Table Of ContentIgor V. Novozhilov
Fractional Analysis
Methods of Motion Decomposition
1997
Birkhauser
Boston • Basel • Berlin
Igor V. Novozhilov
Department of Mechanics
and Mathematics
Moscow State University
Moscow 119899
Russia
Library of Congress Cataloging-in-Publication Data
Novozhilov, I. V. (Igor' Vasil' evich)
[Fraktsionny'i analiz. English]
Fractional analysis : methods of motion decomposition /
I. V. Novozhilov.
p. cm.
Includes bibliographical references and index.
ISBN-13 :978-1-4612-8667-7 e-ISBN-13:978-1-46 I 2-4130-0
DOl: I 0.1 007/978-1-4612-4130-0
1. Differential equations--Numerical solutions. 2. Differentiable
dynamical systems. 3. Approximation theory. I. Title.
QA372.N6813 1997
531' .11 • 01515352--DC21 96-49997
CIP
O'JJ
Printed on acid-free paper ®
© 1997 Birkhauser Boston Birkhiiuser H(p)
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9 8 7 6 5 4 3 2 1
Contents
I Dimensional analysis and small parameters 1
1 Dimensional analysis . . . . . . . . . . . . . . 1
1.1 The main concepts of dimensional analysis 1
1.2 'fransformations in dimensional analysis 4
2 Introduction of small parameters . . . . . . . . . 6
2.1 Normalization of equations of motion .. 6
2.2 Variants of small parameter introduction 14
2.3 Regular and singular perturbations with respect to
the small parameter . . . . . . . . . . . . . . . . .. 18
2.4 Two types of power series expansion with respect to
a small parameter .............. 21
2.5 Redundancy in methods of approximation . . . . . 23
II Regularly perturbed systems. Expansions of solutions 25
3 The Poincare theorem. The algorithm of expansion . 25
4 Applications of the Poincare theorem. 28
4.1 Stokes' problem. . . . . . . . . . . . . . 28
4.2 Secular terms. . . . . . . . . . . . . . . 30
4.3 Systematic drifts of a gyro in gimbals. Method of
successive approximations. . 34
5 Poincare-Lyapunov method. . . . . . . . . . . . . . . . .. 37
5.1 Algorithm of the method ............... 37
5.2 Examples. Nonisochronism of nonlinear system os-
cillations ...................... 39
III Decomposition of motion in systems with fast phase 43
6 Method of averaging in systems with a single fast phase 44
6.1 Krylov - Bogolyubov equations in standard form 44
6.2 Algorithm of asymptotic expansion ....... 46
6.3 Approximation accuracy. . . . . . . . . . . . . . 51
6.4 Averaging over trajectories of the generating system 52
6.5 Variants of averaging methods ............ 53
vi CONTENTS
7 Applications of the method of averaging . . . . . . . 55
7.1 Free oscillations with friction of various types 55
7.2 Free oscillations of a tube generator 59
8 Method of harmonic linearization . 62
8.1 Foundations of the method . . . . . 62
8.2 Examples. . . . . . . . . . . . . . . . 67
9 Method of averaging in systems with several fast phases 70
9.1 Averaged equations of the first approximation. 70
9.2 Resonances in multifrequency systems . . . . 71
9.3 A veraging algorithm in the case of resonance 72
9.4 Pendulum resonance oscillations ...... 74
9.5 Resonant oscillations with friction ..... 79
10 Averaging in systems without explicit periodicities 82
10.1 Volosov averaging scheme . . . . . . . . . . 82
10.2 Separation of characteristic motions of an oscillator
with high friction. . . . . . . . . . . . . . . . . . .. 83
IV Decomposition of motion in systems with boundary layer 87
11 Tikhonov theorem . . . . . . . . . 87
11.1 Introductory considerations . . . . . . . . . . . . .. 87
11.2 Tikhonov theorem . . . . . . . . . . . . . . . . . .. 89
11.3 Decomposition of motion on an infinite time interval 93
12 Application of the Tikhonov theorem . . . . . . . . 95
12.1 Quasistatic motions of mechanical systems 95
12.2 The method of "frozen coefficients" . . . . . 98
12.3 The limit model for a double pendulum of high stiffness 100
12.4 Relaxation oscillations of the valve generator . . .. 104
13 Asymptotic expansion of solutions for systems with a bound-
ary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.1 Algorithm of expansion . . . . . . . . . . . . . . .. 107
13.2 Asymptotic expansions for the Stokes problem . .. 113
13.3 Asymptotic expansions on the problem of pendulum
motion in a medium of high viscosity ........ 115
13.4 Decomposition of motions of a railway car in magnetic
suspension. . . . . . . . . . . . 117
V Decomposition of motion in systems with discontinuous
characteristics 123
14 Definition of a solution in discontinuity points ....... . 123
15 Examples............................ 126
15.1 Relay control of angular motion of spacecraft. Sliding
mode ...................... . 126
15.2 Disc rolling motion with Coulomb friction . . . 129
15.3 Relaxation oscillations of the Froude pendulum 133
CONTENTS
VB
VI Correctness of limit models 137
16 Limit model of holonomic constraint (absolutely rigid body) 137
16.1 Conditions for correctness of the model in statically
definable and indefinable cases 137
16.2 Examples........................ 144
17 Limit model of kinematic constraints . . . . . . . . . . . .. 147
17.1 Conditions of model correctness in kinematically de
finable and indefinable cases. . . . . . . . . . . . .. 147
17.2 Change of kinematic constraints for rolling of a braked
wheel .......................... 153
17.3 Kinematic indefinability in a rolling rail car problem 156
18 Limit model of servoconstraint . . . . . . . . . . . . . . .. 163
18.1 Conditions of servoconstraint realizability . . . . .. 163
18.2 Realization of servoconstraints, defining the manip
ulator extremity motion . . . . . . . . . . . . . . .. 167
19 Precession and nutation models in gyro theory ....... 173
19.1 Correctness conditions for an extended precession model173
19.2 Precession model for a gyrotachometer . . . . . . .. 178
19.3 Precession model of a three-axis force gyrostabilizer 179
19.4 Two-step method for stability approval of the nuta-
tion model for a three-axis gyrostabilizer. . . . .. 183
20 Mathematical model of a "man - artificial-kidney" system 186
21 Approximate models of an aircraft motion ..... 194
21.1 Models of zeroth approximation with respect
to small parameters .... 194
21.2 Refined models of motion . . . . . . . . . . . 201
22 Automobile motion decomposition . . . . . . . . . . 206
22.1 Structure of automobile motion partial models 207
22.2 Mathematical model of rolling for a deformed wheel.
Are nonlinear nonholonomic constraints possible? . . 211
References 223
Author Index 229
Subject Index 230
Preface
This book considers methods of approximate analysis of mechanical, elec
tromechanical, and other systems described by ordinary differential equa
tions.
Modern mathematical modeling of sophisticated mechanical systems
consists of several stages: first, construction of a mechanical model, and
then writing appropriate equations and their analytical or numerical ex
amination. Usually, this procedure is repeated several times. Even if an
initial model correctly reflects the main properties of a phenomenon, it de
scribes, as a rule, many unnecessary details that make equations of motion
too complicated. As experience and experimental data are accumulated,
the researcher considers simpler models and simplifies the equations. Thus
some terms are discarded, the order of the equations is lowered, and so
on. This process requires time, experimentation, and the researcher's intu
ition. A good example of such a semi-experimental way of simplifying is a
gyroscopic precession equation. Formal mathematical proofs of its admis
sibility appeared some several decades after its successful introduction in
engineering calculations.
Applied mathematics now has at its disposal many methods of approxi
mate analysis of differential equations. Application of these methods could
shorten and formalize the procedure of simplifying the equations and, thus,
of constructing approximate motion models.
Wide application of the methods into practice is hindered by the fol
lowing.
1. Descriptions of various approximate methods are scattered over the
mathematical literature. The researcher, as a rule, does not know what
method is most suitable for a specific case.
2. The preceding methods are essentially variants of the method of
small parameter. Equations of systems for the most part do not contain
small parameters. The problem arises of reducing the initial equations to a
form containing small parameters so that mathematical formalism can be
applied to them. The problem is solved using similitude theory.
The two stages, introduction of small parameter and application of
mathematical tools, together form a procedure which has been termed frac-
x
tional analysis. Fractional analysis singles out the main motion components
and small "additives" to them, and separates slow and fast components, in
short, separates the motion on big and small fractions.
Such "fractioning" facilitates the numerical analysis of dynamic sys
tems. It is known that one of the difficulties of numerical analysis is
separation of eigenfrequencies of a system when one has to take into ac
count high-frequency components over large time intervals with a small
integration step. Methods of fractional analysis make it possible to obtain
approximate equations that describe fast and slow components of motion
separately. Each of these equations can be solved numerically on its own
time scale.
The book presents basics of fractional analysis. It can be of interest
to students, postgraduates, and researchers who deal with analytical and
numerical analysis of equations of motion of dynamic systems. The book
surveys the variety of existing methods of small parameters and shows
how and where to use them. The author tries to present the material in
an easy-to-understand way, omitting proofs of mathematical theorems and
simplifying mathematical formulations.
The content of the book was certainly affected by the author's taste
and experience. He has been dealing mainly with "rough" systems that
are presented ambiguously by mechanical models, are difficult to be de
scribed formally, and possess high damping. Approximate analysis of such
objects is preferably done by the method of successive approximations,
boundary-layer methods, and, to a lesser extent, by the method of averag
ing. The author does not deal with pure Hamiltonian mechanics with its
celestial-mechanical and trajectory-ballistic applications. Hence the bulk
of resonance situations of modern methods of averaging are outside the
scope of the book.
The book is to fill the gap between mathematical formalism and ap
plications. Hence emphasis is on examples, mainly from the mechanics of
controlled motion. In them, the whole cycle of fractional analysis is per
formed, from writing initial equations and introducing small parameters to
constructing models and estimating their accuracy. The author's own re
sults on construction and analysis of approximate mathematical models in
the mechanics of gyroscopes, transport, robots, and the like, are presented.
CHAPTER I
Dimensional analysis and
small parameters
1 Dimensional analysis
1.1 The main concepts of dimensional analysis
Scientific disciplines deal with quantities that are by nature qualitatively
different and that are measured with the help of numbers.
In analytical description qualitative concepts are represented by differ
ent names or labels, that is, dimensions. For example, there exist quantities
with dimensions such as "length," "mass," "energy," and so on. To make a
qualitative idea of a physical characteristic quantitative, a unit and proce
dure of measurement must be defined. Thus, the meter (or its fraction) is a
unit of length. Therefore, the length of a beam is measured by the number
of times a given scale unit (that is, the meter and its fractions) can be laid
off along the beam.
Let a length of a beam and a unit of length be denoted by Land L*,
respectively. Then L = L*f, where Land L* are quantities with dimension
"length," and f is a numerical measure (that is, relative number) that
determines the length of a beam for given L*. An area S, velocity V,
time T, and any other dimensional quantity can be determined similarly.
Therefore,
... , (1.1)
where L*, S*, V*, T*, ... are units of measurement, and f, s, v, t, ... are nu
merical measures of corresponding quantities.
Here and in the remainder of this treatment quantities with dimensions
are denoted by capital letters and their numerical measures are denoted by
small letters.
2 CH. I. DIMENSIONAL ANALYSIS AND SMALL PARAMETERS
In any specific problem a set of quantities, variables, and constants is
used. The problem of choosing these quantities is related to the problem
of choosing a mathematical model proper for the phenomenon under con
sideration. If a model is already determined, that is, all the equations and
initial and boundary conditions are already written, then this set consists
of all the notations involved in the model representation.
The quantities mentioned previously must satisfy a number of base re
lations, which were established for problems with the same set of variables
and constants. These relations are usually written as definitions of one
quantity by means of other ones.
In mechanics the following base definitions are usually accepted: defi
nition of an area of a square (S) via its side length (L); definition of speed
velocity (V) as the rate of change of length with respect to time; definition
of acceleration W as the rate of change of velocity with respect to time;
Newton's Second Law, which is often considered as a definition of a force
F by means of acceleration and mass M; and so on.
The preceding relations can be written as
dL dV
V = dT' W = dT' F = MW, .... (1.2)
From the point of view of dimensional analysis, the quantities L, S, T, ...
and relations (1.2) comply with two almost trivial postulates [3,4].
1. Quantities L, S, V, ... themselves do not depend on the choice of units
of measurement and the measurement procedure.
2. Only quantities with the same dimensions can be used in such rela
tions as equality or inequality.
