Theoretical and Applied Mechanics: Proceedings of the 13th International Congress of Theoretical and Applied Mechanics, Moskow University, August 21–16, 1972

374 Pages·1973·26.79 MB·English

by  Arthur E. Bryson Jr. (auth.)

THEORETICAL AND
APPLIED MECHANICS
PROCEEDINGS OF
THE THIRTEENTH INTERNATIONAL CONGRESS OF
THEORETICAL AND APPLIED MECHANICS
MOSCOW UNIVERSITY
AUGUST 21-26, 1972
EDITORS
E. BECKER . G. K. MIKHAILOV
SPRINGER·VERLAG BERLIN· HEIDELBERG· NEW YORK 1973
Dr. rer. nat. ERNS:C BECKER Professor GLEB K. MIKHAILOV
Professor an der USSR National Committee on
Technischen Hochschule Theoretical and Applied Mechanics
Darmstadt Moscow
With 203 Figures
ISBN-13:978-3-642-65592-0 e-ISBN-13:978-3-642-65590-6
DOl: 10 .1007/978-3-642-65590-6
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 machine or similar means,
and storage in data banks.
Under § 64 of the German Copyright Law where copies are made for other than private use, a fee Is payable to the publisher,
the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag, Berlin· Heidelberg 1973.
Softcover reprint of the hardcover 1st edition 1973
Library of Congress Catalog Card Number 58-2749
The use of registered names, trademarks, etc. In this publication does not imply, even in the absence of a specific statement,
that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Preface
The Thirteenth International Congress of Theoretical and Applied Mechanics was held in
Moscow from Monday, 21 August, to Saturday, 26 August 1972. About 2500 participants from
37 countries all over the world attended the congress that was convened by the Congress
Committee of the International Union of Theoretical and Applied Mechanics. The local or
ganization lay in the hands of the Organizing Committee, established by the USSR National
Committee on Theoretical and Applied Mechanics. The USSR Academy of Sciences rendered
partial financial help to the organization of th8 congress. The Organizing Committee was assisted
by the Institute of Problems of Mechanics of the USSR Academy of Sciences, by the Research
Institute for Mechanics of Moscow University, and by the Computing Center and the Institute
of Applied Mathematics of the USSR Academy of Sciences. The Bureau of IUTAM had
allocated a considerable sum for partial financial support of young scientists attending the
congress.
The Thirteenth Congress was officially opened on Monday morning at the Kremlin Palace
of Congresses by Academician N. 1. Muskhelishvili, President of the Congress, and Professor
W. T. Koiter, President of IUTAM. Greeting addresses were offered by:
Mr. K. N. Rudnev, Minister, member of the Council of Ministers of the USSR,
Academician M. V. Keldysh, President of the USSR Academy of Sciences,
Mr. L. N. Efremov, First Deputy Chairman of the State Committee for Science and
Technology under the Council of Ministers of the USSR,
Mr. V. P. Isaev, First Deputy Chairman of the Executive Committee of the Moscow City
Council of Deputies.
At the opening ceremony, Academician L.1. Sedov presented to Academician N. 1. Muskhe
lishvili the book "Continuum Mechanics and Related Problems of Analysis-To the 80th
Birthday of N. 1. Muskhelishvili." The book contains 67 papers that were dedicated to Aca
demician Muskhelishvili by scientists from sixteen countries.-Professor Koiter informed the
participants of the opening ceremony that the Congress Committee had decided to send a tele
gram with greetings to Professor C. B. Biezeno, Delft, who was President of the First Congress,
held at Delft 48 years ago. He also greeted the Secretary of that First Congress, Professor
J. M. Burgers, who was among the participants of the Thirteenth Congress.
Immediately after the opening ceremony, the first general lecture of the congress was
presented by Academician V. V. Novozhilov. The following scientific events all took place at
the main building of Moscow University on the Lenin Hills. The Thirteenth Congress was
officially closed on Saturday afternoon at the aula of Moscow University. On that occasion
members of the Congress Committee expressed their gratitude and praise to the Organizing
Committee for the competent arrangement of the Thirteenth Congress, which was, by number
of attendants, the biggest ever held. The audience was informed that the General Assembly
of IUTAM had elected, in its meeting on 24 August 1972, Professor H. G6rtler President of
IUTAM. According to the statutes of IUTAM, Professor W. T. Koiter thereby became Vice
President, and Professor M. Roy retired from the vice-presidency. Professor D. C. Drucker
was elected Treasurer of IUTAM. The Academy of Sciences of the German Democratic Republic
was admitted, by vote of the General Assembly, as an adhering organization of IUTAM.
Furthermore, Professors G. K. Batchelor, N. J. Hoff, and W. T. Koiter ended their term as
IV Preface
members of the Executive Committee of the Congress Committee. Professors B. A. Boley,
W. Fiszdon, and N. I. Muskhelishvili became new members of the Executive Committee;
Professor H. G6rtler will be ex officio President of the Committee. Finally, it was disclosed
that the Congress Committee had gratefully accepted the invitation by the Section for Mechanics
of the Division of Applied Sciences of the Royal Institute of Engineers (The Netherlands) to
hold the Fourteenth Congress at Delft during the week commencing on 30 August 1976.
Five general lectures on topics of broad interest and twenty sectional lectures on more
specialized topics, each of one hour duration, were presented at the Thirteenth Congress upon
invitation by the Congress Committee. In addition, 225 contributed papers were presented
in six parallel sessions. The authors' names and the titles of the contributed papers are listed
at the end of this volume. The number of contributed papers had to be strictly limited in
order to allow not less than thirty minutes for presentation and discussion of each paper without
unduly increasing the number of parallel sessions. Therefore an International Papers Committee,
established by the Congress Committee, selected the contributed papers on the basis of extended
summaries submitted by the authors. The decisions of the Committee were strictly guided
by considerations of the scientific merit of a paper. On the average, the number of papers
submitted was about six times as high as the number that could be accepted. This made the
task of the International Papers Committee rather difficult, and a large number of valuable
papers had to be rejected. The work of the International Papers Committee was alleviated by
National Papers Committees of USSR, USA, UK, France, and the Federal Republic of Germany.
These national committees preselected the papers submitted by authors from their countries.
The authors of contributed papers are free to publish their manuscripts in journals of their
own choice.
This volume contains the five general lectures and seventeen, out of twenty, sectional
lectures. The following sectional lectures are, upon the authors' request, not included in this
volume:
G. F. Carrier (Cambridge, Mass.): The dynamics of atmospheric vortices;
S. S. Grigoryan (Moscow): Some problems of mechanics in oil industry;
L. E. Scriven (Minneapolis, Minn.): Molecules, particles, interfaces and continua.
Professor Carrier has informed us that his manuscript should not be included because he has
not yet completed the quantitative aspects of those parts of the study which are not already
published.
In order to make all contributions to this volume accessible to as many readers as possible,
all articles are in English. Six manuscripts that were originally submitted in Russian had
therefore to be translated.
As for two previous congresses, Springer-Verlag has again taken up the task of publishing
the Proceedings of the Thirteenth Congress. We appreciate the competence and helpfulness of
Springer-Verlag in their coopeIation with us.
Darmstadt and Moscow, June 1973
Ernst Becker Gleb K. Mikhailov
Opening ceremony at the Kremlin Palace of Congresses; Academician M. V. Keldysh at the speakers
rostrum.
Academician N. 1. Muskhelishvili delivering opening address.
International Union of Theoretical and Applied Mechanics
Congress Committee
J. Ackeret F. K. G. Odqvist
*G. K. Batchelor H. Parku8
*
E. Becker - Secretary M.Roy
B. A. Boley W. Rubinowicz
J. M. Burgers L. 1. Sedov
A. Caquot H. Solberg
S. Goldstein I. Tani
H. Gortler G. I. Taylor
*N. J. Hoff G. Temple
*
W. T. Koiter - President W. Weibull
N. 1. Muskhelishvili
(* members of the Executive Committee).
USSR National Committee on Theoretical and Applied Mechanics
Organizing Committee
N. I. Muskhelishvili - Chairman S. G. Korneev
L. I. Sedov - Deputy Chairman M. A. Lavrent'e v
G. G. Chakhmakhchev - Deputy Chairman L. G. Loitsyanskii
G. K. Mikhailov - Secretary-General M. M. Maslov
N. Kh. Arutyunyan M. D. Millionshchikov
G. G. Ohernyi D. E. Okhotsimskii
A. A. Dorodnitsyn B. N. Petrov
A. Yu.Ishlinsky G. S. Pisarenko
M. V. Keldysh G. N. Savin
V. V. Struminskii
V. V. Sychev
List of Authors of Papers in this Volume
General Lectures Institute of Technology Cambridge, Mass. 02139,
USA, and Division of Mechanics, The Royal Institute
Prof. Arthur E. Bry8on, Chairman, Department of of Technology, Stockholm 70, Sweden.
Applied Mechanics, School of Engineering, Stanford, Prof. H. W. Liepmann, Graduate Aeronautical La
Cal. 94305, USA. boratories, California Institute of Technology, Pasa
Prof. B. M. Fraeij8 de Veubeke, Laboratoire des dena, Cal. 91109, USA.
Techniques Aeronautiques et Spatiales, Universite Prof. M. S. Longuet-Higgin8, Department of Applied
de Liege, 75, Rue du Val-Benoit Liege, Belgium. Mathematics and Theoretical Physics, University
Prof. Sir M. J. Lighthill, University of Cambridge, of Cambridge, Silver Street, Cambridge CB 3 9 EW,
Department of Applied Mathematics and Theoreti UK, and Institute of Oceanographic Sciences,
cal Physics, Silver Street, Cambridge CB 3 9 EW, Wormley, Godalming, U.K.
UK. Prof. E. H. Man8/ield, Structures Department, Pro
Prof. V. V. Novozhilov, Faculty of Mathematics and curement Executive, Ministry of Defence, Royal
Mechanics, University of Leningrad, B. 0.10 line 33, Aircraft Establishment, Farnborough, Hants, Eng
Leningrad B 178, USSR. land.
Prof. Dr. K. Wieghardt, Institut fur Schiffbau der Prof. J. R. Melcher, University of Cambridge, De
Universitat Hamburg, 2 Hamburg 33, Lammersieht partment of Applied Mathematics and Theoretical
90. Physics, Silver Street, Cambridge CB 3 9 EW, UK.
Prof. F. I. Niord8on, The Technical University of
Sectional Lectures Denmark, Department of Solid Mechanics, Building
404, 2800 Lyngby, Denmark.
Prof. A. A. Deriba8, Novosibirsk, USSR. Dr. P. Peder8en, The Technical University of Den
Dr. H. L. Gold8mith, University Medical Clinic, The mark, Department of Solid Mechanics, Building 404,
Montreal General Hospital, 1650 Cedar Avenue, 2800 Lyngby, Denmark.
Montreal 109, Quebec, Canada. Dr. J. R. Philip, Chief of Division CSIRO, Division
Prof. J.-P. Guiraud, Universite de Paris, Faculte des of Environmental Mechanics, P. O. Box 821, Can
Science Mecanique TMorique, Tour 66-9. Quai berra City, A.C.T., 2601, Australia.
Saint-Bernard Paris 5, France. Dr.-Ing. E. h. J. C. Rotta, Deutsche Forschungs
Prof. N. J. Hoff, Department of Aeronautics and und Versuchsanstalt fur Luft- und Raumfahrt E.V.,
Astronautics, Stanford University, Stanford, Cal. Aerodynamische Versuchsanstalt, BunsenstraBe 10,
94305, USA. D-34 Gottingen, FRG.
Prof. A. Yu. 18hlin8ky, Director, Institute for Prob Prof. L. I. Sedov, Moscow, USSR.
lems in Mechanics, Academy of Sciences of the USSR Prof. J. G. Simmond8, University of Virginia, Char
Moscow A-40, Leningrader Avenue 7, USSR. lottesville, Virginia 22901, USA.
Prof. W. T. Koiter, Technische Hogeschool, Mekel Prof. 1. N. Vekua, Rector, Tbilisi State University,
weg 2, Delft, Netherlands. I, Chavchavadze Avenue, Tbilisi, Georgian SSR,
Prof. M. Landahl, Department of Aeronautics and USSR.
Astronautics, School of Engineering, Massachussetts Prof. I. I. Vorovich, Rostov-on-Don, USSR.
Contents
General Lectures
Bryson, Arthur E.: Control theory for random systems 1
0
Fraeijs de Veubeke, B.: The numerical analysis of structures 20
Lighthill, James: Aquatic animal locomotion 29
0 0 0 0 0 0 0
Novozhilov, Vo Vo: Perspective in phenomenological approach to the problem of fracture 47
Wieghardt, K.: Ship hydrodynamics 56
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Sectional I,ectures
Deribas, Ao Ao: Some phenomena accompanying high-speed collision of solids 72
Goldsmith, Harry Lo: The microrheology of human erythrocyte suspensions 85
Guiraud, Jo-Po: Gas dynamics from the point of view of kinetic theory 104
Hoff, Nicholas Jo: Creep buckling of plates and shells 124
0 0 0 0
Ishlinsky, Ao YUo: On the dynamics of a system of rigid bodies 141
Koiter, Wo To, Simmonds, Jo Go: Foundations of shell theory 150
Landahl, Mo To: Drag reduction by polymer addition 177
0 0 0 0 0
Liepmann, Ho Wo: Experimental fluid mechanics: the impact of modern instrumentation 200
Longuet-Higgins, Mo So: The mechanics of the sud zone 213
Mansfield, E. Ho: Inverse methods in two-dimensional elasticity 229
0 0
Melcher, Jo R.: Electrohydrodynamics 240
0 0 0 0 0 0 0 0 0 0 0 0 0
Niordson, Fo 1., Pedersen, Po: A review of optimal structural design 264
Philip, Jo R.: Flow in porous media 279
0 0 0 0 0 0 0 0 0 0 0 0 0 0
Rotta, Jo Co: Turbulent shear layer predicition on the basis of the transport equations for
the Reynolds stresses 295
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Sedov, Lo: High speed water motion 309
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Vekua,1. No: On two ways of constructing the theory of elastic shells 322
0 0 0 0 0 0 0 0
Vorovich, 1. 1.: The problem of non-uniqueness and stability in the non-linear mechanics
of continuous media 340
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Contributed papers, presented at the 13th International Congress of Theoretical and
Applied Mechanics 358
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Control theory for random systems
Arthur E. Bryson, Jr.
~anford, CA, USA
I. Introduction
Automatic control is used in connection with goal-oriented, man-made systems. There are
several, usually overlapping, reasons for using automatic control: (a) to relieve human operators
from tedious, repetitive tasks (e.g., dial telephones, traffic signals, automatic elevators, auto
pilots), (b) to speed up and/or lower costs of production processes (e.g., chemical process control,
automatic cutting and drilling machines), and (c) to control rapidly-changing or complicated
systems accurately and safely (e.g., guidance and control of spacecraft, automatic landing of
aircraft).
There are two distinctly different types of automatic control: programmed (or open-loop)
control and feedback (or closed-loop) control (see Fig. 1). The difference is that the output affects
the signals to the actuator when using feedback control but does not affect them when using
programmed control. Typical programmed control systems are those used with traffic signals
or automatic washing machines where a sequence of events is scheduled by a clock-timer.
Examples of simple feedback control systems are the temperature controllers used in buildings,
ovens, and hot water heaters, where the temperature error (the difference between sensed and
desired temperature) is used with some controller logic, to switch on and off the heating ele
ment.
Disturbances
Output
I
_________
l_~_~ J
- r
Feedback loop
Disturbances
Fig.I. Block diagram; open-loop (programmed) control has no feedback loop; closed-loop· control has feed
back loop.
The purpose of feedback control is to reduce to acceptable levels the effects of unpredictable
disturbances and uncertain parameters, using acceptable levels of control action. Thus, it is
important for the designer of a feedback control system to have reasonably accurate models
of (a) the system to be controlled and (b) the expected disturbances. Until recently, engineers
had better tools for modeling the system than they had for modeling the expected disturbances;
they used only rather simple descriptions such as constant values and impulse functions (or
integrals of the latter-step functions, ramp functions, etc.). While such descriptions are often
adequate to yield satisfactory designs when only one disturbance source acts on the system,
more realistic descriptions are needed when several disturbance sources act on the system
simultaneously.
2 Arthur E. Bryson, Jr.
Many systems of practical interest can be modeled by linear ordinary differential equations.
A convenient form for such a model is a set of coupled first-order ordinary equations:
x = F(t) x + G(t) u + T(t) w; x(to) = xo' ( 1.1)
Y = M(t) x, ~ = Nx(tf), (1.2)
+
z = H(t) x v, (1.3)
where
x(t) = n-vector of state variables,
u(t) = m-vector of control variables,
w(t) = r-vector of process disturbance variables,
~ = s-vector of terminal state parameters (s < n),
<
y(t) = p-vector of output variables (p n),
z(t) = q-vector of measurement variables,
v(t) = q-vector of measurement disturbance variables,
F(t) = n X n open-loop dynamics matrix,
G(t) = n X m control distribution matrix,
T(t) = n X r disturbance distribution matrix,
M(t) = pXn output distribution matrix,
N = s X n terminal state distribution matrix, and
H(t) = q X n measurement distribution matrix.
In most problems, the vectors are deviations from nominal values and the matrix elements are
partial derivatives evaluated on the nominal "path".
The control problem is to design a dynamic system (a "compensator") with measurements, z,
as input and controls, u, as output, so that y(t) and ~ of the controlled system are kept acceptably
close to zero in the presence of anticipated disturbances w, v, and xo' while using acceptable amounts
of control, u.
During the last ten years it has become clear that excellent compensators can be designed
in two completely separate steps: (a) a statefeedback controller, u = -Ox, is designed to minimize
an appropriate quadratic error criterion, assuming the availability of perfect measurements of
the system state variables and no disturbances, i.e., z = x, v = 0, w = 0; (b) a measurement
feedback estimator (or filter) is designed to yield maximum likelihood estimates of the state
variables of the system, i, from the available sensor measurements, z, taking into account
w, v, xo' and assuming perfect knowledge of the control variables, u(t). The state-feedback
controller is then used with the state estimates from the filter, i.e., u = -O(t) i. The resulting
control system forms a "dynamic compensator" (see Fig.2) which minimizes the quadratic
error criterion on the average (Joseph and Tou (1961, Gunckel and Franklin (1963), and Potter
(1964)1.
Actuator
signals utI
Sensor
signals z[t
Fig. 2. Block diagram; dynamic compensator formed from filter and state feedback gains.
In Section II we discuss state feedback controller design, and in Section IV we discuss the
design of state estimators (measurement-feedback filters). Section III is a review of the concept
of a Gauss-Markov random process which forms the basis of the filter design. Section V discusses
the behavior of systems controlled by such filter/state-feedback compensators.
1 Joseph and Tau, as well as Gunckel and Franklin, showed this for Gauss-Markov sequences (discrete
step processes); Potter extended the proof to continuous Gauss-Markov processes.