OPTIMIZATION OF CONTINUOUS ONE-DIMENSIONAL STRUCTURES UNDER STEADY HAFSMONIC EXCITATION+

Erwin H. Johnson* Ames Research Center, NASA, Moffett Field, Ca. 94035

Paulo Rizzi** Dept. of Aeronautics and Astronautics, Stanford University, Stanford, Ca. 94305

Solly A. Segenreich*** COPPE-Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

Holt Ashley++ Dept. of Aeronautics and Astronautics, Stanford University, Stanford, Ca. 94305

Abstract

To illustrate the minimum-weight design of one- dimensional, elastic structures under dynamic exci- tation, methods from optimal control theory are applied to the cantilever bar driven sinusoidally by an axial force at its tip. Other directly analogous problems are identified, and closely related cases are discussed along with practical applications. Realistic constraints are enforced during the opti- mizations: maximum allowable stress amplitude at any point along the bar, and minimum cross-sectional area. In the absence of damping, the design space may contain many disjoint feasible regions, and mul- tiple optima can exist. This novel feature is examined, first by reference to a simple example with just two element areas to determine. Solutions are worked out in detail for continuous bars, with the excitation frequency less than, then greater than, the fundamental free-vibration frequency. The latter results overcome a limitation inherent in previous analyses. Above a certain excitation fre- quency, two or more arcs with different constraints characterize the optimal designs; a concentrated tip mass is also needed in some cases. Free vibrations of the optipal designs are analyzed. A study of the forced-response mode shapes, along with parallel solutions to the same problem made with finite ele- ments, furnish background for a critique of the dif- ficulties in going to even higher frequencies than those successfully treated here.

I. Introduction and Summary of Previous Research

This paper examines some questions relating to the optimal design of continuous, one-dimensional structures driven by harmonically oscillating loads. The subject matter differs in two respects from most research being done in the rapidly expanding field of structural optimization: (1) the examples studied are subjected to dynamic excitation, without limits on the frequency of the loading, and (2) the emphasis is on continuous structural representation, SO that differential equations rather than finite elements are used in searching for the optimum.

The time-dependent forcing produces distributed inertia load's. which can have important effects on

+This research was supported in part by the Air Force Office of Scientific Research (contract AFOSR 74-2712) and in part by the National Aeronautics and Space Administration Grant NGL-05-020-243. The authors are indebted to Professor John V. Breakwell, Stanford University, for valuable discussions of the mathematics of optimization.

*NRC Postdoctoral Fellow; Dr. Johnson wishes to acknowledge the'generous support of the Fannie and John Hertz Foundation during his doctoral studies.

**Graduate Student.

++professor. ***Assistant Professor.

the solutions. Furthermore, the possible resonances associated with sinusoidal motion give rise to dis- connected feasible regions in the design space and to numerous local optima - phenomena that have rarely received systematic attention in the past.

The use of differential equations naturally draws on the calculus of variations as a tool for formulating necessary conditions and enforcing con- straints. These methods have been developed exten- sively in the field of optimal control theory (cf. Bryson and H o , ~ Ch. 2, 3, etc.) and are sufficiently well known that they require no elaboration here. Instead, examples are presented to illustrate some interesting features that harmonic excitation intro- duces into minimum-weight design. It is believed that the cases chosen are merely typical and that many other one-dimensional structures can be syn- thesized in an analogous fashion.

With regard to related previous work, the reader need not be reminded that structural optimiz- ation has now been an active field for nearly two decades nor that much of its literature is focussed on refining computer algorithms to perform the numerical search for optimal solutions. A notable survey paper published by Sheu and Prager2 in 1968 cited 146 publications, including prior surveys; more than the usual amount of attention was paid by them to continuous (as contrasted with discrete) structural representations. Two more recent sym- posium proceedings are those edited by Gellatly3 and S~hrnit,~ but, as in Ref. 2, the bulk of their contents deals with purely static loadings. Pierson's' is the only known review paper that emphasizes free and forced dynamic conditions. It contains an apparently exhaustive list of 61 cita- tions, divided between those investigations that place constraints on structural natural frequencies and those - fewer in number - where some property of the forced dynamic response is constrained. 1972, Pierson states his conclusions that the field is "still in its infancy" and that "there is a not- able lack of published work" in the latter category of problems. Both observations are valid today.

As of

Among relevant papers whose subjects do not closely overlap the present work are those of Fox and Kapoor6 and B r a ~ h . ~ approximate optimal solutions for simple structures loaded by step, impulsive, or other transient forces. To avoid the greater complication of deal- ing with the exact time-dependent response, limits were placed on estimated upper bounds. Wolf8 developed an algorithm to compute "fully stressed" designs of one-dimensional configurations Although such designs are known to be optimal for determinate structures that are statically loaded, it cannot be proved that the same holds true for dyn-amic cases. Indeed, some counterexamples are described in what follows.

These early studies found

Levy and

R8hrleg has applied the

473

fully stressed concept to proportion an AEROS satel- lite structure under dynamic conditions. assumption that phis approach is sufficiently accu- rate for practical purposes does often seem justifiable.

His

Another relevant application concerns the design of civil structures to withstand earthquake motions. Examples are found in papers by Solnes and Holst,lo Nigam and Narayanan," and Kat0 et a1.12 The diversity of methods used in this area, some involving a reduction to equivalent static con- straints, suggests it may offer fertile ground for further research m d systematization. Thus the optimality-criterion technique of Venkayya and Khot,13 although originally developed for aircraft structures, might prove useful for earthquake problems.

The series of investigations most directly antecedent to the present one are those of 1cerman,l4 Mroz,15 and Plaut.16 All employ energy methods con- nected with Rayleigh's inequality to design minimum- weight structures under simple harmonic excitation. A common feature of their approaches is that the forcing frequency is limited from above by the fund- amental natural frequency of the optimal arrangement. Two contributions of the present study are the removal of this limitation and the imposition of conditions deemed more realistic than those in Refs. 14- 16. Section I11 treats the associated question of disjoint feasible regions in design space, giving references to where these have been encountered.

The "technology transfer" of methods from opti- mal comtrol theory and trajectory analysis to minimum-weight design of continuous structures was pioneered at Stanford by one of the present authors and his co-workers. For example, Ashley and McIntosh1' introduced constraints on natural fre- quencies, as well as aeroelastic eigenvalues like divergence or flutter speed. Weisshaar * and Armand and Vitte19 systematized the approach, examining constraints on minimum thickness and the addition of nonstructural mass. theory for distributed-parameter systems (i.e., sys- tems described by partial differential equations) to calculate minimum-weight shear plates and sandwich plates with prescribed fundamental frequency and minimum thickness.

Armand" adapted optimal control

Finally, it should be recognized that most of the material in the following sections derives from portions of doctoral theses by two of the authors (Johnson21 and Rizzi"). Reference 21, in particu- lar, uses finite-element structural representations and computer searches based on mathematical program- ming to find a variety of optimal one-dimensional structures subjected to harmonic or random forcing.

11. Statement of Focal Problem and Its Analogs

To introduce somewhat more realistic design cri- teria than related previous investigations, the fol- lowing example is selected for study here: the cantilever bar in Fig. l(a) undergoes steady harmonic response to an oscillating axial load at its tip. A limit is imposed on the maximum amplitude of oscil-

iwet latory stress u(x)e , and a "minimum-gauge" con- straint is placed on the cross-sectional area A(x). The optimal area distribution is sought in the sense of minimum volume (or weight). When the effects of transient response to load startup can be neglected,

this problem statement is believed closer to what a designer would encounter than, for example, the effective limitation on tip amplitude of Icerman14 or the "minimum dynamic compliance'' used by Mroz.

When writing differential equations and bound-

MT 2 0 ary conditions for a continuum model of the bar, allowance is made for a concentrated mass at the tip. Under some circumstances, MT is required as part of the optimal solution, but in vibration-isolation applications it might also rep- resent that portion of the isolated mass supported by the particular mount being designed. After elim- ination of the simple harmonic factor eiwet, the axial displacement amplitude u(x) is governed by the well-known equation:

[EA(x)u'(x)]' + o:pA(x)u(x) = 0 (1)

Here E and p are, respectively, Young's modulus and the density of the homogeneous, elastic material; the "prime" denotes differentiation with respect to x. The boundary conditions are

(2) u(0) = 0

FA(!L)u'(P.) = P + Fiyw:u(P.)

Let umax be the maximum allowable stress amplitude in tension or compression. Then the dimensionless variables and parameters

s = X/L v(s) = u/P. a(s) = Aumax/P

( 3 )

transform Eqs. (1) to (3) into

(5)

v(0) = 0 (6 )

(7 )

In these terms, a constrained optimal design a(s), would be a dimensionless area distribution

which constitutes a local or global minimum of

.T = aT + J f a(s)ds (8)

while satisfying Eqs. (5), (6), (71, and the inequality constraints

Aminurnax 1 E - A a(s) 6

Here A,,,in is the minimum allowable area; 6 = P/(knumax) is a convenient parameter of the problem, which couples the stress and area limita- tions with the tip-force amplitude.

474

I n p r e p a r a t i o n f o r an a t t a c k by t h e methods of optimal c o n t r o l theory , one must express Eq. (5) i n the " s t a t e vec to r" form of a p a i r of f i r s t - o r d e r equat ions . The d e f i n i t i o n s

XI = v

dv x2 = a - d s

thus l e a d t o kl = x 2 / a

= -2ct2ax1

The boundary c o n d i t i o n s and c o n s t r a i n t s are

- a ( s ) + 6 1 0

Two o t h e r s t r u c t u r a l c o n f i g u r a t i o n s occur i n v i b r a t i o n i s o l a t i o n , f o r c e t r ansmiss ion and r e l a t e d t echno log ies whose d e s i g n f o r minimum weight i s mathemat ical ly analogous t o t h e ba r . Without s e t t i n g down t h e complete e q u a t i o n s , one may c h a r a c t e r i z e them as fo l lows :

(a) C a n t i l e v e r rod undergoing t o r s i o n a l e x c i t a- tion a t i t s f r e e t i p - t h e o u t e r c r o s s- s e c t i o n a l shape and s i z e of t h e rod a r e e s s e n t i a l l y f i x e d , b u t i ts w a l l i s s u f f i c i e n t l y t h i n t h a t t h e t o r s i o n a l r i g i d i t y G J and p o l a r m a s s moment of i n e r t i a Ip are bo th d i r e c t l y p r o p o r t i o n a l t o t h e ( v a r i a b l e ) skin t h i c k n e s s t ( x ) . C o n s t r a i n t s are placed on t h e minimum v a l u e of t and t h e s h e a r- s t r e s s ampli tude. A concen t ra t ed t i p i n e r t i a IpT may be c a r r i e d .

Among o t h e r s , Armand and V i t t e 1 9 and Weisshaar' ' s tudied s i m i l a r op t ima l d e s i g n s wi th l i m i t a t i o n s on the f r e q u e n c i e s of f r e e v i b r a t i o n .

placements a r e c o n t r o l l e d mainly by i t s s h e a r r i g i d i t y . governing d i f f e r e n t i a l equa t ion f o r t h e displacement w(x,t) i s

(b) Stubby c a n t i l e v e r beam whose l a t e r a l d i s -

According t o Timoshenko and Young,23 t h e

-[GK(x)w']' + pA(x)W = 0 (19)

Here t h e "prime" and "dot" denote p a r t i a l d i f f e r e n t i - a t ion wi th r e s p e c t t o x and time t. The q u a n t i t y K(x) i s r e l a t e d t o t h e c r o s s- s e c t i o n a l a r e a bv

where k is a shape f a c t o r , r ang ing from 413 f o r a so l id c i r c l e t o 2 o r 3 f o r I-beam s e c t i o n s . The analogy w i t h Eq. (1) i s obvious; boundary c o n d i t i o n s s imi la r t o Eqs. (2 ) and ( 3 ) would r e s u l t from t i p forcing of a c a n t i l e v e r , p o s s i b l y c a r r y i n g a mass a t its f r e e end. With a p p r o p r i a t e shape s i m i l a r i t y , k could be absorbed i n t o t h e parameter J of Eq. (5) during nond imens iona l i za t ion . t i on mounts a r e b e l i e v e d t o be f a i r l y w e l l approxi- mated by Eq. (19).

Many shock and v ib ra-

111. The Quest ion of M u l t i p l e F e a s i b l e Regions

I n h i s comprehensive, i l l u m i n a t i n g s tudy of minimum-weight frames sub jec ted t o t r a n s i e n t dynamic e x c i t a t i o n , cas si^^^ concludes: "It was found t h a t

t h e des ign space may have d i s j o i n t f e a s i b l e r eg ions t h a t a r e due t o t h e s i n u s o i d a l c o n t r i b u t i o n s t o t h e dependence of t h e dvnamic response f u n c t i o n s on t h e design v a r i a b l e s . Th i s fundamental and important c h a r a c t e r i s t i c had no t been a n t i c i p a t e d , nor was i t p r e v i o u s l v recognized i n t h e s t r u c t u r a l optimiza- t i o n l i t e r a t u r e . " The a u t h o r s b e l i e v e t h e c l a s s of problems t r e a t e d he re t o be unique i n i t s involve- ment wi th such m u l t i p l e , d i s j o i n t f e a s i b l e r eg ions . I n f a c t , t h e r e appear t o be a denumerable i n f i n i t v of r eg ions , each a s s o c i a t e d w i t h an e x c i t a t i o n f r e - quencv we t h a t f a l l s i n t h e range d i < we < wi+l, where w i i s t h e i t h f r e e- v i b r a t i o n frequency of t h e corresponding op t ima l b a r , rod , e t c . The des ign space of t h e cont inuous b a r i s an " i n f i n i t e - dimensional" f u n c t i o n space, and i t i s somehow c h a r a c t e r i z e 6 bv insurmountab! e " b a r r i e r s" between f e a s i b l e r eg ions , a long which re sonan t response occurs nea r each

*

me = o i .

E a r l i e r s o l u t i o n s have avoided t h i s d i f f i c u l t y because t h e i r requirement t h a t we < w 1 e l i m i n a t e s t h e ambiguity. Although o p t i m a l i t y c r i t e r i a l i k e those of Icerman14 and Mrozl' do s e e m t o be bo th necessa ry and s u f f i c i e n t f o r t h e " abso lu te minimum" des igns t h u s produced, t h i s i s accomplished a t t h e p r i c e of r u l i n g out o t h e r p o s s i b l y s u p e r i o r des igns t h a t meet a l l t h e s t a t e d c o n d i t i o n s o t h e r than me < w 1 . adopted h e r e a r e admi t t ed ly weaker i n t h a t i t i s ha rde r t o examine t h e s u f f i c i e n c y o r " g loba l i ty" of t h e s o l u t i o n s ob ta ined from t h e necessa ry c o n d i t i o n s they fu rn i sh . Yet they y i e l d s a t i s f a c t o r y , p r a c t i - c a l des igns t h a t a r e c l e a r l v l o c a l optima and t h a t tempt an eng inee r t o postpone t h e d e t a i l e d examina- t i o n of Suf f i c i ency .

The methods of v a r i a t i o n a l c a l c u l u s

Johnson'' s t u d i e d a case , wi th j u s t two des ign v a r i a b l e s , t h a t i l l u s t r a t e s t h e s i t u a t i o n simply and g r a p h i c a l l y . S l i g h t l y modified t o conform wi th t h e problem i n 5 11, it c o n s i s t s of a b a r wi th two equal- length segments of a r e a s a1 (near t h e c a n t i- l e v e r r o o t ) and a2 (near t h e t i p ) . It i s d r iven a t

i w e t t h e t i p by f o r c e Pe ; a1 and a2 a r e t o b e s e l e c t e d f o r minimum weight , w i th s p e c i f i e d minimum gauges and c o n s t r a i n t s on maximum a l lowab le s t r e s s amplitude. S t a r t i n g from d i f f e r e n t i a l equa t ions of t h e uniform b a r i n each segment, t h e problem can b e solved e x a c t l y . The s t r a i g h t f o r w a r d mathematical d e t a i l s a r e omit ted here .

F igures 2 (a ) and (b) d e p i c t des ign spaces f o r t h e two-segment b a r a t two v a l u e s of t h e fo rc ing- f requency parameter u from Eq. ( 4 ) . F e a s i b l e por- t i o n s of t h e s e spaces a r e l e f t unshaded; t o t a l weight ( p r o p o r t i o n a l t o al+a2) i s cons tan t a long t h e 450 s l o p i n g l i n e s used t o shade t h e i n f e a s i b l e r eg ions . The r e q u i r e d minimum a r e a s a r e a1 = a2 = 0.2. Eviden t ly , a 50% i n c r e a s e i n u h a s moved t h e " g loba l optimum" from t h e v e r t e x of t h e right- hand r e g i o n t o t h a t of t h e second r e g i o n i n t h e upper lef t- hand corner . Furthermore, t h e p ropor t ions of t h e minimum-weight b a r have changed r a d i c a l l v , from one having n e a r l y uniform c r o s s s e c t i o n t o one wi th a t h i c k t i p s e c t i o n and a minimum-gauge roo t .

The exp lana t ion of t h i s behavior i s t o be found by s tudy ing t h e dimensionless v i b r a t i o n e igenvalues

*Sved and 6 inos2 encountered d i s j o i n t behavior when de_signing oDtimal t r u s s e s under m u l t i p l e s t a t i c loadings . u n i f y i n g t h e des ign space i n such cases .

F a r s h i and Schmit26 suggest a scheme f o r

47 5

ni = oi9. m, i = 1,2, of the system. 2(a), the designs with the fundamental n1 equal to n = 0.707 can all be shown to lie on the dashed straight line:

In Fig.

a1 = 0.298 a2 . (21) X l = 2n2a~2 It proceeds through the middle of the infeasible region and constitutes the "maximum of infeasibil- ity," because along it the resonant response is a '1 a unbounded. Below this line, a1 > n and conversely; each of the associated feasible regions clearly If aT (i.e., MT) is not prescribed to be a con- offers its own optimum. stant or zero, a suecial boundary condition

sign(x2) (28)

A 1 x 2 = - - -

By an extension of the work of Segenreich and 1 - v12a2q (1) - u 2 = 0 (29) Riz~i,'~ it can be proved that segmented cantilevers modeled in this fashion have bounds on their eigen- values. In the present case of two equal segments, determined. From integrations by parts in the vari- these limits are ational process, the following additional boundary

arises, where VI and v2 are constants, to be

conditions are found:

for i = 1,2,3, ... It follows that there will be X1(l) = -v12a2+ (31) two feasible regions for each nonzero excitation frequency. (In the static case, a = 0, the fully h2(1) = v1 (32) stressed solution is the single optimum.) Both sit- uations graphed in Fig. 2 therefore permit two Finally, the nonnegativity conditions on the regions. multipliers are

1x2 I if-<8

u 1 =I 1x2 I In principle, the foregoing line of reasoning

can be extended to one-dimensional discrete models with an arbitrary number of segments. It also moti- vates the hypothesis of a denumerable infinity of local optima for continuous structures. More details I

are given in Ref. 27 and in a recent paper by Johnson.

8 z O , i f - =

0 , if 1/6 < a

u2 = I It might be supposed that the disjoint design ( 2 o , if I/& = a

0 , i f a T > O space results from the omission of any structural damping, and this is strictly true. When damping is provided, the infinitely high "ridges" disappear and the infeasible regions no longer extend to infinity.

However, structural even energy with dissipation, unrealistically it is high found21 amounts that of v z = [ 2 0 , i f + = o

many "pockets of feasibility" remain in the unified design space. The basic problem of singling out a Equations (29) and (35) should be disregarded global optimum from several local candidates persists. aT is not part of the optimization.

IV. Optimization of the Continuous Bar

As set forth, for instance, in 5 2.4, 3.8, and 3.12 of Bryson and H o , l necessary conditions for a minimum of J in Eq. (8) under the constraints and boundary conditions (13) and (14).may be constructed from the Hamiltonian:

Here Xi(s),pi(s) are Lagrange multipliers that introduce the respective constraints factored by them and a(s) is the "control variable." The requirement 6 J = 0 leads to equations of the form

0 , (i = 1,2) - aH xi+-= a xi

which, in the present case, are written:

First-Mode Solutions (3: < CUI)

(33)

(34)

(35)

when

When the forcing frequency is less than the fundamental frequency of the optimal design, it is easily seen that the entire bar must respond in phase with the applied load:

The tip mass will at first be omitted, as will the minimum-gauge constraint (la), for reasons that will become apparent later. The design is assumed to be "fully stressed" over a region or arc near the tiu, with the stress constraint relaxed inboard, i.e.,

(25) where y is a temporarily unknown matching point. On the f u l l y stressed arc,

4

(38)

476

whose integral yields a response mode shape that is just a straight line:

On the unconstrained inboard arc 0 I; s s y, I J ~ = 0, and Eqs. ( 2 6 ) and ( 2 8 ) reduce to

( 3 9 )

( 4 0 )

1 - A 1 - x2 - X2 2a2x1 = 0

a2 ( 4 9 )

and - In terms of the new coordinate s , the tip region lies in 0 2 E 5 $, with

x1 (Y) qJ = 1 - y + 7 ( 4 1 )

Differentiating Eq. ( 3 8 ) and substituting it and ( 4 0 ) into ( 1 4 ) , one derives an equation for the area distribution:

Equations (13), ( 1 4 ) , ( 2 7 ) , and the root boundary conditions are unaltered. It can therefore be reasoned (cf. § I11 c(2) of Johnsonz1) that

d = -2a2sa where C3 and C4 are the values of x2 and X I at s = y. By appropriate eliminations, a simple non- linear differential equation is found to govern XI. With boundarv conditions (15) and k l ( y ) = 5, the solution reads

With boundary condition ( 7 ) solved by

a2 a(:) = e

With p2 = 0, Eqs. ( 2 6 ) and into

1 e x 2 = - - +

which has the solution

and aT = 0, this is

( 4 3 ) 12-52 J

(28) can be manipulated

2a2EX2

( 5 3 ) 6 sinh(fi as)

f i a cosh(fi cry) x1 =

( 4 4 ) From a similar elimination between Eqs. ( 1 3 ) and ( 1 4 ) , the following area distribution can be calculated:

( 4 5 ) The continuities of x1 and X 2 at s = y yield, respectively,

( 5 5 ) Here the error function

c3

C 2

Note that Eq

is used, and C2 is a constant to be determined. Similarly, one can calculate the other Lagrange multiplier,

and

( 5 5 ) allows d~ in (411 to be written . , entirely in terms of a and’y. The relations (51) and ( 5 2 ) lead to

(a2{$2-rx;(Y)/6211) C, = e cosh2(n a y ) ( 5 8 )

According to 5 3.12 in Ref. 1, aH/aa must be continuous across the junction between arcs. With all other variables continuous, it follows that p1

that

2-a J;; e

This function is expres- must be also, whence p1 + 0 at s = y. It follows by the nonnegativity of PI. sible, by substituting the foregoing results for and Xi into Eq. ( 2 6 ) , in the following form: [ (a/B)xl (y)I2

xi

-a2[uJ2-E2I J;; e(a5)2(% t25-$] $Ierf(a$)-erf[ (a/B)xl (y)] 1

2-e -a

e 1 1116 - = a

- [ (a/B)x1 (y) 1 2 ax1 (Y) -a2{$’-[x: ( y ) / B 2 ] } XI (Y) - 25 [erf(T)-erf(u~)] -e -2J2 7 - $ = 0

The extent of the fully stressed arc is controlled

( 4 8 ) ( 5 9 )

Given an excitation frequency parameter a and stress constraint 6 , the transcendental relations

The quantities y, xl(y), and C p remain to be fixed ( 5 3 ) , ( 5 6 ) , and ( 5 9 ) can be solved by trial For the by matching. corresponding y. When a 5 1.0908, this calcula-

tion shows that there is no junction point within

477

length 0 5 y i 1 of the bar. For larger a, the single meaningful root thus obtained is plotted in Fig. 3. It starts from y = 0 with infinite slope dylda and tends asymptotically to y = 1.

Because of the manner in which P and amax enter the problem statement, a single-parameter fam- ily of solutions emerges. Five typical dimension- less area distributions a = Aomax/P are plotted in Fig. 4 , for values of a increasing from the static a = 0 to a = 1.5 where nearly half the bar is not stress constrained. Up to a = 1.0908, however, the often-hypothesized, fully stressed design is indeed optimal. often be quite wide; for instance, it applies for aluminum bars up to about where f is in inches. (For shear beams and rods in torsion, the limiting frequency can be much lower. )

This range of excitation frequencies may

we = 3x105/f rad/sec,

The area distribution of these designs may he summarized as

When Eq. (60) is integrated over the length, the following dimensionless weight (or volume) results:

Equation (61) is plotted vs. ci in Fig. 5 and com- pared, in the higher range, with the corresponding fully stressed designs. between the two is not significant until a exceeds about 1.5, nevertheless the rapidly rising weight of both leads one to suspect that better designs might be discovered than these, requiring as they do large amounts of material to force 01 to exceed we.

Note, from Fig. 4 , that over their entire

Although the separation

length the dynamic solutions display areas larger than what would withstand the same force applied statically ( a = 0). This explains why no minimum- area constraint seemed necessary for the first-mode analysis.

Second-Mode Solutions (a > ai)

As discussed further in 5 V, the response of the bar in the frequency range in antiphase with the applied load, i.e., x l ( s ) 5 0 except possibly in a small region near the tip. It can then be reasoned that a tip mass may be required (aT # 0) to satisfy boundary condition (16). It definitely proves necessary to enforce the minimum- gauge constraint (18) t o avoid a tendency toward vanishing area at the point of zero slope in the response mode x1 ( s ) .

01 < we < w2 must be

Without reproducing all mathematical details (see Ch. 4 , Ref. 22), one determines that the outi- m a l solution consists of two arcs: stress con- straint active in 0 I s 5 y and minimum-area constraint active in y 2 s 5 1. Its characteris- tics are worked out (as above) by first solving in the two ranges separatelv, then fixing several unknown constants by trial-and-error matching of transcendental relations at s = y. Since is retained in Eq. ( 1 6 ) , its optimal determination becomes an organic part of the process.

Figure 6 displays the resulting values of Y, plotted vs. a for a wide selection of minimum- gauge parameters 6 = P/&,inamax. For too large a values, at any given 6, the Lagrange multiplier ~2 is no longer positive throughout the entire tip arc. This limitation is expressed by the broken line in Fig. 6. Its onset indicates that more than two arcs are needed in the optimal design - a situation discussed further below.

With y specified, the dimensionless area distribution is

whereas the tip-mass parameter is

From Eqs. (62) and (63), one may compute the total weight parameter:

Figure 7 plots the optimal a(s) (and aT on the right-hand scale) for two 6 values and CY = 1.0, 1.3. Perhaps most interesting are the Fig. 8 data on dimensionless total weight, Eq. ( 6 4 ) . Here the first-mode curve from Fig. 5 is also repro- duced. When a < 0.63, the latter proves superior to any possible second-mode solution. At higher forcing frequencies, on the other hand, a selection of designs with w1 < we is available which can save substantial weight.

A surprising mathematical feature of the second- mode solutions is the discontinuous behavior of Lagrange multipliers and ~2 at the matching point s = y. discovery that 1 ~ 1 drops from a finite positive value to zero when passing outboard across s = y, whereas to make H and aH/aa (Eqs. (23) and ( 2 6 ) ) urouerly continuous. It was subsequently observed that the last two terms in Eq. (23) might as well have been written

An error was first suspected with the

1~2 jumps from zero to just the size needed

47%

and might have been i n t e r p r e t e d a s a s i n g l e ad jo ined c o n s t r a i n t of t h e g e n e r a l form

The d i s c o n t i n u i t y i n p i t s e l f i s then found t o disappear . A t s = y , t h e r e a r e s l o p e d i s c o n t i n u i- t i e s i n p ( s ) , a ( s ) , and t h e f u n c t i o n f , b u t such phenomena a r e a c c e p t a b l e and o f t e n encountered i n control- theory a p p l i c a t i o n s .

V. Normal Modes of t h e Optimal Designs; R e l a t i o n s h i p Between Forced and Free-Mode Shapes

F u r t h e r examinat ion of f r e e and f o r c e d motions of t h e cont inuous b a r i s u s e f u l , both t o i n t e r p r e t the fo rego ing r e s u l t s and t o a s s i s t w i t h s p e c u l a t i o n on how they might be extended. From E q s . ( 5 ) , ( 6 ) , and ( 7 ) , one s e e s t h a t t h e i t h normal mode and frequency ( a i = w i ~ m ) must s a t i s f y

sub jec t t o t h e boundary c o n d i t i o n s

u . ( 0 ) = 0

Only i f a i > c1 can Eq. ( 7 4 ) be s a t i s f i e d , s i n c e o the rwise every term i n t h e summation would be p o s i t i v e .

I n Fig. 9 , "1, " 2 , and a3 a r e given a s func- t i o n s of a f o r t h e f i rs t- mode opt imal ba r s . These numbers were e s t ima ted from E q . ( 7 4 ) f o r a 5 1.0908 and computed numer ica l ly from Eqs. ( 6 5 ) t o ( 6 7 ) a t h igher f o r c i n g f r equenc iee . One observes t h e a n t i c i p a t e d behav io r , ever- heavier b a r s be ing fo rced i n c r e a s i n g l y c l o s e r t o resonance wi th t h e fundamental mode a s a becomes u n n a t u r a l l y l a r g e f o r t h e s e first-mode des igns .

I n Ch. 4 of R i z z i , 2 2 a similar c a l c u l a t i o n i s c a r r i e d ou t f o r t h e second-mode ba r s . S o l u t i o n s a r e f i r s t w r i t t e n f o r t h e two d i f f e r e n t l y con- s t r a i n e d segments, whereupon matching a t s = y produces a f a i r l y simple t r a n s c e n d e n t a l r e l a t i o n f o r q. Some r e s u l t s f o r t h e f i r s t two n a t u r a l f r e q u e n c i e s appear i n Fig . 10 , w i t h t h e minimum- gauge parameter chosen a s 5 = 10. I n t h i s example, i t i s seen, a s expected, t h a t t h e f o r c i n g f r e- quency always f a l l s between t h e fundamental w1 and t h e f i r s t harmonic w2. A s i n F ig . 9 , t h e r e i s a tendency wi th i n c r e a s i n g a f o r t h e "optimal" n a t u r a l f r e q u e n c i e s t o be d r i v e n away f,rom t h e resonance l i n e . But he re t h e u n d e s i r a b l e asymp- t o t i c convergence of cr t o a2 would no t set i n u n t i l t h e former approached twice t h e magnitude covered by t h e Fig. 10 c a l c u l a t i o n s .

C u r i o s i t y , i f no t a need f o r p r a c t i c a l app l i ca- ( 6 7 ) t i o n s , impels one t o i n v e s t i g a t e no t only t h e

behav io r of t h e second-mode s o l u t i o n s t o t h e r i e h t of t h e l i m i t i n Fig. 8 bu t a l s o what might happen i n r anges a i < a < c1 where i > 1. Consider-

A v a r i a b l e change t = as t r ans fo rms Eqs . ( 6 5 ) t o ( 6 7 ) i n t o i+l '

where

5 2 d a du i

d t +- - - a dt dt + 2rliUi = 0

When t h e a r e a d i s t r i b u t i o n

a b l e l i g h t can be thrown on such q u e s t i o n s by means of a modest ex tens ion of t h e Sturm-Liouville analy- sis, which determines t h e r e l a t i o n s h i p s between t h e fo rced mode shapes u (x ) and t h e i r a d j a c e n t normal

mode a n a l y s i s i s givenm by Morse and Feshbach.30

( 6 8 )

(69) modes u i ( x ) . An e x c e l l e n t summary of t h e normal

for t h e f u l l y s t r e s s e d op t ima l s o l u t i o n ( a 1.0908) i s i n s e r t e d i n t o Eq. (68) , one i s l e d t o t h e equa t ion

du

d t i i i i d 2u - - 2 t - + 2 n . u = 0 d t 2

( 7 3 )

Using t h e i r aDproach, R i z z i 2 2 (Ch. 4 ) proves t h a t t h e z e r o s ( i f any) of u (x ) a r e b racke ted by t h e nodes quenc ies b racke t t h e f o r c i n g f requency we. Th i s f i n d i n g i s i l l u s t r a t e d i n Fig . 11 f o r t h e f i r s t - and second-mode opt imal des igns . Since u l ( x ) has no nodes and ze ro s l o p e a t i t s unloaded t i p , t h e f i rs t- mode resuonse must a l s o be e n t i r e l y above t h e x a x i s . I t s p o s i t i v e t i p s lope i s necessa ry t o counterbalance t h e e f f e c t of t h e app l i ed o s c i l l a t o r y fo rce .

u i = 0 of t h o s e normal modes whose f r e-

A s f o r t h e second-mode s o l u t i o n , whose f r e- quency f a l l s i n w 1 < we < w2, it may have zero o r one node, bu t t h e l a t te r must f a l l c l o s e r t o t h e t i p than t h e s i n g l e node t h a t u2(x) has t o have.30

which i s so lved by n o n i n t e g r a l H e r m i t e f u n c t i o n s (see, e.g., Murphy,29 p . 322). A s worked o u t i n Ch.

of Ref . 2 2 , condition (69) eliminates one of the two independent s o l u t i o n s , whereas Eq. (70) ( f o r aT = 0) y i e l d s t h e c h a r a c t e r i s t i c r e l a t i o n

'The a u t h o r s must t a k e excep t ion t o one s t a t e - ment i n t h e o the rwise l u c i d t r ea tmen t of Ref. 30: i n t h e d i s c u s s i o n of " s i n u s o i d a l behavior ," i t is n o t t r u e t h a t , when t h e c o e f f i c i e n t of t h e u n d i f f e r - e n t i a t e d modal f u n c t i o n i n t h e aovernina d i f f e r e n - - t i a l equa t ion i s p o s i t i v e , t h i s func t ion always " curves toward t h e ax i s ." A s a counterexample, t h e first-mode opt imal s o l u t i o n ( 4 0 ) of ( 5 1 , where 2 a 2 a ( s ) i s c e r t a i n l y g r e a t e r than ze ro , i s a s t r a i g h t l i n e w i t h no c u r v a t u r e a t a l l . The hyper- h o l i c - s i n e mode of Eq. ( 5 3 ) a c t u a l l y cu rves away

( 7 4 ) from i t s a x i s .

[02-a:] [3a2-ai]. 2 . . [ 0 = . + Z 2 n [ n= 1 (2111 !

479

This relationship is shown in Fig. 11, as is another fact proved by Rizzi: u(x) must start from x = 0 with negative slope and therefore remain below the x axis until outboard of the node location. If this mode has zero nodes, its tip slope may be positive (as in the figure) or negative. Incidentally, this last observation explains the need for a tip mass as part of all the second-mode solutions worked out quantitatively here. It turns out that their mode shapes have both u ' (L) and u(L) negative. Consequently, the only way to ensure compatibility between the negative left-hand side of boundary condition (3) and the positive P on the right is to add the tive and which must be larger in magnitude than P. For a values to the right of the dashed boundary in Fig. 8, it is hypothesized that MT will some- times be unnecessary.

the second response mode

% term, which is nega-

The foregoing offers a key to the difficulties that may be encountered when seeking optimal designs for the continuous bar at higher frequencies. Since each noma2 mode shape U~+~(X) is known to have one more node than its predecessor, the forced-response mode shapes will also be progressively more sinuous. Thus, for w? < we < w3, there will be one or two nodal points; for 0 3 < we < 0 4 , there will be two or three, etc. This has the effect that the optimal solutions will be composed of increasing numbers of differently constrained arcs, with increasingly com- plicated algebraic manipulations demanded by the matching of their various properties at the points of junction. To illustrate, for a > 1.0908, the first-mode solution of § IV consists of a fully stressed arc outboard of an unconstrained arc. Its corresponding u(x) consists of a rectilinear seg- ment outboard of a hyperbolic-sine segment, so that du/dx grows monotonically from root to tip; obvi- ously no node would be possible. solution is a minimum-area arc outboard of a fully stressed one, i.e., a sine curve matched to the end of a straight line. Evidence is presented in 9 VI that, for higher values of a, even this second mode may involve a hyperbolic segment outboard of a sinusoidal segment outboard of a straight line. In regions of maximum slope, du/dx, the full-stress constraint tries to be active, whereas in "loop" regions where du/dx is near zero the minimum area appears. It follows that three arcs might be char- acteristic of a mode with one node, five or six arcs for two nodes, and so on up the scale. In view of the difficulty with matching just two arcs, one's disinclination to tackle higher modes by purelv analytical means is formidable.

The second-mode

VI. Optimal Bar Treated by Finite Elements

By way of a parallel, alternative method for investigating minimum-weight designs, a finite- element approximation to the axially forced bar was subjected to finite-state optimization. As described, e.g., in Ch. 10 of Ref. 31, the bar can be divided into n uniform elements of equal length !L/n (Fig. l(b)), and its equations of sinusoidal response can be written symbolically as

(75)

Here [K] and [MI are stiffness and inertia matrices; Iu? is a column matrix of the successive displace- ments u1, u2, . . . at the outer ends of the ele- ments; and the applied-force-amplitude column matrix {p} contains zeros, except in the last tip row. [K] and [MI involve the areas of the elements, regarded

as a vector of design variables. the ith element is proportional

it is a simnle matter t o quantify maximum stress and minimum area. function for oatimization becomes

n J = C A i

i= 1

Since stress in to bi - Ui-J Y

the constraints on The objective the sum

or its dimensionless equivalent, with possible allowance for a concentrated tip mass.

The full problem has been set up in Ref. 22 by standard methods of constrained algebraic oDtimiza- tion. Solutions were found by means of an extremely efficient optimality algorithm, adapted from Kiusalaas. 3 2 mathematical details, some results are given here.

Without a full presentation of the

To begin with, it is remarked that, when n is large enough to ensure convergence, the finite- element calculations tend to confirm verv accurately the continuum designs of § IV. Figure 12 exempli- fies this conclusion by comparing the discrete and continuous optimal distributions of dimensionless area a for a = 1.0 and 6 = 10. The former were computed with n = 20 elements. Although no special provision was made for a tiu mass, there is seen to be a large concentration in the outboard segment of the bar. Indeed, when multiplied by the dimension- less element length 0.05, the dimensionless tip area 11.695 vields an effective aT = 0.585; this checks quite reasonably the more exact value of 0.560.

Then seeking to isolate first- or second-mode solutions in the finite-element calculations, one must carefully select the initial design vectors. The optimality criterion scheme proceeds through the infeasible region and is therefore unaware of the design space's multiple connectivity, treating the uroblem merely as one with several optima. It is believed that higher response modes could be

i+i handled by imposing the constraint

on the natural frequencies. But these higher fre- quencies seem to be verv sensitive to design-vector changes, and the present method therefore might fail. Perhaps an unconstrained search procedure (e.g., CassisZ4), with large penalties attached to the frequencv constraints, has a greater chance of success.

ai < a < a

Figure 13 plots the second-mode a-distribution, computed by finite elements with n = 20, for a = 2.0 and 6 = 10 (amin = 0.1). As shown in Fig. 6, this case falls well into the parameter ranges where the two-arc continuum solutions of 9 IV are nonoptimal. The dimensionless weight of 0.342 does, however, appear to lie roughly on an extrapolation of the curve in Fig. 6. mass concentration, but the indication is that an actual "IT at the tip is no longer needed for optimality .

There is still an outboard

The curve in Fig. 14 is the response mode u(x) corresponding to the weight distribution (Fig. 131, with each of the 20 element displacements approxi- mated by a straight line. As well as can be deter- mined from the discretized solution, this is a case of three matched arcs. Over the segment marked A , u(x) is a perfect straight line since the stress con"sraint is active. The elements between (x/L> = 0.45 and 0.75 are at amin, so that in the arc approximatelv defined by C the minimum-gauge

480

constraint applies. Here u(x) has the expected sinusoidal shape and goes through a "loop ." Neither constraint is active on arc D, but the length is too short to ascertain whether u(x) is behaving like the hyperbolic function previously encountered. Finally, the single elements B', C', and A' are arbitrarily designated as two matching zones plus a zone for adjustment to the tip boundary condition. They would be expected to shrink to points as n + m.

If it were attempted to solve for the continu- ous bar corresponding to Figs. 13 and 1 4 , it is clear how the three arcs would have to be specified. The matching point locations s = y1 and y2 would be among the 10 or more unknowns to be found from a set of transcendental matching relations. The greatest practical difficulty will arise in the unconstrained arc D. Its system of governing differential equa- tions resembles (13), ( 1 4 ) , ( 2 7 ) , ( 4 9 1 , and ( 5 0 ) . Because simple, homogeneous boundary conditions no longer apply, however, convenient relations like (51) and ( 5 2 ) cannot be assumed. The authors are not yet in a position to say whether an analytical solution to this nonlinear system exists, or whether it would be necessary to resort to numerical integra- tions over the unconstrained arc as part of an elab- orate trial-and-error process. In either event, a very tedious computation would be called for, and one reluctantly concludes that finite elements are the "path of least resistance." Indeed, the obser- vations at the end of 5 V relating to higher-mode designs are strongly reinforced.

VII. Discussion and Conclusions

Although this paper deals primarily with explo- ration of the current limits of function-space oDti- mization as related to one-dimensional elastic con- tinua, the question of practical applications needs to be addressed. In seeking them, one turns to structures intended for dynamic force transmission, vibration isolation, and shock mounting. In many earth-bound devices, the motivation toward saving weight hardly exists, and designers may be expected to stick, as in the past, with uniform rods, beams, bars, etc. Not so for aeronautical and space vehicles, where every Newton of weight may count. There are other surface and marine transportation systems where weight minimization may be required; submarine propeller shafts are an obvious possibility.

To be rigorous, the present designs seem lim-

(a) The applied tip force or torque is trans- ited to the following circum, p t ances :

mitted from its source through an essentially mass- less medium, like a soft spring.

(b) Sinusoidal loads predominate over any start- ing or stopping transients.

(c) The internal stresses due to any static loadings are outweighed by the dynamic stresses.

These conditions do not occur together in many practical situations. On the other hand, (b) and (c) will often apply to a shear-beam or torsional vibration isolator, which supports an engine or machine tool containing some reciprocating source of excitation. In "soft mounting," some very large vibratory stresses may be superimposed on the essen- tially static stress due to the weight andlor they may be in another direction. Then it appears that the present designs might be used, with only the addition of a prescribed tip mass MT. So far as the mathematics is concerned, such a modification is almost trivial.

This suggested extension is one of several that come to mind, wherein continuous structures are involved and the methods borrowed from optimal con- trol theory will retain their potency. Other bound- ary conditions such as simple support may be invoked. Distributed external loads are encountered nearly as often as concentrated ones. There are tractable examples, such as thick-walled or solid rods, where the design variable does not enter certain equations linearly, as a(s) does here. Finally, as demon- strated with free-vibration and aeroelastic con- straints by Weisshaar, tial equations that govern beamlike structures may also be successfully tackled, albeit numerical integration then becomes inevitable.

the fourth-order differen-

Some direct conclusions from what has already

(a) Variational calculus techniques, as adapted been done are:

from the automatic control literature, provide a valuable tool to design one-dimensional structures under certain classes of dynamic loading. Although not as effective as, e.g., energy methods in provid- ing s u f f i c i e n t conditions for global optimalitv, they enable one to penetrate barriers like that which hitherto limited we < w1.

(b) The natural frequencies of the' structure play a central role in creating many local optima and, in the absence of damping, multiple feasible regions. tify the global best design, solution methods must be capable of finding and computing more than one.

(c) It is often necessary to specify minimum- gauge constraints to ensure meaningful optima.

(d) A concentrated mass must sometimes be included in a design to satisfy a boundary condi- tion, although it is not felt that they will be required at interior points.

arcs characterize the optimal solution, the contin- uum approach may be impractical. challenge for the future. Finite-element approxima- tions then offer the only alternative. Special care must be taken both in modeling and in the opti- mal search; frequency-range constraints will be needed to isolate various local optima associated with higher forcing frequency.

Since there are no c( p r i o r i ways to iden-

(e) When more than two different constrained

This remains a

VIII. References

IRryson, A. E. and Ho, Y.-C., Applied Optimal Control, Rev. printing, Hemisphere Publishing Corn., Washington, D. C., 1 9 7 5 .

*Sheu, C. Y. and Prager, W., "Recent Develop- ments in Oatimal Structural Design," AopZied Week- anics Reviews, Vol. 2 1 , No. 10, Oct. 1 9 6 8 , pp. 985- 992.

3Gellatlv, R. A., Editor, Structural Outirniza- t ion , AGARD Lecture Series 7 0 , HamDton, Va., Oct. 1974.

ation Symposium, American Society of Vechanical Engineers, New York, N.Y., Nov. 1974.

Spierson, B. L., "A Survey of Optimal Struc- tural Design IJnder Dynamic Constraints ,I1 I n t e r n - t i o n a l Journal *or !lumerical fitetkods i n E?gineerini;, Vol. 4 , NO. 4 , July-Aug. 1 9 7 2 , pp. 491- 499.

6Fox, R. L. and Kapoor, Y. P., "Structural Optimization in the Dynamics Response Regime: Computational ApDroach," 41AA Journal, Vol. 8 , No.

'Rrach, R. M., "Optimum Design of Beams for Sud- den-Loadings ,I1 Proceedings ASCE, Journal o f the En& neemha Mechanics Diu., Vol. 9 4 , No. EM6, Dec. 1 9 6 8 ,

'Schmit, L. A., Jr., Editor, Structuraz O p t i m i z -

A

1 0 , Oct. 1 9 7 0 , pp. 1798-1904.

PV. 1395- 1407.

481

8Levy, H. J. and Wolf, B. Y., "Fully Stressed Dynamically Loaded Structures," ASME Publication 74-WAIDE-19, ASME Winter Annual Neeting, Nov. 1974.

3R5hrle, H., "Structural Dynamic Optimization of Satellites," Proceedings of t h e Th i rd S y m p o s i m on Large S tmctures f o r Manned Spacecra f t , Frascati, Italy, Oct. 1973.

Framed Structures Under Earthquake Loads," 5th Norld Conference on Earthquake Enpheer<np, Rome, June 1973 (Preprint Paper 376).

Optimization in Aseismic Design," 5th World Confer- ence on Earthquake Engineering, Rome, .June 1973 re rint Paper 374). (' P2Kato , B., Nakamura, V., and Anraku, H., "Opti- mum Earthquake Design of Shear Buildings," ASCE Pro- ceedings, Journal of t h e Engineering b!echanics D i v - is ion, V o l . 98, No. EM&, Aug. 1972, pp. 891-910.

13Venkayya, V. B. and Khot, N. S., "Design of Optimum Structures to Impulse Tvne Loading," A T A A Joum,al, V o l . 13, No. 8, Aug. 1975, PD. 989-994.

Given Dynamic Deflection," Internat ionaZ <Tournal o f Solids and S t r u c t u r e s , Vol. 5, No. 5, "lay 1969, pp.

1'3Solnes, ,J. and Holst, 0. L., "Optimization of

IlNigarn, P?. C. and Narayanan, S., "Structural

141cerman, L. J., "Optimal Structural Design for

473-490. 15Mroz, Z., "Optimal Design of Elastic Struc-

tures Subjected to Dynamic, Harmonically Varving Loads ," Z e i t s c h r i , f t .ftr Lngewandte ?4atkematik und Mechanik, Vol. 50, No. 5, May 1970, pp. 303-309.

16Plaut, R. H., "Optimal Structural Design for Given Deflection Under Periodic Loading," 5uarterlu of Applied Ilathernatics, Vol. 29, No. 2, Julv 1971, pp. 315-318.

I7Ashley, H. and McIntosh, S. C., Jr., "Applica- tion of Aeroelastic Constraints in Structural Optimi- zation," Proceedings of t h e i%el.ftk I n t e r n a t i o n a l Congress of Applied Yechanics, Springer-Verlag , Berlin, 1969.

18Weisshaar, T. A., "An Application of Control Theorv Methods to the Optimization of Structures Having Dynamic or Aeroelastic Constraints," SUDAAR 412, Department of Aeronautics and Astronautics, Stanford Univ., Oct. 1970.

of Aeroelastic Optimization and Some Applications to Continuous Systems," SUDAAR 390, Department of Aero- nautics and Astronautics, Stanford Univ., Jan. 1970.

"Armand, J. P. and Vitte, W. J., "Foundations

20Armand, J. P., "Applications of the Theory of Optimal Control of Distributed-Parameter Svstems to Structural Optimization," CR-2044, Tune 1972, NASA.

Undergoing Harmonic or Stochastic Excitation," Ph.D. Dissertation, Department of Aeronautics and Astro- nautics, Stanford Univ., June 1975 (also NASA

"Johnson, E. H., "Optimization of Structures

CR-142,936). 22Rizzi, P., "The Optimization of Structures

with Complex Constraints Via a General Optimality Criteria Yethod," Ph.D. Thesis, Department of Aero- nautics and Astronautics, Stanford Univ., 1976.

S t r e n y t k o f MaterioLs, 4th Ed., Van Nostrand Co., New York, 1962, D. 225.

Subjected to Dynamic Loads," UCLA-ENG-7451, UCLA, Schoo! of Engineering and Apnlied Science, Tune 1974.

"Sved, G. and Ginos, Z., "Structural Optimiza- tion under YultiDle Loading," in terna t iora i ' fTournal o f fineekanical Sciewe, v o l . 10, 1968, PO. 803-805.

Z6Farshi, R. and Schmit, L. A., Jr., "Minimum Weight Design of Stress Limited Trusses," Droceed- ings oc t h e A Y E , ,Tourno1 of t h e ' tructura? %ixs ion , vel. 100, ST1, Jan. 1974, DD. 97-107.

27Segenreich, S . A. and Rizzi, P., "Properties of Axial or Torsional Free-Vibrational Frequencies of Rods," A I A A Journal , Trol. 13, No. 8, Aug. 1975, OD. 1111-1112.

28Johnson, E. H . , "Disjoint Design Spaces in the Optimization of Harmonicallv Excited Structures," AIA.1 J o u ~ ~ G ~ , Vol. 14, No. 1, Jan. 1976 (to aDpear).

29Vurphy, G. I t . , Ordinarzz D i f f e r e n t i a l Yuuations and T h e i r S o l u t i o n s , Van Nostrand Reinhold Co., Y.Y., 1960.

30Morse, P. M. and Feshbach, H., ,bfethods o" Theore t i ca l Physics, YcCraw-Hill Book Co., Inc.,

"Przemieniecki, J. S., Theory o r Vantriz Stmc-

23Timoshenko, S. and Young, D. H., E l w e n t s o f

24Cassis, J. H., "Optimum Design of Structures

N.Y. 1953, DP. 719-724.

t u r a l Analusis , McGraw-Hill Book Co., Inc., N.Y., 1968.

Structures via Optimalitv Criteria," TN D-7115, Dec. 1972, NASA.

32Kiusalaas, J., "Minimum Weight Design of

t- 1 (a) CONTINUOUS AREA DISTRIBUTION

hl h2 lb) FINITE ELEMENT REPRESENTATION.

Fig. 1 Two idealizations of a cantilever, elastic bar of length E , excited in axial vibration by a sinusoidally varying force at the free end. A concentrated mass MT may be attached at the tip. For finite-element modeling, the bar is divided into n uni- form segments of equal length.

Fig. 2

.

o 2 0 4 0 60 a o

Illustrating the feasible design spaces (unshaded) for a simple, two-element bar force? from its tip at the two indicated values of dimensionless frequency a; a1 and a2 are dimensionless areas adjacent to the root and tip, respectively.

482

1.0

0.8

0.6

7 ; - 0.4

0.2

0 1 .o 2.0 3.0 4.0

(Y

I

-

- 2 a

k I

0 - Y 2

m w 0

- -

P

P

- 5 5

- 2

- 2 W

2

-

I I

Fig. 3 Matching point s = y between the uncon- strained and fully stressed arcs, plotted vs. frequency parameter a for the first- mode designs.

12.0 I I I I

4.0 -

2.0 - (r = 0.8 1 = o

20.0

16.0

12.0

Y X

i’ 1

8.0

4.0

0.0

, t = o ) = o I I I 1 I I I

0 0.2 0.4 0.6 0.8

- I I 1 I I

I I I

I - I

- - - - - - - - - - - - - - - - -

1 1 1 I l l I l l I 1 1 I 1 1

0.4 0.8 1.2 1.6 2.0

.Y/l

Fig. 4 Dimensionless cross-sectional area distribu- tions of the first-mode optimal designs, plotted vs. distance along the bar for five values of frequency parameter a .

- OPTIMAL

- _ _ - FULLY STRESSED

Ly

Fig. 5 Dimensional total weight (or volume) of the first-mode optimal designs as a function of a. Above a = 1.0908, the full$ stressed design is shown for comparison.

I

0 0.5 1 .o 1.5

Ly

1

Fig. 6 Matching point s = y between the fully stressed and minimum-area arcs, plotted vs. a values of parameter 6.

for the second-mode designs at eight

483

0.4 0.8 I I I I

/ /

/ /

/ I I I I

0.6

0.5

a, 0.4

0.3

0.2

1-1 0.0 0 0.2 0.4 0.6 0.8 1 .o

.tA

Fig. 7 Dimensionless cross-sectional area distri- butions of the second-mode optimal designs, plotted vs. distance along the bar for two values each of ci and 6. The correspond- ing dimensionless tip masses aT are indi- cated on the right-hand ordinate.

0 0.5 1 .o 1.5 2.0

a Fig. 8 Dimensionless total weight (or volume) of

the second-mode optimal designs as a func- tion of ci and 6. The first-mode curve from Fig. 5 is shown dotted. To the right of the dash-dot line, the two-arc solutions of § IV are no longer optimal.

0

Fig. 9 First three dimensionless natural frequen- cies of the first-mode optimal designs, plotted vs. a.

2.5 I

cy

Fig. 10 First two dimensionless natural frequencies * of the second-mode optimal designs, plotted

vs. a .

484

_ _- - FIRST MODE----- _- -

Fig. 11 Qualitative relationships between (1) the first two normal mode shapes of a canti- lever bar and (2) possible first- and second-mode forced response amplitudes u(x) due to tip excitation.

0.20 I I I I

- --- EXACT SOLUTION (aT = VTomax /PZ= 0.560)

1 ./

1.

1.

0.

&a P

0.

0.

0.

0.0 0.2 0.4 0.6 0.8

Fig. 12 Typical optimal solution for the bar approximated by 20 finite elements, com- pared with the corresponding continuum solution; CL = 1.0 and 6 = 10.

I I

J

1 I I I I I I I '

.+l

Fig. 1 3 Finite-element optimal solution for n = 20, a= 2.0, and 6 = 10.

0

-0.2

-0.4

-0.6

Fig. 14 Mode shape of forced response for the Fig. 1 3 design (meaning of segments A , B', etc. discussed in text).

485