A note on unimodal and bimodal optimal design of vibrating compressed columns

lnt. J. Mech. 5c£ Vol. 23, pp. 11-16 Persamon Press Ltd., 1981. Printed in Great Brittin

A NOTE ON UNIMODAL AND BIMODAL OPTIMAL DESIGN O F VIBRATING C O M P R E S S E D C O L U M N S ANTONI GAJEWSKI Instituteof Physics,Technical Universityof Cracow, ul. Podchor~'ych I, 30-084 Krak6w, Poland (Received 31 January 1980: in revisedform 26 May 1980)

Summary--The bimodaloptimizationwith respect to the frequency of transverse vibrations under axial compression is demonstrated on the simple Pragers' model. This model consists of rigid segments that are joined by elastic hingesto each other and to the supports and point masses which are situated in the inner hinges. I. INTRODUCTION Recently Prager and Prager[6] have illustrated the characteristic aspects of optimal design of columns, using a discrete model which consists of rigid segments that are joined by elastic hinges to each ot.her and to the supports. It has been shown that the "optimal design" realizing the maximal buckling load can be obtained analytically w h e n the elastic clampings are weak, but nonanalytically w h e n they are strong. Moreover, in the second case the buckling load is a double eigenvalue, associated with the symmetric and antisymmetric modes of buckling simultaneously. Necessity of the bimodal formulation of the optimization problem of clamped, compressed columns has been noticed for the first time by Olhoff and Rasmussen [5]. T h e optimality condition, derived in this case, leads to a nonlinear integro-differential eigenvalue problem and differs essentially from the investigated previously cases. The bimodal optimization appears, in an even more natural way in the cases of optimal design of arches and frames subjected to buckling or vibrating. The unimodal and bimodal optimal design of funicular arches were treated by B l a c h u t and Gajewski[1], who found the optimal shapes of circular arches for a maximal buckling pressure or for a maximal first f r e q u e n c y of vibrations under constant loading. According to the steepness of the arc, the optimal design has been calculated from the unimodal or, if it was necessary, bimodal formulation. T h e bimodal or even multimodal optimization were also presented in the paper of Masur and Mr6z[4], who derived nonstationary optimality condition, valid in nonconservative cases of loading. B t a c h u t and Gajewski [2] derived the same optimality condition on the basis of Pontryagin's maximum principle and used it to optimal design of a cantilever c o m p r e s s e d by the follower force. The aim of this paper is use of the simple Pragers' model to demonstrate (without complicated calculations) the bimodal optimization in a more general case, namely with respect to the f r e q u e n c y of transverse vibrations under axial compression.

2. EQUATIONS OF M O T I O N We shallinvestigatea small,linearvibrationsof the model, which consistsof fiverigid,masslessrods of the length I,joinedby the elastichingesto each otherand to the supports.The pointmasses are situatedin the inner hinges (Fig. I) and the system is compressed by the constantaxialforces P, actingat the ends. Taking intoaccount the notations,presentedin Fig. I,and assuming thatthe angles a~ are small (a~"~ I), we calculate: (a) potential energy of the hinges U = ~ .. bi(~ - ai-I)2. 11

(I)

12

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p

P /.DI

I

m2

"

f/"/3

~3

Flo. !. (b) w o r k of the external f o r c e

(2)

Lp = 1 PI ~ a, 2. 3'

i=0

(c) kinetic e n e r g y of the s y s t e m (point m a s s e s )

1 T = ~ , . o mi~, 2.

(3)

After s u b s t i t u t i o n the following e x p r e s s i o n to e q u a t i o n s (!), (2) a n d (3):

1

Otl •ffi 7 (Yl+t -- Yl),

i ffi 0-5,

(4)

we get the L a g r a n g i a n function:

1 1 P L = ~ ~4, mlyf'" - ~ ~ bi(Yi+t - 2yi + yi_=)2 + ~-~ ~i=0 (y~÷=_ y~)2

(5)

In the a b o v e f o r m u l a e we h a v e to a s s u m e : Y0 -- Ys -- Y6 --- 0,

3~0= )~ = 0,

m0 = ms = 0.

T h e Lagrange e q u a t i o n s lead to f o u r linear differential e q u a t i o n s of m o t i o n (small vibrations): mk~k + ~ [bk-t(yk -- 2y~-i + Yk-2) -- 2bk(y~+l - 2y~ + Yk-I) P + bk.t(yk.., -- 2Yk*l + Yk)] +'7- (Y~-~ -- 2Yk + Yk*t) = 0,

f o r k = I-4.

(6)

U n d e r the a s s u m p t i o n that m a s s e s and stiffnesses of hinges are s y m m e t r i c a l with r e s p e c t to the centre of the s y s t e m , i.e. b 0 = b~,

bl = b4,

b2 = b3,

ml = m4.

m: = m~,

(7)

and after introduction the n e w i n d e p e n d e n t variables: ~ = YI+Y4, = Y,_+ Y3,

g~ = Y I - Y 4 ~k = Y: - Y~,

(8)

the set of f o u r e q u a t i o n s (6) can be s e p a r a t e d into t w o i n d e p e n d e n t sets, describing the s y m m e t r i c a l (~0 = ~ = 0) or antisymmetrical (~ = 11 = 0) vibrations of the model:

{

ml12~ + [(b0 + 461 + b2) - 2PI]~ - [(2bt + b:) - P I ] ~ = 0 m212i~ - [(2bl + b,.) - PI]~ + [(bl + b:) - PI]~ = 0

(9)

mll2ip + [(bo + 4hi - b,) - 2 P / I , - [(2bt + 362) - P I ] $ = 0 m212~ - [(2bl + 3b2)'- Pl](p + [(bl + 9b2) - 3PI]~b = 0

(IO)

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A note on vibrating compressed columns

13

3. O P T I M A L DESIGN In order to describe the real column in an approximate way, we assume that the point masses are proportional to a certain power of stiffnesses of the hinges: (11)

mk ~ Ak ~ ( E I k ) " ~ bk*,

where Ak and ( E I D denote the cross sectional area and the stiffness of the real rod. According to the above proportionality we shall apply the following equation: (12)

mt= m*(bdb*)',

where m* and b* are certain reference mass and stiffness respectively, and the exponent r is equal, in most cases to 1, I/2 or !/3, [3]. We shall also assume that the given " c o s t " of the system is equal to the sum of masses mt and m2 and equal to m*. Consequently we obtain from (12): bl" + b2" = b*'.

(13)

We look for the design (i e. the values of b~ and b2), satisfying the condition (13), that has the maximal values of the lowest frequency o f vibrations. Introducing the following substitutions: =~e ~,

-~=vtoe ~ ,

¢,=¢oe ~,

~=~oe ~

(14)

and convenient dimensionless variables: q = bJb*,

fl = P I I b * ,

(15)

1"~= m*12to21b *

we obtain from equations (9) and (10), in the simplest case K = 1, two quadratic equations for frequencies of the symmetric and antisymmetric vibration modes: cl(1 - c l ) f l s 2 - [1 + coo - cl) + 3cl(! - cl) - t](2 - cl)]fls + [/]2 _ (1 + Co+ cl)~O + co+ cl(l - c,)] = 0

(16)

c1(I ' cl)[IA2 - [! + c o o - c 0 + 1 lc,(1 - c,) - / ] ( 2 + c0]EIA + [5/] 2 - (15 + 3co - 5cl)/3 + coo - 8c0 + 25c1(1 - c0] = 0.

(17)

If ~r~$ < 0 o r ~'~A• 0, the motion o f the system is unstable, while for l l s = flA = 0, equations (16) and (17) enable u s - t o calculate the critical forces, associated with the symmetric and antimmetric modes of buckling [6]. The squares of the two lowest frequencies of vibrations are presented in Figs. 2--4, for various values of the support stiffnesses co and compressive force ~8.

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FIG. 4.

It is seen that for the small values of support stiffnesses (for example Co= 0.2.5, Fig. 2), maximal value of the lowest frequency (symmetric mode) is always less than second frequency (antimmetric mode). However, for greater value of co(for example co = O. 75, Fig. 3) and for suf~ciently large compressive force /3, (for example/3 = 0 . 7 ) the maximal value of frequency f~s is reached for cj = 0 . 1 , while for this value of ct the system loses its stability according to the antimmetric mode (1"1~= - 0.6). Hence, the optimal solution: cl = 0 . 1 5 is connected with the double eigenvalue fl~ = fls. This characteristic feature of the problem is seen much better in Fig. 4, where, for certain values of ~, the lower frequency of the symmetric vibrations reaches its maximal value for cs = 0, while the system can already lose its stability according to the second mode. The optimal values of cl correspond also to the double eigenvalue of ~ which belong to the curve ABCD. Additionally, to emphasize the similarity of the results for simple Pragers' model and for real column, the diagrams of compressive forces vs squares of dimensionless frequencies fl, for optimal "shapes" and

15

A note on vibrating compressed columns

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for two different values of clamping rigidities, are presented in Fig. 5. Comparing them with the corresponding results obtained for real cantilever by B/achut and Gajewski [2], we may see that Prager's model generally preserves the characteristic features of the optimization problems for real columns. 4. FORMULATION OF THE UNIMODAL AND BIMODAL OPTIMAL DESIGN OF VIBRATING, COMPRESSED COLUMN The exact formulation of the problem can be briefly presented on the base of the Pontryagin maximum principle. The equations of small vibrations of the elastically clamped-clamped compressed column of total volume V and with similar cross-sections have the form (after separation of the space and time variables): y~ = (p,,

~', ffi _ ~ / ~

2,

Miffi Q~+~,p,, QI = -~,.~by,.

(18)

Subscript i = 1, 2 correspond to the symmetric or antisymmetric modes of vibrations respectively. They are distinguished by the boundary conditions: yl(O) = O, ~(I/2) ffi O, Ml(O)+-~l(O)ffiO,

Ql(1/2)ffiO,

for i ffi I (symmetric mode)

(19)

y2(O) ffi O, y2(1/2)ffi O,

M~(O)+ 1 ¢~(0) ffi O, M2(I/2) = O. for i = 2 (antisymmetric mode)

(20)

16

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In equations (18)-(20) the following dimensionless quantities are introduced: pp

_

t3 = e - - U ~ ,

x = ~/l,

pl

:

-

I

~ - e--~-~o, , c = - ~ ,

y = w/l,

ab(x) = Atx)/Ao.

(21)

The parameter ~" characterizes the same rigidities of clampings (~" = 0 for clamped-clamped, whereas ~ ' ~ for hinged-hinged columns). Defining now: (22)

Ao = I//1

the dimensionless shape d~(x) must be normalized according to the condition: f [ O(x) dx = 1.

(23)

One can easily show that in this ~:onservative problem, both, the conjugate differential equations and corresponding boundary conditions are the same as the state equations (18) and boundary conditions (19) or (20). Assuming as a cost function the total volume of the column V and as constraints /3 = const, and fl = const., we obtain the optimality conditions: (i). for unimodal formulation:

f 2M? ]"'

d~(x) = L ~ j

for i = 1,2

(24)

(2). for bimodai formulation: ~,(x)

{A 2[I-~)MI2+/.LMz 2] ~m + (! - ~)l~,yl ~+ p.l'lzyz2J '

(25)

In addition, the optimal configuration ¢b(x) derived from (24) or (25), can satisfy geometrical constraints of the form:
(26)

and must satisfy the normalization condition (23) determined the constant k. In the case of bimodal formulation a constant ~ (0 ~ ~ ~ I) have to be calculated by equating the symmetric and antisymmetric mode fre~lucncies: 1"1,= 122. In order to obtain the solution of the problem formulated above, one can use an iterative procedure suggested by Grinev and Filippov, which was succcsfully used in the papers [I] and [2]. However. the calculations have not been performed here. 5. CONCLUSIONS

In conclusion, it seems worth emphasizing that the bimodal optimization of real columns for maximum frequency of vibrations ought to be used only for compressive forces close to the buckling loads and sufficiently small values of parameter ~. Although for the model of column adopted here, the bimodal optimization is also necessary for a small compressive force (for instance/3 = 0. l, Fig. 4 and curve CD in Fig. 5), it seems that for real columns this situation will not appear. The bimodal optimization with respectto the lowest frequency of free vibrations will be probably indispensable for shallow arches and portal frames [1,6]. REFERENCES [I] J. BEACHUTand A. GAJEWSKI, On unimodal and bimodal optimal design of funicular arches. Int. 3.. Solids Struct., (to be published). [2] J. B£ACHUT and A. GAIEWSKI, A unified approach to optimal design of columns, Solid Mech. Arch., to be published. [3l A. GJOEWSKI and M. ZYCZKOWSKL Optimal design of elastic columns subject to the general conservative behaviour of loading. J. o f Appl. Math. Phys. Z A M P , 21(5), 806-818 (1970). [4] E. F. MAsuR and Z. MR6Z, On non-stationary optimality conditions in structural design. Int. Y. Solids Struct. 15, 503-512 (1979). [5] N. OLHOFF and S. H. RASMUSSEN,On single and bimodal optimum buckling loads of clamped columns. D C A M M , Report No. 111, (1976), or Int. J. Solids Struct. 13, 605-614 (1977). [6] S. PRAGERand W. PRAGER, A note on optimal design of columns. Int. J. Mech. Sci. 21,249--251 (1979).