Table Of ContentDifferential Equations
Theory, Numerics and Applications
Differential Equations
Theory, Numerics and Applications
Proceedings of the ICDE '96
held in Bandung, Indonesia
Edited by
E. van Groesen
Faculty of Applied Mathematics,
University of Twente,
Enschede, The Netherlands
and
E. Soewono
Department of Mathematics,
Institut Teknologi,
Bandung, Indonesia
..
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6168-1 ISBN 978-94-011-5157-3 (eBook)
DOI 10.1007/978-94-011-5157-3
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@ Springer Science+Business Media Dordrecht 1997
Originally published by Kluwer Academic Publishers 1997
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TABLE OF CONTENTS
PREFACE vii
PART I
Invited Lectures(in alphabetical order of the authors)
A.H.P. van der Burgh: Pammetric Excitation in Mechanical Systems 3
H.C.Chang, E.A. Demekhin & E. Kalaidin: Genemtion and
Suppression of Radiation by Solitary Pulses 17
H. Djojodihardjo: Buffeting Problems modelling by use of Lifting
Surface Methods in Unsteady Aerodynamics with Sepamtion 51
R. Grimshaw: The Influence of an External Force on a Solitary Wave 89
A.J. Hermans: An Asymptotic Method in Ship Hydrodynamics 103
J.C. Saut: Non Classical Solitary Waves 125
N.J.Shankar: Mathematical Modelling of Tidal Motion in Regional
Waters of Singapore 139
J.W. Reyn: Phase Portmits of Quadmtic Systems 169
PART II
Contributed papers (in alphabetical order of the first authors)
Andonowati: Microwave Heating: Critical Dependence on Data and
Pammeters 189
P. Astuti, M. Corless & D. Williamson: On the Convergence of
Sampled Data Nonlinear System 201
F.P.H. van Beckum, M. Muksar & E. Soewono : Nonlinear Galerkin
Methods for Hamiltonian Systems; Comparison of Two
Approximation Techniques 211
G.J. Boertjens &W.T. van Horssen :On Internal Resonances For a
Weakly Nonlinear Beam Equation 221
E. Cahyono, E. van Groesen, E. Soewono & S. Subarinah : Genus-two
Solutions to the Kadomtsev-Petviashvili Equation 233
v
vi
D. Chandra, T. Nusantara, M.A. Imron & Basuki : On the Formation
of Hotspot in Microwave Heating 245
E. van Groesen & W.L. Ijzerman: Model Manifolds for Analysing
Data of Wave Splitting 257
S. Hadi & S. Permono: Modelling of Sediment Transport and
Bathymetric Change in Lemah Abang, Jepara Coastal Waters 269
M.J. Huiskes & A.H.P. van der Burgh: A Double Pendulum in a
Wind Field 281
A.A. Iskandar: Solution of the Affine Toda Equations as an Initial
Value Problem 301
A.I.B.MD. Ismail & R.A. Falconer: Mathematical Modelling of
Chlorine Decay in a Contact Tank 311
J. Molenaar, J. de Weger, D. Binks & V.D. Water: The Dynamics
of a Near-Grazing Oscillator 325
S.M. Nababan: Oscillation Criteria for Second Order Nonlinear
Differential Equations 335
B. Soenarko: Radiation and Diffraction of Sound from Bodies
Mounted on an Infinite BafJle Using Boundary Element Methods 345
Tran Hung Thao: A Differential Equation for Filtering of a
Stochastic Dynamical System 355
Triyanta: General Theory of Relativity Without Christoffel Symbols 361
L.H. Wiryanto & E.O. Tuck: A Back-turning Jet Formed by a
Uniform Shallow Stream Hitting a Vertical Wall 371
N. Yaacob & B. Sanugi: A Fifth-order Five-stage RK-Method Based
on Harmonic Mean 381
PREFACE
The International Conference on Differential Equations, theor·y, nu
merics and applications(ICDE'96-Bandung) was held successfully at the
West Aula of Institut TeknoIogi Bandung on September 29 - October 2,
1996, hosted by the Center of Mathematics and the Department of Mathe
matics ITB. This was the first international conference on differential equa
tions in the region and attended by participants from 12 countries: Aus
tralia, Cambodia, Hong Kong, France, IndonE'sia, Malaysia, Netherlands,
Philippine, Thailand, Singapore, USA and Vietnam.
We would like to express our gratitude to the following organizations and
institution: Directorate General of Higher Education of Indonesia (through
Center Grant Project), Institut Teknologi Bandung, UNESCO (through
Participating Programme and ROSTSEA Programme), European Economic
Community(through the Joint Research Project between the Faculty of Ap
plied Mathematics, Universiteit Twente and the Department of Mathemat
ics, Institut Teknologi Bandung), South East Asian Mathematical Society
(SEAMS), French Embassy in Jakarta and local sponsors for their generolls
support which have made this conference possible.
Editors
vii
Part I: Invited Lectures
PARAMETRIC EXCITATION IN MECHANICAL SYSTEMS
A.H.P. VAN DER BURGH
Delft University of Technology
Faculty of Technical Mathematics and Informatics
P.O. Box 5031, 2600 GA Delft, The Netherlands
Abstract. Parametric excitation in a mechnical system may occur if a pa
rameter of the system becomes time-dependent. The mathematical model
for this type of excitation is characterized by terms in the differential equa
tions which have time-dependent coefficients. A standard example of an
equation which displays parametric excitation is the Mathieu equation. In
this paper two systems will be considered in more detail: a pendulum and
a stretched string both with varying length. The attention will be focused
to the (unstability) of the equilibrium position (trivial solution). Also finite
amplitude motions will be considered, for the description of which however
additional nonlinear terms are taken into account.
1. Introduction
Although the concept of parametric excitation seems not to be very well
known among scientists in the exact sciences, almost everybody has expe
rience with paramet.ric excitation. The mechanism that. causes unstability
of the equilibrium position of a swing, the moving body of a child on the
swing, is a practical example of parametric excitation to which can be re
ferred as a phenomenon which almost everybody has experienced in his life.
From mechnical point of view a person moving vertically on a swing may be
viewed as a pendulum with a variable length. As the lengt.h is a parameter
of a pendulum one may identify a pendulum with a varying length as an
example of a system wit.h parametric excitation. In the real-life situation it
is well-known that an appropriate vertical motion of the body of a perSOll
on the swing will cause an unstability of the equilibrium position and will
hence induce motion of the swing. In section 3. it will be shown that the
Mathieu equation is an adequate model for the study of the stability of
3
E. van Groesen et al. (eds.), Differential Equations Theory, Numerics and Applications
© Kluwer Academic Publishers 1997
4 A.H.P. VAN DER BURGH
the equilibrium position of this system. A second example of a system that
displays parametic excitation is related to Melde's [4.] experiment and is
discussed in section 4. This example is interesting because it concerns a
continuo~ system which is still not understood very well nowadays. Melde
performed experiments with a string of silk, fixed on one side at the end
of the prong of a tuning fork and on the other side to a peg, allowing to
vary the tension in the string by turning the peg. The position of the peg
and the tuning fork is such that the oscillating tuning fork is only able to
induce pure horizontal deflections in the string. Melde observed that for a
critical value of the tension in the string the longitudinal horizontal exci
tation by the tuning fork induced a vertical, that is a transverse motion
of the string. A motivation to study this problem is that adequate model
equations for the study of Melde's phenomenon seem not available in the
literature. Moreover it is believed that the understanding of Melde's ex
periment is of relevance for the understanding of wave propagation in the
hull of a ship and in walls and floors of buildings. For convenience of the
reader some qualitative and quantitative methods and results needed for
the study of parametric excitation are presented in section 2.
2. Some theory
The most simple equation which displays parametric excitation is the Math
ieu equation:
it + (8 + oScos2t)u = 0, (1)
where u = u(t) and 8 and are real parameters.
oS
A Mathieu equation with a cos wt forcing term is by no means more general
than (1). By an appropriate time and parameter transformation one easily
shows that limiting the attention to w = 2 is no restriction of the generality.
One easily verifies that u == 0 is a solution of (1). Depending on the values
of 8 and this solution mayor may not be stable. This can be made clear
oS
by application of a theorem by Floquet.
If we set u = Xl and u = X2 then (1) can be written as:
= = ( ~ ~).
where x (Xl, X2)T and A2(t) -8 _ cos 2t
Instead of (2) we consider now the system
x
= A(t)x, (3)
where A(t) is now a continuous T-periodic n x n-matrix.
PARAMETRIC EXCITATION IN MECHANICAL SYSTEMS 5
Theorem (Floquet):
Each fundamental matrix 4»(t) of (3) can be written as:
~(t) = P(t)eBt,
where P{t) is T-periodic and B a constant n x n-matrix.
Proof: If 4»{t) is a fundamental matrix of (3) then 4»{t + T) is also a
fundamental matrix.
4»(t) and 4»{t + T) are linearly dependent as they contain both n linearly
independent solutions of (3).
This means that there exists a nonsingular constant matrix C such that
4»(t + T) = 4»(t).C.
Let B a constant n x n-matrix be defined by
C = eBT.
Now we show that P(t) = 4»{t)e-Bt is T-periodic.
P(t + T) = 4»{t + T)-B(HT) = 4»{t).Ce-B(t+T) =
= 4»(t)e-Bt = P(t)
Some remarks:
- The matrix C and its eigenvalues are usually called the monodromy
matrix respectively the characteristic (Floquet) multipliers of equation
(3).
From Floquet's theorem it follows that the general solution of (3) can
be written as:
x(t) = 4»(t)c = P(t)eBtc,
where cERn.
It is not difficult to show that x(t + T) = P(t)eBtCc.
Let now c be an eigenvector of C: c = Cj with Cc = AjCj. Then it
follows that:
x{t + T) = AjX(t).
= ,n,
From this property it can be understood that Aj ... j 1, ... the
eigenvalues of C are called the Floquet multipliers of equation (3).
- Suppose that P{t) would be known. Then it follows that x = P(t)y
transforms equation (3) in:
y=By
which is an equation with constant coefficients.
