Asymptotic analysis of wind-induced vibrations

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus,

Prof.drs. P.A. Schenck,

in het openbaar te verdedigen ten overstaan van een commissie door het College van Dekanen

daartoe aangewezen

op donderdag 9 november 1989 te 14.00 uur

door

CLEMENS GEORGES ANDREAS VAN DER BEEK,

wiskundig doctorandus.

TR diss

1765

Promotiecommissie: Prof.dr.ir. P.G. Bakker Prof.dr. B.L.J. Braaksma Dr.ir. A.H.P. van der Burgh Dr.ir. E.W.C, van Groesen Prof.dr. P.B. Hagedom Prof.dr.ir J.W. Reyn Prof.dr. M. Roseau

Dr.ir. A.H.P. van der Burgh heeft als toegevoegd promotor in hoge mate bijgedragen aan het tot stand komen van het proefschrift. Het College van Dekanen heeft hem als zodanig aangewezen.

INTRODUCTION 1

CHAPTER 1

NORMAL FORMS AND PERIODIC SOLUTIONS IN THE THEORY OF NON-LINEAR OSCILLATIONS.

EXISTENCE AND ASYMPTOTIC THEORY. 7

1.1 Introduction 8 1.2 Normal form and normal form transformation 15

1.3 Existence and stability of periodic solutions 21 1.4 Theory of asymptotic approximations for initial value problems 24

1.5 Theory of higher order asymptotic approximations for

periodic solutions 27 1.6 A simple model for the dynamics of an oscillator with

one-degree-of-freedom in a uniform windCeld 33

1.7 Concluding remarks 39

Appendix 41

CHAPTER 2

ON THE PERIODIC WIND-INDUCED VIBRATIONS OF AN

OSCILLATOR WITH TWO-DEGREES-OF-FREEDOM. 49

2.1 Introduction 50 2.2 The oscillator and the equations of motion 52

CHAPTER 3

ANALYSIS OF A SYSTEM OF TWO WEAKLY NON-LINEAR COUPLED HARMONIC OSCILLATORS ARISING FORM

THE FIELD OF WIND-INDUCED VIBRATIONS. 69

3.1 Introduction 70 3.2 Preliminaries 72 3.3 The 2:1 resonance 75 3.4 The 1:2 resonance 84 3.5 The 1:1 resonance 97 3.6 Concluding remarks 103

Appendix Dynamics of an oscillator with

two-degrees-of-freedom in a uniform windDelds 105

CHAPTER4

DYNAMICS OF AN OSCILLATOR WITH

THREE-DEGREES-OF-FREEDOM IN A UNIFORM WINDFIELD. 115

4.1 Introduction 116 4.2 Description of the oscillator and derivation of the

equations of motion 117 4.3 Analysis of the model equations for non-resonance

up to order 3 129 4.4 Concluding remarks 135

NORMAL FORMS FOR WEAKLY NON-LINEAR PERTURBED

WAVE EQUATIONS. 137

5.1 Introduction 138 5.2 Normal form and normal form transformation 141

5.3 Some remarks on the analysis of the P.D.E. by

using truncated normal forms 150 5.4 The relation between the normal form for P.D.E and O.D.E. 154

5.5 Application of the normal form concept to the Rayleigh

wave equation 161 5.6 Concluding remarks 170 REFERENCES 173 ACKNOWLEDGEMENT 179 SUMMARY 181 SAMENVATTING 183 CURRICULUM VITAE 185

INTRODUCTION

Several problems describing the wind-induced vibrations of an object in a windfield give rise to differential equations which involve a small parameter t. In these differential equations, the unperturbed differential equations (i.e. i = 0) model the dynamics of the object in absence of a windfield and are usually linear. On the other hand, the (small) perturbation terms, which represent the influence of the windfield on the dynamics of the object consist of linear as well as non-linear terms. In this thesis two types of differential equations are theoretically studied (chapter 1 and chapter 5), whereas in chapter 1, 2, 3 and 4 several models are presented and the resulting equations of motion are analyzed.

In chapter 1 a system of (autonomous) ordinary differential equations is discussed in which the unperturbed differential equation is a system of fully resonant harmonic oscillators i.e.

i = Ax+ef(x,e), x e n cK2n, (l)

x(0) = xo, (2)

where e is a small parameter, A a matrix with purely imaginary, rational and nonzero eigenvalues, fl an open and bounded domain and f a vectorfunction. Furthermore f satisfies certain regularity conditions, which are mentioned in chapter 1.

a weakly non-linear perturbed wave equation is discussed:

"t l-u x x= ff(x> u>ut > V £)> 0 < X < T , t > 0, (3) u(x, 0) = ipi{x, e), 0 < x < T, (4)

ut(x, 0) = <?2(x, t), 0 < x < T, (5)

u(0, t) = u(?r, t) = 0, t > 0. (6) Here is ( a small parameter and f, ip, and tp-i satisfy certain regularity conditions,

which are mentioned in chapter 5.

For both types of differential equations a method of normal forms is presented in order to classify and analyze these differential equations. The basic idea is to transform (1) resp. (3) into a so called normal from by using autonomous transformations. Normal form here means that the resulting equations (the normalized differential equations) have a perturbation term, which is invariant under the fundamental solution of the unperturbed equations (i.e. ( = 0 in (1) resp. (3)). One could say that by normalization only those parts of the perturbation term remain, that cause a behavior of the solutions of (1) resp. (3), which is different from the behavior of the solutions of the unperturbed differential equation.

The type of equations, as discussed in chapter 1, occurs frequently in the literature. One could say that the two methods, that are most frequently used for this type of differential equations are the method of averaging [11,38,39,43] and the method of normal forms [9,12,24]. In this thesis is chosen for the method of normal forms, or to be more precise for an adapted method of normal forms. This has been done, since the method of averaging is basically a method developed for non-autonomous differential equations and therefore not immediately applicable to the type of differential equations discussed here. Although it is possible to

transform the systems discussed here into non-autonomous systems for which the method may be applicable this has not been done because of theoretical as well as practical reasons (see section 1.1 of chapter 1 for more details). Concerning the method of normal forms, it may be pointed out that it is here not only presented as a local classification theory, which is its main application in the literature, but also as a non-local classification and bifurcation method for the type of equations discussed here (it will be shown that the method can not only be used to classify differential equations but also to obtain existence and stability of periodic solutions as well as for the calculation of asymptotic approximations).

This however, requires a modification of the usual method of normal forms, which is done in chapter 1.

The method, that is used in chapter 5 to analyze the second type of differential equations, an initial-boundary value problem for a weakly non-linear perturbed wave equation, can be regarded as a straight forward extension of the method of normal forms for ordinary differential equations as presented in chapter 1. On the use of the method of normal forms for partial differential equations, a paper of Shatah may be mentioned ([40]), where it is used to simplify an initial-boundary value problem for the Klein-Gorden equation with a quadra­ tic non-linearity. Concerning the results found here, one could say that it might be used as an alternative for the two-time scales method ([16,19,27,28]). In addition it may be pointed out that this method can also be used to classify differential equations since, as in the case of the ordinary differential equations, it transforms the P.D.E. into a form (the so called normal form) which has a certain structure.

In chapter 1, 2, 3 and 4 several models, describing the wind-induced vibrations of a cylinder with a ridge attached to it in a windfield, are presented

and the resulting equations of motion are studied by using the theory as presented in chapter 1. The models and the resulting differential equations that are discussed in these chapters are not only interesting from the mathematical point of view, but also motivated by a phenomenon occurring in real life. Overhead transmission lines on which ice has accreted may have cross sectional shapes that are aerodynamical unstable to transverse disturbances in a windfield. The evolution, in time, from this unstable equilibrium position may result in galloping: a large amplitude oscillation with low frequency (< 1 Hz). This very complicated phenomenon of galloping of overhead transmission lines, which involves the aeroelastic interaction of longitudinal, transversal and torsional oscillations of a continuous system is far from being understood. From the modeling point of view, one faces various problems. On the one hand one has to model the complicated dynamics of the transmission lines and the influence of the windforces acting on the overhead transmission lines whereas on the other hand one wants to obtain a system of differential equations which can be analyzed. An additional problem is that it seems hard to reproduce this galloping phenomenon for overhead transmission lines on laboratory scale (by using this approach, one could hope to isolate the physical interesting parameters). Therefore in this thesis a simple modelation of the dynamics of the cable is used (a cylinder-spring system).

In chapter 1 a simple model is discussed describing the wind-induced vibrations of a cylinder with a ridge attached to it (representing the cable with ice accretion) hung from springs with one-degree-of-freedom in a uniform windfield. From the practical point of view the main interest here is to investigate the influence of the windforces and to see if a phenomenon related to the galloping phenomenon can occur. Here (and henceforth) the assumption is

made that the aerodynamical forces are quasi-steady (for the study of galloping there is no disagreement in the literature on this approach ([34,35])), whereas a modeling of the equations of motion is used as in [10,13].

Next, in chapter 2, 3 and 4 two other models are presented and analyzed, which have two resp. three-degrees-of-freedom. The model discussed in chapter 2 describes the wind-induced vibrations of the cylinder with ridge with two-degrees-of-freedom: oscillations in and perpendicular to the direction of the uniform winddeld. Furthermore, for a number of cases, existence and stability of periodic solutions of the differential equations is established. In chapter 3 the equations, as derived in chapter 2, are discussed in more detail and an interpre­ tation of the (mathematical) results is given in order to describe the dynamics of the oscillator.

In chapter 4, a model is presented with three-degrees-of-freedom in a uniform windfield; in addition to the oscillations in and perpendicular to the direction of the uniform windfield, now also rotational oscillations are possible. Furthermore, for a number of cases, the dynamics of the oscillator is analyzed.

Observe that in this way the influence of the several degrees-of-freedom can be investigated.

Finally, in chapter 5, an equation is discussed (the Rayleigh wave equation), which describes the phenomenon of galloping by assuming that the dynamics of the transmission lines may be modeled by a stretched string with one-degree-of-freedom (see for instance [27,31]). This equation, which is studied in detail in the literature by using the two-scales time method [16,26,27,28,31], is analyzed here by using the method of normal forms. In addition the results found here (for the Rayleigh wave equation), are compared with the results found in chapter 1 (there the dynamics of the transmission lines was modeled by a cylinder-spring system,

leading to an O.D.E., the Rayleigh equation).

Finally it may be mentioned that, in order to obtain several results in this thesis, use has been made of the Computer-Algebra system Macsyma ([30,36]); not only a computer program has been written to actually calculate the normal forms for the ordinary differential equations, but also the various model equations have been obtained by using Macsyma.

CHAPTER 1

NORMAL FORMS AND PERIODIC SOLUTIONS IN THE THEORY OF NON-LINEAR OSCILLATIONS.

EXISTENCE AND ASYMPTOTIC THEORY.*)

Abstract

A concept of normal forms is presented to analyze a class of O.D.E.'s arising from the field of non-linear oscillations. An algorithm for the calculation of the normalized differential equations is presented as well as theorems on the existence, uniqueness and stability of periodic solutions and their calculation in an approximative way by using the normalized differential equations.

As an application of the theory a simple model is discussed describing the wind-induced vibrations of an oscillator with one-degree-of-freedom in a uniform windfield. The necessary calculations to perform the analysis have been done by an implementation of the algorithms for the calculation of higher order normal forms, as presented here, on the Computer-Algebra system Macsyma.

1.1. Introduction.

In this chapter normal forms are used for the study of periodic solutions of equations of the form

x = A x + r f ( x , f ) x e R2 n. (1.1.1)

tA

Here A is a constant matrix, t H e T - periodic for some T > 0, c a small parameter, |e| < e « 1 and f(0,t) = 0. This study includes existence, uniqueness and stability of periodic solutions as well as asymptotic approxima­ tions of the periodic solutions and their periods. The method used in this paper is based on a combination of principles of the classical concept of normal forms and the method of averaging. In order to make clear what is meant with a combina­ tion of principles, first both methods are briefly discussed.

In the literature (see [9] or [24]) the theory of normal forms is usually presented in the following context (in this paper referred to as the classical concept): Consider differential equations of the type

x = (j)(x) x e Rp, (1.1.2)

where <j) is a vectorfield, analytical in x and <j>(0) = 0.

The basic philosophy behind the theory is to eliminate as many as possible non-linear terms in (1.1.2) by making use of the linear part of (1.1.2) and a non-linear near-identity transformation x = y + P(y); the resulting equation is the normalized differential equation and the remaining non-linear terms are the so called resonant terms. By using power series the normalized differential equation can be (formally) calculated by the following procedure:

First $ is expanded in a Taylor series about x = 0 i.e.:

x = A x + £ L(x) (1.1.3) k>2 K

and the (unknown) P(y) is written as a (formal) power series:

x = y + E P,(y). (1.1.4) k>2 K

Here A is the Jacobian of (j) in the point zero and (f>. resp. P, are homogeneous vectorpolynomials of degree k.

Next (1.1.4) is substituted into (1.1.3), from which it follows that y satisfies the equation:

y = Ay + E [ A P - M - DP.(y)Ay + Uy)] =: Ay + E $°(y). (1.1.5)

k>2L K K J k > 2 K

Here is (j>„ = <]>„ and L (k>2) is determined by P„,..., P. , and ^ ■■-, <J>.. Finally the coefficients of the formal power series of the transformations P are calculated by substitution of a general (unevaluated) P, into

AP, (y) - DP, (y)Ay + <My) and choosing the coefficients in such away that as many as possible terms vanish. Observe that this is an iterative process; P„ can be calculated directly since $„ = fa whereas P, for k > 2 is determined by (j), and hence by P2, ..., Pk l and 4>2, ..., ^

It should be pointed out that the procedure as sketched above is formal and local: convergence of the power series in (1.1.3) does not imply that the power series in (1.1.4) and (1.1.5) are convergent (in general they do not, see [12]) and if they are convergent only local convergence can be established i.e. for |y| (and hence | x | ) sufficiently small.

For the equations considered here i.e. (1.1-1), this classical concept of normal forms has the following disadvantages or limitations:

the classical normal form theory is a local theory,

absence of an (efficient) algorithm for the computation of P-,

the normal form theory, as known in the literature, has still a number of formal aspects; little is known about the use of normal forms in

relation to the existence of periodic solutions and asymptotic approxi­ mation theory.

With the method of averaging (see for instance [11] or [39]) differential equations of the type

x = eh(x,t,f), x 6 Rp, (1.1.6)

are studied and analyzed; here is t e ( - «0, O a small parameter (0 <i << 1)

and h(x,t,t) a vectorfield, T - periodic with respect to t for some T > 0. First a non-autonomous T - periodic transformation of the form

x = £ + tUjU.t) + (\(i,i) + ... + <kukU,t), (1.1.7)

is applied, which transforms (1.1.6) into the averaged equation of which all terms up to 0(< ) are autonomous:

I = AjCfl +

h

(0 + ... + c\(0 + £

R(?,t,0- (1-1-8)

Next the non-autonomous part is truncated and the remaining equation i.e.

(=^(0 + ^2(0 +- + \ ( 0 (11-9)

is analyzed. Finally by using an asymptotic theory one may show that solutions of (1.1.9) can be used as approximations for the solutions of (1.1.6).

In the context of the equations considered here i.e. (1.1.1), the averaging method has the following disadvantages or limitations:

in the averaging approach one has to transform the autonomous equation (1.1.1) into a non-autonomous equation of the standard form (1.1.6). Not only from a theoretical point of view but also from a practical point of view this is a disadvantage: the transformation from the autonomous equation to the non-autonomous is not uniquely determined and may introduce singularities in the resulting nonautonomous equation. To illustrate this consider the following. As it is known from the literature (see for instance

[43]) critical points of h.(£) give rise to periodic solutions of (1.1.6) if the Jacobian of h. in that point is nonsingular. Most transformations used in the literature to put (1.1.1) into the standard form for averaging involve an explicit t dependence and hence averaging has to be carried out with respect to t (t is the so called fast variable). In that case however, it is shown here (see Remark 1.3.1) that the Jacobian is always singular and so no conclusions can be drawn with respect to existence of periodic solutions. Moreover these transformations force (1.1.6) to be T - periodic, which is of course not very interesting since in general the period of solutions of (1.1.1) depends on (. For n = 1 this problem can be solved by introducing polar-coordinates, but for n > 1 this introduces another kind of singularities in the equation (1.1.6). Since the introduction of these singularities is only a consequence of the transformation from (1.1.1) to (1.1.6) this should be avoided,

although the method of averaging provides formulas which might be easily implemented in a Computer-Algebra system there is still little known about the general structure and properties of these equations for k > 1 and n > 1. This may cause problems if higher order averaged equations which are calculated by symbolic manipulation have to be simplified in an efficient way (recall that in symbolic manipulation one faces not only the computa­ tional problem, but also the problem of simplification).

In the combination method presented here advantages of both methods are combined and some extensions are made. First observe that since the classical concept of normalization is a local theory, one may introduce in this context a small parameter by the scaling x -• /xx and y -> fiy (0 < fi « I) in (1.1.3) and (1.1.4), which yields:

i = Ax+ £ /ik ^ . ( i ) (1.1.10)

k>2 K

and

x=y+ E Mk_1Pk(3/). (1.1.11)

k>2 K

Arguments like "for |x| and |y| sufficiently small" may now be replaced by "for fi sufficiently small" and the iterative process for the elimination of nonlinear terms can be done by considering increasing powers of /i instead of increasing degree of the polynomials; (1.1.10) and (1.1.11) represent the classical concept of normalization in terms of a small parameter. It may be clear that, since all coefficients are independent of /i, this leads to the same result as when (1.1.5) is scaled,

y=Ay+ E ^ " ^ ( y ) . (1-1.12) k>2 *

In several applications, for instance in the Geld of non-linear vibrations, one encounters differential equations of the type (1.1.1). The small parameter in those cases is usually introduced by taking the magnitudes of the coefficients in the differential equation into account rather than by scaling. Hence the term

k-1

corresponding to e is not necessarily a homogeneous vectorpolynomial of degree k and therefore the classical normal form concept can not be applied by simply setting \L = e in (1.1.10). On the other hand the method of averaging may­ be applicable since, besides some regularity conditions, no restrictions are imposed on the terms in the expansion with respect to c. The method presented here generalizes the concept of reducing an autonomous O.D.E. by autonomous transformations (the normalization) to vectorfields with a more general structure (as in the method of averaging). In addition a non-local result can be obtained, since one now can use the small parameter i instead of local arguments to

establish convergence. Finally it may be mentioned that although (1.1.1) is Hamiltonian for e = 0 (the unperturbed system), it is not assumed that system (1.1.1) is Hamiltonian for t i 0. However, if (1.1.1) is also Hamiltonian for e i 0 one may also apply the KAM theory (see for instance [1] or [2]), from which one may obtain additional information on the behavior of the solutions.

The chapter is organized as follows:

For the development of a combination method, in section 1.2 a definition of the normal vectorfield and the normalized differential equation is introduced which avoids the polynomial characterization of the classical concept. Moreover simple formulas are derived for the calculation of the normal vectorfield and the corresponding transformation. As basis for the normal form definition an invariance principle for vectorCelds is used. This principle was introduced by the author in [8], but as was pointed out to the author [44] this principle was also used in [18] to characterize normal forms in the classical (local) theory and was also mentioned in [17]. However, here it is used for the development of a perturbation method and presented in a more general context. In section 1.3 a theorem is derived on the existence and stability of periodic solutions on the basis of truncated normal forms and in section 1.4 and 1.5 it is shown how (truncated) higher order normal forms can be used to obtain higher order asymptotic approximations of solutions of initial value problems and for periodic solutions respectively. The reason for presenting these theorems based on truncated normal forms instead of considering the complete normal form is that, as already mentioned, it is not likely that a complete normalization leads to a convergent system and moreover that it is (practically) impossible to carry out the calculations. However, by using a Computer-Algebra system like Macsyma (see [30] or [36]) and the algorithms presented here, one can normalize a system up to

a high degree (see also [3]). In section 1.6 it is shown that the dynamics of an oscillator with one-degree-of-freedom in a uniform windfield can be modeled with a modiGed Raleigh equation and the theory derived in the previous sections is used to analyze this equation. Finally some concluding remarks are made in section 1.7.

1.2. Normal form and normal form transformation.

In this section differential equations of the form x = Ax + ef(x,e), x e D,

are considered, which satisfy the so called N -conditions (m > 1): (1) 0 € D C R2 n (n 6 W), D open and bounded,

(2) e e (-£0,e0) C K, 0 < eQ « 1, N in ( 3 ) A = _• Ai = 0 1 - a2 0 and (j- > 0, (1.2.1) ( 4 ) w1/ wie < | f o r a l l i e { l , . . . , n } ,

(5) the domain D has the property that if x e D then e ^Ax e D for all <p e R,

(6) f 6 Cm + 1( D x (-*0,£0),B2 n), £(0,0 = 0 for aU | c| < tQ.

Remarks

(1.2.1)'The matrix A is independent of t. However, f may contain linear terms in

x. (1.2.2) Observe t h a t e * ^ = S wk A = k>0 k ! ipA. <pkr and

e =

cos(Wjp) w- sin(u.<p) 'W- sin(u>- <p) cos(Wj^)

(1.2.3) The condition (4) implies that t - e is periodic; T is the primitive period.

(1.2.4) Although for a general domain D, 0 e D C R n, condition (5) does not

hold, there is always a subdomain of D which satisfies this condition.

To introduce and to show which róle the normal forms used here can play in finding periodic solutions, the differential equation

E « A £ + d ° ( f l , £ e D (1.2.2) is considered, where f° satisfies the following condition:

fO(e¥>Afl = g^Ajo^j for a„ ^ 6 „ a n d ^ e D (1.2.3)

By using the transformation £ = e y the following differential equation for y holds:

y = ff°(y). (1.2.4)

Non-trivial critical points of (1.2.4) induce T -periodic solutions of (1.2.2): y 6 D\{0} and f°(y0) = 0 implies that t -< e y is a non-trivial T - periodic

solution of (1.2.2).

In what follows property (1.2.3) is used to define a normal form for (1.2.1) and it is shown how this differential equation can be put into normal form.

Definition 1.2.1

A vectorfield f° e C°(D,IR n) is called a normal vectorfield (with respect to A) if it

is invariant under the flow produced by A, i.e.: f°(e^Ax) = evAP{x) for all x e D and <p € R.

Definition 1.2.2

th

The system x = Ax + ef(x,£) is in m -order normal form (m > 1) if

m _{i—I m}

f(x,f) = E e f?(x) + e g(x,£), where the f? are normal vectorGelds. i = l ' '

If may be clear that in general (1.2.1) is not in m order normal form. In order to establish this one may use a near-identity transformation. To be more precise:

Theorem 1.2.3

Consider the system (1.2.1) i.e.: x = Ax + ff(x,«)

and suppose that the N -conditions hold. W i t h ( l < k < m ) : 1

a"-

o

1 f V ^ f j e ^ x ^

P

the following holds:

(1) f», Pk e Cm + 1-k( D , K2"), f^(0) = Pk(0) = 0,

(2) f£ is a normal vectorfield, (3) application of the transformation

leads, for i small enough, to:

e = A É + e E M W f l + e m + 1g ( 6 0 , £ € D . | e | < eo (1.2.6)

k=l K °

(the differential equation is in m -order normal form). Here is g e Cl(Dx(-eQ,eQ), 0)2n), g(0,t) = 0 for all 11 | < tQ.

Proof:

see the appendix

a

Remark (1.2.5)

For the study of initial value problems i.e.:

x = Ax + ef(x,e); x(0) = xQ, (1.2.7)

it is attractive to use the transformation x = £ + tP(Q where

T T

P(*):=4-f °e-V\,0)d^4[ V V ' V ) ^ .

% % Since P(x ) = 0 and the first order normal form of (1.2.7) associated with this

9

transformation is the same (up to order £ ) as the one given in theorem 1.2.3, one may conclude that the initial value problem (1.2.7) leads to the following initial value problem for the first order normal form:

Hence the initial values for x and £ are the same.

Remark (1.2.6)

Although theorem 1.2.3 is based on the fact that w-/w, 6 Q it can easily be adapted for the non-rational frequency-ratios which satisfy u- = til, + eS- and uju. € H by simply putting the nonrational part {(6) in the perturbation term.

It is of interest to mention the following equivalent relations for normal vectorfields: (as was pointed out to the author ([44]) an analogous result can be found in [18]):

Theorem 1.2-4

Suppose that f° e C (D, K n) and A and D have the properties as mentioned in

the Nm-conditions. Then the following statements are equivalent:

(1) Af°(x) - Df°(x)Ax = 0 for all x € D,

&h

T

(4) f°(x) = -L I °e~vkf,(e'pAx)dip for all x e D.

% Proof: ( l ) - ( 2 ) : For ip 6 R and x € D:

^[e^V(e^x)

By using the fact that e^ x e D and (1) if follows that Af°(e^Ax) - D f ^ e ^ x J A e ^ x = 0 and thus (2).

( 2 ) - ( 3 ) :

Integration of (2) leads to e-^ f°(eV x) = c for every <p e R.

By substituting ip = 0 one finds then that c = f°(x) and thus: f ^ e ^ x ) = evAf°(x) for every ip 6 R and x e D.

(3) -* (4): trivial. ( 4 ) - ( l ) :

Af°(x) - Df°(x)Ax = -^r- I

V

1 f

°d

dip = -- T A T A e ° f°(e ° x)-f°(x) dp = m 0. Remark 1.2.7

Note that (1), (2) and (3) are also equivalent for an arbitrary matrix.

One may thus conclude that definition (1.2.1) of a normal vectorfield also leads to a Kernel-Image decomposition, which is often used for the classical concept of normalization (see for instance [24]). On the other hand it also leads to a formulation in terms of averaged functions, which enables one to consider more general vectorfields as in the classical normal form concept. Hence the method used here can be regarded as a combination of principles.

1.3. Existence and stability of periodic solutions.

In this section a theorem for the existence and stability of periodic solution of system (1.2.1), on basis of the first order normal form is given. In order to formulate this theorem two transformations have to be applied. The first and most obvious one is (1.2.5) (for m = 1) which puts by virtue of theorem 1.2.3 the system (1.2.1) into first order normal form:

( = A£ + rf»(£) + e2g(É,e), ( e D a n d c f (-1Q,10). (1.3.1)

As afore mentioned the transformation y(t) = e~ £(t) may be used to establish periodic solution with period T . In general however, when periodic solutions exist for system (1.3.1) their periods depend on t. Assuming that a period is

T

smooth with respect to e one may write this period, say T , as T = ■_ , where TJ depends continuous on t and lim CTJ = 0 . This motivates the

6 £-1 0 '

introduction of the second transformation (variation of constants and period): t ( l - « l ) A

tft) = e y(t), ,( £« ! (1.3.2)

which leads to the differential equation:

y(t) = cF°(y,r,e) + £2g(t,y, V ) - (1.3.3)

Here is:

V°(y,V ) := V Ay + f°(y); F° is still a normal vectorfield,

' £-t(l-£7/)A t(l-tr,)k

g ( t . y > V ): = e g(e y'e

Theorem 1.3.1

Suppose that system (1.2.1) satisfies the N -conditions and that there exists a ( y % ) € D x R and a i 6 {l,...,2n} such that with F° as defined above, the following conditions hold:

( i ) y o ' ° . F ° ( y > o ) =

°-3(F°, , F ° ) (2) the Jacobian matrix is regular.

ö('?,y1,-,yi o.1>yi o + 1,.,y2 n) (y0,u0)

Then there exists for every e, c sufficiently small, an isolated periodic solution of (1.2.1). For this periodic solution, say t - xf(t) with period T , the following

estimate holds:

x£(t)-e °; y ° | = O ( 0 _{for 0 < t < T}£.

and

The condition (1) is a necessary condition i.e. if F°(y,r/) i 0 for all (y, r?) e D x R, then there is no periodic solution of (1.2.1), in D, with smooth period (with respect to t) for i sufficiently small.

Furthermore this periodic solution is asymptotically orbitally stable, resp. o

unstable if with 7j(y) = f? (y)/a (a = w- y9 i . if i0 = 2j0 and a = -y„. if

' o Jo ^Jo^i i j o

io = 2j0-l) and F-(y) = F?(r/(y),y), the real parts of the eigenvalues of the

Jacobian matrix e 'o ' 'o

one of the eigenvalues has a positive real part. Proof:

See the appendix

Remarks:

(1.3.1) In theorems on the existence of periodic solutions one often uses a variation with respect to all space-variables (see for instance [43]). In this theorem variation with respect to one space-variable is replaced by a variation with respect to the period. In this case this is necessary since the Jacobian of F with respect to all space-variables in the point (y ,J? ),say DF°(y°,7?o) is always singular:

D F ° ( y >o) A y ° = A Fo( y >o) = 0.

Hence one may conclude that the method of averaging applied to the class of equations considered here, averaging with respect to t always leads to a singular Jacobian.

(1.3.2) From the theorem one learns that condition (2) is only satisfied if a t 0. This can be used if one has to choose i .

(1.3.3) Since the periodic solution found on basis of the theorem is isolated, it may be called a limit cycle. As follows from the proof, this is also a simple limit cycle.

1.4. Theory of asymptotic approximations for initial value problems.

When equation (1.2.1) has been normalized up to a certain order, the question may be asked if and how the solutions of the normalized differential equation can be used to approximate solutions of the original differential equation. To answer some of these questions for the initial value problem consider:

x = Ax + rf(x,e), (l-l.l) x(0) = x0.

Suppose that the Nm-conditions hold. The m - order normal form may now be

calculated by using the transformation (1.2.5) i.e.:

x = f +£p1( e ) + . . . + «mPm( e ) . (1.4.2)

The differential equation for £ then becomes:

£ = A( + £m' f U 0 + £m + 1g t t . 0 . l < m , < m (1.4.3)

where f° is a normalized vectorfield i.e. f°(«*aA^«) = e ^ f ^ . t ) for all $ € D and

v e « (

U.<) = fm ( 0 + ■■■+ <

'f

(0; mi

y

e

order normal vectorfields vanish). Truncation of the non-normalized rest term and application of the transformation £(t) = e y(t) leads to the differential equation:

y = 6m'f°(y,0- (1-4-4)

Then the following holds:

Theorem I.4.I

Suppose that y is a solution of the initial value problem: y = em«f»(y,0

y(0) = y0

lxo - ( y o+ f Pl ( y0) + - + f m pm ( y o ) ) l = ° (f ')■

By setting ?<t) = etAy(t), x(t) = ?(t) + e P ^ H ... + « ^ ( H O )

(O < k2 < m, k2 = O means x(t) = ?(t)) and with t - x(t) the solution of (1.4.1),

the following estimate holds:

I x(t) -x(t) | = 0(ck|) + 0(fk 2 + 1) + 0{tm+l~u) on time-scale 0(e~")

(1 < i / < m , ) .

Proof:

Suppose that t -> x(t) is a solution of (1.4.1). By using the transformation (1.4.2) and y(t) = e £(t) if follows that t - y(t) satisfies:

y = £m|f°(y.0 + em+1g(y,t,e), with g(y,t,e) = e-tAg(etAy,£).

As estimate for y and y one finds that:

+ e

y(t)-y(t) | < | y(0)-y(0) | + f m' | f [f°(y(t),£)-f°(y(t),£)

+ 1| g(y(0,M)dt|.

■0

dt I +

This leads, by using Gronwalls lemma, to:

| y ( t ) - y ( t ) | = O(|y(0)-y(0)|) + O ( um + 1) _Lte

m ,

here is L the Lipschitz-constant of f° in D and use has be made of the fact that g is bounded in D. Furthermore observe that:

| y ( 0 ) - y ( 0 ) | = O ( ek l) ,

(follows from the estimate for x and y ) U ( t ) - ? ( t ) I - I et Ay(t)-et Ay(t) | = | y(t)-y(t) |,

I x(t) - x(t) | = | e(t) - J(t) + tpiUW) ~ Pi(?(*))) + + 'k 2(Pk 2(^(t))-Pk 2(?(t)))| +

< c u ( t ) - ? ( t ) | + 0 (t k 2 + 1)

for some constant C (independent of £ and | ) . Using these estimates one finds that:

| x(t) - x(t) | = 0 ( ck 2 + 1) + [ 0(<k') + 0 ( um + 1) ] eL t £ m'

and thus:

| x(t) - x ( t ) | = 0 ( «k 2 + 1) + 0(fk') + 0 (É m + 1 _ I /) on time-scale 0 ( r " ) ,

for 1 < v< mi,

which completes the proof.

□

Remarks:

(I.4.I) Since theorem 1.4.1 concerns a priori estimates, the interval of existence of solutions of (1.2.1) may also be extended from time-scale 0 ( {- ) to

o(r

).

(1.4-2)ln order to avoid unnecessary calculations k, and k? should be chosen in such away that k, = ka+1 = m+l-i/; in particular note that in order to get an optimal approximation one does not need to choose k2 = m;

1.5. Theory of higher order asymptotic approximations for periodic solutions.

In this section higher order asymptotic approximations for a class of periodic solutions of (1.2.1) are presented and it is shown how the time-scale of validity can be extended. This extension (classically one only establishes approximations on a 0(1) time-scale) may be of interest when one wants to know on what time-scale the periodic solution and the corresponding approximations are in phase.

Consider the differential equation:

x = Ax + ef(x,c) (1.5.1) and suppose that the N -conditions hold.

Observe that if t -. x£(t) is a T - periodic solution of (1.5.1), application of the

transformations:

x£(t) = ?£(t) + e P j ^ t ) ) + ... + emPm(4£(t)) (1.5.2)

and

t(l-£7J )A

ee(t) = e £ y£(t) (T€ = T0/ ( l - a ,f) ) (1.5.3)

leads to the conclusion that t -• ye(t) is a T - periodic solution of

y = tF°(y,V) + em+1I(t,y>V£)' (L 5 4)

here is

Fo(y,v£) = '?€Ay + £ m r l f 0( >r-£)

-t(l-en)A t(l-£B)A (1.5.5)

g(t.y.»?£.«) = e g(e y,e).

In what follows it will be shown that a certain class of periodic solutions of (1.5.1) can be reduced by the transformations (1.5.2) and (1.5.3) to critical points of (1.5.4) and asymptotic approximations of the periodic solutions of (1.5.1) can be

It is assumed that there exists a family of solutions of (1.5.1), x , which satisfy the following conditions (the ERm-conditions):

1. for every «, 0 < 111 < 7 , x£ is a T - periodic solution of (1.5.1)

ERm 2. x£ as well as n (T = Tn/(l-«? )) have continuous derivatives with

respect to t up to order m.

Remark (1.5.1)

If in theorem 1.3.1 the N -conditions are replaced by the Nm-conditions,

periodic solutions found on basis of that theorem satisfy the ER -conditions.

Due to the regularity of the problem the corresponding periodic solutions of (1.5.4), say t - y (t), have also continuous derivatives up to order m. Hence one may set for 0 < N < m-1:

yS = /(0)

o

)

»

= U

+0(e

)

(the first N terms of the expansion with respect to i) Then the following holds:

Lemma 1.5.1

Suppose that (1.5.1) satisfies the Nm-conditions (m > 1) and there is a family of

solutions for which the ER -conditions hold. With y a corresponding periodic solution of (1.5.4) and y' and TV. as defined above, the following estimates hold for 0 < N < m-1:

F ° ( y % , e ) = 0 + O ( eN + 1) (1.5.6)

and

| ye( t ) - y ^ l = o (f N + 1) o < t < T£. (1.5.7)

Proof:

The case N = 0 follows from theorem 1.3.1 and theorem 1.4.1.

To prove N = 1 first observe that due to the regularity of ye(t) one may write:

y£(t) = y0(t)+ <y.(t,c)- (1.5.8)

Since (1.5.7) holds for N = 0, yQ(t) = y° and thus (1.5.8) becomes:

y£(t) = y°+ «yi(t,e). (1.5.9)

Substitution of (1.5.9) into (1.5.4) and expansion with respect to e leads to: y i(t,0 = FO( y ° ,V) + O(f).

Using the fact that (1.5.6) holds for N = 0 results in yi(t,e) - 0(e).

One may thus conclude that yi(t,e) = yi(0,0) + 0(f) and hence (1.5.9) becomes

y£(t) = y°+ cyi(o.o) + o(e2) = yJ + o(£ 2) (1.5.10)

which proves (1.5.6) for N = 1.

To obtain (1.5.7) for N = 1 observe that

T T

0 = y

(T

) - y

(0) = e f V ^ t ) ,

) d l +

f W ( t ) , V«)* =

Finally, by using the principle of induction and the ideas presented above one can show that (1.5.6) and (1.5.7) hold for arbitrary N.

a

t N

Since y (t) is T - periodic and y time-independent it is clear that the estimate (1.5.7) also holds for arbitrary t and thus:

Corollary 1.5.2

With the same assumptions as in lemma 1.5.1 one finds:

I y£W - y o I = ° (f N + 1) for all t € R. (1.5.11)

□

Using this corollary one obtains the following higher order asymptotic approximation theorem for the corresponding periodic solutions of (1.5.1).

Theorem 1.5.S

Suppose that (1.5.1) satisfies the Nm-conditions and that xf is an element of a

family of solutions of (1.5.1) for which the ERm-conditions hold. With y and

7k. as in lemma 1.5.1 define (0 < N, M < m-1):

I (t) - et ( 1~£ 7 / M )\N

Then the following estimate holds:

|x£(t) - xN M( t ) | = 0 (e N + 1) + 0 ( fM + 2 _ , /) on a time-scale 0 ( t-" )

Proof:

As approximation for £ and ~£„ „ one finds that

e

t ( l - ^£) A t(l-cr/M)A t(l-£7?M)A t(l-f77M)A N

y£( t ) - e " y£(t)| + |e m y'(t)-e ™ y"|<

.l^ ^ | | | | I . .- ( , « ^) A| | | ^ ) | + l | . t ( ^ ) A | | ^ ) ^ | .

t ( l - « i ) A t(l-t>?M)A

Next observe that | |e | = | |e | = 1, |y (t)| is uniformly bounded with respect to t and 111 - e e M 11 = 0 ( UM + Z) .

Together with corollary 1.5.2 this reduces the above estimate to i £e( t ) - W 0 l = O ( eN + 1) + O (u M + 2)

or

I ^ W - ^ N . M W I = ° (f N + l) + 0 ( 6M + 2 _ I /) on a time-scale 0(f)

for 1 < u< M+l.

The estimate for x£ can now be obtained by application of the transformation

(1.5.2).

[]

To obtain the analog for corollary 1.5.2 (a uniform estimate with respect to t) one simply replaces 7?w by n and finds:

Corollary 1.5.4

With the same assumptions and notation as in theorem 5.3 define: t(l-«7e)A N

' o

and ( 0 < N < m - l ) *N(t) = ?£(t) + £P , ( ^ ( t ) ) + .- + <NPN( ^ W )

Then following estimate holds:

| x£( t ) - x ^ ( t ) | = 0 ( fN + 1) for all t 6 R

u Remarks

(1.5.1) For the practical application of this approximation theory observe that lemma 1.5.1 provides a straight forward way to calculate the approxima­ tions for the periodic solutions; once having established the existence of the periodic solutions of (1.5.1) satisfying the ER -conditions, lemma 1.5.1 defines an algorithm to calculate the approximations.

(1.5.2) For the actual calculation of the zeros of F° observe that since F° is a normal vectorfield, F 0 ^ . ^ ) = 0 implies that F ^ e ^ ' o . ^ O = 0 for

every <p e IR. This is of course a consequence of the fact that (1.5.1) is autonomous:

if t H x(t) is a periodic solution of (1.5.1) then is also t -• x(t+v?) a periodic solution (for every <p e IR)). Hence the solution of (1.5.6) is not unique. This non-uniqueness may be avoided by simply setting one component of the initial value y (0) equal to zero.

(1.5.3)h is easy to verify that if the conditions of theorem (1.3.1) are satisfied and in addition the righthand-side of (1.2.1) is analytical with respect to all variables, the periodic solutions are analytical with respect to c. Hence one may write xe(t) = x\-(l) + £ R-WCM) where R^, is an analytical function with respect to e (for e sufficiently small). However, it is still an open question if there exists an t > 0, independent of N, such that R^, is analytical for |«| < c (this is of course related to the question of convergence of the normal form transformation for m -> m).

1.6. A simple model for the dynamics of an oscillator with one-degrec-of-frcedom in a uniform windfield.

In this section a mathematical model, describing the flow-induced vibrations of an oscillator with one-degree-of-freedom in a uniform windfield, is presented and the resulting equations are analyzed.

The oscillator basically consists of a rigid cylinder with a small ridge and four springs providing elasticity. The oscillator and the frame in which the oscillator is mounted is sketched in figure 1.6.1. The cylinder with ridge is rigidly attached to two shafts. These two shafts can simultaneously move within two air-bearings in the z-direction. The air-bearings are subsequently rigidly attached to the side-walls; the springs provide the restoring forces in the z-direction. Notice that the oscil­ lator has one-degree-of-freedom Figure 1.6.1 (torsional vibrations of the cylinder and vibrations perpendicular to the picture-plane (the xz-plane) are excluded). If this construction is put into a uniform windfield v , v = v e (v

' r m ' CD QD y V OD

> 0), aerodynamic forces will start to act on the cylinder and galloping may occur. s u p p o r t s to the <H w a l l ( " a i r -bearings

To derive the equations of motion observe that since only oscillations in the z-direction are considered one may restrict one's attention to a cross section of the cylinder. In Ggure 1.6.2 such a cross section is sketched. Although the cross section is not circular, due to the ridge, it is assumed that there is still an axis of symmetry; a (the static angle of attack) is the angle between the axis of symmetry and the inco­ ming windfield v (the angle a is positive anti-clockwise). The aerodynamic forces, the drag D = DeD and the lift L = Le, are also

sketched in figure 1.6.2; eD and e,

are unit-vectors such that e^ has the same direction as the virtual windvelocity v , v = v - ze and e. is obtained by rotating e^ over an angle x/2 in anti-clockwise direction. It is easy to verify that the equations of motion become:

mz + (s + êaz)z = Dsin(v?) + Lcos(v?). (1.6.1) Here is z the displacement of the cylinder (z = 0 corresponds to the equilibrium

position of the cylinder in absence of the aerodynamic forces), m the total mass of the construction that can move (cylinder, ridge, springs and shafts), y the angle between e and er> (positive in anti-clockwise direction) and s +eaz the total stiffness of the spring (a weakly nonlinear spring, 0 < è < < 1 and a = 0(1)). The magnitude of the aerodynamic forces, D and L, may be expressed as:

od „ , . v . 2 Figure 1.6.2

D = ^ CD( a ) v ^ ,

Here is p the density of air, d the diameter of the cylinder, v = | v | , Cp(a), resp. C, (a), the quasi-steady aerodynamic drag, resp lift, coefficient and a the angle between the axis of symmetry and v (positive anti-clockwise).

The aerodynamic drag and lift coefficient may be obtained from wind-tunnel experiments; typical results are sketched in figure 1.6.3.

C

( a )

-0.5

Figure 1.6.S

Q (a) According to the den Hartog

criterion galloping may occur if

CD(Q)

+ k

i>)

(linear instability of the equilibrium position).

Since Cp,(a) is always positive one can restrict one's attention to the case where -^-C, (a), < 0.

s Hence the drag and lift coefficients curves are approximated by:

c

(")

=

c

,

CL(a) = CL( a - a0) + CL( « - a0

where Cr, > 0 a constant and Do

where C, < 0 and C, > 0 are constants.

Note that:

v? = v2 + z2, S 0

cos(ip) = v /v sin(ip) = - z/v and tan(v) = - z/v , a = <p + Qg.

that in the case |aQ - a \ = 0(7), the equation of motion becomes (terms of

fourth and higher order in z are neglected):

z + (so + ëaZ)z = - 4 [ - v J CD o+ CL i) z +

-Oc^ + c^ + ecgv;

^]

= Axt (Aj =

in

h-Introduction of the dimensionless variables Z and r defined by z(t) = A„Z(r) and

) leads to the -2m ( Cn + CT )v

( 3 Cn + CT + 6Cr )s

' D o L i L 3 ' o

differential equation (dot stands now for differentiation with respect to T):

Z + Z + taZ2 = e2(l-Z2/3)Z, (1.6.3)

where t =

-pdv ( CD + C, ) __ 2

-!— , a = eaA„/(fmA,) and it is assumed that 2mA

pA v

c/e = 0(1) (it follows from physical observations that *m\W is small,

CD + C, < 0 (the den Hartog criterion) and 3CD + C, + 6C, > 0).

To analyze (1.6.3), a modified Raleigh equation, the equation is first written as a system of first order O.D.E. ( x. = Z, x„ = Z):

X2 . = 0 1 -1 0. xl x2 - 1 0 2 ax, -fe2 (l-x^/3)x2 (1.6.4)

Next the normal form of (1.6.4) is calculated (up to order 7) by setting:

* * € + i P

( 0 + - + «

P

(0 (1-6.5)

(P- as defined in theorem 1.2.3). This leads to the following equations:

'( = H + <2f°(£,«) + 0(£8), with: (1.6.6) I°({.O = fi0(r,O + Ö(r,e)

<1

'-«+4

, / 3 2 , 1539-14644a4 4 _1701-34124a46^ 4 + ^ 128r + 82941 r 663552 r >( '

(0/r c\ _ 5 a22 , , -864+432r2-(3140a4+81)r4 , 2 ,

12(t'l) ~ 12 r + l 6912 >e +

, / -231360a2r2+180960a2r4-( 1309900a6^16869a2)A .4

+ ^ 1658880 ' ■

Using this normal form and theorem 1.3.1, one can show that within a sufficient large but bounded domain there exists, provided t is sufficiently small, exactly one periodic solution of (1.6.4) and hence (1.6.3). Furthermore, by using the

2 2

Lyapunov-function L(£.,£„) = £i + £o, one can show that all solutions within this domain tend to the periodic solution for r-> o.

To find higher order approximations for the periodic solution one sets: (l-tn )TA

£(r) = e e y(r) (1.6.7)

and finds:

y = e F ° ( y , V ) + 0({ 8) (L 6 8)

The higher order approximations are now found by calculating the critical point of F°(y,T](,t) (see theorem 1.5.3).

For the frequency (ai = -m^) one finds that:

. „ . 5a2 2 27+3620a4 4 826300a6+9183a2 6 , n, 7,

« « « l - ^ - l — J - * Ï32 c Ï2WÖ £ +° (f) '

The approximation for the periodic solution of equation (1.6.3), say Z£(T), can

now be calculated up to order 5 by using the transformations (1.6.7) and (1.6.5). One finds:

Z0(T) = 2 cos(r), Zj(f) = -2a + 2§cos(2f),

2

Z2( T ) = ja2cos(r) - Jsin(T) + j^in(3f) + ^ O S ( 3 T ) ,

+ Ï7jsin(4r),

„ ,-% 54+5327a4 ,-x 347a2. ,-.. . 6 4 a2. , „ - , . 9+242a4 /,-■> ,

Z4ST'= — 4 1 2 — C 0 S(T) _T 3 2 s l n(T)+ m-51"^ + — T F r -0 0^3 7" ) +

- J j j * Bïn(3f) + 2 9 » ^ cos(5f) + g g s\n(5r),

•, ,ix 35a 21205a5 , 4a „ „ „ / ^ , 13096a3-,-■> 2253a-395300a° .0-x , Z5 ^ = " " 8 S I T- + 4 T ^0 S(r) + - 2 1 3 5 S i n<T) Ï296Ü C O s ( 2 r ) +

11989a3. / „ - , 4 a „ , , - , . 8 a3. ,„-., , 7461a+26200a5 , , - \ ,

- "3240— S l n(2 r) " 3 5C O S^3 T) + W*iaW + T86ÖÖ C O S(4 T> +

55a3.:_„ix 9819a-700a5.„/ eix , 797a3_:_,..

+ m Sin(4r)- 4 4 5 3 6 0 0 cos(6r) + m^^(6i

1.7. Concluding remarks.

In this chapter an attempt has been made to answer some of the questions concerning the use and applicability of normal form theory for analyzing the type of differential equations considered here. By using an invariance principle as normal form definition, algorithms are derived to calculate not only the normal form, but also the normal form transformation by making use of averaging principles. Note that these formulas make it possible to consider a more general type of O.D.E. as often used in the literature, since they avoid the usual polynomial characterization. Moreover these algorithms are easy to implement on a C.A.-system like Macsyma. In fact a computerprogram has be written by using Macsyma to calculate the normal forms in IR (for arbitrary n) up to order k (k arbitrary) (more details or a copy of this program can be obtained from the author). The theorem on the existence of periodic solutions can be regarded as an extented form of the theorem found in [38]. Concerning the use of higher order normal forms one can say that with respect to periodic solutions they can be used to obtain a more accurate approximation for the periodic solution and the period. Therefore this may be regarded as an alternative for and generalization of the Poincare-Lindstedt method (the method presented here may also be used for n > 1). Moreover it is shown that these approximations are also valid on larger time-scales. As an application a weakly non-linear oscillator in a uniform windfield has be considered and it is shown that the dynamics of the oscillator can be modeled with a modified Raleigh equation. Finally it can be mentioned that the theory derived in this paper also may be used for a class of autonomous equations where some of the eigenvalues of the unperturbed system are zero i.e.

x = Ax + ff(x,x,e), x e R ,

x = €g(x,x,(), x E Rq, (1.7.1)

(the matrix A as defined in the N -conditions),

or for certain nonautonomous equations i.e.

x = Ax + rf(x,sin(o;ot+a),«), x e R2n, (1.7.2)

(the matrix A as defined in the N -conditions and u /u. € Q).

That the theory also may be applied to (1.7.1) follows from the observation that the fundamental solution of (1.7.1) for e = 0 is periodic, whereas (1.7.2) can be put into the standard form (1.2.1) by replacing sin(w t + a) by x„ , ,(t) and

2

adding the differential equations x„ , , = xonj.o anc* x9n+2 = ~wox2n+l t 0

Appendix.

In this appendix the proofs of theorem 1.2.3 and 1.3.1 are given.

Proof of theorem 1.2.3:

Since f e Cm + 1( D x (-£0lto), K2") and f(0,0) = 0 it is clear that (1) is satisfied. In

order to prove (2) let s e K and x e D. Then: T

-sAfOc„sA \ 1 f „-sA -wKr /.ipA„sA >, e i[(e x) = y e e v fk(e^ e x)d<p.

%

By introducing the variable T = s + ip one finds:

„s+T„ . 0 e ^ e 5 Ax) = - ^ 1 f -TA[ I TA i, i f -TAC /„rA ■,, ,

Y~ e fk(e x)dr = ->r- e fk(e x ) d r + s s s+T T L f VT Afk( er Ax ) d r + 4 - f Vr Afk( er Ax ) d r . O Jrr 0 Jn

+

Finally observe that r -> e is T - periodic, so the equation becomes: t*tH*Ax) = - f e -Mfk( e ^Ax ) d r + 4 - e ^Af . ( er Ax ) d r + f°(x) =

oJs oJQ

which proves (2).

To show that (3) holds note that, as is easily verified, substitution of the transformation (1.2.5) into equation (1.2.1) results in:

i = A( + e ? ek-»[ A Pk« ) -DPk(flAÉ + fk(fl ] + cm+ië((,c),

with Pk, Ï, and g as in the theorem. It should be pointed out that in order to

establish existence of the inverse of I + £DP(£,c) and to assure that for all £, x = £ + fP(£,f) remains in D one may have to choose a smaller to and a

subdomain of D instead of D.

Remains to show that APfc(0 - DPk(OA£ + ffc(0 =

f£(0-Observe that since f? is a normal vectorCeld one may write:

T r

k(e)= 1 f ° J e ^Afk( e ^A0 - e ^Af ^A0

0 Jn

&tp, and thus:

T r APk(fl-DPk(0A{ = 4 - f %fc - e ^ f / o + e A r t o Jf, dp = 1 _{- e - ^ ( e *A0 + e-*Atf(e*A0}

+

i f V

" 4 J e-^f^fldp.

-which completes the proof.

Proof of theorem 1.8.1: Necessity.

Suppose there exists for every t > 0 (e sufficient small) a T - periodic solution, say t -i xc(t), of (1.2.1) with smooth period. By using the transformation xf(t) =

that y£ satisfies:

yf(t) = <F°(ye(t),nt) + ^ ( t ^ t ^ e )

(F° and gas in (1.3.3)).

Since x is a T£-periodic it follows that y (T£) = y (0) and thus for e t 0:

T T 0 = - ~ (y'(Tf) " y£(°)) = [ fF°(y€(t),^)dt + t f £g(t,y£(t),7/£,()dt.

^0 ''O Since (1.2.1) satisfies the N -conditions and rj depends continuously on t, y°:= lim ye(0) and n := lim 77 exist. From this one can conclude that (see

0 « - 0 ° (- 0 f

the above equation): .T

0 = ƒ ° F ° ( y >0) d t = ToF°(y°,r,o), and thus:

^°(y>

) - *

In other words, there exists a (y,r/) e D x IR such that F°(y,r;) = 0. This proves the necessity.

Existence:

Suppose that condition (1) and (2) hold. Note that since 1.2.1 is autonomous it is sufficient (and necessary) for the existence of a periodic solution to find a T > 0 and a solution of (1.2.1), say t - x6(t), such that x£(T ) = xe(0). By using again

the above transformations one has to show that there exists a solution of: yf(t) = eF°(y\V() + £2g(t,yé, V ) ,

T

satisfying yf(T£) - ye(0), T£= - j - 2 ^ - > 0.

Now introduce the following notation:

CO y? = (

r 4n>'

(.) p-1: R2""1 . R2 n ; p"1(y1 y^) = {j^^fy^***)

(.)p(D) = { y6f l (2 n-1:p-1( y ) 6 D } c K2 n-1

and define:

G : p ( D ) x K x ( - 70, 70) - I R2 n;

T„

G(y,u,0 = ƒ [F0(ye(t,p"1(y)).'?) + «Kt,ye(t,p-1(y),»7,£)]dt =

_ye(T0/(i-t7?),p-1(7))-yt(o,P"1(y))

here is t - y£(t,p_1(y)) the solution of (1.3.3) with y'(0,p_1(y)) = p- 1(y).

Observe that G(y>| €) = T / V ^ y » + 0(e).

Due to the regularity of the righthand-side of (1.3.3) it is standard (see [25]) that G 6 C1(P( D ) x R x ( - 7o, 7o) , R2 n) .

From condition (1) one learns that G(p(y ),;/ ,0) = 0. Furthermore if follows from condition (2) that the implicit-function theorem can be applied and thus there exists for every e, e sufficiently small, an unique (y',n ) £ P(D) x R (in a 0(e) neighborhood of (p(y°),w )) such that G(y~!,7? ,e) = 0.

T.

By setting y j = p '(yj). T £ = !_",, ( > 0) and with t - yf(t,yj) the solution

of (1.3.3) satisfying yf(0, y*) = y*, one finds that y£(T(, y*) = ylQ. Hence the

solution of (1.2.1) with initial value x* := y* + «P(VQ). say t - xe(t), is T

-periodic. That this is an isolated periodic solution of (1.2.1) follows from the uniqueness in the implicit-function theorem; in fact there is no other periodic solution of (1.2.1), with a period close to T , in a 0(e) neighborhood of the orbit.

t(l-en )A

The estimate | x£(t) - e ° y* | = 0(e) for 0 < t < T£ follows from the

2

fact that n is differentiate with respect to e and lim r\ = i\ .

£ £ - 0 £ °

Stability:

First note that it is sufficient to prove that the periodic solution of the first order normal form i.e.

è€(t) = A£((t) + 6 f V ( t ) ) 4- €2g(e£(t),€)

Ée(0) = y0,

is asymptotically orbitally stable resp. unstable. Next observe that it is sufficient to have asymptotically orbitally stability, resp instability, by showing that the Poincaré-Return map of a solution of (1.3.1) (for a specified cross plane) in the neighborhood of the periodic solution converges, resp. diverges. We claim that the hyperplane V := {(Yp-Jon) 6 K " : v; = Y? ) i s a c r o s s P, a n e a t t n e Po i n t YQ' I f

o o

one uses the normal-vector n of V, n = (0,..,0,1,0,..,0), one finds that:

*e(0).n = Ay°n + 0(e), so V is a cross plane if Ay°n i 0. Now suppose that

Ay°n = 0. With i = 2j for some j € {l,..,n} (the case iQ = 2jQ - 1 is analog) it

follows that 0 = Ay°n = -w. y° ,, so yS- , = 0. Since F° is a normal

0 Jo %~v % l

vectorfield one finds that (theorem 1.2.4): DF°(y>0)Ay« = A F ° ( y >0) = 0.

Since y2 • _. = 0, the i - component of Ay° is zero and it is easy to verify that:

^

V-

?

-l.

,/?

l-.

2n)

ö l y p - y j _p v ,y. + 1, . . , y2 n) (y°0,v0)

, A y ° = 0.

On the other hand it follows from condition (1) that Ay f 0 and so one can conclude that the above matrix is singular. This however contradicts with condition (2), so V is a cross plane.

Hence one may consider the perturbed initial value problem: ^ = A ^ + ^ ) + e 2 g ( ^ , £ ) where:

£ » = yo + M

Let T be the time that t H ^ ( t ) returns to the plane V. In order to obtain asymptotically orbitally stability, resp. instability, one has to show that

^ ( T ) = y + pH with | 6 | = 1 and n < p for every 8 and n sufficient small, resp. '8=8 and and /x > \i for every n, p sufficient small and some 8.

Due to the regularity of the problem one finds that T « T + 0(f//). By defining:

T„ -tfl-en )A

one finds that t -> yf//(t) satisfies:

y ^ =

F

( y ^

)

I ( t , y ^ ,

0

y"*(o) = y^ + M-Integration leads to:

^ e y - >-o - y

"(V - y

(

<)

* » - y

(°)

.T_. T

f

[F°(y^(t),

)-F

(y

(t),^)]dt-

r V(y

(t),^) +

i

T

£

g"(t,y

(t),

) ] d t + e

f "fg(t,y^t),i, 0 -g(t,y

(t),

)

+

- • - ■ - t • |

'0 By using the following estimates:

( 3 ) r ?6= r ?o+ O ( 0 ,

(4)y£ M(t)-y£(t) = ^ + 0 (f / i) f o r O < t < T ^ ,

(5) y€(t) = yj + 0(e) for 0 < t < max(Tf/i,Tf),

one finds that:

F°(y£(t).'7 )+ eg(*iy ( t ) . O d t ■ ° (£M ) (use (1) and the boundedness

' T

e

T

e/ir

of the integral).

[

[g(t,y

(t),

) -g(t,y

(t)

£)]dt =

''O T

f

[F°(y

\t),r, ) - F°(y

(t)

)] dt = T

DF°(y>

)M +

•'o

To ( " e , - " e ) C( y> o ) + <X«* + ° < A

If one now uses the fact that the i -component of 5 as well as Ti equals zero it follows that » - n = /<VF°(y°).« / a + O M + 0((?) (a = u?. y° if i = 2j0

tfi e o Jo ^Jo i

and a = - y „. if i0 = 2j0—1). Furthermore p(6) and p(3) mav be used instead of 6

and 5 and one can thus conclude that: fflp(S) = [ l 4 B]/IP(Ó) + 0(e2/*) +0(e/ i 2),

afFp.-.P. j , F . + 1» « ^ 2 n )

where B

=

f T

o ws—r^r—y—r

In the case that all eigenvalues of B have negative real parts one can thus conclude that /z < fi for every 6 and /J sufficient small and thus stability. In the case that there is at least one eigenvalue with real part positive one takes for 6 the corresponding eigenvector and observes that /i > n for every n, /i sufficient small.

D y0

CHAPTER 2

ON THE PERIODIC WIND-INDUCED VIBRATIONS OF AN OSCILLATOR WITH TWO-DEGREES-OF-FREEDOM *)

Abstract.

In this chapter the dynamics of an oscillator with two-degrees-of-freedom in a steady flow is studied. Principles from the theory of galloping are used to derive the equations of motion. The normal forms for the equations of motion for a number of interesting cases are presented and the existence of periodic solutions and their stability is established. Formulas, which may be used to calculate amplitudes and periods in an approximative way, are presented.

This chapter is a revised version of a paper ([8]) by the author of this thesis and A.II.P. van der Burgh.

2.1. Introduction.

Overhead transmission lines on which ice has accreted may have cross sectional shapes that are aerodynamically unstable to transverse disturbances in a windfield. The evolution from this unstable equilibrium position may result in galloping: a large amplitude oscillation with low frequency (< 1 Hz). The very complicated phenomenon of galloping of overhead transmission lines which involves the aeroelastic interaction of longitudinal, transversal and torsional oscillations of a continuous system is far from being understood.

For an interesting survey paper on wind-induced vibrations of overhead transmission lines the reader is referred to [41]. In this chapter a simple oscillator, which has some relation with this galloping problem is introduced. At this stage the oscillator is a theoretical model, that is, no experimental prototype has been developed yet: however, it is believed that the ideas presented here, may be used for the actual development of an experimental model. The oscillator, as sketched in figure 2.2.1, consists of a rigid circular cylinder with a small ridge and a number of springs mounted in a frame. The oscillator is constructed in such a way that the cylinder-spring system has two-degrees-of-freedom, i.e. oscillations in and perpendicular to the direction of the air flow; both modes of vibration are decoupled in the absence of the air flow. A more detailed description of the oscillator is given in section 2.2. The oscillator may be considered as an extension of the one-degree-of-freedom oscillator introduced and studied theoretically and experimentally in a windtunnel in [34] and [35], where only oscillations perpendicular to the direction of the air flow are possible.

This chapter is organized as follows:

derived. The assumption is made that the aerodynamic forces are quasi-steady which implies that they can be derived from static force measurements (in a steady flow). For the study of galloping there is no disagreement in the literature (e.g. [34,35]) on the use of a quasi-steady theory. The mathematical modeling of the oscillator is analogous to the model of a swinging-spring oscillator introduced in [13]. This oscillator consists of a cylinder, hung from springs, such that spring and pendulum oscillations may be carried out. However, in the equations of motion of the oscillator as considered here a new parameter representing the position of the ice accretion (ridge) is introduced. Finally, by truncating the equations of motion, the so called model equations are obtained. In section 2.3 periodic solutions for the model equations in a number of interesting cases are established and criteria for the stability of these periodic solutions are given (based on the theory in chapter 1). The chapter ends with some concluding remarks. With respect to the calculations of the normal forms a computer program has been written by making use of the Computer-Algebra Macsyma ([30]) which has the capability of manipulating algebraic expressions involving unevaluated variables.

2.2. The oscillator and the equations of motion.

In this section the oscillator, as sketched in Cgure 2.2.1 and 2.2.2, is described in more detail. In addition, the equations of motion will be derived.

The oscillator, as sketched in figure 2.2.1, consists of a rigid circular cylinder with