On normal mode vibrations of nonlinear conservative systems
Citation for published version (APA):Varst, van der, P. G. T. (1982). On normal mode vibrations of nonlinear conservative systems. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR108689
DOI:
10.6100/IR108689
Document status and date: Published: 01/01/1982 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
ON NORMAL MODE VIBRATIONS
OF NONLINEAR CONSERVATIVE SYSTEMS
DISSERTATIE DRUKKERIJ
ON NORMAL MODE VIBRATIONS
OF NONLINEAR CONSERVATIVE SYSTEMS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. S. T. M. ACKERMANS, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
DINSDAG 30 NOVEMBER 1982 TE 16.00 UUR
DOOR
PAULUS GERARDUS THEODORUS VAN DER VARST
GEBOREN TE POSTERHOL T (L)Dit proefschrift is goedgekeurd door de promotoren:
Prof. Dr. lr. D.H. van Campen en
CONTENTS
ABSTRACT v
INTRODUCTION
1.1 General part
1.2 Relevance of normal modes to systems in which damping
and periodic forcing are included 7
2 THE NORMAL MODE THEORY 15
2.1 Introduction to chapter 2 15
2.2 Precise description of the system and other preliminaries 16 2.3 The definition of normal modes 19 2.4 Modal orbit and rest points 25 2.5 Geometrical and dynamical aspects of the motion 27 2.6 Similar and nonsimilar normal modes 29 2.7 A variation principle for the modal orbit 33
2.7.1 Introductory remarks 33
2.7.2 Derivation of Jacobi's principle of least action 35 2.7.3 A variation principle for modal orbits 40 2.8 Survey of the existing literature on normal modes 43
3 COMPUTATION OF NORMAL MODES 49
3.1 Introductory treatment of computational methods 49 3.2 The method of the first maximum of the orbit potential 51 3.2.1 Foundation of the method 51
3.2.2 A continuation algorithm 55
3.2.3 Numerical aspects 57
3.2.4 Examples 61
3.3 Computation of the normal orbit using finite elements 67
3.3.1 General part 67
3.3.2 The constrained algebraic variational problem 70
3.3.3 Numerical aspects 71
3.4 Normal modes as solutions of a nonlinear eigenvalue
3,4,1 The eigenvalue problem with boundary values 75 3.4.2 Discrete approximation of the eigenvalue problem 77 3.4.3 Arc-continuation methods; general part 78 3.4.4 An arc-continuation method applied to ~<z,A.)=O
80
3,4,5 Initial points and directions for thearc-continuation algorithm 3.4.6 Numerical aspects 3.4.7 Examples
3.5 Comparison of the three methods
4 LINEARLY SEPARABLE SYSTEMS 4.1 Introduction
4.2 Separating conditions 4.2.1 General
4.2.2 Separation using two transformations 4.3 Normal modes in linearly separable systems
4.4 Physical interpretation of the separation condition 4.5 Example
4.6 Influence of damping
4,7 Linearly separable systems and systems of the Liouville type
5 ALMOST SEPARABLE SYSTEMS 5.1 Introduction
5,2 Approximating an almost separable system 5.2.1 General
5.2.2 Sets K and Ke
5.2.3 The constrained minimization problems 5.2.4 The metrics PI and P2
5.3 Methods to calculate higher-order corrections
6 APPLICATION OF THE NORMAL MODE THEORY 6.1 Introduction
6.2 Ball bearings as elastic elements
6.3 Lagrangian of the yokes, the blades and the tube 6.3.1 The yokes 6.3.2 The blades 82 85 89 101 103 103 106 106 108 Ill 117 122 126 127 131 Ill 133 133 135 141 143 146 151 151 154 157 157 158
6.3.3 The tube 162
6.4 The final equations 164
6.5 Numerical results 167
6.5.1 General part 167
6.5.2 Accuracy 168
6.5.3 Remarks 169
7 SUMMARY AND CONCLUSIONS 181
APPENDICES 185
REFERENCES 205
SAMENVATTING 219
ABSTRACT
In mechanical engineering practice. dynamic systems are mostly modelled in such way that the resulting equations are linear. Next, the behaviour of the system is analysed by determining the eigen-vibrations (eigenfrequencies and eigenvectors). One is thus able to make statements about the conditions at which resonance occurs. Besides that,the eigenfrequencies and eigenvectors can be used to characterize the general motion of the system.
In Rosenberg's theory on normal modes the notion of eigenvibrations is generalized in such a way that also the nonlinear case is covered. It is found (chapter I) that these normal modes are also relevant infindingthe conditions at which resonance occurs.
Chapter 2 treats·the properties of normal modes, The question, how these modes can be calculated, is dealth with in chapter 3. Three methods are treated. It is found that one method is particularly useful. In this method the modes are calculated by solving a discretized eigenvalue problem with boundary conditions. Combined with an arc-continuation method one is thus able to calculate the periods as function of the energy, the amplitudes as function of the period etc.
Chapter 4 deals with the question as to what extent normal modes can be used to characterize the general motion of the system. It is found that in some cases this is possible indeed, as the equations of motions can be completely decoupled by applying a linear coordinate
transformation. Systems having this convenient property .. are called: linearly separable systems. Chapter 5 deals with the question how a nonseparable system can be approximated systematically by a separable one. Starting with a solution of the approximated system. one next can try to obtain approximations of the original non-separable system. Two methods to achieve this are also discussed in. chapter 5.
In chapter 6 an application of the normal mode theory is treated. For a continuous system, the equations of motion are approximated - in the usual manner - by a set of ordinary differential equations
describing the motion of a system with two degrees of freedom. Of the system thus obtained. the normal modes are determined numerically.
The final chapter (ch. 7) summarizes the main conclusions which can be drawn for the previous chapters.
CHAPTER I INTRODUCTION
1.1 General part
The subject of the present thesis is the theory of normal mode vibrations in nonlinear conservative systems having two or more degrees of freedom.
The study originated from the consideration that there is now every reason to apply nonlinear theories to practical problems as encountered in mechanical engineering. In the "statical" parts of mechanics (elasticity, plasticity, etc.) such theories are already applied on a rather large, and rapidly growing scale. This trend is less clear in the "dynamic" parts of
mechanics, such as vibration analysis of machinery and structures. Although in this case, where systems with many degrees of freedom are involved, sometimes nonlinear instead of linear equations are used, these equations are merely solved numerically for particular cases.
As not much insight is gained by proceeding in this manner one can safely say that linear theory is still predominant. However, it is to be expected that the days of linear theory are numbered since, owing to ever increasing demands on precision, speed and performance, linear theory will gradually cease to apply to the conditions under which dynamic systems are operated.
An important aspect of the dynamical behaviour of systems is the occurrence of resonance under some circumstances. A dynamic system is said to be in resonance if it reacts "excessively" to some time-dependent external cause. (This cause will henceforth be called: external forcing). Resonance phenomena are important for various reasons, which generally depend on the systems under consideration.
A saxophone or a wireless, for example is based on the occurrence of resonance because otherwise they simply would not work.
In many other cases however resonance phenomena are to be avoided because they can prevent the system from performing as required,
and give rise to unacceptable working conditions in factories, to excessive wear or even breakdown of parts of the system.
Whether resonance occurs or not depends on the properties of the system, on those of the external forcing and on the initial conditions. These aspects should therefore all be taken into account. In spite of this fact, the systems studied here differ in two respects from those very frequently encountered in practise. In this study we consider systems with equations of motion
given by
0 (1.1.1)
(Then degrees of freedom x
1, ••• , n are brought together in column x. The n*U matrix M is real, symmetric and positive. The function Vis the potential of the system). First note that no external forcing is present because the right-hand side of (1.1.1) is zero: the system is a freely vibrating one. Second, note that no damping terms are incorporated.
Freely vibrating systems are not uncommon, but in general external forcing is present to do work and replenish (on the average) energy loss from damping which is always present to some extent. The nearly typical dynamic systems encountered in mechanics are forced and damped ones. Once started, they eventually reach a steady state, so that what has to be established is whether periodic forcing can trigger resonances or not.
This matter being as it is, it is naturally desirable to know whether knowledge of freely vibrating, undamped systems has any relevancy to the resonance behaviour of those systems encountered in practise. If the system is linear, i.e F
=
Kx(K
=
KT>O:n*n matrix), it is known that knowledge of the freely undamped motion, characterized by the so-called natural frequencies and eigenvectors, is of interest in describing the motion of the forced and (slightly) damped system. Slight damping is in fact often present in practise. However, in designing dynamic systems much effort is often directed to minimizing energy losses arising from damping.The vibrations described by the natural frequencies and eigen-vectors, that is the eigenvibrations, natural vibrations or normal modes, are in fact special periodic solutions of the equations of motion. The question thus arises to the extent that such "natural" vibrations also exist for nonlinear systems, and as to what knowledge of the behaviour of the forced and damped system can be derived from knowledge of these "natural" vibrations. I t turns out that in the nonlinear, undamped, freely vibrating case too, these periodic solutions (provided all velocities vanish at the same time *))offer useful information about the behaviour of a system with damping and external forcing. This is demonstrated in section 1.2 of this chapter.
The normal modes or "natural" vibrations of the freely vibrating undamped system (1.1.1) are the subject of this thesis.
Rosenberg's theory concerning these vibrations is the starting point.(Detailed references will be given later). Although Rosenberg based the development of his theory on the ideas of other researchers, he was -to our knowledge- the first to study normal mode vibrations over a longer periode of time, thus building up a useful theory.
In this theory normal modes are so defined that the natural vibrations of the linear theory emerge as a special case.
Chapter 2 deals with the theory of normal mode vibration.
After describing the type of system of interest, special attention is given to the definition of normal modes. This is done not only because Rosenberg's definition contains some hidden as-sumptions and unwanted properties but also to make as clear as possible how the definition differs from that known from the linear case.
Next the properties of normal modes are considered, the most important one being that the modes depend on the energy of the system. Some new theorems concerning these properties are proved.
An extensive discussion of Jacobi's principle of least action is also included in chapter 2, as it can be applied to the
calculation of normal modes. However, in order to do so Rosenberg's method of using this variation principle requires modification. The necessary changes are also discussed.
The closing section of chapter 2 surveys the existing literature on normal modes.
Chapter 3 concentrates on the question of how normal modes can be calculated. Three methodes will be dealt with.
Each is based on different aspects of normal modes. The first method, basically a shooting method, calculates initial conditions, so that the equations of motion together with these initial
condition are solved by normal modes. The method is called the method of the first maximum of the orbit potential, because, if this maximum equals the total energy present in the system, the shooting has been successful. In this respect the method differs from known shooting methods. The second method computes the shape of the modal orbit (= the orbit in the configuration space along which the system travels in the course of time). Using Jacobi's principle of least action, finite element approximations of the modal orbit are calculated.
In the third and last method a nonlinear eigenvalue-boundary
value problem is solved by discretizing the equations. T.he eigenvalue is the period of the normal modes. Hence, in this method both
the periods and the coordinates (as function of time) are calculated. Using an arc continuation algorithm this method is capable of calculating amplitude-period relations.
After the chapter 2 and 3 the attention is shifted from questions, such as the nature of normal modes and how they can be calculated, to an entirely different problem, namely what can be said about the motion of the system in general if the normal mode vibrations are known. In het linear case it is known that the general
solution (apart from some undetermined constants which have te be fixed by specifying the initial conditions) is determined, at least in principle, once the eigenvectors and frequencies are known. Apart from that, the systems themselves are in fact completely determined by these quantities because two systems
are dynamically equal if they have the same frequencies and eigenvectors, even though their respective mass matrix M and stiffness matrix K are different. Because these properties are based on the principle of superposition, valid in linear
systems, and this principle does not apply to nonlinear systems, it can be expected that a relation between the general motion and the normal modes -if it exists at all- will be very complicated, Nevertheless, it is found that in some cases such a relation not only exists but is also fairly simple.
Chapter 4 deals with a class of systems, the so-called linearly separable systems, for which the relation mentioned above is a linear one. Conditions are derived, making it possible to decide whether a given system belongs to this class or not. It is found that decision can be made using only the linearized system.
Chapter 5 deals with "almost separable" systems. The question primarily considered in this chapter is how a system, which is not linearly separable, can be approximated systematically by a separable one. This is basically a minimisation problem, the solution to which depends on the norm chosen for measuring the difference between the given (nonseparable) and the approximated (separable) system.
Two norms for measuring this difference are discussed.
The underlying idea of this chapter is to use the solution to the approximated system as a lowest order approximation of that of the given system. As separable systems are much easier to understand than nonseparable ones, it is felt that, in many cases, loss of accuracy (by studying an approximation instead of the "real thing") is more than compensated for by the gain in simplicity. It is always possible to improve the lowest-order approximation with the aid of perturbation techniques or some iterative method. In some cases it is even necessary. This will be the case if questions of stability are concerned, as in separable systems normal modes are energetically isolated, whereas in most other cases normal modes do not have this
Chapter 6 contains an application of the normal mode theory to a system met with in practise.
The final chapter 7 summarizes the main conclusions which can be drawn from the chapters 2-6.
In concluding this section three remarks have to be added: 1. Stability of normal modes, though very important, wilL
not be discussed. The main reason for this is that it was felt that a thorough discussion of stability is a problem in itself, to be tackled at a later stage.
2. As already stated, systems encountered in practice are always damped and in most cases periodically forced. After these systems are set in motion, which has to be done at one time or another, the motion evolves to a steady one.
This steady stateisa limit cycle of the system. Generally it is only possible to attain a particular limit cycle if the initial condition be in the so-called domain of atraction. Although a normal mode can (at least approximately) coincide with a limit cycle of a forced and damped system, the theory does not furnisch any knowledge about the shape and magnitude of the domain of atrection corresponding to it.
3. Normal modes are solution of the equations of motion for special initial conditions.
Therefore, the system vibrates only in a normal mode if the initial conditions are chosen accordingly. For this reason initial conditions are almost as important as the equations of motion, and it should be kept in mind that the equations and the initial conditions determine the motion.
This applies to linear and nonlinear systems.
In the linear case, due to the validity of the superposition principle, it is easy to determin how a change of the initial conditions changes the motion. Probably this is the reason why in practice not much attention is paid to initial conditions. In nonlinear theories however the initial conditions should be given as much attention as they (out of necessity) deserve.
1.2 Relevance of normal modes to systems in which damping and periodic forcing are included
In the preceding section it has been claimed that knowledge of the freely vibrating, undamped system is relevant to the resonance behaviour of those systems met with in practice, that satisfy the sole restriction that the damping forces are small compared with inertia and elastic forces.
To substantiate this assertion, let us first consider a nonlinear system with one degree of freedom; the classical example of the harmonically forced Duffing system with linear viscous damping.
w 0 3
X + e X + X + e:Bx ea cos wt, B>O O<e <<I
The equation is written in this form to express that both the external forcing and the nonlinear part of the elastic force are of the same low order as the damping. The amplitude-response curves of this system are shown in figure 1.1
1 Fig 1.1. backbone curve 2 w +
Response ouwea and backbone aul"Ve for the damped Duffing system wlth hamonic forcing. Diffe:t'ent vaZuee of a3 but e and B fked.
The dashed line, called the backbone curve, represents the amplitude-period relation of the undamped freely vibrating system, i.e. the system with equation of motion.
(1.2.1)
The dash-dot-da.sh line is the locus of extremal values of the amplitude of x. This line, which has the same shape as the backbone curve is shifted to the left over a "distance"
!e:
2• For this reason, provided the damping coefficient e; is small,the backbone curve represents a useful approximation of the locus of extremal values of the amplitude x.
Hence, knowledge of the freely vibrating system is in fact relevant to the resonance behaviour of the linearly viscously damped Duffing system with harmonic forcing.
Having thus indicated the line of thought we have in mind, let us now proceed to consider the forced system
!(t} (1.2.2)
By definition, the function !(t) is said to be strictly natural
f orc1ng funct1on . . *) • 1f:
o the solution to (1.2.2) is periodic;
o !(t)
=
constant*
~(~(t}) (!(t) and E(~(t)) similar. Now let E;;(t)be some periodic solution (periode T0 ) of the unforced system (1.1.1). Defining the strictly natural forcing function
*) The concept of natural forcing is taken from Harvey (1958). In treating Duffing's equation such forcings are sometimes called "elliptic" forcing. See Hsu (1959) and Ojalvo and Bleckman (1961). Note that harmonic forcing functions are the natural ones for linear systems. The concept did not receive much attention in literature. (However see also Iwan (1969) and Bangen (1971).) This is unfortunate because the concept is very useful in studying forced systems, particularly if the last defining property is relaxed to f G (x,i) (G is some suitable function). Without proof
we remark-that the concept can be used to deal with harmonically forced linear systems, to show the existence of subharmonic vibration of Duffing1s equation and to demonstrate the
stabilizing effect of periodic forcing on the inverted pendulum or Duffing's equation with a softening spring.
f (t) by -n
(I .2.3)
it follows that the function
x = t(t):= i(at) (1.2.4)
solves the system (1.2.2) if f(t) = f (t). The function -n x(t) is periodic, with periode T = T /a. Comparing (1.2.4)
- 0
and (1.2.3) shows that if a+ I (i.e. T + T) then the amplitude 0
of !n becomes infinitesimally small, whereas the amplitude of ~
does not change at all. Hence the forced system is at resonance. Calculating the n amplituds of ~ as functions of the period T and plotting the results in an (n+l) -dimensional space
0
yields a response curve of the undamped, but forced system (1.2.2). Note that this curve can be found by using the equations of motion of the unforced system (1.1.1) only. As systems with several degrees of freedom have more than one periodic solution*)
phase
Fig. 1. 2.
Baakbone curves of a system with two degrees of freedom.
V(x,y) •
iw
2(x2 + y2) +~a
2(x-y)
2 + !S(x-y) 4, S > 0, M • I.*) Just as in the linear case, one expects that, for each value of the energy, there are at least as many periodic solutions as there are degrees of freedom.
for each value of the energy, a number of backbone curves are thus obtained. Figure (1.2.2) contains the backbone curve of a nonlinear system with two degrees of freedom.
The backbone curves in this case also represent approximations of curves of extremal values of the amplitudes of ~ if the
system is lightly damped. This will be proved later. To indic~te
how this will be done we consider again Duffing's 'equation
co 0 3
X + EX + X + E6X
=
f(t)Let ~(t) be any periodic solution (with period T
0) of the
undamped freely vibrating system (1.2.1). If we now take as forcing function
(') d/d(at)
then the damped and forced Duffing system is solved by x(t)
=
~(t):= ;(at). Suppose that~ reaches at t= t* a local
• ( *) 0 ( * . 2 ( )
extremum, that l.s cp t = a and cp t ) = 0. By expandmg f t in a series of powers in E it follows that f2 reaches a local
extremum if
in wich
(E) I
- (l-a2 )
E + Be: 2 + O(E ) 3
The function o(e:) is the phase shift caused by the damping. B is some function which is of no concern here because it contributes in the final analysis only to those terms of f2 which are of higher order. Setting t
=
t* + o(e:) in f2(t) yields an expression for the squared amplitude f2(t*+o) of the forcing as a function of E and the frequency w (= 1/T = a/T )0
of the forcing. If now, taking E fixed, the frequency w is varied it is found that the amplitude reaches a local minimum if (up to second order in E) the frequency w is chosen as
w wr
=
a /T r o ar + •••• (w r 2
As the amplitude a of the responce is constant, it follows that the quotient of a and lf<t* + o)l reaches a maximum if
w = w • Note that we obtained the same results as those r
using perturbation techniques.
The analysis given above can be generalized to the case that there is more than one degree of freedom and more general damping terms as those considered so far. To confirm this consider the system as given below.
( 1.2.5)
The damping matrix D is symmetric and positive definite
Again, let i(t) be any periodic solution of the undamped freely vibrating system. Kntirely in the spirit of the method of natural forcing we now take as forcing function
f = f -p in which
2
(1-a )E(~(at)) + EaD(i(at), ~'(at))i'(at) (1.2.6)
(I) + d/d(et.t)
Note that, for sufficiently small e, f -p can be considered as a perturbed strictly natural forcing function. Again we see that
solves the system of equations (1.2.3) with forcing (1.2.6). Now, let t* be any instant of time for which
f
vanishes, so that0
*
P.(t )
=
Q
*
*
Furthermore, let !:= !(t ). At t all components of
1
reach extremal values. Hence I l!(t)l 12:=tT!
reaches a local maximum(aTa). If no damping were present (E=O), the same would hold
f~rl
If I I -p 2=
f Tf • However, the damping slightly perturbes -p -p not only the shape of the forcing function, but also causes a slight phase shift which is generally different for. each compont f. of f • As a result, the components of f reach1 ~ ~ .
their respective extrema at different times. For this reason it makes sense to search for extremal values of h(t;E):= f Tf •
-p -p A necessary condition which must then apply is
h :== ilh- 0
t
at
From (1,2.6) we have
h(t;e:,a)
*
*
Because i'(at ) =
Q
we have ht(t ; O,a) • 0, Nowin which the matrix J is the Jacobian matrix of
!•
so that*
From (1.2.8) we conclude that htt(t ; 0, ) ~ 0 if
(1.2.8)
Assuming that these conditions apply it then follows from the implicit function theorem that a function o(E) and an interval (-e:
It is found that
o(r::) (1.2.9)
in which
B is a complicated expression. Its precise form is of no concern because B contributes only to those terms of h(t*+o(E:),r::,a) which are of third or higher order in E:. On setting t=t*+ o(e:) in expression (1.2.7); expanding h(t*+ o(e:);e,a) in a series with powers of 8, and retaining only those terms which are of second order in 8 we obtain
h(t* + 0(8),8,a) (1.2.10)
The function h(t*+ o(e;),e,et) expresses how extremal values of fTf change in the neighbourhood of t
*
if the damping is varied. -p-pWe now seek to determine a value of a in such a way that h is minimal (at least locally) if -for fixed values of 8 and a- it
is considered as a function of a. This amounts to searching for
a "frequency" w (:=1/T, T=period) of the forcing so that the "amplitude" h of f -p is minimal because a is connected with w by:
T
0 (I)
a = = -T
T0 : period of the solution ~(t) (~: vector of amplitudes)
of the freely vibrating undamped system. Note that (w
0, ~) E Backbone curve
T : period of the forcing of the damped system.
Differentiating (1.2.10) with respect to a gives that h assumes
a local minimum if, up to the considered order
a
=
a r 2 2 ec
1 (~) I ' : T : : -4C2(~)~ (!)!(!)2
As the difference between wr
=
ar w0 and w0 is of the order E ,
the point (wr,~) lies in the vicinity of the backbone curve if the damping coefficient is sufficiently small. For this reason, the backbone curve represents an approximation of the points (w, ~) for which the ratio
I
1~1l/lh
is maximal. As a damped system is said to be at resonance, precisely if this ratio peaksconsidered as function of the forcing frequency - - we
conclude that knowledge of de backbone curve is indeed relevant when studying the resonance behaviour of the forced (slightly) damped system. Note that results obtained here are quite analogous to those of the Duffing system with linear viscous damping
CHAPTER 2. THE NORMAL MODE THEORY.
2.1. Introduction to chapter 2
Thb chapter deals with the major aspects and problems of the normal mode theory. After a description of the system (section 2.2), the definition of the concept of normal modes will be treated in section 2.3. Starting with the definition of Rosenberg, it will be shown that this definition has to be modified since :
o the concept as used by Rosenberg is not invariant for rotations of the coordinate system;
o There are assumptions concerning the behaviour of the potential V hidden in Rosenberg's definition.
Section 2.4 dicusses properties of the modal orbits of normal modes. These modal orbits are in fact the nonlinear counterpart of the eigenvectors, known from the linear theory. The differential equations for these orbits are discussed in section 2,5. Apart from geometrical aspects (orbits) the motion of the system has also dynamical aspects. The most important of these aspects are the period-energy re~ations
of the normal modes. These are also found in section 2.5.
A classification of normal,modes in similar and nonsimilar modes is given in section 2,6, Two useful theorems on similar normal modes are proved.
Section 2.7 contains an extensive discussion of Jacobi's principle of least action. As this principle will be used in a later chapter to calculate finite element approximation of the modal orbits, a thorough understanding of this principle is indispensable. Rosenberg's method of calculating modal orbits with Jacobi's principle will be discussed. It will be found that this method does not work in some cases. An alternative method is presented. After the first six sections the reader will know enough to appreciate a survey of the existing literature on normal modes. This survey is given in section
2.8, in which the following questions will be discussed in particular I) What kind of systems are treated?
2) What types of modes (similar, nonsimilar) have been discussed? 3) How many normal modes are to be expected?
4) What is known about stability of normal modes?
5) What is the relevancy of normal mode theory for the general motion of freely vibrating systems and the steady state solution of systems with external forcing1
6) What numerical methods for calculating normal modes·are available? 7) Has the theory been applied to the systems encountered in practice?
2.2. Precise description of the system and other preliminaries
The system considered is a dynamic system, with equations of motion given by
in which x :c col(x
1(t), ••• ,x (t)), x.(t) are the coordinates;
- n 1
t c time ; (O)
=
d/dtM
=
MT 0 constant and real symmetric n*n matrix V = potentialo/ox := col(o/oxl, ••• ,o/ox}
- n
(2.2.1)
Many systems encountered in engineering and physics can be modelled by equations of this type. Typical examples are systems of point mass and (massless) springs. Very often equations (2.2.1) also describe the (approximate) motion of continuous systems.
Throughout this study it will be assumed that all quantities and consequently all equations are dimensionless,Nevertheless, all quantities will be denoted by physical names. The matrix M and the real scalar t will be referred to as mass matrix and time
respective-ly. The gradient of V will be called elastic forces and the quantity E defined by
is denoted as energy.
In order to make more detailed statements concerning the system it is necessary to restrict the class of possible potentials. The first assumption is that the elastic force is continuous, and that in a neighbourhood of the origin even the second partial derivation of V exist; or:
a) V(x) is at least once continuously differentiable and az;/ax.ax. exists in some neighbourhood of
Q.
1 J
The second assumption is that at ~Q the elastic potential V attains a local isolated minimum. This assumption ensures that ~Q (system at rest) is a Lyapunov stable solution of (2.2.1) (Dym (1974)).
Without loss of generality we may take v(Q)•Q so that the precise statement of this assumption reads as follows
b) there is a neighbourhood E of the origin in which and: o V(~) ~ 0 ! € E
o V(x) 0 and
av/a! -
Q
if and only if ~QAs normal mode solutions are special periodic solutions, the last assumption is motivated by the wish to ensure the existence of at least one periodic solution (A discussion of the existence of such solutions will be given in a later section of this chapter). According to a result obtained by Rabinowitz {1978) the following assumption ensures the existence of at least one periodical solution ·of the equations (2.2.1):
c) There is a real positive scalar E
0,possibly infinite.such that
on the pointset E(E
0) defined by:
E(E ) := {x € R
I
V(x) s E ; OEE, E simply connected}-the following conditions apply cl: Assumption b) holds with E=I:(E
0)
c2: I:(E
0) is convex ;
c3: for 0 < E < E
0 the surfaces V(!)
=
E w~th ! E I:(E0)bound convex regions I:(E) so that:
0
0
I:(E
1) ci:(E2) i f E1 <E2 (<E0)
if E1 < E
0, E2 < E0 and E1 ~ E2 the surfaces V(!)=E1 (! E I:(E
0) ) and V(!)=E2 (! E I:(E0) ) have no point in
common.
This asspumption implies that V is strictly increasing on the set I:(E
0) , that is
X E I:(E ) \{6}
- 0
-In many papers on normal modes, the additional assumption is made that the potential is point-symmetric: V(!)
=
V(-!)• Although this is in practice indeed frequently the case, it also often occurs that V is not point-symmetric. Unless explicitely otherwise stated, we will in general have V(!) ~ V(-!)•To conclude this section some remarks on the notation and the symbols (which are used) is necessary.
a) Throughout this work the symbols !•n,t,M,V,K and E denote the same quantities, namely o column of coordinates, -o number of coordinates, o time, o mass matrix o elastic potential
o stiffness matrix of the linearized system, and
o energy respectively.
The symbol T is used to denote the kinetic energy and the trans-pose of a matrix. Base vectors in the n-dimensional space Rn are written as ~i' and ~i := col(O, •••••• ,O,I,O, ••• ,O)
The symbol
o ..
denotes the Kronecker delta. ~]b) Other symbols as those mentioned above, can denote different quantities if they appear in different sections. The meaning of these symbols is defined in the text as soon as they appear.
c) Time derivatives will be denoted by a dot placed above the function
.
~
(For example ! and !·
d) Equations are numbered in the following manner Equation (~.S.y)
(~.S.y) denotes equation ~ Y of section a.a.
2.3. The definition of normal modes
The concept of normal modes of a nonlinear system was first used by Poschl (1952), (1953) and (1954). Kauderer (1958) also used this concept. However, it seems that Rosenberg (1960) was the first to define normal modes more or less rigorously. Therefore, the definition of Rosenberg will be given below,
Consider first a freely vibrating linear system with coordinates x
=
col (x1, •• ,xn)' mass matrix M and stiffness matrix K (K=KT>O). Such a system is said to vibrate in a normal mode if the solution !(t) is of the type
!(t) • A cos(oot + $)~ (A,~ real scalars) (2.3.1)
where oo and ~ are solutions of the eigenvalue probl~
u
=
coll(u1, ••• ,u)- n
(In linear systems normal modes are often defined with the aid of this (linear) eigenvalue problem).
Note that the solution (2.3.1) is a special solution of the equations of motion. This solution is special as
solution is possible; 2) The solution is harmonic;
3) All coordinates have the same minimal period as the vectorfunction !(t);
4) All coordinates vanish at the same time and the velocities vanish simultaneously;
5) The coordinates are linearly related to each other (i.e. if x:=x1, t~O then xi=x*(ui/u1) for i=2, ••• ,n).
The last three properties which can serve as a definition of normal modes of linear systems, have been generalized by Rosenberg in such a way that the nonlinear case*) is also covered (Rosenberg (1966-a) p 162)**>. To do so he first defines "vibrations in unison" as follows.
Definition
A vibration in unison is a motion ~(t) such that:
o xi(t)
=
xi(t+T) i=l,2, ••• ,n;o I f t is any instant of time there is a single t in
0
t ~ t < t +
IT
0
so that
i=l,2, ••• ,n
o If t is any instant of time there is a single t
1 ~ t0 in t ~ t
1 < t + iT so that
i=l,2, ••. tn
o Let l (fixed be any one of the subscripts i=I,2, ••• ,n such that
*) Rosenberg considers only systems with point-symmetric potential (V(x)=V(-x)). This is reflected in the definition of vibrations-in-Unison: Restrictions of this kind are not necessary. For example see the-paper of Haughton (1967) in which a system with nonsymmetric potential is considered.
**) See also Rosenberg (1960) which contains very illuminating remarks concerning the difference between linear and nonlinear systems and how this difference affects the definition of normal modes.
xz.(t) ~ 0. Then every xi can be written as a single valued function of xz.:
x.
=
f. (xz)1 1 i=1,2, ••• ,n
(2.3.2) These relations hold for all times t, i.e. x.(t)
=
f.(x7(t)).1 1 "
Next he defines normal modes as the vibrations-in-unison of the freely vibrating system.
Definition Rosenberg (1966-a)
A normal ~ is any solution of the equations (2.2.1) which is also a vibration in unison.
To facilitate the understanding of this definition consider the system as shown in fig. 2.1.a. A homogeneous disk is connected with the inertia space by means of two identical and linear springs. If the disk is at rest the springs are unstretched, The system, has three degrees of freedom. The coordinates are chosen as shown in fig. 2.l.b. Because of this symmetry in the system three possible
(a) (b) (c) (d) (e)
Fig. 2.1.
P"lan.ar> motion of a disk IJYlth tr»o identiicaZ linear sp:roings.
a) System at rest;
b) System 1.n motion. Coordtnates x, y, 4J;
a) Normal. mode l(transZationaZ.); d) Norma Z. mode 2 ( " ) ;
motions can be identified immediately. The translational motions are normal modes for all values of the total energy. The rotational motion is only a normal mode if the energy is below some threshold-value E
0 because this motion is not periodic if the energy exceeds
this threshold. The table below lists the initial conditions and allowed values for the energy so that the system vibrates in one of the three modes.
mode I x(O)
~(0)
! y(O) y(O) $(0);(o)
EI ~o 0 0 0 0 0 all I 0 ~o 0 0 0 0 values of I ~o ~o 0 0 0 0 E 2 0 0 ~o 0 0 0 allowed 2 0 0 0 ~o 0 0 2 0 0 ~o ~o 0 0 3 0 0 0 0 ~o 0 3 0 0 0 0 0 ~o 0 < E < E 0 3 0 0 I 0 0 ~o ~o
Looking carefully at Rosenberg's definition shows that the definition has two defects:
a) It contains hidden assumptions on the potential V.
b) It is not invariant for linear transformations of the coordinates. To demonstrate the first point let us assume that the system vibrates in a normal mode. Without loss of generality we may assume that
!<O)
= Q• Let t 1>0 be the first time at which !(t1)=Q.
As the systemis autonomous, and no damping is present, the system is invariant for time inversion. Therefore we also have
As the system vibrates in a normal mode we must have
As the zeros of
!
are also spread at time intervals ofIT,
we find that the time interval between a zero of ~ and the next or preceeding zero of!
is always a quarter period. Normal modes of systems with point-symmetric potential indeed have this property. Periodic vibrations of systems with a potential which is not point-symmetric0
will in general not have the property that the zeros of ~ and ~ are spaced at time intervals of !T. Nevertheless it may well happen (examples are easy to construct) that these periodic vibrations have the following properties:
o all coordinates are zero at the same time; o all velocities vanish simultaneouly;
o every coordinate xi can be written as a single-valued function of one coordinate, say x1•
From a physical point of view only these three properties are essential to the notion of vibrations-in-unison. One would also like to call such periodic vibration normal modes. The definition of Rosenberg therefore has to be altered. It is found that only the second property of Rosenberg's definition requires alteration. This property should read:
Between two successive zeros of
!
there is a single time t0 such
that
x. (t ) = 0
1 0 i•t,2, ... ,n
To demonstrate that Rosenberg's definition is not invariant for linear transformations consider the following example
"" co
X + X
=
0 y + 9y=
0With initial conditions so chosen that
x(t) • sin t y(t) = 0,01 sin 3t
This solution is ~ a normal mode according to Rosenberg's definition. Next apply the following coordinate transformation
I]
!We then find that
~(t) =~{sin
t + 0,01 sin 3t}=~sin
t (1,03- 0,04 sin2t)n(t)
=
~
{sin t - 0,01 sin 3t} =~
sin t {0,97 + 0,04 sin2t) The function i(t) is a normal mode in the sense of Rosenberg.Obviously in some cases it seems possible to generate· (or annihilate) normal modes simply by applying linear transformations. This state of affairs is not satisfactory. Invariance (at least to some extent) for coordinate transformation should be required. In order to make things not too complicated we require invariance of the definition for linear orthogonal coordinate transformations because only with these
.
*>
. **>
f .transformations 1s the shape of the orb1t o the mot1on preserved.
Combining all changes needed, we arrive at the new definition of vibrations-in-unison:
Definition
A periodic solution !(t) of the equations of motion is a vibration · in unison if there exists an orthogonal transformation T so that the vector function
!
defined be!
= TX has the properties o .f(t) = f(t+T)0
0
If t is any instant of time, there is a single time t
0 so that 0
9i(t0) (i=l,2, ••• ,n) and t ~ t
0 ~ t +,T.
In the time interval between two successive zeros of
t
each (transformed) coordinate9.
vanishes only once. This happens1
for all of these coordinates at the same time.
o Let
Z
(fixed) be any of the subscripts i=1,2, •• n, so that9z(t)~O.Then every $i can be written as a single-valued function of
9z•
that isi=l,2, ••• ,n
These relations· are valid for all times; i.e. ~.(t) = f.(~z(t))
1 1
*) Shape (of a curve) has to be understood in a purely geometrical sense. In R3, for example, the shape is determined by specifying curvature and torsion as functions of arc length.
**)
Orbit is the curve in the configuration space along which the system travels in the course of time.If this definition of vibrations in unison is used, Rosenberg's definition of normal modes still applies.
Remarks
a) The relations f. are called modal relations;
l.
b) The fourth requirement follows from the first three. It could be dropped. However, as the functions fi will play an important role in the theory (because they directly represent geometrical aspects of the motion), the fourth requirement has not been dropped.
2.4. Modal orbit and rest points
Let !(t) be any solution of the equations of motion for some initial values !(0) and
!<o).
The set{!
I
x(t) E configuration space; t ~ 0}is called the orbit (associated with the initial values !(0), !(O))
The orbit of a normal mode solution will be called: modal orbit. As the energy is a conserved quantity, that is
oT o
i
!M!
+ V(!) = E = constant (2.4.1) all orbits with a fixed value of the energy E remain within the point set for all times.Because M > 0 the boundary V(!)=E of this set can only be reached at • 0
those times for wh1.ch ~Q· Such points on an orbit are called: rest points. Modal orbits, by definition, have rest points. Assume that the system for t=t has a rest point x , i.e.
r -r
~(t }
- r 0
With the aid of the equations of motion and the Taylor series
expansion of ~ around t it is easy to prove that at rest points the
- r
at :x=x
- -r
with: o n(x ) unit normal to the surface V(:)=E - -r
av
Iv
1 n(x ) = { ( - ) / - } - -r 3x X :x=x - - - -ro ~(!r) unit vector tangent to the orbit
v(x) =lim {~(t)/j~(t)j}
- r t+t -
-r
(2.4.2)
(It should be noted that equation (2.4.2) also has to apply to rest points of systems for which M=M(!)).
If M=l the condition (2.4.2) states that at a rest point the orbit intersects the surface V(!)=E orthogonally. For normal, modes
condition (2.4.2) is in fact a transversality condition because - in the variation principle for modal orbits - it arises from the
constraint that a rest point has to lie on the surface V(!)=E. (See section 2.7 and, for example Rosenberg (1962-a)).
The modal orbit is a simple*) curve which passes through the origin. The rest points are the end points of this curve. (In ref, Rosenberg (1966-a) p 185-187, some geometrical properties of modal orbits are derived. The results apply to a system with two degrees of freedom). In figure 2.2 a possible modal orbit, rest points, etc. are drawn for a system with two degrees of freedom. In the course of time the system travels along the modal orbit.
point
1 - - - t -modal
orbit
- X
Fig. 2. 2.
Modal orobit of a system hlith thlo degroees of fr>eedom
*) Simple curves are curves which are not closed and which contain no loops.
2.5. Geometrical and dynamical aspects of the motion
The solution !(t) of the equations of motion contains both geometrical and dynamical aspects. By this we mean the following. Let the orbit be described by some function g of the arc lengths, that is g • g(s). As the system moves along the orbit, the arc length s must depend on the timet; s=s(t). Then !(t)=g(s(t)).
The function g(s) represents the geometrical aspects of the motion (shape and orientation of the orbit), whereas s(t) represent the dynamical ones (for example, s(t) is the speed with which the system moves along the orbit).
It is worthwhile to separate these aspects. This is for example always done in the linear case because the eigenvectors determine the direction of the (straight*)> modal orbits and the eigenvalues
determine the frequencies.
The possiblity of such a separation is present in the definition of the normal modes because the functions fi of equation (2,3.2) represent only geometrical aspects.
It is possible to derive a set of differential equations for the functions f.(x) or any other purely geometrical representation (for
l.
example ~!(s) where s is the arc length). This can be done by using either a variation principle or the equations of motion as a starting point. The variation principle will be discussed separately in
section 2,7, Kauderer (1958) and Kinney((l965) p 123-126) derived differential equations (starting with the equations of motion) for systems with M-I and two and n degrees of freedom,
In appendix I these differential equations for the are derived for the case that M~I. Some properties are also treated in this appendix.
respectively, functions f. (x)
l.
of these equations As the assumption M•I involves no loss of generality**) and the equations for the modal orbit in this case are simpler in structure,
*) In linear systems it is always assumed that modal orbits are straight because a normal mode solution of the type !"'! exp(iwt) is. used.
**) Most papers on normal modes consider only systems with M•I. This can always be achieved by means of a transformation to new coordinates
n:=
Mi!• The transformed system is sometimes called pseudosystem. The equations then describe the motion of unit . masses moving in a n-dimensional space,it will henceforth be asswned that M=I. SU1lllllarizing the main results of Appendix I we define:
X (J) S X S X (2}
r r
( '} := d/dx
and arrive at the following differential equations for f.(x)
1
(2.5.1)
With respect to the interval (xr (l} ,xr(2)) for the independent variable the following can be said. Equations (2.5.1) apply to those time
intervals where x=x
1(t) can be inverted. In general this is only possible locally. For normal modes however the situation is quite different, because x
1 can be inverted at any time interval that starts when the sytem is at a restpoint and ends, when the system arrives at
the next restpoint. Denoting these times by tr (l) and tr(2}, respectively we have •x(t(l)) I r X (t (2)) I r with with
i
(t (1)) 1 ri
(t (2)) 1 r 0= 0
Once the functions fi(x) are known, the function x(t) and the period T can be calculated. As
!-i!'
it follows from the energy equation thato2 X
=
Hence 2(E - V(£(x))n
<!.'
(x)) t(x)- t(xr (I)) "'t(x) - tr (I)n<!'
<~>>
I
2(E-V(,!:(~)))}
T 2 t(x (2)) - t(x (J) 2 r X (2) • 2 ( dU n<!'(l;;))
}l
(1) 2(E-V(f(l;;))) X -rThe solution x as a function of time reads
!(t) !(x(t))
Remark
If £(x) is known the dynamical equations read
The first equation of this set reads:
00
X+
(av
ax
>
=
0x=f(x) I
-The remaining (n-1) equations are , "" ' ' ( )o2 (av) f. (x)x + f. . x x ;;- f( ) 1 1 oX. x= X 1 -0 1=2,3, ••• ,n (2.5.2) (2.5.3) (2.5.4)
As each of these equations depends on only one unknown (namely x) Rosenberg ((1960), (1962-a)) speaks of decoupled equations. This is somewhat misleading*) because no decoupling is carried out. Indeed (as it should be) the equations (2.5.4) are merely different
representations of (2,5,3), This can be proved by means of equations (2.5.1), As this proof is rather elementary it has been omitted.
2.6. Similar and nonsimilar normal modes
Similar normal modes are, by definition, those which have a straight modal orbit. Nonsimilar normal modes therefore have a modal orbit which is curved.
*) For example, see the note by Henry(l962). The comment (Rosenberg (1962-b)) of Rosenberg on this note shows that he was aware of the fact that no decoupling is achieved.
In this section attention will be almost entirely focused on similar normal modes. The reason for this is that the differential equations (2.5.1) for nonsimilar normal modes are so difficult that few general results are available*), For similar normal modes however the
equations (2.5,1) reduce to purely algebraic equations, because f''=O and !'=constant. Indeed, for such an orbit we must have £(x) =X!= x col(J,a2,,,,,an), As !''=Q and 0=(!'•!') > I the equations (2.5.1) reduce to:
(2. 6. I)
This equation states in fact that the modal orbit of any similar normal mode is an orthogonal trajectory of the surfaces V(~)=constant
(Rosenberg (1962-a)). Conversely, only straight orthogonal trajectories of these equipotential surfaces can, at least in principle, be orbits of similar normal modes, The restriction in principle is needed because not every motion along a straight orbit
(which must be an orthogonal trajectory) is always periodic. However, for systems of the type considered here we have
THEOREM 2.1. If V behaves as assumed in section 2.2., then every motion along a str~ight orthogonal trajectory is necessarily a similar normal mode for all values of the energy E below~· provided !(0) € E(E).
Proof: Let the conditions as stated above be satisfied. The straight orbit can be written as ~s(t)! in which s is a scalar function and ! a constant vector, Then
0
Assuming that _aT_a=l and defining ~(s):=aT(cV/cx)
- - ~S!
this equation reads
""
s + ~(s) = 0
*) There are some general results. These will be discussed in section 2.8 where a survey of the existing literature on normal modes will be given.
Because of conservation of energy and the initial condition for !• we have
s(t)! € E(E) V t
This implies (due to the properties of V on E(E)) that s(t) lies (for all times) in some neighbourhood I s of zero:
s(t) € I V t
s (2.6.2)
I :={sis . :>s:Ss ;s .«O;s >0; V(s . a)=V(s a)=E} s m1n max m1n max m1n-
max-By virtue of the assumed properties of V on E(E) we also have s<j>(s) • (xT
av)
> 0 V s € I \{0}- a! ~s! s (2.6.3)
Combining (2.6.2) and (2,6.3) we have o s(t) periodic with period T ;
o zeros of ~ _.are spaced at time intervals !T ;
o between two successive zeros of : the function s assumes once the value zero •
Hence x
=
s(t)! is a similar normal mode.0
An example of a motion along an orthogonal trajectory of the.
equipotential surfaces V=constant, which ceases to be a similar normal mode if the energy exceeds some threshold value, is furnished by the purely rotational motion of the system considered in section 2.3.
A second property of similar normal modes is expressed in the following
THEOREM 2.2. The modal orbits of similar normal modes always lie along eigenvectors of the linearized system,
Proof: It suffices to show that orthogonal trajectories of the
surfaces V(:)=constant lie along eigenvectors of the linearized system. Let X! be any orthogonal trajectory, Because of (2.6.1)
we must have
av
(xa)av
(xa)- - a = -
-ax
1 -a:
(2.6.4)Let K be the stiffness matrix of the linearized system
writing
av
0~ =k:
+ ~<=> and K :'" {k 1, ••• ,k } - -nequation (2.6.4) can be written as
(2,6.5)
The functions ~i(!) have the property
~. (x)
lim ~=0
llxij+O
11!11i•l,2, ... ,n
because V is twice differentiable in some neighbourhood of
Q
and K is the matrix of second derivatives evaluated at
Q·
Therefore, dividing (2.6.5) by x and taking the limit x +0 we obtainHence, ! must be an eigenvector (See also Appendix I for some observations 9n this eigenvalue problem),
0
The truth of the theorem can be infered from the observation that a straight modal orbit does not depend on the energy. The direction of
such an orbit does not change if E + 0. However, if E is sufficiently
small, the linearized system approximates the actual system. Therefore it is no surprise that the straight orbits of the actual system coincide with those of the linearized system,
Theorem 2.2. can be used to make estimations on the.number of similar normal modes, because as a corollary of this theorem we have
COROLLARY: If the stiffness matrix of the linearized system is non-degenerated, there are at most n similar normal modes. Degeneracy of the stiffness matrix is a necessary condition for the existence of more than n similar normal modes. The proof of this corollary is straightforward.
Apart from their shape there is a marked difference between similar and nonsimilar normal modes, because the influence of energy changes is quite different. This can be infered from equations (2.6.1). As the lefthand side (which contains the energy) of (2,5.1) does not vanish identically for nonsimilar normal modes, it has to be expected that both the shape and the orientation of the modal orbits change as a function of the energy E. This is not the case for similar normal modes as the equations (2.6.1) for the modal orbit do not explicitly contain the energy E.
The eigenvibrations of linear systems are always similar normal modes. However, in linear systems nonsimilar normal modes also can occur.
(See Appendix II where a linear system with nonsimilar normal modes is discussed).
2.7. A variation principle for the modal orbit
2.7.1. Introductory remarks
As mentioned before equations (2.5.1) can be derived from a variation principle called Jacobi's principle of least action. As this principle will be used to calculate finite element approximations of the modal orbit (See section 3.3) it will be discussed in this section, as there are some problems connected with it which need some attention.
Following Arnold (1978) Jacobi's principle of least action reads: Define on the set E(E) the Riemannian metric dp by
s = arc length
Then the trajectories of the system with kinetic energy
<>To o2 • •
T =
!
! !=
~ s , potent1al energy V(!) and total .energy E w1ll be geodesic lines of the metric dp. The orbit is thus a solution of the variation problemJ :=
]ds
i2(E - V(!)) + stat (2. 7.1)T
!s !3"' 1 (constraint)
The constraint is necessary to ensure that the parameter s is indeed the arc length. Note that (2.7.1) can be written in the parameter invariant form
The parameter u is arbitrary.
In case the orbit is represented in the form used so far, i.e.
! = [:
l
f
= [
f2]
f(x) fn
Then the principle reads as
Jdx F(x,f,f') + stat
F := 1{2(E- V(x,f)}{l + f'Tf'}
The equations (2.3.5) are equivalent*) to the Euler-Lagrange equations aF d aF •
0
a'f -
dxaf'
(2.7.2) (2.7.3) (2.7.4)*)Derivation of equations (2.5.1) from (2.7.3) is not straightforward. In order to yield equations (2.5.1), starting from (2.7.4), a relation has to be used which only applies to solutions of (2.7.4)
There are some problems connected with the variation principle (2.7.2). 1) It is not very clear on which domain D(J) the functional J should be defined, This is a serious problem because the metric dp degenerates on the surface V(!)=E; two distinct points on this surface have a "distance" zero. Consultation of textbooks (Whittaker (19 I 7), Birkhoff
(1927), Pars (1965)p 543-548, Rosenberg (1977), Arnold (1978), Abraham & Marsden (1978)) on this matter only confuses*) things or leaves the questions unanswered,
The derivation of Jacobi's principle of least action as given by Funk (1962) provides the clue as to the answer of our question, although Funk does not deal with the above-mentioned problem either. This will be discussed in section 2.7.2.
2) In order to characterize the orbits of normal modes, Rosenberg ((1960)0(1962-a)) uses some particular version of Jacobi's principle. As it can be demonstrated that the variational problem as used by Rosenberg does not have a solution in some cases, suitable changes have to be made. This demonstration and the discussion of the necessary changes will be given in section 2.7.3.
2.7.2, Derivation of Jacobi's principle of least action
Starting point is Hamilton's principle of least action tl A= fdt {I~Tx- V(!)} t 0 t fdt
L(!,!)
t 0 ! € DA := {! E RnI
x(t )=x - 0 0 (2.7.5) i=1,2,., ,n}*) Deriving·Jacobi's principle of least action usually starts with
the so-called Maupertuis' principle of least .action. In this principle, curves- all with the same value of the energy E-are compared. As a result, different curves must in general have a different parameterization. This fact is probably the basis for the confusion mentioned above. The difficulty is not new. Arnold
(1978) cites a statement of Jacobi on this subject: "In abnost
all te~tbooks~ even the best~ this p~naiple is pPesented so that
it is impossible to undePstand" and he adds (rather sarcastically) "I have not ahosen to bPeak UJith tPadition". Abraham and Marsden (1978), who give a rather complicated version of the variation principle, remark: "The authoPs like many othePs (we u>ePe happy to
