- siva,
(6)
w~,(z,y) = - w . ~ ( z , y ) = " w ~ ;
(7)
(c) antisymmetric displacement function on Type A plates, symmetric on Type B plates wA,(x, y) = - w ~ ( x , y) -- ^ w~,
(8)
Wn,(z,y) = Wn2(z,y)= s w , ;
(9)
(d) antisymmetric displacement function Type A and B plates W . , ( x , y ) = w . . ( x , y ) = ^WA,
(lO)
W.,(z,y) = vG.(z,y) = ^ w . ;
(ii)
MODESOFA FOURPLATESTRUCTURE
261
where
sWA=Ao(COs~tX+Kcoshot2a)sinm---7-sinr ^WA=Ao
sine, a + Ksinhe2
sm--~-- sm cot,
(12)
sWB=Ao,(COS3,d + L cosh f12~) z\ s i nm~y - - ~ sin oot, =
smb
smoJt.
(13)
Application of displacement, slope and moment continuity conditions at plate intersections permits determination of the various constants and substitution of the four sets of displacement functions into equation (I) yields four independent sets of solutions:
(15)
(c)
CpFI(OB,6B) = --FI(pI2B,aa), CpF2(QB, ~n) = Ft(pI2B,6A), CpFt(fiB, fiB) = F2( pfIB, aA),
(d)
Cp F2(f~B, ~SB)= --F2(pOB, ,5,,),
(17)
(a) (b)
(14)
(16)
whcre
[
__r
(~ +_X_~)l/2]
Ft(f2,3)= (-Q-6)t/Ztan ('Q ~
+(O+6)t/2tanh
_ ~),,2
F2(Q, 6)= (Q--6)I/2cot (IQ ~
(~Q+fi)t/2c~
-
-
+0a/2]
2
,
(18)
J'
(19)
(pA t.,,/"" ' 6A--
(21)
\o.) 38 -- ( ' ~ ) 2 and p has been defined as
(23)
thefrequency compatibilityconstantfor plates A and B, (24)
17= O-B _-d--.i \,B ~
tB D,,]
,
(25)
262
A.L. ABRAHAMSON
and C as the moment compatibility constant for plates A and B, C = Dad. DB a
(26)
A list of expressions for the displacement function constants in terms of the above parameters is given in Appendix I. 3. SOLUTION OF THE FOUR TRANSCENDENTAL EQUATIONS GOVERNING LATERAL PLATE VIBRATION OF A FOUR PLATE STRUCTURE The simplest method by which these equations may be solved is by applying a carefully constrained Newton-Raphson iteration. Thus, for example, consider equation (14). Rewrite this in the form CpF,(f2B, 6B) + FI(pQs, 6a) = e(f2B), (27) where the solution of equation (27) occurs when the error sum is zero: i.e.,
e(f2n) = 0.
(28)
Increments (6f2a's) may be computed by assuming that e is a linear function of f2B:
6f28 =
~e/Of2n "
(29)
There are several other factors to consider, however, for equations of the type (14)-07). (a) For any given combination of C, p, 6.4 and fib there exists an infinity o f solutions for 12B and, due to rounding errors if left unconstrained, the process may converge to any randomly selected solution. Indeed for some combinations the process may not converge at all. (b) For some combinations off2 and 6, FI or/72 may tend to -t-oo. On a digital computer, rounding errors, under these circumstances, become large and computation meaningless, although a precise analytical solution may exist. (c) Computational rounding errors result from expressions of the form (29) if the denominator tends to zero. Thus it is clear that carefully devised constraints must be placed on each iteration if consistent meaningful solutions are to be obtained. In order to determine what form the constraints must take it is necessary to examine closely the common functions FI and Fz. Since f2 is directly proportional to frequency, only those values of f2 are o f interest where 0 < f2 ~< co (in practice, attention is limited to the first few solutions since higher order structural modes are rarely excited with sufficient amplitude to be of importance). There are three specific types of parameter to determine: (a) poles of Fa and F2 (i.e., values off2 where F1 o r F 2 tend to +co); (b) zeroes of F~ and F2 (i.e., values off2 where F1 or F2 tend to zero); (c) range of F~ and F2 between all their respective limits whether poles or zeroes. When these parameters have been determined they can be applied to constrain the iterative process in order to obtain convergence to a predetermined solution.
(a) Sohttion of equation
(14): cpF~(f2a, 6n) + Fl(pf2a, 6a) = 0
Since both c and p are positive we require F~(f20,fa) and Fl(pf2a, fa) to be zero or of opposite sign for a solution to occur: 0 < FI(f2B, 6s) < +oo
when (szB),_I < ~n < (sPo),
MODES OF A FOUR PLATE STRUCTURE
263
and -oo < Fdpf2~, 6a) <~0
when (sPa),. < pI28 -<<(sza)"
(n, m positive integers) where, for plates B, the nth symmetric pole and zero are written as (sPe)~ and (szs)~, and the nth antisymmetric pole and zero as (gPe)~ and (AZe)~, respectively. There is another way in which F~(f2n,6e) and F~(pf2e,6a) may have opposite signs: - ~ < Fl(f2e, 68) < 0
when (sPe)~ < [2e < (sze),,
and 0 < F~(pf28, 6a) < +oo
when (sZa)m_~ < pf2B < (sPa)re.
Since we know precisely the bounds within which either the first or the second 0~, m)th solution must lie, it is a simple matter to iterate for a solution using the constrained NewtonRaphson technique outlined in Appendix II. (b) Solution of equation (15): CpF2(f2n, 6a) - Fa(pt2n, 6a) = 0 Proceeding as before we find that On must satisfy both the following conditions for the first (n, m)th solution: (AZB)n-I ~'~ ~'~B< (APB)n [F2(~'~B,
6B) --
ve]
and (spa),.
(sza),"
< 12B < ~ P
[Fl(pg2e, 6a) -- re]; P
and, for the second (n, m)th solution, ("PB)~ < ~,, < ( " z ~ ) ,
[F2(12e, 6s) + ve]
and (SZa)m-x (sPA),. [F~(pl2s, 6a) + re]. -
coshC~[((2a 6 a )av-+2 /6A)tt2/2] 2 ] [ ( ( 2 cosh (f2a + }~a)v2x - 9x
mrcy . • cos--~-- sin ogt. Applying the condition for simple supports, that the bending moments at x = +a]2 are zero, we find that the following condition must hold:
~,, = [20~ - 89 ~]~ + 6,, (where n = positive integer), which is identical to the condition for poles of the symmetric characteristic function F1(0,6). It may similarly be shown that in the case of a general
264
A.L. ABRAHAMSON
antisymmetric type function for plate A~ the condition for simple supports at x=-)-#/2 reduces to the condition that 0.4 = (2nn) 2 + fA, which is idcntical to the condition for poles of the antisymmetric characteristic function F2(O, 3). We thus have the useful result that "'poles of the characteristic functions F~(~2,3) and F2(~, 3) correspond to simple supports (or zero bending moment) at the A-B plate joints". Applying the condition for fully fixed supports, that slopes at x = +a]2 are zero, we find that the following condition must hold: (f2a - fa) t/2 tan
+ (f2a + 6a) t/2 tanh 2
= 0. 2
This is identical to the equation giving zeroes of the characteristic function Ft(f2,6). It may similarly be shown that in the case of a general antisymmetric type function for plate A, the condition for simple supports at x = 4-a]2 reduces to the condition tl]at -
(~r
--
-
t~A)1/2
fA) 1/2 cot (f2a 2
- (QA + fA) 1/2 coth
(QA + fA) 1/2 2
0,
which is identical to the equation giving solutions for zeroes of the antisymmetric characteristic function F2(f2, f). We thus have a further result that "zeroes of the characteristic fimctions Ft(f2,f) and F2(f2,f) correspond to fully fixed supports (or zero slope) at the A-B plate ~ints".
5. A GRAPHICAL METHOD FOR ESTIMATING SOLUTIONS OF THE FOUR TRANSCENDENTAL EQUATIONS GOVERNING LATERAL PLATE VIBRATION OF A FOUR PLATE STRUCTURE It has been shown in section 3 that the bounds within which solutions of the four transcendental equations governing lateral plate vibration of a four plate structure may occur are given by poles and zeroes o f either of two characteristic functions. The particular choice of characteristic function depends on whether the plate under consideration is undergoing vibration in a mode symmetric or antisymmetric about a line through its centre parallel to the plate joints. Further it has been shown that these poles and zeroes are related to easily definable physical concepts, when the platejoints behave as though they were simple supports or fully fixed supports respectively. Poles and zeroes are uniquely defined for any plate, by specification of 6, and are plotted vs. plate aspect ratio (R = a/b or d/b) in Figure 2. If we plot the poles and zeroes o f plates A, for a particular f = fa, along the horizontal axis and the poles and zeroes of plates B, for a particular f = fiB, along the vertical axis of a Cartesian coordinate system then the areas within which solutions may occur are immediately clear. From equations (14)-(17) we may conclude that (1) if f2,~ lies on a pole, the only possible values off2B which will give solutions are poles unless C is very large; (2) iff2B lies on a pole, the only possible values off2a which will give solutions are poles unless C is zero; (3) ift2a lies on a zero, the only possible values of OB which will give solutions are zeroes unless C is zero;
265
MODES OF A FOUR PLATE STRUCTURE
(4) if C2B lies on a zero, the only possible values of 12A which will give solutions are zeroes unless C is very large. In general, the constraints on the iteration processes define the possible solution areas. These are shown as shaded areas on Figure 3. They take the same form regardless which t
I
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I
-- - - AApZ44~.
I
~
9s ~
j
.
.
.
.
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.
,
o.o
I000
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,
,.o
Plol'e a s p e c t r a t i o (R)
.
.
.
.
,
,
,
o
{R,a.a/b,F?a'd/b}
Figure 2. Variation of poles and zeroes with plate aspect ratio. ( , symmetric displacement functions, , antisymmetric displacement functions)9 (N.B. sZo = AZo = 0 for all R.)
combination of "symmetric/antisymmetric" displacement functions is being considered on adjacent plates, due to the similarity in the constraint equations. For any particular set of plates A and B we must have a unique frequency compatibility constant (p) as defined in equations (24) and (25). 5.1. RELATION BETWEEN ~-2 AND MODE ORDER It is useful to note the relation between f2 and the order of the mode it defines. " O r d e r " will be interpreted here as meaning the number of half waves making up the displacement function. It may be seen that the following is true for a symmetric displacement function: for sz,_l < f2 < sz,, mode order = 2(n - 89
(n = positive integer);
and for an antisymmetric displacement function the mode order is given by for AZ,_t < I2 ~
266
A . L . ABRAHAMSON
Some interesting conclusions regarding m o d e order m a y be drawn from Figure 3, as follows. (i) The mode occurring at the lowest frequency is always a mode o f order " o n e " on each plate. (ii) When the value o f p becomes large or small then the mode order on one pair o f plate s will increase more slowlY than the mode order on the other pair. CI
o
r
g
e
~
C "small
C large P83
-
,
,
.
2,3]
PB2 . . . . . . .
, . . . .
'3,3 ,xx Clarge
9. . . . . t2.2:
3,2
3,~
z.=o
P~,
zA,
P~z ~ 2
~3"A~
Figure 3. Possible solution areas for {A, B} s i , $2, $3, $4 indicate solution areas for a particular p. Mode order = no. of displacement half-waves = 2(n - 89 symmetric function = 2.n antisymmetric function. Numbers in brackets indicate the (i,j)th solution, where n = i for plates A and n = j for plates B.
5.2. SOLUTIONSFOR 6,i = 6B ~- 0 To illustrate the precise f o r m w h i c h solutions assume, FORTRANc o m p u t e r programs were written to produce graphical records for a quasi-continuous range in the band 1 0 - 1 < (12a, f2B)<10 a. Each graphical record was obtained for a particular set o f (6`1,3B) values and the m o m e n t compatibility constant C was adjusted to give suitable line densities over each plot. Labels for C are attached to each line. Examples for 6`1 = 3B = 0 are given (Figures 4-7), where there are o f course four plots representing solutions o f the four transcendental equations. Solutions for other values o f 6a and 6 B are o f generally similar form within the appropriate permissible areas. 6. F U R T H E R APPLICATIONS OF T H E F O U R PLATE M O D E L 6.1. EXTENSIONOF ANALYSISTO AN INFINITE MULTI-PLATESTRUCTURE It is possible to extend the applicability o f the four plate structure to yield the principal modes o f an infinite multi-plate structure with identical bays (Figure 8). Since the structure is infinite the modal density is, o f course, also infinite, but we m a y determine the modes
MODES OF A FOUR PLATE STRUCTURE
267
tO 3
iO-I
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,
,
,
,
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, I
,
,
, , l i l t
f iO
1
i
i
illll
i iO 2
I
I f r l l l iO 3
.q,, Figure 4. Symmetric displacement function on plates A and B. diA = c~n= O; RL = RB = 0. The area enclosed by the dotted line is shown enlarged in Figure 4(a).
Figure 4(a) Part o f Figure 4 enlarged.
268
A.L. ABRAHAMSON i
10 s
i
i
i
i
J i j
i
,
*,,I
t
*
,
i , * ,
I
,
,
,
,
i i l ~
I
50
1:
0 02 50 002 50 0 02 10 2
50 IO 2
0 02
,o
i0-1
i I0 -I
,
i
f
i
f
ii
t
i
I
i
i
f
i
i i i I
~
~
i
II i
I0
ill1
i
i
10 2
i
i
i
ii 10 a
Figure 5. Symmetric displacement function on plates A, antisymmetric displacement function on plates B. diA = t~B = 0; R,t = RB = 0. The area enclosed by the dotted line is shown enlarged in Figure 5(a).
] 50
50
Figure 5(a). Part of Figure 5 enlarged.
50
I
269
MODES OF A FOUR PLATE STRUCTURE 103
f
5o O o2
5o
I
O O2 50
~0
002
2
I0z i
\ \ \
IO
OOZ
ol
iO-i
~A Figure 6. Antisymmetric displacement function on plates ,4, symmetric displacement function on plates B. ~A = ~B = 0; R,~ = R~ = 0. The area enclosed by the dotted line is shown enlarged in Figure 6(a).
.
.
.
.
5O
50
Figure 6(a). Part of Figure 6 enlarged.
~O
i
270
A. L. ABRAHAMSON 6
I0 3
I
J
I
I
I
I
50 0 02 ~0 0 02 50 J-O 0 02 i0 z 50 20 002
0-5
,o
I0 -I
I
IO
I0 z
I0 ~
Figure 7. Antisymmetric displacement function on plates A and B. 6,t = 6B = 0; R,t = Re = 0. The area enclosed by the dotted line is shown enlarged in Figure 7(a).
o O2 0
o
0 02
~
o
0O2
Figure 7(a). Part of Figure 7 enlarged.
MODES OF A FOUR PLATE STRUCTURE
271
in which each bay behaves in an identical manner (zero or ~ out of phase) with relative ease since it is only necessary to analyse one bay [6]. We will define these modes as principal modes. The previous four plate analysis is applicable with minor alterations. The moment compatibility constant, C, now becomes, 2DA d C~
Daa
while other relations remain unchanged. It is also important to note that, under our definition of principal modes, the moment equations may assume yet another form, giving additional
. -
~
~'2 ~, .42 ~ .""
Sm i pe ls u p p o r t ~ ~ ',
i
'
Figure 8. Infinitely long multi-plate structure with identical bays.
modes in which the top and bottom multisupported plates vibrate independently of each other, and there is no rib motion at all. The motion is similar to that in which each top and bottom plate section is fully fixed at its joints with other plates. 6.2. EXTENSION OF ANALYSIS TO INCORPORATE PLATES WITH UNI-DIRECTIONAL STIFFENERS It is possible to apply the analysis in a limited manner to plates with uni-directional attached or integrated stiffeners, or corrugations in a direction perpendicular to the plate joints. The manner in which application is achieved is based on the assumption that plates of this nature may be treated as though they were comprised of a set of unconnected parallel beams. Tile problem thus reduces t o a four beam structure where, of course, 6a = 6B = O. The exact solutions for this case, also, are plotted in Figures 4-7. The application is approximate since no stiffener or plate torsion characteristics are accounted for and can be considered to yield only the stiffener bending modes for m = 1 with any accuracy. Some modifications are necessary to previously defined relations. These concern definitions of"equivalent flexural rigidities and densities" for particular structural configurations. Two cases, (i) and (ii), are detailed below. (i) A plate with uni-directional stiffeners
E1 Oequlvalent = O p i a t e + - - ,/-
pA (pt),,v,i,,l,nt = (pt)pla,, 4 - - - , r
where r is the stiffener spacing, I = (moment of inertia of tee-section about tee-section neutral axis) - ( m o m e n t of inertia of plate section about plate neutral axis), A is the cross-sectional area of stiffener and other parameters are as defined in equations (2).
272
A.L. ABRAHAMSON
(ii) A corrugated web
EI Dequlval9
= --, r
pA (Pt)equtvalcn t --
r
,
where r is the corrugation repeat length, I is the moment of inertia of one corrugation repeat section and A is the cross-sectional area of one corrugation repeat section. 6.3.
EXTENSION OF ANALYSIS TO A SINGLE PLATE WITH A ROTATIONAL SPRING ALONG TWO OPPOSITE EDGES
It is often tile case that the stiffness/mass ratio of supporting members is so large that they may effectively be regarded as rotational springs (Figure 9). It is easily shown that this situation results naturally as a special case of the four plate structure analysis. z A
C .........
- -.-.-.'.-.-.'+"
======================================= :.: : : :.: :.:.:.: :.:.:+::: ::::;:; ;::::;:; ( )
-
-
-
,I x
Figure 9. Plate supported by rotational springs. The cylindrical bars are infinitely stiff in bending. The rotational spring constant is K.
The moment compatibility constant C under these circumstances becomes C=
2D aK
'
while the frequency compatibility constant, p = 12a/f2B, tends to infinity as 12o tends to zero. As examples, asymptotic values o f f2a as f2B tends to zero may be obtained from Figures 4 and 6. 6.4. EXTENSION OF ANALYSIS TO AN INFINITE MULTI-SUPPORTED PLATE WITH A ROTATIONAL SPRING AT EACH SUPPORT
In the same way that it was possible to extend the analysis of a four plate structure to yield the principal modes of an infinite multiplate structure, it is readily possible to extend the single plate case described in section 6(c). Upon applying the same definition of principal modes, the expression for the moment compatibility constant C now becomes 4D C
~
.
aK 7. CONCLUSIONS A method of deriving the natural frequencies and normal modes of a four-plate box structure has been described. Although somewhat cumbersome in derivation and solution, it provides a clear insight of the form which solutions may assume, being an "exact" approach which proceeds directly from the differential equations.
MODES OF A FOUR PLATE STRUCTURE
273
Since many flight vehicle structures are of the type discussed in section 6(b), the curves o f 6 a = 6B = 0 in Figures 4-7 yield solutions directly, and could be used in the feasibility and preliminary design phases of flight vehicle development. In the case of rectangular ducts the assumption of simple supports at y = +b/2 is particularly useful since coincidence excitation is achieved by a pressure wavelength of 2rob. The graphical method outlined in section 5 and Figure 4 permits ready derivation of mode orders and approximate frequencies for the general case. More accurate estimates may be obtained by using the curves of 6.4 = cSB= 0 as interpolation guides. A more comprehensive set o f interpolation guides is given in reference [2]. The possible application of the four plate model to a wide range ofstructural configurations has been illustrated and increases the usefulness o f the analysis. ACKNOWLEDGMENTS The work reported here has been sponsored by the Ministry of Defense Procurement Executive under Contract No. KS/I/0632/CB.43A2. The author would like to thank his employers, the British Aircraft Corporation Limited, for permission to publish this work. REFERENCES 1. B. L. CLARKSONand A. L. ABRAHAMSON1970 Conference on Current Developments in Sonic Fatigue, Institute of Sound and Vibration Research, University of Southampton, Paper N. The response of skin-rib structures to jet noise. 2. A. L. ABRAHAMSON 1972 British Aircraft Corporation Acoustics Report 364, Contract No. KS/I/0632/CB.43A2. Structural response to aero-acoustic noise. 3. G. B. WARBURTON1954 Proceedings of the h~stitute of Mechanical Enghleers 168, 371-381. The vibration of rectangular plates. 4. A. G. R. THOXtSON1972 AGARDograph No. 162. Acoustic fatigue design data Part 1. 5. A. G. R. THOMSONand R. F. LA~IBERT1972 AGARDograph No. 162. Acoustic fatigue design data Part 2. 6. C. A. MERCER and C. SEAVEY1967 Journal of Sotmd and Vibration 6, 149-162. Prediction of natural frequencies and normal modes of skin-stringer panel rows.
APPENDIX I EXPRESSIONS FOR DISPLACEMENT FUNCTION CONSTANTS
The displacement functions of equations (4)-0 1) are defined within an arbitrary amplitude. Here they are normalized by assuming. A o = I. The other parameters become (a) = d 7
I---a, sin (cq/2) + K~z2sinh (~2/2)] a [-fl~ sin (fld2) + Lfl2 sinh (f12/2)J' cos
K=
L=
(~1/2)
cosh (cq/2) ' cos (fl,/2) cosh (fl2/2);
274
n.L. ABRAHAMSON
(b) [--~l sin (~,/2)__+_+K~2 sinh (a212)] a [fl,cos(fld2) + L f l 2 c o s h ( B J 2 ) J '
-- _ d
7-K=
cos (~d2) cosh (c~2/2)'
L
sin (fl,/2) sinh (flJ2)'
(r
K=
sin (0q]2) sinh (c~J2) ' cosflJ2 cosh (fl,/2);
(d) d [~, cos (~z112)+ K~2cosh (~212)] ~' = - a [fl, co-----~(fl,12~ $ Lfl2 cosh
(flj2~J'
K=
sin (~,12) sinh (aJ2)' sin (fl112) sinh (fl2]2)
'
and ~1 = (~A - 6A) "2, or2 = (g2 ~ + 6A) '1", p, = ( ~ - 68)" 2, p~ = ( ~ + 6~) 'n.
275
MODES OF A FOUR PLATE STRUCTURE
A P P E N D I X II LOGIC FOR CONSTRAINED N-DIMENSIONAL NEWTON-RAPItSON ITERATION l Set up predetermined increment reduction facility as a series tending to zero (typically < I0 terms); set up permissible ranges of variables; estimate initial values for variables; set error sum equal to a very large number
L Test whether current values of variables lie within their
[
I
Yes
No
I I Compute error sum; compute partial derivatives of error sum with respect to each of the N variables
I
I Test whether current error sum is less than previously computed error sum .9 I
I
Yes
No
I
I
Add one to increment reduction counter I i
I
Test whether increment reduction counter is greater than number of terms in predetermined series ? I
I
Yes
No
I As a function of the increment reduction counter and the predetermined increment reduction series, reduce increment set, and compute new current values of variables from previously computed values and reduced increment set I Test whether mod(error sum) and mod(all increments) are sufficiently small to cease iteration .9 I
I
No
Yes
I Set increment reduction counter = zero; rename current values of variables as "previously computed values of variables", solve set of N linear equations to determine increments; compute current values of variables using increments and previously computed values of variables
I
Test whether maximum permissible number t2f cycles have been ] performed ? I
No (
]
I
[Indicate: no solution I
I
[ Indicate: solution obtained
NATURAL FREQUENCIES AND NORMAL MODES OF A FOUR PLATE STRUCTURE A. L. ABRAHAMSONt British Aircraft CorporationLimited, lYeybridge, Surrey KT13 0SF, England
(Received 18 October 1972, and in revisedform 7 February 1973) A simple model of a four plate structure is assumed and joint continuity conditions enable solution of the differential equations in terms of a single transcendental equation, involving non-dimensional parameters, for each of the four symmetry cases. The bounds within which solutions must fall are defined in terms of the natural frequencies of the composite plates with fully-fixed or simple supports at joints. Solutions are given for the case of plates with zero aspect ratio.
1. INTRODUCTION It is of great importance to the structural designer to have at his disposal a rapid means of estimating natural frequencies and normal modes of built-up structures [1, 2]. Most previous work, aimed at the production of data sheets, has considered elementary structures. For example, Warburt0n [3] treated the vibration of thin plates with various combinations of edge conditions. Recently [4, 5], design data sheets for obtaining the natural frequencies of skin-stringer 'arrays and singly curved plates, or sandwich panels, have been compiled. This paper is concerned with a built-up, four plate structure (Figure 1), and attempts to develop a design procedure for it. Z
T w~l t
I
I1'
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"b
-
"
Y
-'
Figure 1. Model of a four plate structure. The following nomenclature and simplifications are applied: (1) plate in x-y plane at z = d]2 - p l a t e in x - y plane at z = -d/2; these two plates will be known as Type-A plates; (2) plate in z-y plane at x = a/2 - plate in z-y plane at x = -a/2; these two plates will be known as Type-B plates; t Now with Wyle Laboratories, 3200 Magruder Boulevard, Hampton, Virginia 23366, U.S.A. 259
260
A.L. ABRAHAMSON
(3) all plates have simple supports at y = • (4) only rotation (that is no translation) is allowed for plate joints; (5) the angles of plate joints are assumed to remain equal to rc[2 radians throughout the vibratory motion; (6) deflections are assumed to be small; thus simple bending theory is applicable. Although the assumptions appear restrictive the model has a wide range of applications from skin-rib-spar structures, to fairings, pods, and rectangular ducts. 2. PARTITIONING OF THE PROBLEM The classical differential equation of motion for the transverse displacement W o f a plate is pt V' W + - ~ i f ' = 0, (1) where p is the mass density of the plate material, t is the thickness of the plate and D is the flexural rigidity of the plate. Also Et a D = 12(1 - v2) ' (2) where E is the Young's modulus of the plate material and v is its Poisson's ratio. Dots indicate differentiation with respect to time and V4 = Vz V2 where V2 is the Laplacian operator. In Cartesian coordinates V4 is given by V4
04
04
+ ~
+
2 ~
.
(3)
The solution of equation (1) for the rectangular plates A (Figure l) with simple supports at y = • and identical boundary conditions at x --- a]2 and x = -a]2 may be separated into two orthogonal displacement functions with opposite symmetries about the line x = 0. The same type of conditions also apply to plates B. Thus there are four independent sets of displacement functions for the four plate structure shown in Figure I : (a) symmetric displacement function on Type A and B plates WA,(x,y) = - W A . ( x , y ) =sW~,
(4)
W.,(z,y) = - W..(z,y) = s W . ;
(5)
(b) symmetric displacement function on Type A plates, antisymmetric on Type B plates
Wa,(x,y) = Wa2(x,y) -- siva,
(6)
w~,(z,y) = - w . ~ ( z , y ) = " w ~ ;
(7)
(c) antisymmetric displacement function on Type A plates, symmetric on Type B plates wA,(x, y) = - w ~ ( x , y) -- ^ w~,
(8)
Wn,(z,y) = Wn2(z,y)= s w , ;
(9)
(d) antisymmetric displacement function Type A and B plates W . , ( x , y ) = w . . ( x , y ) = ^WA,
(lO)
W.,(z,y) = vG.(z,y) = ^ w . ;
(ii)
MODESOFA FOURPLATESTRUCTURE
261
where
sWA=Ao(COs~tX+Kcoshot2a)sinm---7-sinr ^WA=Ao
sine, a + Ksinhe2
sm--~-- sm cot,
(12)
sWB=Ao,(COS3,d + L cosh f12~) z\ s i nm~y - - ~ sin oot, =
smb
smoJt.
(13)
Application of displacement, slope and moment continuity conditions at plate intersections permits determination of the various constants and substitution of the four sets of displacement functions into equation (I) yields four independent sets of solutions:
(15)
(c)
CpFI(OB,6B) = --FI(pI2B,aa), CpF2(QB, ~n) = Ft(pI2B,6A), CpFt(fiB, fiB) = F2( pfIB, aA),
(d)
Cp F2(f~B, ~SB)= --F2(pOB, ,5,,),
(17)
(a) (b)
(14)
(16)
whcre
[
__r
(~ +_X_~)l/2]
Ft(f2,3)= (-Q-6)t/Ztan ('Q ~
+(O+6)t/2tanh
_ ~),,2
F2(Q, 6)= (Q--6)I/2cot (IQ ~
(~Q+fi)t/2c~
-
-
+0a/2]
2
,
(18)
J'
(19)
(pA t.,,/"" ' 6A--
(21)
\o.) 38 -- ( ' ~ ) 2 and p has been defined as
(23)
thefrequency compatibilityconstantfor plates A and B, (24)
17= O-B _-d--.i \,B ~
tB D,,]
,
(25)
262
A.L. ABRAHAMSON
and C as the moment compatibility constant for plates A and B, C = Dad. DB a
(26)
A list of expressions for the displacement function constants in terms of the above parameters is given in Appendix I. 3. SOLUTION OF THE FOUR TRANSCENDENTAL EQUATIONS GOVERNING LATERAL PLATE VIBRATION OF A FOUR PLATE STRUCTURE The simplest method by which these equations may be solved is by applying a carefully constrained Newton-Raphson iteration. Thus, for example, consider equation (14). Rewrite this in the form CpF,(f2B, 6B) + FI(pQs, 6a) = e(f2B), (27) where the solution of equation (27) occurs when the error sum is zero: i.e.,
e(f2n) = 0.
(28)
Increments (6f2a's) may be computed by assuming that e is a linear function of f2B:
6f28 =
~e/Of2n "
(29)
There are several other factors to consider, however, for equations of the type (14)-07). (a) For any given combination of C, p, 6.4 and fib there exists an infinity o f solutions for 12B and, due to rounding errors if left unconstrained, the process may converge to any randomly selected solution. Indeed for some combinations the process may not converge at all. (b) For some combinations off2 and 6, FI or/72 may tend to -t-oo. On a digital computer, rounding errors, under these circumstances, become large and computation meaningless, although a precise analytical solution may exist. (c) Computational rounding errors result from expressions of the form (29) if the denominator tends to zero. Thus it is clear that carefully devised constraints must be placed on each iteration if consistent meaningful solutions are to be obtained. In order to determine what form the constraints must take it is necessary to examine closely the common functions FI and Fz. Since f2 is directly proportional to frequency, only those values of f2 are o f interest where 0 < f2 ~< co (in practice, attention is limited to the first few solutions since higher order structural modes are rarely excited with sufficient amplitude to be of importance). There are three specific types of parameter to determine: (a) poles of Fa and F2 (i.e., values off2 where F1 o r F 2 tend to +co); (b) zeroes of F~ and F2 (i.e., values off2 where F1 or F2 tend to zero); (c) range of F~ and F2 between all their respective limits whether poles or zeroes. When these parameters have been determined they can be applied to constrain the iterative process in order to obtain convergence to a predetermined solution.
(a) Sohttion of equation
(14): cpF~(f2a, 6n) + Fl(pf2a, 6a) = 0
Since both c and p are positive we require F~(f20,fa) and Fl(pf2a, fa) to be zero or of opposite sign for a solution to occur: 0 < FI(f2B, 6s) < +oo
when (szB),_I < ~n < (sPo),
MODES OF A FOUR PLATE STRUCTURE
263
and -oo < Fdpf2~, 6a) <~0
when (sPa),. < pI28 -<<(sza)"
(n, m positive integers) where, for plates B, the nth symmetric pole and zero are written as (sPe)~ and (szs)~, and the nth antisymmetric pole and zero as (gPe)~ and (AZe)~, respectively. There is another way in which F~(f2n,6e) and F~(pf2e,6a) may have opposite signs: - ~ < Fl(f2e, 68) < 0
when (sPe)~ < [2e < (sze),,
and 0 < F~(pf28, 6a) < +oo
when (sZa)m_~ < pf2B < (sPa)re.
Since we know precisely the bounds within which either the first or the second 0~, m)th solution must lie, it is a simple matter to iterate for a solution using the constrained NewtonRaphson technique outlined in Appendix II. (b) Solution of equation (15): CpF2(f2n, 6a) - Fa(pt2n, 6a) = 0 Proceeding as before we find that On must satisfy both the following conditions for the first (n, m)th solution: (AZB)n-I ~'~ ~'~B< (APB)n [F2(~'~B,
6B) --
ve]
and (spa),.
(sza),"
< 12B < ~ P
[Fl(pg2e, 6a) -- re]; P
and, for the second (n, m)th solution, ("PB)~ < ~,, < ( " z ~ ) ,
[F2(12e, 6s) + ve]
and (SZa)m-x (sPA),. [F~(pl2s, 6a) + re]. -
coshC~[((2a 6 a )av-+2 /6A)tt2/2] 2 ] [ ( ( 2 cosh (f2a + }~a)v2x - 9x
mrcy . • cos--~-- sin ogt. Applying the condition for simple supports, that the bending moments at x = +a]2 are zero, we find that the following condition must hold:
~,, = [20~ - 89 ~]~ + 6,, (where n = positive integer), which is identical to the condition for poles of the symmetric characteristic function F1(0,6). It may similarly be shown that in the case of a general
264
A.L. ABRAHAMSON
antisymmetric type function for plate A~ the condition for simple supports at x=-)-#/2 reduces to the condition that 0.4 = (2nn) 2 + fA, which is idcntical to the condition for poles of the antisymmetric characteristic function F2(O, 3). We thus have the useful result that "'poles of the characteristic functions F~(~2,3) and F2(~, 3) correspond to simple supports (or zero bending moment) at the A-B plate joints". Applying the condition for fully fixed supports, that slopes at x = +a]2 are zero, we find that the following condition must hold: (f2a - fa) t/2 tan
+ (f2a + 6a) t/2 tanh 2
= 0. 2
This is identical to the equation giving zeroes of the characteristic function Ft(f2,6). It may similarly be shown that in the case of a general antisymmetric type function for plate A, the condition for simple supports at x = 4-a]2 reduces to the condition tl]at -
(~r
--
-
t~A)1/2
fA) 1/2 cot (f2a 2
- (QA + fA) 1/2 coth
(QA + fA) 1/2 2
0,
which is identical to the equation giving solutions for zeroes of the antisymmetric characteristic function F2(f2, f). We thus have a further result that "zeroes of the characteristic fimctions Ft(f2,f) and F2(f2,f) correspond to fully fixed supports (or zero slope) at the A-B plate ~ints".
5. A GRAPHICAL METHOD FOR ESTIMATING SOLUTIONS OF THE FOUR TRANSCENDENTAL EQUATIONS GOVERNING LATERAL PLATE VIBRATION OF A FOUR PLATE STRUCTURE It has been shown in section 3 that the bounds within which solutions of the four transcendental equations governing lateral plate vibration of a four plate structure may occur are given by poles and zeroes o f either of two characteristic functions. The particular choice of characteristic function depends on whether the plate under consideration is undergoing vibration in a mode symmetric or antisymmetric about a line through its centre parallel to the plate joints. Further it has been shown that these poles and zeroes are related to easily definable physical concepts, when the platejoints behave as though they were simple supports or fully fixed supports respectively. Poles and zeroes are uniquely defined for any plate, by specification of 6, and are plotted vs. plate aspect ratio (R = a/b or d/b) in Figure 2. If we plot the poles and zeroes o f plates A, for a particular f = fa, along the horizontal axis and the poles and zeroes of plates B, for a particular f = fiB, along the vertical axis of a Cartesian coordinate system then the areas within which solutions may occur are immediately clear. From equations (14)-(17) we may conclude that (1) if f2,~ lies on a pole, the only possible values off2B which will give solutions are poles unless C is very large; (2) iff2B lies on a pole, the only possible values off2a which will give solutions are poles unless C is zero; (3) ift2a lies on a zero, the only possible values of OB which will give solutions are zeroes unless C is zero;
265
MODES OF A FOUR PLATE STRUCTURE
(4) if C2B lies on a zero, the only possible values of 12A which will give solutions are zeroes unless C is very large. In general, the constraints on the iteration processes define the possible solution areas. These are shown as shaded areas on Figure 3. They take the same form regardless which t
I
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I
-- - - AApZ44~.
I
~
9s ~
j
.
.
.
.
.
.
,
o.o
I000
I
_
,
,.o
Plol'e a s p e c t r a t i o (R)
.
.
.
.
,
,
,
o
{R,a.a/b,F?a'd/b}
Figure 2. Variation of poles and zeroes with plate aspect ratio. ( , symmetric displacement functions, , antisymmetric displacement functions)9 (N.B. sZo = AZo = 0 for all R.)
combination of "symmetric/antisymmetric" displacement functions is being considered on adjacent plates, due to the similarity in the constraint equations. For any particular set of plates A and B we must have a unique frequency compatibility constant (p) as defined in equations (24) and (25). 5.1. RELATION BETWEEN ~-2 AND MODE ORDER It is useful to note the relation between f2 and the order of the mode it defines. " O r d e r " will be interpreted here as meaning the number of half waves making up the displacement function. It may be seen that the following is true for a symmetric displacement function: for sz,_l < f2 < sz,, mode order = 2(n - 89
(n = positive integer);
and for an antisymmetric displacement function the mode order is given by for AZ,_t < I2 ~
266
A . L . ABRAHAMSON
Some interesting conclusions regarding m o d e order m a y be drawn from Figure 3, as follows. (i) The mode occurring at the lowest frequency is always a mode o f order " o n e " on each plate. (ii) When the value o f p becomes large or small then the mode order on one pair o f plate s will increase more slowlY than the mode order on the other pair. CI
o
r
g
e
~
C "small
C large P83
-
,
,
.
2,3]
PB2 . . . . . . .
, . . . .
'3,3 ,xx Clarge
9. . . . . t2.2:
3,2
3,~
z.=o
P~,
zA,
P~z ~ 2
~3"A~
Figure 3. Possible solution areas for {A, B} s i , $2, $3, $4 indicate solution areas for a particular p. Mode order = no. of displacement half-waves = 2(n - 89 symmetric function = 2.n antisymmetric function. Numbers in brackets indicate the (i,j)th solution, where n = i for plates A and n = j for plates B.
5.2. SOLUTIONSFOR 6,i = 6B ~- 0 To illustrate the precise f o r m w h i c h solutions assume, FORTRANc o m p u t e r programs were written to produce graphical records for a quasi-continuous range in the band 1 0 - 1 < (12a, f2B)<10 a. Each graphical record was obtained for a particular set o f (6`1,3B) values and the m o m e n t compatibility constant C was adjusted to give suitable line densities over each plot. Labels for C are attached to each line. Examples for 6`1 = 3B = 0 are given (Figures 4-7), where there are o f course four plots representing solutions o f the four transcendental equations. Solutions for other values o f 6a and 6 B are o f generally similar form within the appropriate permissible areas. 6. F U R T H E R APPLICATIONS OF T H E F O U R PLATE M O D E L 6.1. EXTENSIONOF ANALYSISTO AN INFINITE MULTI-PLATESTRUCTURE It is possible to extend the applicability o f the four plate structure to yield the principal modes o f an infinite multi-plate structure with identical bays (Figure 8). Since the structure is infinite the modal density is, o f course, also infinite, but we m a y determine the modes
MODES OF A FOUR PLATE STRUCTURE
267
tO 3
iO-I
, IO -I
,
,
,
,
, , ,1
, I
,
,
, , l i l t
f iO
1
i
i
illll
i iO 2
I
I f r l l l iO 3
.q,, Figure 4. Symmetric displacement function on plates A and B. diA = c~n= O; RL = RB = 0. The area enclosed by the dotted line is shown enlarged in Figure 4(a).
Figure 4(a) Part o f Figure 4 enlarged.
268
A.L. ABRAHAMSON i
10 s
i
i
i
i
J i j
i
,
*,,I
t
*
,
i , * ,
I
,
,
,
,
i i l ~
I
50
1:
0 02 50 002 50 0 02 10 2
50 IO 2
0 02
,o
i0-1
i I0 -I
,
i
f
i
f
ii
t
i
I
i
i
f
i
i i i I
~
~
i
II i
I0
ill1
i
i
10 2
i
i
i
ii 10 a
Figure 5. Symmetric displacement function on plates A, antisymmetric displacement function on plates B. diA = t~B = 0; R,t = RB = 0. The area enclosed by the dotted line is shown enlarged in Figure 5(a).
] 50
50
Figure 5(a). Part of Figure 5 enlarged.
50
I
269
MODES OF A FOUR PLATE STRUCTURE 103
f
5o O o2
5o
I
O O2 50
~0
002
2
I0z i
\ \ \
IO
OOZ
ol
iO-i
~A Figure 6. Antisymmetric displacement function on plates ,4, symmetric displacement function on plates B. ~A = ~B = 0; R,~ = R~ = 0. The area enclosed by the dotted line is shown enlarged in Figure 6(a).
.
.
.
.
5O
50
Figure 6(a). Part of Figure 6 enlarged.
~O
i
270
A. L. ABRAHAMSON 6
I0 3
I
J
I
I
I
I
50 0 02 ~0 0 02 50 J-O 0 02 i0 z 50 20 002
0-5
,o
I0 -I
I
IO
I0 z
I0 ~
Figure 7. Antisymmetric displacement function on plates A and B. 6,t = 6B = 0; R,t = Re = 0. The area enclosed by the dotted line is shown enlarged in Figure 7(a).
o O2 0
o
0 02
~
o
0O2
Figure 7(a). Part of Figure 7 enlarged.
MODES OF A FOUR PLATE STRUCTURE
271
in which each bay behaves in an identical manner (zero or ~ out of phase) with relative ease since it is only necessary to analyse one bay [6]. We will define these modes as principal modes. The previous four plate analysis is applicable with minor alterations. The moment compatibility constant, C, now becomes, 2DA d C~
Daa
while other relations remain unchanged. It is also important to note that, under our definition of principal modes, the moment equations may assume yet another form, giving additional
. -
~
~'2 ~, .42 ~ .""
Sm i pe ls u p p o r t ~ ~ ',
i
'
Figure 8. Infinitely long multi-plate structure with identical bays.
modes in which the top and bottom multisupported plates vibrate independently of each other, and there is no rib motion at all. The motion is similar to that in which each top and bottom plate section is fully fixed at its joints with other plates. 6.2. EXTENSION OF ANALYSIS TO INCORPORATE PLATES WITH UNI-DIRECTIONAL STIFFENERS It is possible to apply the analysis in a limited manner to plates with uni-directional attached or integrated stiffeners, or corrugations in a direction perpendicular to the plate joints. The manner in which application is achieved is based on the assumption that plates of this nature may be treated as though they were comprised of a set of unconnected parallel beams. Tile problem thus reduces t o a four beam structure where, of course, 6a = 6B = O. The exact solutions for this case, also, are plotted in Figures 4-7. The application is approximate since no stiffener or plate torsion characteristics are accounted for and can be considered to yield only the stiffener bending modes for m = 1 with any accuracy. Some modifications are necessary to previously defined relations. These concern definitions of"equivalent flexural rigidities and densities" for particular structural configurations. Two cases, (i) and (ii), are detailed below. (i) A plate with uni-directional stiffeners
E1 Oequlvalent = O p i a t e + - - ,/-
pA (pt),,v,i,,l,nt = (pt)pla,, 4 - - - , r
where r is the stiffener spacing, I = (moment of inertia of tee-section about tee-section neutral axis) - ( m o m e n t of inertia of plate section about plate neutral axis), A is the cross-sectional area of stiffener and other parameters are as defined in equations (2).
272
A.L. ABRAHAMSON
(ii) A corrugated web
EI Dequlval9
= --, r
pA (Pt)equtvalcn t --
r
,
where r is the corrugation repeat length, I is the moment of inertia of one corrugation repeat section and A is the cross-sectional area of one corrugation repeat section. 6.3.
EXTENSION OF ANALYSIS TO A SINGLE PLATE WITH A ROTATIONAL SPRING ALONG TWO OPPOSITE EDGES
It is often tile case that the stiffness/mass ratio of supporting members is so large that they may effectively be regarded as rotational springs (Figure 9). It is easily shown that this situation results naturally as a special case of the four plate structure analysis. z A
C .........
- -.-.-.'.-.-.'+"
======================================= :.: : : :.: :.:.:.: :.:.:+::: ::::;:; ;::::;:; ( )
-
-
-
,I x
Figure 9. Plate supported by rotational springs. The cylindrical bars are infinitely stiff in bending. The rotational spring constant is K.
The moment compatibility constant C under these circumstances becomes C=
2D aK
'
while the frequency compatibility constant, p = 12a/f2B, tends to infinity as 12o tends to zero. As examples, asymptotic values o f f2a as f2B tends to zero may be obtained from Figures 4 and 6. 6.4. EXTENSION OF ANALYSIS TO AN INFINITE MULTI-SUPPORTED PLATE WITH A ROTATIONAL SPRING AT EACH SUPPORT
In the same way that it was possible to extend the analysis of a four plate structure to yield the principal modes of an infinite multiplate structure, it is readily possible to extend the single plate case described in section 6(c). Upon applying the same definition of principal modes, the expression for the moment compatibility constant C now becomes 4D C
~
.
aK 7. CONCLUSIONS A method of deriving the natural frequencies and normal modes of a four-plate box structure has been described. Although somewhat cumbersome in derivation and solution, it provides a clear insight of the form which solutions may assume, being an "exact" approach which proceeds directly from the differential equations.
MODES OF A FOUR PLATE STRUCTURE
273
Since many flight vehicle structures are of the type discussed in section 6(b), the curves o f 6 a = 6B = 0 in Figures 4-7 yield solutions directly, and could be used in the feasibility and preliminary design phases of flight vehicle development. In the case of rectangular ducts the assumption of simple supports at y = +b/2 is particularly useful since coincidence excitation is achieved by a pressure wavelength of 2rob. The graphical method outlined in section 5 and Figure 4 permits ready derivation of mode orders and approximate frequencies for the general case. More accurate estimates may be obtained by using the curves of 6.4 = cSB= 0 as interpolation guides. A more comprehensive set o f interpolation guides is given in reference [2]. The possible application of the four plate model to a wide range ofstructural configurations has been illustrated and increases the usefulness o f the analysis. ACKNOWLEDGMENTS The work reported here has been sponsored by the Ministry of Defense Procurement Executive under Contract No. KS/I/0632/CB.43A2. The author would like to thank his employers, the British Aircraft Corporation Limited, for permission to publish this work. REFERENCES 1. B. L. CLARKSONand A. L. ABRAHAMSON1970 Conference on Current Developments in Sonic Fatigue, Institute of Sound and Vibration Research, University of Southampton, Paper N. The response of skin-rib structures to jet noise. 2. A. L. ABRAHAMSON 1972 British Aircraft Corporation Acoustics Report 364, Contract No. KS/I/0632/CB.43A2. Structural response to aero-acoustic noise. 3. G. B. WARBURTON1954 Proceedings of the h~stitute of Mechanical Enghleers 168, 371-381. The vibration of rectangular plates. 4. A. G. R. THOXtSON1972 AGARDograph No. 162. Acoustic fatigue design data Part 1. 5. A. G. R. THOMSONand R. F. LA~IBERT1972 AGARDograph No. 162. Acoustic fatigue design data Part 2. 6. C. A. MERCER and C. SEAVEY1967 Journal of Sotmd and Vibration 6, 149-162. Prediction of natural frequencies and normal modes of skin-stringer panel rows.
APPENDIX I EXPRESSIONS FOR DISPLACEMENT FUNCTION CONSTANTS
The displacement functions of equations (4)-0 1) are defined within an arbitrary amplitude. Here they are normalized by assuming. A o = I. The other parameters become (a) = d 7
I---a, sin (cq/2) + K~z2sinh (~2/2)] a [-fl~ sin (fld2) + Lfl2 sinh (f12/2)J' cos
K=
L=
(~1/2)
cosh (cq/2) ' cos (fl,/2) cosh (fl2/2);
274
n.L. ABRAHAMSON
(b) [--~l sin (~,/2)__+_+K~2 sinh (a212)] a [fl,cos(fld2) + L f l 2 c o s h ( B J 2 ) J '
-- _ d
7-K=
cos (~d2) cosh (c~2/2)'
L
sin (fl,/2) sinh (flJ2)'
(r
K=
sin (0q]2) sinh (c~J2) ' cosflJ2 cosh (fl,/2);
(d) d [~, cos (~z112)+ K~2cosh (~212)] ~' = - a [fl, co-----~(fl,12~ $ Lfl2 cosh
(flj2~J'
K=
sin (~,12) sinh (aJ2)' sin (fl112) sinh (fl2]2)
'
and ~1 = (~A - 6A) "2, or2 = (g2 ~ + 6A) '1", p, = ( ~ - 68)" 2, p~ = ( ~ + 6~) 'n.
275
MODES OF A FOUR PLATE STRUCTURE
A P P E N D I X II LOGIC FOR CONSTRAINED N-DIMENSIONAL NEWTON-RAPItSON ITERATION l Set up predetermined increment reduction facility as a series tending to zero (typically < I0 terms); set up permissible ranges of variables; estimate initial values for variables; set error sum equal to a very large number
L Test whether current values of variables lie within their
[
I
Yes
No
I I Compute error sum; compute partial derivatives of error sum with respect to each of the N variables
I
I Test whether current error sum is less than previously computed error sum .9 I
I
Yes
No
I
I
Add one to increment reduction counter I i
I
Test whether increment reduction counter is greater than number of terms in predetermined series ? I
I
Yes
No
I As a function of the increment reduction counter and the predetermined increment reduction series, reduce increment set, and compute new current values of variables from previously computed values and reduced increment set I Test whether mod(error sum) and mod(all increments) are sufficiently small to cease iteration .9 I
I
No
Yes
I Set increment reduction counter = zero; rename current values of variables as "previously computed values of variables", solve set of N linear equations to determine increments; compute current values of variables using increments and previously computed values of variables
I
Test whether maximum permissible number t2f cycles have been ] performed ? I
No (
]
I
[Indicate: no solution I
I
[ Indicate: solution obtained