Lecture Notesin PhysicsEdited by J. Ehlers, Munchen, K. Hepp, Zurich andH. A. WeidenmUller, HeidelbergManaging Editor: W. Beiglbock, Heidelberg

21

Optimization and StabilityProblems inContinuum Mechanics

Lectures Presented at the Symposium on Optimizationand Stability Problems in Continuum MechanicsLos Angeles, California, August 24, 1971Edited by P. K. C. WangUniversity of California, Los Angeles, CNUSA

Springer-VerlagBerlin Heidelberg New York 1973

ISBN 3-540-06214-9 Springer-Verlag Berlin . Heidelberg . New YorkISBN 0-387-06214-9 Springer-Verlag New York Heidelberg' Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machineor similar means, and storage in data banks.Under 54 of the German Copyright Law where copies are made for otherthe amount of the fee to be determined by agreement with the publisher.

by Springer Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-78080. Printed in Germany.

Offsetprinting and bookbinding: Julius Beltz, Hemsbach/Bergstr.

CONTENTS

PART 1. OPTIMIZATION PROBLEMS

H.Halkin: The Method of Dubovitskii-Milyutin in MathematicalProgramming .... 1

R.T.Shield: Optimum Design of Structures Through VariationalPrinciples ...... 13

T.Y.Wu ,A.T.Chwang and P.K.C.Wang: Optimization Problems inHydrofoil Propulsion ..... 38

PART 2. STABILITY PROBLEMS

E.F.Infante: Stability Theory for General Dynamical Systemsand Some Applications .... 63

E.M.Barston: Stability of Dissipative Systems with Applicationsto Fluids and Magnetofluids .... 83

PREFACE

The five papers in this volume represent expanded versions of the

l~ctures presented at the Symposium on Optimization and Stability Prob-

lems in Continuum Mechanics at the University of Southern California,

Los Angeles, August 24,1971. The Symposium was held in conjunction withthe Western Applied Mechanics Conference sponsored by the Applied Mechan-

ics Division of the American Society of Mechanical Engineers with the co-

operation of the University of Southern California.

The objectives of this Symposium were twofold,namely,to introduce re-cent results in general optimization and stability theories which have

potential applications to continuum mechanical systems and to present

new results dealing with specific classes of systems. It is felt that

there is a wealth of new and interesting optimization and stability prob-

lems in continuum mechanics. Hopefully, these lectures will help to

stimulate further research in this relatively new area.

The idea for this Symposium was originally conceived by Professor

C.S.Hsu of the University of California,Berkeley, who also presided over

the Stability Session of this Symposium. Professor G.H.Hegemier of the

University of California, San Diego, served as the Co-Chairman of this

Symposium.

Los Angeles,CaliforniaApril,1972

P.K.C.Wang

*THE METHOD OF DUBOVITSKII-MILYUTIN IN MATHEMATICAL PROGRAMMING

**Hubert Halkin

Department of MathematicsUniversity of California at San Diego

La Jolla,California

1. INTRODUCTION

I want to give here a brief description of a very attractive formalism in optimi-

zation theory: the method of Dubovitskii and Milyutin [1] and the relate some recent

extensions of that method, Halkin [2], with the necessary condition of Fritz John[3].

The first step in the method of Dubovitskii-Milyutin is to notice that, in any

optimization problem, to say that some solution is optimal is equivalent of saying

that a certain family of sets {Si: i E I} have no points in common i. e. ni E:. I Si=ll.Consider for example the optimization problem consisting in minimizing a function f

over all point of the plane R2 where a function g is nonpositive. To say that anA. 2 A

element x 1n R with g(x)

2In the simple example given above. if f and g are differentiable at x, we could con-

sider the convex sets

E 2...." A"x R .f(x)+(x-x)grad f(x)

32mize f(t)=t subject to the constraint g(t)=t

4Let us show that in the case of~ sets ~l and ~2' this new definition of separation

coincides with the classical definition of separation. If ~l and ~2 are (classically)

vector p such that sup E ~ p.x ~ inf E ~ p.x.x 1 x 2

w2 (x)=p'x - sUPxE~lP'x, we thus obtainIf we let wI (x)=-p'x + sUPxE ~/'x and

(i) wI + w2= 0,

(it) w.(x) ~ 0 for all iE{l,2} and all xE~.,1 1

separated, then there exists a nonzero

(iii) both WI and w2 are nonconstant.

In other words, ~l and ~2 are separated according to the new definition. Conversely,

if ~l and ~2 are separated according to the new definition, i.e. if for some WI (x)=

(i) WI + w2=0,

(it) W. (x) :;, 0 for all iE{l,2} and all xE~.,1 1

(iii) at least one of the vectors PI or P2 is different from zero.

Then, by (i), we have P2 = -PI' a 2= -al and, by (iii), we have P2= -PI ~ O. If we

let p = P2 = -PI' we obtain sUPxE ~/.x ~ infx E~/'x and the two sets ~l and ~2 areseparated according to the classical definitions.

We know that two disjoint nonempty convex sets in Rn can be separated, but it isnot correct to say that two separated convex sets in Rn are disjoint. For instance

2the sets ~l = {(xl ,x2): xl~O} and ~2 ={(xl ,x2): xl~O} in the plane R are not dis-

joint, since ~l n ~2 ={(xl ,x2): xl=O} ~~ but they are separated since for p=(l,O)

~O, we have sUPXE~lx,P = infxE~2x.P' If either ~l or ~2 is open however, we know

that the fact that ~l and ~2 are separated will imply that ~l and ~2 are disjoint.The same result can be extended to several convex sets in the following manner:

Theorem 2.1. If {~. : iEI} is a finite family of nonempty convex sets in Rn such1

that niEI ~i is empty, then the family {~i ; iEI} can be separated, Converse-ly, if Wi : i E: I} is a fin:j.. te separated family of convex sets in Rn and if at

most one of them fails to be open, then ni I ~i is empty.The proof of Theorem 2.1 can be found in Balkin [2].

In several applications, we shall assume that 0 E IT. for each i E 1. In1

that case, if the family W.: iEI}1

of

5convex sets is separated by the family {w.: i E I} of affine functions, then we shall1

have w.(O)~ for each iEI and Z'EI w.(O)=O which imply w.(O)=O for each iEI and111 1hence the functions w. are not only affine but linear,i.e. of the form w.(x)~p . x.1 1 1We can thus state that a family {rl.:iEI} of convex sets with OE'?f for each iEI

1 1

is separated if and only if there exists a finite set of vectors {p.:iEI} such that1

(ii) p . x ~ 0 whenever iE I and x E: rl.,1 1

In mathematical program-

(iii) p. :f 0 for some i E 1.1

3. CONVEX APPROXIMATIONS OF SETS

We shall consider three different types of convex approximations of sets;(i)the in-

terior convex approximation (ii)the tangent convex approximation. and (iii) the simpli-

cialk convex approximation where k is a positive integer.

ming, the two concepts of interior convex approximation (asociated with the objectivefunction and the inequality constraints) and of tangent convex approximation (asso-

ciated with the equality constraints) are the most useful. The simplicialk convex

approximation is used chiefly in optimal control theory and is associated with oper-

ator constraints (i.e. when one requires a trajectory to satisfy some differentialequations). however, in the case k~l,i.e. in the case of the simpliciall convex ap-proximation, this concept is also used in mathematical programming under the form of

the Abadie Sequential Constraint Qualification, Abadie [4].We should normally give all the definitions under the form: the set rl is an inte-

rior (resp. tangent or simplicialk convex approximation around a point ~ to a set S,if ... For the sake of simplicity of notation, we shall give all those definitions

. "-

with respect to the p01nt x=O. To go back to the general case, we shall use the fol-

lowing convention: the set rl is an interior (resp. tangent or simplicialk ) convex

approximation around a point ~ to the set S, if the set rl-~ is an interior (resp.tangent or simplicialk convex approximation to the set S-~). If A is the set in Rnand a is a vector in Rn , then we use the notation A-a to denote the set {x-a:x EA}.

Let us specify some further notations. If xERn , then [x] will be the Euclidean

length of x. If AERn, then coA will be the convex hull of A. A set {xl' . , xl} in

Rn is said to be in general position if the vectors x2-xl,x3-xl, ,Xt-Xl are linear-

ly independent.

6Definition 3.1. A subset ~ of Rn is an interior convex approximation to a subset Sof Rn if (i) ~ is open, (ii) ~ is convex, (iii) oEIT and (iv) for all iE~ there existsan E>O such that flxES whenever !x-i!

7is possible the function describing those constraints are not "smooth" enough to ap-

ply the concepts of interior convex approximation and/or tangent convex approximation.

This is the reason why it is convenient to keep operator constraints under their giv-

en forms and to define a special type of convex approxiamtion adapted to those opera-

tor constraints. This special type of convex approximation is the simplicialk convex

approximation. In optimal control theory, the simplicialk convex approximation ~ tothe set S will be the set of all solutions of a certain linearization of the given

family of ordinary differential equations. For more details, see Halkin [5] and

Balkin-Neustadt [6].4. LuE THiOREM OF DUBOVITSKII M~D MILYUTIN

~h 4 1 - t S S S S b subsets of Rn such that ('\+.1 S '" As-L eorem . Le _~, . , -1' 0' 1 e I 11--~ i=w,sume that we have convex sets ~ .. '~l such that~. is an interior convex approxima-

-~ 1

tion to 5i for each i--~, .. ,O and such that ~l is a simpliciall convex approximationThen, the sets ~ ""'~l are disjoint and hence separated.

-~

The proof of Theorem 4.1 is particularly simple.

joint, then there exists an

If the sets ~ .. '~l are not dis--~

element x which belongs to each of them. Since for each

i E. {-~ a} the set~. is an interior convex approximation to the set S . we know1 1

that there exists an 1.>0 such that nxEs. whenever Ix-xl

8all zero. such that

(i) A.:S 0 for each iE:{-]1 . 0}.1

(ii) A..(~) ~ 0 for each i E{-]1 -l}.1 1

(iii) '? A. grad . (~). (x-~):;;;, 0 for all x E:Jl 1""-]1 1 1Proof of Theorem 4.2: Without loss of generality. we assume that ~=O and that O(O)=0. For each i.{-]10} .let S.~{x:.(x)

9that AO~O. If we would know that AO

10

theory of necessary conditions for optimization problems with equality and operator

constraints ( and also inequality constraints, but they never present any difficulti-

es) which will be independent of any sort of constraint qualifications.

7. T~ CASE OF I~EQUALITY!EQUALITY AND OPERATOR CONSTRAINTS

The central part of this section is the following result.

Theorem 7.1. If 1={-]1, ... ,m"l-l} and if {s .. :iEI} and {S""2.:iEI} are families of1 1

subsets of Rn such that (i)(\iEISi=qJ, (ii) for i=-]1, .. ,O, the set S""2 i is an interior

convex approximation to the set S., (iii) for i=l, . ,m, the set S""2. is a tangent con-1 1

vex approximation to the set Si' and (iv) S""2m+l is an (m+l)-convex approximation to

the set Sm+l' then the family Wi:iE I} is separated,

We remark immediately that in the case m=O, Theorem 7.1 coincides with Theorem

4,1. From the counterexample given in Section 6, we know that under the assumptions

of Theorem 7.1, it would be incorrect to say (as in Theorem 4,1) that niEI Si=qJ im-plies that (IiEI S""2 i =qJ ,but we can still assert that the family {S""2 i :iE:I} is separat-

ed and this last statement is all that is needed to obtained appropriate necessary

conditions. The proof of Theorem 7,1, given in Balkin [2], makes a critical use of

Brouwer Fixed Point Theorem.

Let us now assume that we are faced with the following optimization problem:given

a subset S of Rn and functions , , defined over Rn , find an element ~Rn-]1 m

which minimizes O(~) subject to(a) the inequality constraints .(x)~o for i=-]1, . ,-l,

1

(S) the equality constraints .(x)=O for i=l . m.1

(y) the operator constraint ~Es.l Ab .. h nm+.1 S .-_'"The optimality of such e ement x can e expressed by writ1ng t at ~1=-P. 1

where the sets S_]1'" "Sm+l are defined by

S.",{x:xERn..(x):;:O} for i=-]1 .. -l.1 1

SO={x:xERn . (x)

11

sets ~ , . ,~ by the relations-11 m~ "" {x:XERn,cr,(~) + grad cr,(~)'(x-~)

12

[3] John,F., Extremum Problems with Inequalities as Subsidiary Conditions, in "Studi-es and Essays:Courant Anniversary Volume", (K.O.Friedricks,O.E.Neugebauer,and J.J.Stoker, (eds.,pp.l87-204)Interscience Publishers)New York,l948.

[4] Abadie,J., On the Kuhn-Tucker Theorem, in "Nonlinear Programming",J.Abadie(ed.),pp.l9-36,North-Holland,l967.

[5] Ralkin)H., Optimal Control as Programming in Infinite Dimensional Spaces. in"C.I.M.E.:Calculus of Variations.Classical and Modern",pp.l79-192,Eidizioni Cre-monese)Roma,l966.

[6] Halkin.H. and Neustadt.L.W.) Control as Programming in General Normed LinearSpaces) Lecture Notes in Operations Research and Mathematical Economics.SpringerVerlag,ll)l969.23-40.

OPTIMUM DESIGN OF STRUCTURESTHROUGH VARIATIONAL PRINCIPLES

by RICHARD T. SHIELD

Department of Theoretical and Applied MechanicsUniversity of Illinois, Urbana, U. S. A.

1. INTRODUCTION

The application of the calculus of variations to the design of structures for minimum

volume leads to necessary conditions for the structural volume to be stationary, and a local

or global minimum is not guaranteed. However, if an appropriate variational principle ap-

plies for the class of structures under consideration, design criteria can be established

which lead to structures of minimum volume. A direct design procedure was first given by

Michell [1] for framed structures composed of a material of limited strength. For per-

fectly-plastic structures, direct design procedures were introduced by Drucker and Shield

[2,3,4] and here the upper bound theorem of limit analysis provided the appropriate varia-

tional principle. For elastic structures, variational principles can provide direct design

methods for design for a given stiffness, for a given buckling load or for given fundamental

frequency of vibration (see Prager and Taylor [5] and Shield and Prager [6]). The majorpart of this paper surveys the direct design procedures which have been developed through

the use of variational principles.

Section 2 describes the procedures for minimum-volume design of structures of per-

fectly -plastic materials which are required to carry a given set of loads. Section 3 dis-

cusses uniform strength designs in which the structural material is required to be stressed

within a certain range under a given system of loads. The stress range may be chosen to

14

ensure that the stresses remain in the elastic range, for example, or to ensure that an appre-

ciable amount of creep will not occur. Section 4 treats elastic design for a given stiffness in

order to illustrate the design procedures for elastic structures. Minimum-volume framed

structures of material of limited strength in tension and compression are considered in Sec-

tion 5. The Michell design method fails when kinematic constraints are present (except when

the tensile and compressive strengths are equal) but an alternative approach [7] does not

have this limitation. An example is given to show that minimum-volumE7 frames are not nec-

essarily unique, and some new plane structures of the Michell type are described, including

the layout for pure bending.

This paper is not intended to provide a comprehensive survey of the literature on opti-

mum design. The reader will find additional references in [7,8,9,10].

2, PLASTIC DESIGN OF STRUCTURES

In this section we discuss the optimum design of structures composed of perfectly-

plastic materials. A restricted formulation of the problem is indicated in Figure 1. It has

the advantage of ensuring that the structure will consist of the usual structural elements,

such as frames, plates and shells, and the limitations also increase the chances of deter-

mining the optimum solution. We suppose that the structure is to have a prescribed middle

surface A. The loading is prescribed and is distributed over A and its boundary. At sup-

ports either the components of displacement and rotation are prescribed to be zero or the

corresponding components of edge traction and moment are given. For a solid shell, prob-

lem (i), the structure is formed by placing a certain thickness h of a given material at

points of the middle surface. For a sandwich type structure, problem (ii), we suppose that

the shell has a core of prescribed thickness H. The core carries shear force only, and

bending moments and force resultants are carried by membrane stresses in two thin identi-

cal face sheets of thickness h. In both cases we wish to design the shell, that is choose h,

so that the shell is just at collapse under the given loading and is optimum for a given crite-rion, Here we design so that the volume

V =1hdA

is minimized but the methods extend readily to minimization of the functional

15

Figure 1. Solid and sandwich shells

) hf (X) dA,

where f (X) is a non -negative function of position over A. The extension allows for minimum-

weight design when different materials are to be used in different parts of A, and also admits

design for minimum moment of inertia about an axis.

The generalized stresses, such as bending moments and stress resultants across a

section, will be denoted by Ql' Q2' , QN or simply Qno Yielding can occur in the shell

when the stresses ~ lie on the yield surface

F (~; h) := 0

in N-dimensional stress space. For stress states Q represented by points on the yieldn

(1)

surface, purely plastic strain rates are possible; and we use ~ (n := 1, 2, '0, N) to denote

the generalized strain rates, such as rates of curvature change and extension of the middle

surface. The vector representing q is normal to the yield surface at regular points, whilen

at singular points the vector lies between adjacent normals. For a convex yield surface,the plastic rate DA of dissipation of energy per unit area of the middle surface is then

uniquely determined by the values of ~,

For the solid shell, DAis quadratic in h in general; while for the sandwich shell, DAis

(2)

16

directly proportional to h. Shear forces have little influence on yielding, even for highly

localized loading [11], and they are not included in the generalized stresses ~.

The theorems of limit analysis [12,13] can be used to provide information about the

volume of a design which can carry the loads. The appropriate theorems are the following:

Lower-bound theorem. If the applied loads can be carried by an equilibrium distribution of

moments and force resultants Q in the shell which are at or below yield, the loading is atn .

or below the collapse loading.

Upper-bound theorem. If the applied loads are such that a deformation of the shell can be

found for which the rate at which the applied loads do work exceeds the rate of internal

energy dissipation, the loading is above collapse.

The lower-bound theorem can be used to determine upper bounds on the volume of the

minimum-volume design for given loads [3]. If a stress distribution Q over A is in equi-n

librium with the applied loads, we can choose the thickness hs

so that the yield condition (1)

is satisfied everywhere on A. Since the design h is then a permissible design, the minimums

volume Vm must be less than or equal to Vs'

Vm ::: Vs == ) hs d A.

The upper -bound theorem can be used to provide lower bounds for V , as in [3], butm

the theorem also leads to a direct design procedure [4]. For a .shell h which is at ors

below collapse under the applied loads and for any kinematically admissible velocity field

u. (i = 1,2,3) we have1

(' DA (q ; h ) dA - CT. u. dA "= 0,j n s ) 1 1 (3)

for otherwise the use of the upper-bound theorem with the deformation u. would predict that1

the loading exceeds collapse. In (3), T i are the components of the applied loads, the re-

peated index i implies summation over 1, 2, 3, and the integral of T. u. is to include the1 1

rate of work at the edge of the shell. Inequality (3) is a variational principle for the permis-

sible design hs

; equality holds in (3) only when u. is a collapse mode for h. The applica-1 s

tion of this principle to the determination of a minimum-volume design is much more direct

than the use of the calculus of variations. Suppose that a design h for a solid shell, prob-e

lem (i), is just at collapse under the loads in a collapse mode u.; and suppose that a neigh-1

boring design hs

== h + 6 h is also a permissible design. If we neglect second order terms,c

17

the dissipation rate for the shell hs

in the defonnation mode ul is

assuming that DAis continuously differentiable in h. Applying the variational principle (3)to the design h , we conclude that

s

( 8 DJ0 h ahA dA ~ 0because equality holds in (3) for the design h. It now follows that if the design h is such

c c

that

over A then

~ ohdA 2: 0,so that the design h provides a relative minimum for the volume of permissible designs.

c

Thus, assuming that the neglect of the second -order terms is permissible, the variational

(4)

(5)

principle leads to a direct design procedure. Mroz [14] has given an example in which the

application of (4) leads only to a stationary value for the volume.

The case when DAis directly proportional to h, as for the ideal sandwich shell of

problem (ii), is simpler and a stronger result is possible. We again suppose that the design

h is at collapse in a deformation mode u., and we use the mode u. in the variational prin-c 1 1

ciple (3)

Since

for another permissible design h. We obtains

~ DA (~; hs ) dA 2: ~ Ti ui dA = ) DA (~; hc ) dA.

DA (q ; h )n s

we can conclude that if the design h is such thatc

= constant (6)

over A, then

18

Not only does the condition (6) lead to an absolute minimum for the design volume but the

condition (6) is much easier to use than condition (4) because (6) does not involve the designthickness directly.

In order to illustrate the use of these design methods we consider the minimum-volume

design of a circular plate with a built-in edge under uniform pressure loading on its upper

face (for problems involving other symmetric and non -symmetric pressure distributions see

[15,16]). When the Tresca yield condition is assumed, the yield condition on the radial bend-

ing moment M and the circumferential bending moment N is the familiar hexagon in (M, N)

space,

max (IMI ' INI ' 1M - NI) = M o'

2in which Mo

= (J H h for the sandwich plate and M ;:;: i (J h for the solid plate. The cur-000

vature rates K, >-- in the radial and circumferential directions are derived from the downward

deflection rate w of the middle surface through

1 dwr dr

where r measures distance from the center.

For the sandwich plate, the design criterion (6) does not involve the design thickness h,and it is readily found that (6) can only be satisfied for a finite range of r when either

(i) M=N=M,o

or (ii) M=M,o

N = O.

For these regimes, condition (6) reduces to

(i) K + >-- = 0., or (ii) K = o.,

where 0. is a constant when the core thickness H is constant. The curvature rates K, >--

must have the same sign in regime (i) while in regime (ii) they are of opposite sign and

IKI 2:: I >--1 For a plate with a built-in edge, regime (i) with the positive sign will apply ina central region r :S a and regime (ii) with the negative sign will apply in the remaining por-

tion a :S r :S R. At the built-in edge wand d w /d r are zero and d w /d r is zero at the cen-

ter. In order to have wand d w/d r continuous at the junction r = a, it is found that we

19

(0 )

( b)

R

Figure 2. Minimum -volume designs for built-in circular plateunder uniform pressure (a) sandwich plate (b) solid plate

must have a ;:; 2 R/3, and at this radius M must vanish because K changes sign. With M

;:; 0 at r ;:; 2 R/3, the moment distribution in the two regions of the plate, and hence the

thickness h;:; lMI /(5 H, can now be found from equilibrium. The design is indicated ino

Figure 2 (a).

For the solid plate, we again assume that M and N have the fully plastic value M foro

for r s: a and that N is zero for a s: r s: R, with M := 0 at r ;;;; a. From equilibrium the

moment distribution and therefore the thickness h(r),

can be found for a general value of a. In order to satisfy the design criterion (4) we musthave

h h

in the inner and outer regions, respectively. The continuity of dw /d r at r a leads to

(see [15])a~ rdr) h (r)o

R

;:; a ~a

drh(r) ,

20

and this equation serves to determine the junction radius a. For uniform pressure loadinga = 0.664 R and the design is as indicated in Figure 2 (b).

Minimum -volume design for other one -dimensional situations, such as symmetrically

loaded circular cylindrical shells of sandwich type [4], is also relatively straight-forward,

but design problems which are two-dimensional can be much more difficult to treat. So far

designs for non-circular plates with built-in edges have only been obtained by an inverse

method [17].

The design procedure can be modified to include body forces (such as weight) which act

only when material is present (see [4]). Also the procedure has been extended to the design

of multi -purpose structures which are to support different systems of loads at different times

[18], and to the quasi-static design of structures under moving loads [19].

3. UNIFORM STRENGTH DESIGNS

The methods for plastic design have been extended [7] to materials which are not per-

fectly plastic, so that design limitations other than plastic collapse are involved. For ex-

ample, we may use a work-hardening material and in order to minimize the possibility of

fracture we may wish to design the shell so that it remains elastic everywhere. For another

material, we may wish to keep the stresses below the level at which an appreciable amount

of creep will occur. In both cases there will be a limiting surface in stress space to restrict

the stress states in a section of the shell. We shall say that a design is a uniform strength

design for the given loading if the stresses everywhere in the shell are on the limiting sur-

face. We seek the uniform strength design which has least volume.

Figure 3 indicates a limiting surface in generalized stress space. We suppose that the

surface is given by

(7)

where L is a known function, and we assume that the surface is convex. In purely elastic

design, for example, the limiting surface is determined by those values ~ for which the

yield limit is reached in the outer fibers of the shell.

Consider an infinitesimal virtual deformation of the shell defined by middle surface

displacements v. and associated generalized strains e We shall say that a virtual defor-1 n

21

Figure 3, Limiting surface in generalized stress space

mation with strains e is compatible with a limiting stress state Q if the vector represent-n n

ing e is normal to the limiting surface at the stress point Q , Figure 3. At a singular pointn n

of the limiting surface the strain vectors representing compatible deformations lie in the fan

bounded by adjacent normals. For a given convex limiting surface, the virtual work WA in

a compatible virtual deformation is then uniquely determined by the virtual strains en'

WA = Q e ;; WA (e ; h),n n m

in which the repeated index n implies summation over 1, 2, , N. Moreover, for any others

stress state Q inside or on the limiting surface we haven

QS e S; W (e . h)n n A m'

with equality only if e are also compatible with QS.n n

The approach to minimum-volume design for uniform strength is similar to that for

( 8)

plastic design, and we shall treat the case when WA is directly proportional to the design

thickness h. If the shell h has limiting stresses under the loads and if there is an admis-c

sible compatible virtual deformation vi' then by virtual work we have

lTi vi dA = ~WA (en; hc ) dA. (9)For any other shell h

swith stresses ~ at or below the limiting values we can use (8) and

22

virtual work to deduce that

where v. is any admissible virtual displacement. We therefore have the variational principle1

( WA (e ; h ) dA - (T. v. d A 2:: 0} n s j 1 1 (10)

for permissible designs and equality holds only when h is a uniform strength design and v.s 1

is compatible with the stresses ~ in the shell. We now use the compatible virtual deforma-tion v. for the design h in the variational principle (10) and with (9) we derive

1 c

Because WA is proportional to h, we see that if the design hc

is such that

'" constant

over A, then the volume of h will be an absolute minimum for all permissible designs.c

For the solid shell the design criterion is

constant

over the shell.

Uniform strength designs have been discussed by Save [20].

4. ELASTIC DESIGN FOR GIVEN STIFFNESS

Direct design methods can be developed in the same way for other problems of optimum

design provided that a suitable variational principle holds for the structure under investiga-

tion. This can be the case in the minimum -volume design of an elastic structure which is to

have a given stiffness under a given set of loads (or, equivalently, elastic design for maxi -

mum stiffness with a given volume of material). Other examples are minimum -volume design

for a given buckling load or for a given fundamental frequency of vibration. Techniques for

design problems such as these that have been developed in a unified way by Prager and Taylor

[5]. Here we outline the procedure in the case of elastic design for a given stiffness.

23

For an elastic shell there is a strain -energy function EA' per unit area of the middle

surface. which is uniquely determined by the generalized strains q derived from middle sur-n

face displacements ui . The strain energy also depends on the design thickness h so that we

write it as EA (Clu; h). The potential energy U is defined as

U {u*; h} == 1EA (~; h) dA - ~ Ti ul' dA.where the integral of T. u. represents all the virtual work of the prescribed loads including

1 1

the edge loading and where u: is a displacement field which satisfies any imposed displace-1

ment conditions. When EA is a positive definite quadratic function of the strains, the Prin-

ciple of Minimum Potential Energy holds. The principle states that the potential energy U is

minimized by the actual displacements u. produced by the loads.1

U{u*;hr ~ Ufu;h}.We now define the compliance of the shell for the given loads to be twice the total strain-

energy of the shell and we note that

2JEA (Clu; h) dA ::: ) Ti ui dA.For two designs hand h with the same compliance. we have

s

(11)

where qS are the strains for the design h. The inequality in (11) follows from the Principlen s

of Minimum Potential Energy applied to the design hs

' When EA is directly proportional to

h. we see from (11) that in designing for a given compliance, the design with EA/h constant

will have least volume. For other types of shells the procedure would be to design so that

8 EA J8 h is constant over the shell, and the design would provide a relative minimum for the

volume of permissible designs.

As a simple example. suppose we have an elastic beam of length 2 , which is built-in

at both ends and which has a transverse point load P at the center. We wish to design the

beam so that the central deflection does not exceed 6 and such that the beam has minimum

volume. For a beam of the sandwich type. minimizing the volume is the same as minimizing

the integral of the bending stiffness over the beam. If two beams with stiffnesses s and s

have the same central deflection 6 under the load. they have the same compliance PO and in

24

the same way that (11) was derived we can use the Principle of Minimum Potential Energy to

get

r- 2 r 2J s K dx::: ) s K dX,where K is the curvature of the design s under the load P and x measures distance from one

end. We now see that the design s will have least volume if IKI is constant. In order to

satisfy the constraints at the ends, the deflection with constant lKI must have inflection points

at the quarter points x == /2, 3 /2. Since the moment M == s K must vanish at the quarter

points where K changes sign, the moment distribution is now statically determinate and M (x)

and therefore s (x) can be found.The design procedure obtained from the Principle of Minimum Potential Energy applies

for design with given compliance. However, the design criterion does not always coincide

with the compliance. Thus if we have a distributed load over the built-in beam and we wish to

limit the central deflection as before, the compliance will not be known in advance. Similarly,

if we have an off-center point load P at the section x == x and we wish to limit the maximumo

deflection of the beam, the compliance is P u , where u is the deflection at x == x and is000

not necessarily the maximum deflection. These design problems can be approached by using

a variational principle of a different type called the Principle of Stationary Mutual Potential

Energy [6]. Let ui and ui be two middle surface displacement fields for a design of thick-

ness h and let qn' ~ and ~, Qn' respectively be the associated generalized strains and

stresses. We define the mutual strain energy through

NL; Q -

n qn1

NL; Qq.1 n n

For two different sets Ti and Ti of applied loads, the mutual potential energy UM is defined

as

where ut, ut are kinematically admissible displacement fields.If ui and ui are the actual displacements that the loads Ti and Ti , respectively, would

induce in the shell, then

(12)

25

With the use of the Principle of Virtual Work, it can now be shown (see [6] for details) that

(13)

If we apply (13) when ut and u; are neighboring displacements to the actual displacementsui and ui ' the right-hand side will be zero to first order. Thus UM {u~', u"; h} is stationaryat the values u: =' U., u:' ;:;; u., and this is the Principle of Stationary Mutual Potential Energy.

1 1 1 1

Suppose we wish to design a structure so that the transverse deflection at a particular

point Xo

of the mid-surface is of amount 0 under the loads Ti We take the second system

of loads T. to be a single unit concentrated load P acting normal to the middle surface at the1

point Xo

' From (12) we then see that the value of UM fu, u; h} is - Po, so that designswhich satisfy the design criterion will have the same value for UM fu, u; h}. We can there-fore use the Principle of Stationary Mutual Potential Energy in the same way as the Principle

of Minimum Potential Energy was used in design for a given compliance. In this way we find

that the design such that

()Cfh EA (qn' ~; h) ;:;; constant

over the shell will provide a stationary value for the volume for designs which have trans-

verse deflection of amount 0 at the point X o

Applications to the minimum-volume design of beams for given deflections (or rotations)

are described in [6]. Suppose we wish to design a beam of sandwich type and we require the

deflection at the section x ;:;; x to be of amount 0 under a certain system of loads. Let so

and s* be the bending stiffnesses of two designs that satisfy the constraint on the deflection

at x , and let u, u* and U, U*' be the corresponding deflections of these designs under theo

given loads and under a unit concentrated load P at x. From (12) we haveo

where we have identified the bending stiffnesses sand s" with the design thicknesses hand

h", as we may do for sandwich beams. The deflections u, u are kinematically admissible

for the design s~' and if we apply (13) to this design we get

UM {u, u; s'~} - UM {u':', u*; s*};:;; j s':' (K'~ - K) (i

26

UM \ u'~, u"; s*} by UM {u, u; s1in (14) and use the definition of UM we find that\ (s* - s) KK dx == 1s* (K* - K) (K* - i

27

length 2 which is built-in at both ends and is loaded by a uniform pressure p along its

length. We wish to restrict the deflection at the center x = to be of amount 6. For a beam

built-in at both ends, u" (x) must change sign at least twice for otherwise no deflection is

possible; thus M (x) will have at least two zeros. Assuming a symmetrical design, we sup-

pose that the bending moment M (x) is zero at x ;;;; b. If we now consider only designs

for which the stiffness vanishes at x ;;;; b, we have a statically determinate beam and we

can determine the design (17) which has least volume in this class of designs. We can now

choose b SO that the volume will have the least value for all possible designs, and this value

of b is found to be 12.01.

When the loading is not symmetric, the maximum deflection may be off center. Sup-

pose, for example, that we have a simply supported beam of length 2 under a system of

loads which produces a bending moment M (x). We wish to limit the maximum deflection to

an amount 6. Let M (x) be the bending moment distribution caused by a unit point load P at

x = b. The design (17) will then be optimum for a deflection of amount 6 at x ;;;; b. We can

ensure that the section x = b will have the greatest deflection if we choose b so that u' (x) is

zero at x ;;;; b. In order to determine b, we note that if u (b) = 0 and u = 0 at the ends,

then

u (b) ;;;;

and this implies that

b b

\ \ u" (x) d x d Yo Y

2 y~ \ uti (x) dx d y,b b

b~ xKdx;;;;o

2~ .(2 - x) Kdx.b

(18)

Because K Mis1

c (M/M)2, we can write (18) as

2= \ (2 - x) (M/M)! dx,

o

and this equation serves to determine b. To give an example, when the beam is loaded by a

point load P at x = a, the value of b varies from to 1.11 as a varies from to 2 .

28

The procedures described here for elastic design can be extended to design with two or

more constraints on deflection or rotation under a single system of loads [6] and to the design

of multi -purpose structures [6,21].

5. MICHELL STRUCTURES

In formulating the problem of optimum design in Section 2, we assumed that the type of

the structure and the layout, that is the middle surface A, were specified. A less restrictive

fonnulation merely specifies the region in which the given material can be placed and leaves

the type and layout of the structure to be determined. In 1904 Michell published his paper [1]

on the minimum-volume design of framed structures. He specified that the structure should

consist of tie -bars in tension and struts in compression, but the layout of the structure was

not specified. The material to be used allows a maximum tensile stress (Jt and a maximum

compressive stress (J , and for a design which carries the prescribed loads, the minimumc

volume allowable is

v = 2: 2t f /(Jt + 2: 2 f /(J t c c c (19)

Here ft is the tension in any tie-bar of length 2t and fc is the thrust in any strut of length

2. Michell showed that a framed structure will be of minimum volume if there is a virtualc

small deformation of the space such that each tie -bar suffers an extensional strain of amount

e and each strut suffers a compressive strain of amount e and no linear element of space

suffers a strain numerically greater than e, where e is a constant. Note that the actual

deformation of the minimum-volume frame under the loads involves extensional and compres-

sive strains of amounts (Jt/E and (Jc/E, respectively, along the frame elements, where E is

Young's modulus.

In the proof of his results, Michell used a theorem due to Maxwell. By imposing a uni-

form dilatation on the whole of space, Maxwell showed that for all structures under the same

system of applied loads

(The constant is 2: F E, where F is an applied load at a point with position vector !.) How-ever, Maxwell's theorem does not apply to structures with kinematic constraints imposed by

29

support conditions because the reactions at the supports can vary with the structure. An ex-

ception is a structure with one fixed point but in this case the reaction at the support is deter-

mined by overall equilibrium. Because Maxwell's theorem is essential to Michell's proof

when CTt f. CTC ' the design procedure of Michell will not be valid in general when kinematic

constraints are imposed. This limitation on the use of Michell's method does not appear to

have been mentioned explicitly in the literature.

An alternative approach, which does not have the limitation of the Michell method, has

been given by Shield [7]. The procedure is to design a frame compatible with a virtual small

deformation in which the principal strains are of magnitude e/CTt if extensional and of magni-

tude e/CT if compressive, the directions of frame elements coinciding with the principalc

directions of strain as before. The virtual deformation must satisfy any imposed kinematic

constraints. The proof that the procedure leads to a minimum -volume frame is straight-

forward and it makes direct use of the Principle of Virtual Work as in the method of Section 3

for unifonn strength designs. The proof has been repeated by Hemp [22] and by Hegemier

and Prager [23] for the case CTt = CTc (when the Michell method and the alternative method

become identical).

Michell [1] supplied some examples of minimum -volume framed structures and other

examples are given in [22,24,25,26]. Cox [27] has shown that a Michell structure has

greater stiffness under the loads than any other structure which is stressed to the limiting

values CTt and CTc' More recently, Hegemier and Prager [23] have shown that an elastic

frame with a specified stiffness (i. e. compliance) has least volume when it has the layout of

a Michell structure, and this holds also for frames designed for a given stiffness in stationary

creep or for a given fundamental frequency of vibration. In the following we give an example

to show that minimum-volume frames are not necessarily unique, and we describe some new

additions to the list of Michell structures.

The diagram at the top of Figure 4 indicates the layout given by Michell [1] for a single

force applied at the midpoint C of the line A B and balanced by equal parallel forces at A and

B. The struts AD, E B and the curved bar DE carry a uniform compressive force and a

quadrantal fan of tie-bars from C to DE maintains the equilibrium of the curved bar. The

layout is symmetrical about A B with tie -bars replacing struts and vice-versa. The virtual

deformation with principal strains e associated with the layout can be adjusted so that the

30

A B

A B

Figure 4. Load at C supported at A, B

31

displacement is zero at points A and B. If we assume ITt = ITc' we can use this virtual

deformation for the case when we have the same force at C, but now A and B are fixed

points of support. The optimum structure has the same volume as the structure with speci-

fied parallel forces at A, B, but the optimum design is not unique. For example, the load at

C can be carried by a frame entirely above A B as indicated in the middle diagram of Figure

4. An infinity of optimum designs results from arbitrarily assigning a fraction of the load at

C to be carried by a structure above the line A B and the remainder by a structure below the

line AB. We note that if we had specified that the load at C be carried by a beam with center-

line AB and built-in at A and B, the optimum design would have bending moments at A and

B. The Michell structure has no moments at the fixed points A, B.

The minimum -volume design indicated at the bottom of Figure 4 uses the same virtual

deformation with principal strains e, but now it is specified that distributed loads at A and

B balance the load at C.

Figure 5 shows the optimum layout for pure bending. A bending moment at the point A

is to be transmitted to the point B by a framed structure of minimum volume, composed of a

material of limited strength (or the structure has an assigned bending stiffness). In the cir-

cular regions around the points A and B, the tie -bars and struts follow logarithmic spirals.

The spiral regions are connected by a strut G H in compression at the top and a tie -bar carry-

ing the same force at the bottom of the structure. The associated virtual deformation with

principal strains e is, apart from a rigid displacement, purely circumferential in the cir-

cular regions. The regions between the larger circles and the straight-line boundaries (such

as G C, C H) of the upper and lower quadrants which meet at C move as rigid bodies. In the

quadrants meeting at C, the principal strain directions are vertical and horizontal, and the

quadrants deform like a plastic hinge in a beam in pure bending. The total volume of material

required is

M[1 + 2In _a ] [.1-- + l.-] .,;[2r ITt ITo C

Here M is the moment applied at A and B, a is the length of A C or A Band r is the ra-o

dius of small circles at A and B over which the forces equivalent to the moments Mare dis-

tributed.

32

~::Ja.

~o

.....

-::Jo>.o...J

33

Figure 6 indicates the optimum layout when a downward force P is added at the central

point C and upward parallel forces P/2 are added at the points A and B. The moment M

applied at A and B and the force P are related to the angle 2 Q of the fan regions through

4M/pa = cotQ - 1.

As the ratio P1M increases, the angle Q tends to 1T /4 and the structure approaches the

Michell structure for three parallel forces. It may be noted that the moments at A, B are of

opposite sign to those that would be developed at the ends of a built-in beam by a downward

central load. The optimum layout for the case of reversed moments at A and B remains to

be determined. In the particular case when there is no moment across the central section,

that is the case M = P a/2, the optimum layout is as shown in Figure 7. In the associated

virtual deformation, the space outside the circular regions does not move while inside the

circular regions the displacement is purely circumferential.

Acknowledgment. The author would like to thank D. E. Carlson for helpful discussions.

The manuscript was typed by Mrs, R. A, Mathine.

34

---/' ........

/ ""I \\ ( \ /

\1 lj~ f

/1 \/ \ )\

\ I" /" /.......... /'

----

"0Co

c~+-cQ)o

ClC

"0CQ)

CD

Q)~::JCl

LL

35

36

REFERENCES

1. Michell, A. G. M. The Limits of Economy in Frame-Structures. Phil. Mag. ~, 589-597 (1904).

2. Drucker, D. C., and Shield, R. T. Design for Minimum Weight. Proc. 9th Interna-tional Congress of Applied Mechanics, Brussels 1956, pp. 212-222.

3. Drucker, D. C., and Shield, R. T. Bounds on Minimum Weight Design. Q. Appl.Math. 12, 269-281 (1957).

4. Shield, R. T. On the Optimum Design of Shells. J. Appl. Mech. 27, 316-322 (1960).

5. Prager, W., and Taylor, J. E. Problems of Optimal Structural Design. J. Appl. Mech.35, 102-106 (1968).

6. Shield, R. T., and Prager, W. Optimal Structural Design for Given Deflection.J. Appl. Math. Phys. (ZAMP) 21, 513-523 (1970).

7. Shield, R. T. Optimum Design Methods for Structures. Plasticity, Proc. 2nd Symp.Naval Struct. Mechanics, Providence 1960, pp. 580-591.

8. Wasiuly:hski, Z., and Brandt, A. The Present State of Knowledge in the Field of Opti-mum Design of Structures. Appl. Mech. Rev.~, 341-350 (1963).

9. Sheu, C. Y., and Prager, W. Recent Developments in Optimal Structural Design.Appl. Mech. Rev.~, 985-992 (1968).

10. Prager, W. Optimization of Structural Design. J. Optimization Theory and Applic. ~,1-21 (1970).

11. Anderson, C. A., and Shield, R. T. On the Validity of the Plastic Theory of Struc-tures for Collapse under Highly Localized Loading. J. Appl. Mech. 33, 629-636 (1966).

12. Drucker, D. C., Prager, W., and Greenberg, H. J. Extended Limit Design Theoremsfor Continuous Media. Q. Appl. Math. 2, 381-389 (1952).

13. Prager, W. General Theory of Limit Design. Proc. 8th International Congress ofApplied Mechanics, Istanbul 1952.

14. Mr6z, Z. The Load Carrying Capacity and Minimum Weight Design of Annular Plates.Rozpr. InZyn. (Engin. Trans., Warsaw) 114, 605-625 (1958).

15. Onat, E. T., Schumann, W., and Shield, R. T. Design of Circular Plates for MinimumWeight. J. Appl. Math. Phys. (ZAMP) ~, 485-499 (1957).

16. Prager, W., and Shield, R. T. Minimum Weight Design of Circular Plates under Arbi-trary Loading. J. Appl. Math. Phys. (ZAMP) 10, 421-426 (1959).

17. Shield, R. T. Plate Design for Minimum Weight. Q. Appl. Math .!..., 131-144 (1960).18. Shield, R. T. Optimum Design Methods for Multiple Loading. J. Appl. Math. Phys.

(ZAMP) 14, 38-45 (1963).

19. Save, M. A., and Shield, R. T. Minimum -Weight Design of Sandwich Shells Subjectedto Fixed and Moving Loads. Proc. 11th International Congress of Applied Mechanics,Munich 1964, pp. 341-349.

20. Save, M. A. Some Aspects of Minimum-Weight Design. Engineering Plasticity,Cambridge Univ. Press 1968, pp. 611-626.

37

21. Prager, W., and Shield, R. T. Optimal Design of Multi-Purpose Structures. Int. J.Solids Structures !, 469-475 (1968).

22. Hemp, W. S. Studies in the Theory of Michell Structures. Proc. 11th InternationalCongress of Applied Mechanics, Munich 1964, pp. 621-628.

23. Hegemeir, G. A., and Prager, W. On Michell Trusses. Int. J. Mech. Sci. Q, 209-215 (1969).

24. Prager, W. On a Problem of Optimal Design. Non-homogeneity in Elasticity andPlasticity, Pergamon Press 1959, pp. 125-132.

25. Hu, T. C., and Shield, R. T. :Minimum-Volume Design of Discs. J. Appl. Math.Phys. (ZAMP) g, 414-433 (1961).

26. Cox, H. L. The Design of Structures of Least Weight, Pergamon Press 1965.

27. Cox, H. L. The Theory of Design. Aeronautical Research Council Report 19791,Great Britain (1958).

OPTIMIZATION PROBLEMS IN HYDROFOIL PROPULSION':'

Th. Yao-tsu Wu, Allen T. ChwangCalifornia Institute of Technology, Pasadena, California

and Paul K. C. WangUniversity of California at Los Angeles

This paper attempts to apply the principle of control theory to investigate thepossibility of extracting flow energy from a fluid medium by a flexible hydrofoilmoving through a gravity wave in water, or by an airfoil in gust. The presentoptimization consideration has led to the finding that although the flexible hydrofoilmay have an infinite number of degrees of freedom, the optimum shape problem isnevertheless a finite-dimensional one. The optimum shape sought here is the Onewhich minimizes the required power subject to the constraint of fixed thrust. Aprimary step towards the solution is to reduce the problem to one of minimizing afinite quadratic form; after this reduction the solution is determined by the methodof variational calculation of parameter s. It is found that energy extraction isimpossible if the incident flow is uniform, and may be possible when the primaryflow contains a wave component having a longitudinal distribution of the velocitycomponent normal to both the mean direction of flight and the wing span. Whensuch waves of sufficiently large amplitude are present, not only flow energy butalso a net mechanical power can be extracted from the surrounding flow.

':' This paper includes further extension to that which was originally presented atthe Symposium.

39

1. Introduction

SorTIe previous observations on fish swiIllIlling and bird flight seeIll to suggest

that SOIlle species Illay have learned, through experience, to acquire the key tohigh perforIllance by executing the optiIlluIll IlloveIllent that Illay be of great interestto control theory related to fluid Illechanics. An especially intriguing aspect ofthe optiIllization probleIll concerns with the possibility of extracting energy froIllsurrounding flow by an oscillating lifting surface ( such as the fish body and fins,bird wings, and artificial wings like airfoil and hydrofoil) and its associatedeffect on the control of Illotion.

This general probleIll has been explored to various degrees of generality.Based on the approxiIllation of potential flow with sIllall aIllplitude, it has beenfound by Lighthill (1960) for slender bodies, and by Wu (1961) for two-diIllensionalplates, that if the basic flow is uniforIll, energy is always iIllparted by an oscillat-ing wing to the surrounding fluid, and an extraneous Illechanical work IllUSt there-fore be continuously supplied to Illaintain the Illotion. Even though it is iIllpos siblein this case to extract energy froIll the flow field, the highest possible hydro-Illechanical efficiency that can be attained by a wing, subject to delivering a givenforward thrust, can be very high, as found by Wu (1971 b,c) for the two-diIllen-sional plate and a slender lifting surface.

As was subsequently pointed out by Wu (1972), the situation becoIlles drastical-ly different when the basic flow is no longer uniforIll, but contains a wave COIllPO-

nent,such as gravity waves in water, or wavy gust in air. The contention that the

wave energy stored in a fluid IllediuIll can be utilized to assist propulsion has beensuggested by intuitive observations. Sea gulls and pelicans have been observedto skiIll ocean waves over a long distance without Illaking noticeable flappingIllotions (save SOIlle gentle twisting) of their wings. In an extensive study of theIlligrating salIllon, Osborne (1960) found that the increased flow rate in a swollenriver did not slow the salIllon down (for known biocheIllical energy expended duringthe travel) by that Illuch a Illargin as would be predicted by the law of resistancein proportion to the square of their velocity relative to the flowing water. Severalpossible explanations were conjectured by Osborne, including the prospect thatthe flow energy as sociated with the eddies in river could be converted to generatethrust. To explore this possibility Wu (1972) introduced an energy considerationto an ear lie r study of WeinbluIll (1954) on the probleIll of heaving and pitchingof a rigid hydrofoil in regular water waves. It was found that the greatestpossible rate of energy extraction is provided by the OptiIllUIll Illode of heavingand pitching. When waves of sufficiently large aIllplitude are present, not onlyflow energy but also a net Illechanical power can be extracted froIll the wave

field.

40

In the present study this problem is further generalized by allowing the

hydrofoil to be flexible so as to admit an infinitely many degrees of freedom ( ofsmall amplitudes). This general problem merits study for several reasons,First, it is of a theoretical interest to find out how much improvement in the

hydromechanical efficiency and energy extraction can be gained by admitting theadditional degrees of freedom. Second, the results of the present study of energytransfer between an oscillating body and surrounding stream can be useful to thedevelopment of control theory for hydrofoil ships and to the analysis of flutterphenomena. In the case of flutter in a uniform stream, it is usually assumedthat the engine maintains the constant forward speed regardless of the flutter-created inertial drag, In a wavy stream, however, the flutter may create apropulsive thrust, which may amplify further instability and a self -excited fluttermay develop. Some of these aspects have already been observed by K~ssner (1935)and Garrick (1936, 1957); this paper is aimed at the general case of propulsiveenergy balance.

Further, from the standpoint of development of control theory, the presentproblem also merits study in its own right since it presents some new featuresand difficulties that-apparently do not confirm with the known classical cases, Abrief description can be given as follows. Section 2 presents the general (lineariz-ed) theory for a two-dimensional hydrofoil oscillating in waves, which is appliedin Section 3 to the general case of a flexible plate wing. In Section 4 the problemof optimum motion is formulated as to find a hydrofoil profile that minimizes theenergy loss C E subject to a constrained thrust coefficient C T , It is shown thatalthough the flexible hydrofoil may have infinitely many degrees of freedom, C Eand C T can be reduced to quadratic forms of finite dimensions, After this crucialstep the optimization problem reduces to one defined on a three-dimensional vector

space ( Sl' s2' s3)' With this drastic reduction it is possible to show that an optimalsolution does not exist unless appropriate bounds are imposed on the independentvariables SIS, Under this condition the optimal solution is determined and

n

compared with the previous special cases, It is felt that the present method ofsolution is still heuristic, to some extent, for much of the intuitive physicalpicture was relied on for guidance. It is with the hope to stimulate furtherdevelopment of the general theory for this class of control problem that thepresent study is presented before this Symposium.

41

2. Two-dimensional Hydrofoil Oscillating in WavesWith specific applications in view we consider the basic flow to be a sinusoidal

gravity wave of small amplitude in water of finite depth, H, in which a two-dimen-sional hydrofoil of chord 2J. moves horizontally with velocity U while submergedat a mean depth hI underneath the free surface. In terms of the body coordinatesystem (x, y), the wave profile of the basic flow (see Fig. 1) may be written as

[ i (wot - kx ) ]y = hI + Re a e

the wave amplitude, a, being assumed small such that ka 1.velocity (U + u , v ) of the wave field, by clas sical theory, is

o 0

u - iv = A." cos [ k (x +iy + ih 2 ) - w t]o 0 ',' 0

(1 )

The corresponding

(2 )

where h 2 = H - hI is the distance from the bottom and wo is the encounter frequen-

cy

(3)

2w,:' = gk tanh kH

1A,:J a = (2gk/ sinh 2kH)2 (4 )

where g is the gravitational acceleration and in (3), the + sign is for heading sea,and - sign for following the waves. In particular, the y-component wave velocityat the x-axis (which coincides with the mean position of the hydrofoil), denoted byv (x, 0, t) =V (x, t), is

o 0

V ( t)-'A i(wot-kx)ox, _1 0 e , A = A", sinh kh2o ',. (5)

Here and henceforth, the real part of a complex expression will be understood forphysical interpretation.

Since the problem of central interest at hand is to determine the effect of awaving stream on the propulsive performance of a hydrofoil in unsteady motion,we shall further assume, for simplicity, that the hydrofoil is located sufficientlyfar from both the free water surface and solid bottom so as to curtail the compli-cated (but only secondary) corrections due to these boundary effects. This condi-tion would be nearly satisfied if the hydrofoil is at a distance more than two chordsaway from each of these boundaries, that is for the chord 2J. < i max{h1 , h 2 }, thisestimate being inferred by the known results of the steady flow case (see Wu, 1954)which is assumed to remain valid in the unsteady case. As an additional simplify-ing assumption, the ratio E = A /U of the magnitude of the orbital wave velocity

oto the mean free stream velocity is taken to be small so that the x-component

42

y

------t----=--i.~=t=~+I.T-------.......... xh2 = H - hi

+J/~/----r/--:/r-rl'I,..--rl-Ir-?'I 1 1 1 7 7 7 7 7 7 7 777771777177/7777

FIGURE 1

43

orbital velocity, u , may be neglected in comparison with U in formulating theo

present linear theory. Although the simple water wave is chosen as a concreteexample, it makes little difference to the subsequent discussion if other kinds ofwavy streams are considered as long as the transverse velocity of the basic flowcan be represented by equation (5). For arbitrary V{x, t) the result can be obtainedby the Fourier synthesis of this fundamental case.

The hydrofoil (or airfoil) is assumed to be thin, though sufficiently roundedat the leading edge to keep the flow from being separated there. The foil-thicknesseffect is then only secondary and will be further disregarded in this study. Forbrevity, the semi-chord, 1., of the hydrofoil will be normalized to unity as thereference length. The unsteady motion of the hydrofoil assumes the fundamentalform

A iwtY = h{x, t) = h{x) e (-I < x< I) , (6 )

'"where the circular frequency w is arbitrary, and h may be a complex functionof x (with respect to i =N in the time factor). With the resultant flow velocitydenoted by (U + U

o+ u I ' v 0 + vI)' the linearized boundary condition that the flow

be always tangential to the moving body surface requires

(/xl

44

boundary conditions

(Ixl < I, y = ),

(Ixl > I, y = ),

(lOa)

(lOb)

(II)

together with the Kutta condition that cP I =at x = I, and that cP I vanishes atinfinity. Condition (lOa) follows fronl substituting (7a) into the last equation of (9);and (lOb) is a consequence of the pressure being continuous in the flow and thefactthat cP I is odd in y.

The solution to this nlixed-type boundary problenl of cP I is known (see Wu,197Ia); in particular, the value of cPI(x, O ,t) at the plate is given by

+ U (I _ x ) i I r I ( Z )1. ScPI-(x,t) = -Zao I +x :;; ~ 1- x 2. i.(J1( ,t) dS

" .)_1 ~ S - xi.(JI(X,t)=-DS

XV 1 (s,t)dS

-II I 1

ao

= [b I - (bo + bI)@(a)] - [ b I - (bo + bI)@(ao )]

(IZ)

(13 )

Z fTrbn=rrJ

oV(cosB,t)cosnB dB

I Z r Trb =-; V (cosB,t)cosnBdB

n Tr vo 0

(x=cosB, n=O,I,Z, ... ) (14 )

(15 )

@(a) = ~(a) + i "(a) , a.wi/u, a !!!w f./U ,o 0 (16)

a = (oK)i + K 0 =(gf. IUZ )tanh kH, K 5. k1o -, (17 )

Here the integral in equation (II) aSSUnles its Cauchy principal value; J (k) is then

Bessel function of the first kind; @(a) is the Theodorsen function, ~and !!J beingits real and inlaginary parts(for a tabulation of @, see Luke and Dengler, 1951);a is the reduced frequency of the body nlotion, a the wave reduced frequency, both

obeing based on the half-chord f.. The function a (K) in equation (17) is the non-

odinlensional fornl of equation (3). We shall write K as k since f. = I.

The differential lift distribution along the chord is clearly

.., - + +ol-(x, t) = p (x, t) - p (x, t) = ZpcP I (x, t) (Ixl < I) (18 )

The integral representation of the lift L and the nlOnlent M (about the nlid-chord,positive in the nose-up sense) are

L = 5~L(x, t) dx , (19 )

45

M = - 51"r.

46

51 fl(X)dx 5I(~)i g(s)ds =51 gl(x)dx &1 (1- x~)i f_(S)dS-I -I1-s S-x -I -II-S s-x

(27)

where f(x), g(x) are two arbitrary functions, provided they and their derivatives- -f'(x), gl(X) are continuous in -I~x~ I. The mean values of thrust T, power P,

and ener gy 10ssE c an be deduc ed by aver aging T, P, and E over a long timeperiod. Two different cases arise according as w = w or w .4. w

o T 0(i) When w = w , that is when the wing oscillates at the wave encounter

o

frequency, the two motions are correlated. In this case we obtain T as

(28b)

(28a)

dxdS ,V(x,t)~(s,t)S - x

...1.. ...1.. ..,( 1 ,-I( 1 ...I" ..I..T = ~ p Re{ (a + b - (3 )(a'" - b'I"+ (3 'I' )- b R'I' - b I (3 "+ (3 (3 '1 + 2I} ,4 0 0 0 0 d- 0 0

2 51!: 1 (1 - x 2 )iI =~ ~ 2

lf -1 -1 1 - s

where the superscript ':' denotes the complex conjugate, (3 n= df3 n (t)/ dt, and

2 (If(3 n (t) = :rr J h(x, t) cos ne de

o(x=cose, n=0,I,2, ... ), (28c)

hence ~ = iwf3 when h is given by (6). The mean power P can be shown, aftern n

some manipulation, to have the following expression

(29b)

(29a)

av~:'(x, t)/ats - xSl CI(I 2)i-1 .)-1 I ~~2

By substituting in (29b) the relationship

av (x, t)/at = - (w/k) aV lax ,o 0

(see (5), with w = w for the present case), and applying the formula (27), ito

immediately follows that

II = (o)k) I , (29c)

where I is given by (28b). Whence, by (24), E = P - UT, or

E = ~ pU Re{(a + b )(b'I:'- a':') + 2(cr/k - I )I} 4 0 0 0 (30)

We note that, upon substituting (13) in (30), the first term on the right-hand side

47

of (30) involves only the first two Fourier coefficients of V, in the particularcombination of (b

o+ b I ). However, the second term in (30) with I, which results

from the interaction between the wave action and body motion, involves all the

Fourier coefficients of V since the integral I has the following Fourier-Besselexpansion

I = A e -iwto

00

\' (i)n+I J (k) ( b b )~J n n+I- n-I n=I

(31 )

,

We further note that the expressions for T and P involve, in addition to the b s,n

also the first two Fourier coefficients, 13 0 and 13 I' of h(x, t).To facilitate the subsequent consideration of the optimum shape problem, it

is useful to recast the above expressions for T, P, and E in terms of certain inner

products. Let J4.- denote a subset of the complex Hilbert space L 2 [ -I, I ]

and let the inner product between f() and g(.) on J4. be defined by

=.; SI f(x)g'~(x)(1- x 2 ) -i dx =< g,f>';':, (f,ge);l.-)-I

(32a)

(32b)

where the weighting function (I - x 2 )-i is introduced in order to convert theFourier coefficients into the inner product form. Any two functions f, g inJ=l.-will

be said to be orthogonal on Jl- if < f, g > = O.Substituting (13) - (16) and (28c) in (29) - (30), we obtain the mean coefficients

_ - _ I 3of thrust, power, and energy loss, defined by (C p ' C E , C T ) = (P, E, TU)/(~ 'TT'pU i),in terms of the inner products as

(33)

C E= Re { B(cy)l< v, 1 > 12+ 2E (1- 2@)(W I +iW 2)< v, f I> + 2E (CY /k -I )< v, g2 > - 4E 2 W 2 },(34 )

where-iwt

v(x) = e V(x, t)/U , A. _ iwth(x) = e h(x, t) , (35a)

f I (x) = I + x , gI(x)=(I-@)x+@ , (35b)

(CY) = 3'" (CY) + i ~ (CY) , B(CY) = ~ _ ( cg;- 2 + &2 ) , (35c)

(35d)

48

(35e)

In the above, as well as in the sequel, the argument k of the Bessel functions J (k)n

will be understood unless otherwise designated. The mean thrust coefficient issimply (the coefficient form of (24))

(36 )

(37a)

Another flow quantity of interest is the mean leading-edge-suction coefficient,- I 2C S = S/.;- TrpU 1. From (21), (13), (14) we obtain

cs=l@-+2E(W1-iW2)12 ,

where f (x) = xo

(-l~x~1)

As suggested by Lighthill (1969, 1970), the ratio cSI C T provides a measure of therelative strength of the leading-edge suction; moderate and large values of cSI C T(as compared to unity) suggest a tendency that the flow would separate, or stall,near the leading edge (such a category of separated flow would be quite differentfrom the completely wetted flow as as sumed here).

(ii) w f: Wo

- - - In this case the mean product of exp(iwt) and exp(iwot) vanishesas the body motion and wave action become uncorrelated. Consequently the termswhich are linear in E in (33) and (34) drop out of the expressions for C p and C E ;further, W2 in (34) then assumes its value at (j. The corresponding C

slikewiseo .

becomes

(38 )

The result of this case therefore reduces virtually to the case of uniform streamexcept for the additional term ( - 4E 2 W2) in the expression for C E and (4E 2 W2) inCS ' These added terms indicate that energy is invariably being supplied by theprimary wave, through the mechanism of generating a greater leading-edgesuction, at no expense of C p ' It thus follows that for C p fixed, C T becomesgreater and C E smaller (hence higher efficiency) with increasing wave action(greater E W). The energy gain in this case, however, is always accompanied byan appreciable increase in the leading-edge suction, suggesting an easier leading-edge stall. When the suction is required to remain reasonably small, the optimummotion and the corresponding improvement of efficiency are not significantlydifferent from the uniform stream case which has been discussed earlier by Wu

49

(1971b). For this reason this second case will not be further pursued here,

4. The Optimum Motion (w = wo)The present problem of optimum motion is formulated especially to analyze

the interaction between the body motion and wave action; it can be stated asfollows:

Given a reduced frequency CT> 0 (hence also the wave number k, see (3)) and aA

thrust coefficient C T 0> 0, find a velocity profile v, or a hydrofoil profile h in the,set .J:t-- (defined by (32a)) such that C E is minimized subject to the constraint

(39)

assuming that the wing oscillates at the wave encounter frequency.

It is desirable to choose C T (rather than C p or C E ) to be a constrained quan-tity since a constant thrust is required to overcome the (nearly constant) viscousdrag if the uniform forward motion is to be maintained. No additional constraintsare imposed here on the total lift L and moment M for balancing the rectilinearand angular recoils of the flexible plate (see Wu, 1971a, Eqs. (56a, b)); this choiceis made for two reasons. First, when a body structure consists of componentsother than the flexible plate, the recoil consideration must take the motion of the

entire body into account. Second, even when the wing alone comprises a self-propelling body in its entirety, there will still be other degrees of freedom leftto be used to satisfy the recoil conditions, if desired, as we shall see later.

In choosing the independent functionals for the optimization calculation, we

note that only two of C p ' C E ' C T are independent since they are related by (36).There are great advantages in the choice of C p and C E as the independent function-A ~als of v and h because C E ' in particular, does not involve h, and C p is alsosimpler in expression than CT' In the expression (34) for C E ' the first term onthe right-hand side is the same as in the uniform-stream case (see Wu, 1971 b,Eq. (13)); it is always non-negative since B(CT) > 0 for CT> O. The second andthird terms, which are bilinear in e and v, represent the body-wave interaction,The last term, which is proportional to e 2, is solely due to the wave action. Thisresult actually proves the statement that extraction of energy from the surroundingflow by an oscillating flexible wing is impossible if the incident flow is uniform.In the presence of a primary wave, with appropriate v and increasing wave para-meter e , the last three terms in (34) may become negative and numerically solarge as to reduce C E at first, and C p eventually, to negative values, as will beseen later. The case of C p < 0 signifies the operation in which a mechanical poweris received by the body, instead of being consumed by it, through a favorableextraction of the wave energy, In spite of these pos sibilities, we shall still con-tinue to use the Froude efficiency

50

(40a)

as a measure of the hydromechanical performance. Aside from its usual signifi-cance for 0 < ,,< I, now we may have new generalized interpretations as follows:

(i) "> I for

(ii) "< 0 forC E < 0, C p > 0;

C E < C p < O.

(40b)

(40c)

we proceed to recast the inner productA.

By (35a) and (7b), h and v are related by

Another step of primary importance is to choose the independent function forthe optimization calculation. Although either v or h may serve as an independentfunction {since they are related by a differential equation (7b)), the advantage oftaking v is clear, as was noted by Wu (197Ib, section 6) in discussing the optimumshape of a flexible plate oscillating in a uniform stream. As another reason, wenote that in the present formulation, an inner product of h with a given f{.) can beconverted into an equivalent one involving v, whereas the converse is generallyimpossible.

Accepting v as the independent function,A

< gl,h> in (33) in terms of v.

A.(d/dx + ia ) h{x):: v{x) (Ix I < I) , (41 a)

which has the general integral as

(41 b)

"'-

where h_ l is an arbitrary integration constant. Substituting (4Ib) and (35b) inA< gl' h >, and integrating by parts, we obtain

(42a)

where (42b)

(42c)

Consequently (33) becomes

C p :: R e { - i a [ < v, f I > - 2e ( J I + iJ0)] [ < g 3 ' v > + C I - i C 2 ]

+ 2e (a / k) < v, g2 >} . I(33 )

51

INow the expression for Cp in (33) and C E in (34) are both expressed in tenns ofv and contain only three inner products: < v,f1 >, < v,g2>' and < g3' v>.

Since f 1 , g2' g3 are not mutually orthogonal on Jt, we next construct a setof three orthogonal functions, f 1 , f 2 , f 3 say{there being no need here to normalizethem), by the Schmidt scheme:

(43a)

(43b)

(43c)

such that (ilk) (44)

The coefficients a are determined by the orthogonality condition (44) asn

a 1 =< g2,f1 >/ = t< g2,f1 > = t [ 2J 1{k) - iJ2{k)] (45a)

a 2 =/ - a 1< f 1 ' g 2> - a~ < g 2 ' f 1 > + a 1 a;' < f 1 ' f 1 > =< g 2' g 2 > - 3 a 1 a~'2 S1 2 3/2 S1 e - ikS ds 51 eikr] dYj

< g ,g > = (I - x) dx I 12 2 ~ -1 -1 (1_g 2 )2 (oS-x) -1 (l_Yj2)2{Yj_X)

= 1- J~{k) + 2Ji{k) - 2Jo {k)J 2{k) ,

which can be shown by successive interchange of the order of integration and bymaking use of the Poincar~-Bertrandformula, and hence

(45d)

Finally,

52

whereI

N (a) = - [ I - @(o')] J (a) - i@(O')J (a)n n n

(n =0,1.2, .. ). (45f)

(46 )(-I~x~l) ,

The above result can be shown by using the series expansion of g2 and g 3as00

g2(x) = 2isinB)' (-it In(k) sin nB (x=cosB) ,.J

n=l 00g3(x) = 2ie- iO'cosB sinB '\' (i)n[ N (a) _ (i)nN (a)] sin nB (x=cosB) /J non

n=1

This cOlnpletes the determination of a 3 , hence also the orthogonalization.It is now evident that v can be expres sed as

3

v(x) = " B f (x) + vJ..(x) ,/~, n nn=1

where B I S are complex coefficients and v..l.. is any function belonging to the ortho-n

gonal complement of the subspace spanned by {fl , f 2 , f 3 }. that is, < f n , v..l.> = 0for n = 1,2,3. For convenience of the subsequent computations. we introduce thereal parameters I S by

n

= B =2 l+ i 2n n n n n- n (47a)

(47b)

where C I + iC 2 is given by (42c). From (43), (46) and (47). we have

= a~v.fl> + = a~I+iS2) + (3+ iS 4) , = a; (SI+iS2) + a; (S3+iS4) + (S5+iS6) - (C I +iC 2 ).

(48a)

(48b)

ISubstitution of (47). (48) in (33) and (34) yields

(49)

(50)

where

a 2 = Al (0')+ iA2 (a) , a 3 = A3 (a, k) + iA4 (a. k) 2

PI =-A I J o (k)-A2J I (k)+3k JI(k). P2=AIJI(k)-A2Jo(k)-J2(k)/3k.

(51 a)

(SIb)

53

(Sl c)

Q 1 ",(l-2:g;')W1+2~W2+i( ~ -I )J1 'Q2=2~W1-(1-2~)W2-}(~-1)J2' Q3=(a/k)-1,

(Sl d)

(Sl e)

The other coefficients appeared here have been given in (3S), (4S).Equations (49), (SO) show that C p depends on only six real parameters{~1' S2"" ~6}, and C E depends on only three parameters {~1' ~2' ~3}' whileboth C p and C E' hence also C T ' are independent of v.J,.(x). (Note that the orthogonalcOITlpleITlent of the subspace spanned by {fl , f2, f 3} is infinite diITlensional.) Thus, it isclear that the optimization problem posed earlier nOw reduces to one defined ona finite-dimensional vector space.

Before we proceed with our discussion from this approach, further simplifica-tion of the expressions for C p and C E can be gained if we first eliminate the termslinear in ~ I and ~ 2 in (49), (SO) and then reduce the number of quadratic terms in(49) by the following transformation

Sl+ iS 2::: i (A4 -iA3 )[ ~1+iS2+ k(Q1+ iQ 2)] ,S 3+ is 4::: (~ 3 + iG 4) + (A 3+ iA4 )(s S+ is 6)/ A 2 ,SS+iS6=(SS+iS6)+(CS+iC6) ,

where

(S2a)

(S2b)

(S2c)

(S2d)

Then (49) and (SO) reduce to

2 2Cp/a =A2 (sl +S2 )+A(Sl s 3+ s 2S4)+E (ASs3+A6S4 )+E Ao2 2

CE=B(Sl+s2)+2EQ3s3-EQo'

where

AS:::2P3-(A3Q2+A4Q1)/B, A6:::2P4-(A4Q2-A3Q1)/B,

Ao =-2[ Jo+(P3A3+P4A4)/A2] sS+2[ J1+(P3A4-P4A3)/A2] S6

[ ] 2 2 2+4E JoP2+J1P1-A2(JoQ2+J1Q1)/B -EA2 (Q1+ Q2)/B,

2 2 2 2Qo

=2Q 3 (A3sS - A4s6 )/A +4E W +E (Q1 +Q 2 )/B

(S3 )

(S4 )

(SSa)

(SSb)

(SSc)

Thus in the above reduced form, C p depends quadratically on {s I' S 2' S 3' S4 }, C E

54

depends quadratically on {I; 1,1;2} but is independent of 1;4' while both C p and C Edepend linearly on { I; 5' I; 6}, When the primary wave is absent (i. e. e =0), equa-tions (53) and (54), or equivalently (49) and (50), reduce to the case of a flexibleplate in uniform flow treated earlier by Wu (1971 b, see his equations (79) and (80),which involve also six independent parameters). The present result of C p and C Eis also very similar to that for a flat plate oscillating in waves discussed by Wu(1972, see his equations (50), (51) for the four independent parameters proper tothat problem). Like those simpler cases investigated previously, we note that inthe three-dimensional Euclidean space (1;1,1;2,1;3) (i.e. with 1;4,1;5,1;6 held fixed),the C E = const. surfaces are paraboloids of revolution with its generating axis lying

along the I; 3 -axis, while C p = const. surfaces are oblique hyperboloids, who secross -sections with I; 3 = const. planes, if real, are circles.

The optimization problem posed earlier can now be reformulated as follows:-Let R 6 denote the six-dimensional Euclidean space of ordered six-tuples I; =(So I'

1;2,1;3,1;4,1;5,1;6) of real numbers; and let n be a subset of R 6 defined by

(56)

The optimization problem is to find a vector fO E n such that C E( fO ) ~ C E( f)-for all I; E n.

FroITl the known geoITletric properties of constant C p and C E surfaces, and hence

also of CT=C T , d'0 surface, it follows thatnis an unbounded set in R 6 . Consequently,it is possible that the optimization problem may not have a solution. It suffices todemonstrate two such cases. As the first, consider a sequence of points {fk} inthe set Sl c n defined by

(57)

ksuch that Q 31; 3 - - 00 as k -00. It is readily shown that in the set Sl' C T dependson (1;2,1;4,1;~1;6)while C E depends on (1;2,1;3,1;5,1;6); Sl is therefore nonempty.But, since 1;. 's are all constant for j=2,4, 5, 6 and for all k, we immediately see

J -kfrom (54) that the sequence of values CE(I; ) - - 00, as k -00, implying the non-existence of an optimal solution.

As the second example, consider another sequence of points {fP. } in the setS2 Co Q defined by

(58)

such that Q o (I;~, I;~ ) - 00 as P. - 00. It is also easily seen that in the s~P.S2 'C T = C T ( I; I ' I; 2' I; 3' I; 4 ), C E = C E ( I; I ' I; 2' I; 3' I; 5 ) and cons equently C E ( I; ) - - 00while C T remains unchanged as P. -00, implying again the nonexistence of anoptimal solution.

To ensure the existence of an optimal solution that is physically meaningful,

55

'"

we shall minimize C E over a subset n of n which is closed and bounded, i. e.

C T > 0 } ,,0 (59a)

'"

where R 6 denote a bounded subset of R 6 such that6II;~~M

56

where

(64 )

(65)

~ = (CTA + Q ) 1so 0 0 0

T 2 ::: CTA2 - B

C C I s2T,o T,o 0

z:=s/sJ J 0 ( j 1,2, ... 6 ) (66a)(66b)

Equation (65) follows from the definition of z3' z4 and So as given by (62) and(66a), there being two branches of z4 for given z3' with I z3 I~ I.

The three equations (63 )-(65) involve three unknowns, z3' z4' A., and threeparameters, namely C T - the "proportional loading factor", e - the,0

"proportional wave factor", and ~ - the "complementary mode factor" whicho

includes the contribution from the mode s5+is6 and that from the waves.As for the actual calculation of the Lagrange multiplier A., we note that if

equations (64), (65) are substituted for z3' z4 in equation (63), then, upon perfectsquaring, there results an eighth degree algebraic equation for A., of which the realsolutions (appearing always in even number s) are of interest. This equation seemsto be too difficult for analytical solutions; resort was then made to numericalmethods. The method which proved to be successful is as follows. Since thephysically meaningful solutions also require z3' z4 to be both real, equation (65)suggests that z3 can be used effectively for parametric computation of A., with theobvious advantage of having a bounded range - I ~ z3 ~ I. In this parametric form,both equations (63) and (64) are quadratic equations in A., each giving two solutionsof A. in closed form, from which the real solutions of A. satisfying both equationswere determined by Newton's method, using z3 as a parameter. The number ofsolutions depend on the values of CT, C T ' E and ~ ; on occasions as many as,0 0eight real solutions were obtained, and there are cases in which two real solutionsare very clo se to each other. In all the cases tried the two real solutions provid-ing the highest and lowest efficiencies were taken as the desired optimal solutions.

The numerical results of T) , as shown in Figs 2 - 5 for a few representa-max

tive cases, exhibit the following salient features of the optimum solution. For

CT,o and E both as small as 10- 3 and with z5 ::: z6 ::: 0,_ ~he maximum efficiencyis already T) = 1.0 for the reduced frequency CT> 10 The corre sponding

maxresults of the maximum efficiency for a rigid hydrofoil in a uniform streaITl (E ::: 0)and in regular waves (e > 0) are reproduced (see Wu I97Ib, 1972) in Fig. 3 over

- -

a similar range of C T ,although the C T and E in those two cases are defined,0 ,0with reference to the heaving amplitude at mid-chord, whereas the definition ofC T and E in the present case (by referring to S , see (66), whose physical,0 0significance is not quite so simple) is slightly different. With this qualification,a comparison between Fig. 2 and 3 shows that T) is further improved by the

max

57

flexible over the rigid foil, in the frequency range of interest. When C T is kept3 2 ,0at 10- and E alone is increased (by having a stronger wave) to 10- , 1)maxbecomes greater than I, corresponding to the operation in which energy is extract-ed from the surrounding waves, but a power (somewhat smaller than before) isstill required for maintaining the hydrofoil motion. When E is as large as O. I,we see that 1) becomes negative, indicating that both energy and power are

maxsupplied by the exterior wave field. At this high level of E, 1) becomes more

maxnegative as CT is increased to 10- 2 This trend implies that mOre energy and,0power can be extracted from stronger waves at higher loadings. This is a quiteremarkable feature since this trend is reversed from that at smaller E (or weaker

- -2waves, see the curves with E = 10 ).

To summarize the case of Zs = z6 = 0, we note that the overall features of1) of the flexible and rigid hydrofoils are very similar, the difference, after

max _

the two definitions of C T and E are properly reconciled, being rather small.,0Since the salient features of the optimum motion, including the variation of theleading -edge suction Cs ' feathering of the hydrofoil to the trajectory, etc., havebeen thoroughly explored for the rigid plate case (see Wu, 1972), these featureswill not be further pursued here for the flexible plate. However, it must bestressed that the additional degrees of freedom provided by Zs and z6 for flexibleplates can alter the 1) of a flexible plate. As indicated by Figs. 4 and S for

maxtwo typical examples, 1) max for the basic case of z S = z6 = 0 can be increased bysuitable choice of Zs and z6 (within the set R6 of (59)). The trend of the influenceby Zs and z6 on the value of 1)max can be seen clearly through the linear depen-dence of ~ on Zs and z6 (see (63) - (66)). Obviously, 1) will remain

o _ max _ 1

unchanged when the set of parameters (C T , 0 ' E , zs' z6) is replaced by (C T , 0 'E, 0, 0), where

_I

C =T,o (67)

This explains the opposite trend of 1)max from its basic value at Zs = z6 = 0 whena set of nonvanishing Zs and z6 is reversed in sign. By making use of this proper-ty' or equivalently the simple formula (67), the utility and interpretation of Fig. 2is thereby greatly extended.

Acknowledgments

This work was partially sponsored by the National Science Foundation, underGrant GK 10216, and by the Office of Naval Research, under Contract NOOOI4 -67-A0094 -0012.

58

w "W". Zs = 08 "' 0.2. Z6 = 0

CT" =10- 3e"10-3

1.5

,.ok::::::::----r-------.:==:::=========------=~

0.5

-0.5

om

FIGURE 2

59

1.2

EO1.0

- -3CT.=100.8 E =10-3

7W = w.0.6

8 =0.10.4

0.2

-0.2

-0.40.01

FIGURE 3

60

2.0..-------,-----,---,-,--r--.-,--,.-,------,---,---.,.----,r---r--r--r-...,..,

1.8

ero" 10-3 , ;?" 10-3

W = Wo, 23" 0.2

FIGURE 4

- -3 1ero ~ 10 ~ ~ 10-

W~Wo 23 ~0.2

61

FIGURE, 5

62

References

Garrick, I. E. 1936 Propulsion of a flapping and oscillating airfoil. NACA TR567.

Garrick, I. E. 1957 Nonsteady Wing Characteristics. Sect. F., High Speed Aero-dynamics and Jet Propulsion, Vol. 7 (ed.