The Krylov–Bogoliubov and Galerkin Methods for Non-linear Oscillations

8/9/2019 The Krylov–Bogoliubov and Galerkin Methods for Non-linear Oscillations

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Journal of Sound and Vibration (1996) 192(2), 413–438

THE KRY LOV–BOGOLIUBOV AND GALERKIN

METHODS FOR NON -LINEAR OSCILLATIO NS

P. Y, Y. M. D, N. P A. H. S

Faculty of Engineering, The University of Manitoba, Winnipeg, Manitoba,Canada R3T 5V 6

(Received 21 July 1994, and in final form 10 April 1995)

The objective of this paper is to consider the dynamic motions of second order, weaklynon-linear, discrete systems. The main attention is focused on a comparison, for suchsystems, of the method of Krylov–Bogoliubov (KB) and an enhanced Galerkin (EG)method which produce seemingly different solutions. Despite the apparent differences, thetwo methods are shown to give identical first order periodic and quasi-periodic solutions,

and the same stability conditions for internal and external resonances, as well as anon-resonance. The ease of applying one or the other method depends upon whether asystem is resonant and upon the number of participating modes. Both approaches are usedhere to analyze illustrative examples that are pertinent to galloping.

1996 Academic Press Limited

1. INTRODUCTION

There are innumerable engineering problems which can be modelled as a set of secondorder, non-linear, ordinary differential equations. If such equations do not explicitlycontain the independent time variable, the system is called autonomous; otherwise, it iscalled non-autonomous. Examples of autonomous systems are vortex-induced vibrations,

the galloping of tower guys or transmission lines and the free vibrations of a centrifugalpendulum; and examples of non-autonomous systems are vibrations in machines andoscillations in electrical networks which have sources. Although the autonomous systemsappear merely as a special case of more general non-autonomous ones, there exist

fundamental differences between these two systems. For example, autonomous systemsusually have equilibrium solutions, while non-autonomous systems do not. On the otherhand, non-autonomous systems more easily exhibit chaotic motions than autonomoussystems. In general, the analysis of a non-linear dynamic system involves three relatedphases: (i) finding the equilibrium points and their stability; (ii) determining the possibleperiodic and quasi-periodic dynamic motions which bifurcate from an equilibriumstate; and (iii) evaluating the stability of the dynamic solutions. For a non-autonomoussystem, only the second and third phases are involved. The main purpose of this paperis to focus upon the analysis of periodic and quasi-periodic motions. Although such a

system may exhibit chaos for certain parameter values, such motions are not investigatedhere.

To quantitatively analyze the dynamic behavior of a non-linear system, manyapproaches have been developed, among them are the Lindstedt–Poincaré method,

multiple time scaling, harmonic balancing, time-averaging, the Krylov–Bogoliubov–Mitro-polsky (KBM) method, and the Galerkin and Ritz procedures (see, e.g., reference [1], PartII of reference [2] and Chapter 8 of reference [3]). These methods can be applied to both

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autonomous and non-autonomous systems to determine steady state solutions; forexamples, periodic or quasi-periodic motions. The techniques can be divided into twogroups. The first group includes the methods of multiple time scaling, time-averaging andthe KBM approach. This group of methods can be used to derive a set of simple differentialequations, usually described in terms of the amplitudes and phases of motions, which

determine not only approximate steady state solutions but also their stabilities. The secondgroup includes the Lindstedt–Poincaré technique, harmonic balancing, and the Galerkinand Ritz procedures. This group of methods can be employed to directly determineasymptotic periodic and quasi-periodic solutions, but the stability analysis of the solutions

leads to variational equations which require the Floquet theory [1] and is much moreinvolved. It should be pointed out that the Galerkin or Ritz procedure is essentially anaveraging method. In this paper some methods will be chosen from these two groups andtheir advantages combined to overcome their shortcomings.

Time-averaging will be used to find explicit periodic and quasi-periodic solutions andtheir stability conditions. Time averaging procedures [4] can be categorized as ones inwhich an a priori known frequency is used in an assumed approximation, while the otherprocedures assume an unknown frequency. The first approach was developed originallyby Krylov, Bogoliubov and Mitropolsky [5], and is usually called the KBM method. Thisapproach is, indeed, a combination of the averaging and variation of parameters methodand the perturbation method. However, the approach used in this paper is the firstapproximation of Krylov–Bogoliubov (KB) method which uses the averaging and

variation of parameters method only. The second approach is sometimes called the methodof (conventional) Galerkin [6] or Ritz [7]. The KB method results in time averaged,differential equations which can be used to find steady state solutions and their stabilityconditions. The conventional Galerkin’s (CG) approach, conversely, yields time-averaged,algebraic equations from which only the steady state solutions can be determined,regardless of their stability. Although the CG method has been enhanced recently in orderto evaluate the internal resonances of a two-degree-of-freedom (DOF) [8] and a 3-DOF[9, 10] galloping model, no rigorous proof was given to show the equivalence of the KBand the ‘‘alternative Galerkin’s averaging method’’ [9, 10]. This enhanced Galerkin method

will be called the EG method for short, in this paper. Moreover, it had been suggestedpreviously that a solution may be found explicitly for only the internal non-resonance[11–14]. However, internal resonances, in which the ratio of any two natural frequenciesis close to a ratio of two positive integers, are potentially more damaging and, hence, have

more practical interest.Newland [15] showed that the first order, non-resonant amplitude–frequency relations,

derived by using the KB approach with a known frequency and the CG (or Ritz) procedurewith an unknown frequency, give identical results for a 1-DOF autonomous system whichhas no external forcing. However, these first order, steady state solutions can be shownto coincide only if the derived amplitudes are proved identical. Moreover, Newland alsoconsidered an external 1:1 resonance by assuming that only one of the natural frequencies(for more than 1 DOF) is close to the forcing frequency. By presuming the same knownfrequency (that of the forcing) in the first order approximations, the two methods were

demonstrated to produce identical first order, time-averaged algebraic equations. It willbe shown in this paper that, by assuming the same known frequency, the proof of the firstorder equivalence merely requires a co-ordinate transformation. On the other hand, if anunknown frequency is considered in Galerkin’s method, an approach which will be shown

to be very useful for certain internal resonances of multiple DOF systems, the proof of the first order equivalence is more complicated. For clarity, the KB method employs, inthis paper, a known frequency, while an unknown frequency is assumed for the EG


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method. The two methods will be shown rigorously to be equivalent for a general n-DOF,non-linear system. Such a system may involve not only external but also internalresonances, as well as a non-resonance.

A new technique has been developed, based on the KB and Galerkin methods, to findexplicit solutions and their stability conditions, especially for the internal resonances of

a multiple DOF, autonomous system [9, 10]. The CG method was modified, by using timeaveraging, to produce time-averaged, differential equations rather than algebraicequations. This modification is termed the enhanced method of Galerkin (EG method).The proof of the equivalence of the KB and EG methods will be given here and includes

the stability analysis which was excluded from Newland’s work. Although the KB andEG methods will be shown to produce identical first order periodic and quasi-periodicsolutions as well as the same stability conditions, the KB method is preferred fornon-resonance as well as for combinations of the non-resonant and resonant cases in whichthe number of resonant modes is fewer than, say, three. On the other hand, the EG methodmay be utilized advantageously for resonances in which more than two modes are involved.

The next section is devoted first to proving that the KB and EG methods produceidentical first order solutions as well as stability conditions for a general n -DOF, weaklynon-linear system. Then the relations of the KB, EG and CG methods will be shown fromthe results. Applications of the theory to the galloping problem are given subsequently.Two typical resonant cases are detailed. Conclusions are drawn in the last section.

2. EQUIVALENCE OF KB AND GALERKIN METHODS

The motion of an n-DOF, weakly non-linear system is described by

¨ i + 2i i = f i (, j , ̇ j ), i , j = 1, 2, . . . , n, (1)

where the dots indicate differentiations with respect to normalized time , and i and i are the i th normalized co-ordinate and eigenvalue, respectively, of the corresponding( = 0) linear system. The f i is the analogous normalized force, including all the non-linear

terms as well as all the damping terms and is the frequency of the externalforcing function. The parameter (01) represents a weak non-linearity. The internalnon-resonance and resonances of an autonomous system, which does not involve externalforcing functions, will be considered in subsections 2.1–2.4. Then the derivations will be

extended directly in subsection 2.5 to non-autonomous systems. The relations of the KB,EG and CG methods are given in subsection 2.6.

2.1. -

It follows from equation (1) that, when = 0, a response can be written in the form

i () = X i cos (i + i ), i = 1, 2, . . . , n, (2)

where constants X i and i , respectively, represent the i th component’s amplitude and

phase, which are determined from the initial conditions. The solutions for 0 can beconsidered to be a perturbation of solution (2) and, therefore, may be expressed by

i () = X i () cos[i + i ()]X i () cos i (), i = 1, 2, . . . , n. (3a)

Now, the X i () and i () are not constants but are ‘‘slowly varying’’ variables such thattheir derivatives, X i () and i (), are small ( order) terms. Equation (3a) belongs to the


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KB approach because the natural frequencies, i , are known. An alternative expressionfor equation (3a) is given by

i () = Ai () cos i + B i () sin i . (3b)

The solution in the EG approach, on the other hand, can be written as

i () = X *i () cos [*i + *i ()]X *i () cos *i (), i = 1, 2, . . . , n, (4a)

where *i , unlike i , is unknown. For a steady state solution, the amplitudes X i () (orX *i ()) are constants, X i . Moreover, the frequencies ¯ i equal *i for solution (4a), whereasthey equal i + i () for solution (3a). That is, for a steady state solution, i () 0 but *i () = 0. An alternative expression for equation (4a) is

i () = A*i () cos *i + B *i () sin *i . (4b)

The choice between the KB solution (3a) or (3b) (or, alternatively, the EG solution (4a)or (4b)) depends upon the ease of deriving explicit solutions by using the time-averaging

procedure. In fact, equation (3a) (equation (4a)) can be transformed to equation (3b)(equation (4b)).

It will be shown in the following subsections that the two formulations (3a) and (4a)(or (3b) and (4b)) give identical first order approximations.

2.2. -

This case implies that a ratio of any two natural frequencies is not close to a ratio of two positive integers. Suppose that the periodic or quasi-periodic solutions are given by

equation (3a). Differentiate this equation with respect to and let the amplitudes andphases be chosen such that

̇i () = −i X i () sin i () (5)

and

X i () cos i () − X i () i () sin i () = 0. (6)

It should be noted that the last two equations cannot be obtained without assuming thatX i () and i () are varying slowly. Next, differentiate equation (5) with respect to andsubstitute the resulting equation and equation (2) into equation (1), to yield

X i () sin i () + X i () i () cos i () = − i f i [X j () cos j (), − j X j () sin j ()].(7)

Solving equations (6) and (7) for X i () and i () and then employing time-averaging leadsto

X i = − i limT 1T T

0

sin i f i (X j cos j , − j X j sin j ) d (8)


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and

i = − i X i limT 1T T

0

cos i f i (X j cos j , − j X j sin j ) d, (9)

where has been suppressed from X i () and i () for brevity.Following the same procedure, the analogous equations from form (4a) are

X *i = − 1*i limT 1T T

0

sin *i [(*2i − 2i )i + f i (X * j cos * j , −* j X * j sin * j )] d

= − *i limT 1T T

0

sin *i f i (X * j cos * j , −* j X * j sin * j ) d (10)

and

*i = − 1*i X *i

limT

1T T

0

cos *i [(*2i − 2i )i + f i (X * j cos * j , −* j X * j sin * j )] d

= −*2i − 2i

2*i − *i X *i limT 1T

T

0

cos *i f i (X * j cos * j , −* j X * j sin * j ) d. (11)

Generally, explicit integrals cannot be obtained from equations (8)–(11) because the X i (X *i ) and i (*i ) are functions of . However, they may be approximated by assumingthat variables X i (X *i ) and i (*i ) are constants in a period (which implies that the i thcomponent of the motion is periodic) because these variables are varying slowly. Theseapproximate, explicit integrals can be obtained by expanding the f i into multi-variableFourier series. It should be noted that non-zero residuals arise from those terms of the

Fourier series containing cos i (cos *i ) and sin i (sin *i ) (i.e., j = i ) because theycorrespond to the non-resonant case.

2.2.1. Steady state solutions

First, consider the KB solution (3a). The amplitude of the steady state solution, X i , isdetermined by setting X i = 0 in equation (8). This yields

limT

1T

T

0

sin ¯ i f i (X j cos ¯ j , − j X j sin ̄j ) d = 0, (12)

where ¯ i = i + i , i 0. Equation (12) contains n independent, non-linear algebraicequations which can be used to solve the n independent variables X i because the i areknown. Note, however, that the solution may not be unique. Having found X i , the

corresponding frequency, ¯ i , is determined from equation (9) as

¯ i = i − i X i limT 1T T

0

cos ¯ i f i (X j cos ̄j , − j X j sin ̄j ) d. (13)

The last equation is in the form ¯ i = i + O(), where O() represents an order term,as expected.


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Next, the steady states for the EG solution (4a) can be found similarly, but fromequations (10) and (11), by setting X *i = *i =0. This results in

limT

1T

T

0

sin *i f i (X * j cos * j , −* j X * j sin * j ) d = 0 (14)

and

*2i − 2i

2*i = − *i X *i limT 1T

T

0

cos *i f i (X * j cos * j , −* j X * j sin * j ) d. (15)

Here *i has been replaced by *i because *i = 0. The resulting 2n non-linear algebraicequations determine n amplitudes, X i , and n frequencies, *i . However, the equations arecoupled, unlike equations (12) and (13), because the *i are unknown and the functions f i explicitly include *i . The 2n equations need to be solved simultaneously and theirnumber is twice that in the KB method. Therefore, the derivation of explicit solutions byusing the EG method is more complex.

2.2.2. First order equivalence of the steady state solutions

It can be observed from equations (12)–(15) that the two steady state solutions, derivedby using the KB and EG methods, are different. To show their equivalence, first note that(*i − i ) is order because the EG solution (4a) is assumed to be a perturbation of theunperturbed solution having frequency i . Therefore, let *i − i = O() and expand the f i , appearing in equations (14) and (15), with respect to * j . Finally, rewrite equations (14)and (15) in the form

limT

1T

T

0

sin *i f i (X * j cos * j , − j X * j sin * j ) d + O() = 0 (16)

and

*i = i − i X *i 1 + O()2i −1

limT 1T T

0

cos *i

× f i (X * j cos * j , − j X * j sin * j ) d + O()or

*i = i −

i X *i

limT

1T

T

0

cos *i f i (X * j cos * j , − j X * j sin * j ) d + O(2). (17)

It can be seen from equations (16) and (17) that their leading (first) order approximateforms are identical to equations (12) and (13), respectively. Therefore, the KB and EGmethods produce the same first order amplitudes and frequencies for the steady statemotions. In particular,

X *i − X i = O() and *i − ¯ i = O(2). (18)


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2.2.3. First order equivalence of the stability conditions

A stability analysis can be performed on the Jacobian matrix of either equation (8),corresponding to the KB approach, or equation (10) for the EG approach. If all theeigenvalues of the Jacobian matrix, evaluated on the steady state solution, have negativereal parts, then the solution is stable. Otherwise, the solution is unstable. The proof of thefirst order equivalence of the stability conditions arising from the two approaches can beshown straightforwardly by rewriting equations (8) and (10) as

X i = − i limT 1T T

0

sin ¯ i f i (X j cos ¯ j , − j X j sin ̄j ) d (19)

and

X *i = − *i limT 1T T

0

sin *i f i (X * j cos * j , − j X * j sin * j ) d + O(2),

or

X *i = − i limT 1T T

0

sin ¯ i f i (X * j cos ̄j , − j X * j sin ¯ j ) d + O(2). (20)

(Note that equations (12) and (16)–(18) have been used in the derivation.) The last twoequations clearly indicate that the formulations of the first order approximate differentialequations are actually identical for the amplitudes.

2.3.

For simplicity, suppose that the ratios of the frequencies, i , are close to p1 : p2 : p3 : · · · : pn , where pi are positive integers. This implies that

j

i −

p j

pi = O(), for i , j = 1, 2, . . . , n. (21)

Let the frequency, , of the greatest common divisor (GCD) be chosen as

= j / p j , for any j (1, 2, . . . , n), (22)

or, more generally,

= l

j = k

j l

j = k

p j , for any k (1, 2, . . . , n) and l (k, k + 1, . . . , n). (23)

The is a known constant. It will be demonstrated later that the choice of which satisfiesequation (23) leads to the same first order results. (As a matter of fact, other forms of may be defined provided that ( j − p j ), i = 1, 2, . . . , n, are order .) By using equation

(23), equation (1) can be rewitten as ¨ i + ( pi )2i = [( pi )2 − 2i ]i + f i ( j , ̇ j ), i , j = 1, 2, . . . , n. (24)

To apply time-averaging, the term ( pi )2 − 2i appearing on the right of equation (24) mustbe of order . Indeed, it can be shown, by using equations (21) and (23), that( pi )2 − 2i = O(). Furthermore, it can be proved that the difference of any two definedby equation (23) is also of order .


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Based on equation (24), the periodic or quasi-periodic solutions which bifurcate froman equilibrium point can be described by

i () = X i () cos [( pi ) + i ()]X i () cos i (), i = 1, 2, . . . , n. (25)

This form is similar to the KB solution (3a) because the frequencies pi are known.Equations similar to equations (24) and (25) can be written for the EG method as

¨ i + ( pi *)2i = [( pi *)2 − 2i ]i + f i ( j , ̇ j ), i , j = 1, 2, . . . , n (26)

and

i () = X *i () cos [( pi *) + *i ()]X *i () cos *i (), i = 1, 2, . . . , n, (27)

respectively. The GCD frequency, *, is unknown and the ( pi *)2 − 2i , i = 1, 2, . . . , n,are of order .

The KB procedure, which was used to obtain equations (8) and (9) for the non-resonantcase, can be applied again to produce the time averaged, differential equations

X i = −

2 pi 2/

0

sin i f i (X j cos j , − p j X j sin j ) d (28)

and

i = −( pi )2 − 2i

2 pi − 2 pi X i

2/

0

cos i f i (X j cos j , − p j X j sin j ) d. (29)

As in the non-resonant case, the above two integrals can be approximated by assumingthat X i and i are constant in the period, 2/. However, any term appearing in theFourier series of the f i which contains cos j or sin j contributes a non-zero residualbecause this situation corresponds to a resonant case. The extra terms (for j i ) include

not only the amplitudes X j but also the phase differences i

m j j

· · ·

mkk whichsatisfy pi m j p j · · ·mk pk = 0. There are only (n − 1) phase differences that are linearlyindependent. Without loss of generality, let i 1 = p1i − pi 1, i = 2, 3, . . . , n; the otherphase differences can be expressed as linear combinations of i 1. Consequently, i 1 = p1 i − pi 1 can be found from equation (29). Similarly, the time-averaged differentialequations for the EG approach can be written as

X *i = − 2 pi 2/*

0

sin *i f i (X * j cos * j , − p j *X * j sin * j ) d, (30)

*i = −( pi *)2 − 2i

2 pi * − 2 pi X *i

2/*

0

cos *i f i (X * j cos * j , − p j *X * j sin * j ) d, (31)

and then *i 1 = p1 *i − pi *1 .

2.3.1. Steady state solutions

The steady state, KB solutions can be found by setting X i = 0, i = 1, 2, . . . , n, and i 1 = 0, i = 2, 3, . . . , n. The resulting (2n − 1) independent, non-linear, algebraicequations can be used to solve the (2n − 1) independent unknowns, X i and i 1. It should


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be noted that the steady state solutions cannot be obtained by setting X i = i = 0,i = 1, 2, . . . , n, because i 0. Consequently, X i = i 1 = 0 results in

2/

0

sin i f i (X j cos j , − p j X j sin j ) d = 0, i = 1, 2, . . . , n, (32)

and

p2i X i 2/

0

cos i f i (X j cos j , − p j X j sin j ) d

− p21 X 1 2/

0

cos 1 f 1(X j cos j , − p j X j sin j ) d

=

i pi

2

−1 p1

2

(i = 2, 3, . . . , n). (33)Note that (/)[(i / pi )2 − (1/ p1)2] is of order due to equation (21). The frequency of the i th component, X i , is given by

¯ i = pi + i

or

¯ i =( pi )2 + 2i

2 pi −

2 pi X i

2/

0

cos i f i (X j cos j , − p j X j sin j ) d. (34)

Equations (32)–(34) suggest that any chosen from equation (23) leads to identical firstorder solutions because the difference between any two such is of order . Furthermore,note that

pi ̄j − p j ¯ i = pi ( p j + j ) − p j ( pi + i ) = [ pi ( j 1 + p j 1) − p j ( i 1 + pi 1)]/ p1 = 0,

for i , j = 1, 2, . . . , n, so that ¯ 1 : ¯ 2 : · · · : ¯ n = p1 : p2 : · · · : pn . The last ratios imply thatthe solution is, indeed, a periodic motion, as expected. Therefore, a GCD frequency ¯ canbe defined such that ¯ i = pi ¯ .

Next, consider the form of the solutions for the EG approach. Then the steady state

solutions can be obtained from equations (30) and (31) by setting X *i = *i =0. This

results in 2n independent, non-linear algebraic equations for the 2n independent unknownsX *i , i = 1, 2, . . . , n, *i 1 , i = 2, 3, . . . , n, and *. Specifically, the 2n equations can bewritten as

2/*

0

sin *i f i (X * j cos * j , −* j X * j sin * j ) d = 0 (35)


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and

*2i − 2i

2*i = − 2 pi X *i

2/*

0

cos *i f i (X * j cos * j , −* j X * j sin * j ) d, (36)

where *i = pi * is the frequency of the i th amplitude component, X i , because *i = 0.It can be seen for the resonant cases that the two approaches do not lead to a too

different number of governing equations for the steady state solutions. (One produces

(2n − 1) equations and the other gives 2n equations.) However, it will be shown next thata particular choice between the two approaches may make it much easier to find explicit,steady state solutions for certain internal resonances.

2.3.2. First order equivalence of the steady state , internally resonant solutions

To prove the equivalence of the two steady state solutions, first note thati = pi + i ¯ i and pi = i + O(). Rewrite equation (32) as

2/

0 sin pi ¯

f i (X j cos p j

¯

, − p j

X j sin p j

¯

) d

= 0,

which implies that

2/(1+ O()) ¯

0

sin pi ¯ f i [X j cos p j ¯, −( j + O())X j sin p j ¯] d = 0

or

2/ ¯

0

sin pi ¯ f i (X j cos p j ¯, − j X j sin p j ¯ ) d + O() = 0 ,

so that

2

0

sin pi f i (X j cos p j , − j X j sin p j ) d + O() = 0. (37)

Similarly, by using *i = pi * and *i pi * = i + O(), equations (35), (34) and (36)can be rewritten as

2

0

sin pi f i (X * j cos p j , − j X * j sin p j ) d + O() = 0, (38)

¯ i = i −

2i X i 2

0 cos pi f i (X j cos p j , − j X j sin p j ) d + O(2

) (39)

and

*i = i − 2i X *i 2

0

cos pi f i (X * j cos p j , − j X * j sin p j ) d + O(2), (40)


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respectively. A comparison of equations (37) and (38) as well as equations (39) and (40)leads to two sets of identical forms of first order equations. This implies that

X *i − X i = O() and *i − ¯ i = O(2) (i.e., * − ¯ = O(2)), (41)

which are identical to equation (18) for the non-resonant case.

By using equation (41) and comparing the equations obtained by setting i 1 =0 withthose found by letting *i 1 = 0, it can be seen that the two sets of (n − 1) algebraic equationsare identical up to first order. Hence,

*i 1 − i 1 = O() (42)

and the KB and EG methods indeed produce identical first order amplitude components,phase differences and frequencies.

2.3.3. First order equivalence of the stability conditions

The stability conditions for the two resonant, steady state solutions can be determinedfrom the eigenvalues of a (2n −1) ×( 2n − 1) Jacobian matrix. This matrix is derived fromthe differential equations of X i and i 1 for the KB form (25) and from the differentialequations of X *i and *i 1 for the EG form (27). A periodic solution is stable if all the realparts of the non-zero eigenvalues of the Jacobian matrix are negative; otherwise, it isunstable. By following the procedure demonstrating the equivalence of the stabilityconditions for non-resonance, it can be shown that the two first-order (2n −1) ×( 2n − 1)Jacobian matrices are identical. Hence, the first order stability conditions derived from theKB and EG methods are identical.

2.4. -

When one or more natural frequencies correspond to internal resonances while theremaining natural frequencies are non-resonant, the differential equation (1) can berearranged into two subsystems. One subsystem involves only resonant frequencies andthe other incorporates solely non-resonant frequencies. Then the procedures employedpreviously for the non-resonant and resonant cases can be used again. However, the two

subsystems are coupled because of non-linear terms. (On the other hand, the governing,time-averaged equations are coupled only by the amplitudes of motion.) The derivationof the steady state solutions and proof of their first order equivalence are similar to thosegiven in subsections 2.2 and 2.3, so they are omitted. The procedure for handling

combinations of resonant and non-resonant cases will be given in the next section for a3-DOF oscillator.

2.5.

Such a system can be handled by employing the procedure used in the previoussubsection. In addition to the two subsystems which correspond to internal non-resonanceand resonances, a third subsystem may be found. This external resonant subsystem consistsof the system’s natural frequencies which are resonant with the external forcing frequency,. However, they cannot be included in the internal resonant subsystem. The solution

forms given for the internal resonances can be applied here, provided that the knownexternal frequency, , is used instead of the GCD frequency, or *, and the externalresonant natural frequencies are expressed in terms of . This implies that the KB solutionforms (3a) and (3b) are identical to the EG forms (4a) and (4b), respectively, because = * = . Then, the demonstration of the equivalence of the KB and EG methods forthis subsystem is merely the proof of the equivalence of forms (3a) and (3b). It will beshown in the following subsection that these two forms produce identical approximate


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solutions up to any order. Therefore, by using the procedure described in subsections 2.2and 2.3, the steady state solutions and their stability conditions can be obtained, and theirfirst order equivalence can be proved similarly for the whole system. Details are omittedfor brevity.

2.6. ,

2.6.1. The KB and EG methods

The relationship between the KB and EG methods is illustrated in Figure 1, in whichODE, ABE, T.A. and E.Q. are abbreviations for ordinary differential equation, algebraicequation, time-averaged and equivalence, respectively. Although the equivalence of the KBand EG methods has been proved explicitly only for solution forms (3a) and (4a), it canbe deduced from Figure 1 that another three equivalences are also true: viz., equations(3b) (4b), (3b) (4a) as well as (3a) (4b). This is because the relation between thesolution forms (3a) and (3b), or between the solution forms (4a) and (4b), is merely aco-ordinate transformation, given by

Ai = X i cos i , B i = −X i sin i , X i = A2i + B 2i , (43a)

for equations (3a) and (3b), or

A*i = X *i cos *i , B *i = −X *i sin *i , X *i = A*2i + B *2i , (43b)

Figure 1. The equivalence of the KB and EG methods.


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T 1

A comparison of the EG and CG methods

Solution Governing StabilityMethod Frequency form equation analysis

DifferentialEG Unknown (4a) or (4b) equation Available

Known (3a) or (3b)CG

Algebraic NotUnknown (4a) or (4b) equation available

for equations (4a) and (4b). Therefore, the results obtained by using solution form (3a)(or (4a)) are actually identical to those produced by employing solution form (3b) (or (4b)).Note that the most frequently used equivalence is the last one; i.e., (3a) (4b).

2.6.2. The EG and CG methods

The EG method can be used for a stability analysis, unlike the CG method. However,the solution forms of the CG method can have either a known or an unknown frequency,while that of the EG method can have only an unknown frequency. A comparison of thetwo methods is shown in Table 1. The proof of the equivalence of the two methods is givennext.

First, assume that an unknown frequency is given in the solution form of the CGmethod. Consider an alternative set of equations for solving the steady state solutions fromthe EG approach by using solution form (4a). They are found, by setting X *i = *i = 0in equations (10) and (11), to be

limT

1T

T

0

sin (*i + *i )[(*2i − 2

i )i + f i (i , ̇i )] d = 0, i = 1, 2, . . . , n, (44a)

and

limT

1T

T

0

cos (*i + *i )[(*2i − 2

i )i + f i (i , ̇i )] d = 0, i = 1, 2, . . . , n. (44b)

It should be noted that equations (44a) and (44b) apply not only to non-resonance butalso to internal resonance and combinations of the two cases. Multiplying equation (44a)by sin *i and equation (44b) by cos *i , and adding the resulting equations, leads to

limT

1T

T

0

sin *i [(*2i − 2

i )i + f i (i , ̇i )] d = 0, i = 1, 2, . . . , n, (45a)

because the *i , i = 1, 2, . . . , n, are constants. Similarly,

limT

1T

T

0

cos *i [(*2i − 2

i )i + f i (i , ̇i )] d = 0, i = 1, 2, . . . , n , (45b)


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can be found from equations (44a) and (44b). In fact, the relationships between equations(44a) and (44b) to equations (45a) and (45b) can be represented by the simpletransformation matrix

cos *i sin *i −sin *

i cos *i

. (46)

Therefore, the steady state solutions derived from the EG and CG methods are identicalif an unknown frequency is assumed in the CG’s solution form. The same relation (46)can be obtained if solution form (4b) is used.

Next, suppose that a known frequency is used in the solution form of the CG method.It is easy to see that this case is merely the KB method without the stability analysis, but

expressed in the forms (45a) and (45b), where the unknown frequency, *i , is now replacedby the known frequency, i . The same relationship given by equation (46) can be derivedif i is substituted for *i . Thus, the steady state solutions derived from the CG methodwith a known frequency are identical to those obtained by using the KB method.Therefore, the first order equivalence of the KB and EG methods implies that the EG and

CG methods give equivalent first order steady state solutions if a known frequency is usedin the CG method. The relationship between the EG and CG methods is also shown inFigure 1.

2.6.3. The KB and CG methodsFrom the above discussion of the relationship between the EG and CG methods, it can

be deduced directly that the KB and CG methods produce identical or equivalent firstorder steady state solutions when a known or an unknown frequency is assumed in theCG method. The relationship between the KB and CG methods is also given in Figure 1.

Also indicated in Figure 1 are the relationships derived by Newland [15], whodemonstrated the equivalence of the KB and CG methods between: (1) equations (3a) and(4a) for the amplitude–frequency relation of the non-resonance of a 1-DOF system, and(2) equations (3a) and (3b) for a 1 : 1 external resonance.

3. APPLICATIONS

The KB and EG methods will be employed next to analyze a 3-DOF, eccentric modelof galloping.

3.1. 3-

A continuous multi-span, iced transmission line can be reduced to the 3-DOF, discreteoscillator model [9, 10] illustrated in Figure 2, where the overall centre of mass, G, deviates

Figure 2. The 3-DOF oscillator.


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from the centre of the bare conductor due to ice accretion. The motions can be describedby a dimensionless form

[M ]{q̈ } + [C ]{q̇} + [K ]{q} = {F }, (47)

where a dot superscript indicates a derivative with respect to time, , vector {q}T = ( y z ),in which y and z represent the dynamic displacements in the vertical and horizontaldirections, respectively, while is the rotation about the centre of the bare line. [M ], [C ]and [K ] are the mass, damping the stiffness matrices, respectively. {F } is the non-linearaerodynamic force vector, details of which can be found, together with the other matrices,

in reference [9]. It should be noted that, although equation (47) is derived for an electricaltransmission line, the results given in the following sections can be applied to other weaklynon-linear systems which are described by equation (47).

The first step in investigating the feasibility of galloping is to find whether the initialequilibrium state (IES) of equation (47), given by {q} = 0, is stable. If the IES is unstable,the next step is to find solutions bifurcating from the IES and to determine their stabilities.The stability conditions for the IES can be found from the linearized equation of system(47). For a dynamic analysis, on the other hand, equation (47) needs to be transformedto canonical form (1). This transformation is accomplished by first finding the naturalfrequencies of system (47), which may be obtained from the associated free vibrationsystem [M ]{q̈ } + [K L ]{q} = {0}, where the linear parts of the aerodynamic forces havebeen included in [K L ] to incorporate the effect of wind. Having determined the wind-onnatural frequencies, i , introduce the linear transformation {q} = []{} into equation(47), to obtain

¨ i + 2i i = Ri ( j , ̇ j ), i = 1, 2, 3, (48)

where {()}T = [1 2 3] are the principal co-ordinates. [] is a 3×3 matrix havingcolumn vectors, {i }, found from [[K L ] − 2i [M ]]{i } = {0}. The Ri are elements of vector{R} which consist of only damping and aerodynamic forces that are small ( order)compared to the i . These forces are usually approximated by polynomials up to thirdorder so that closed form solutions can be obtained. Thus,

{R} = [P ]{ 2 3}T − []{̇}, (49)

in which is the angle of attack [9], given by

= d 3

i = 1

(3i i + ri ̇i ). (50)

Constant matrices [P ] and [] and constants ri can be found in reference [9].The KB and EG methods described in the previous section will be used to evaluate

two representative internal resonant cases, i.e., 1:3 :0 and 1 :1: 1, where 0 indicates anon-resonant mode.

3.2. 1:3:0 —

This case describes not only the 1:3:0 but also the 3:1:0, 1:0:3, 3:0:1, 0:1:3 and 0:3:1resonances. A 1:3:0 resonance implies that the ratio of the first two natural frequencies,1 and 2, is close to 1/3. Specifically, the difference 31 − 2 is of order . Both the KBand EG methods and the technique developed in references [9, 10] have been used to derive

explicit solutions for this resonance. It has been shown that the KB method finallyproduces a second and a seventh degree polynomial (details will be given later), while theEG approach ultimately yields a 13th degree polynomial. Thus, the solutions derived by


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utilizing the EG approach are more sensitive to small variations in the parameter values,so that the EG approach is not recommended for this case. Indeed it has been found forsome parameter values that although the KB method produces accurate approximations,compared to numerically integrated results, the EG approach is intractable. Therefore,only the KB procedure is detailed here. Apply equation (23) to let

= 14 (1 + 2), 1 = 2 − 21 , 2 = (3)

2 − 22 , (51)

so that equation (48) can be rewritten as

¨ 1 + 21 = R1 + 11, ¨ 2 + (3)22 = R2 + 22, 3 + 23 3 = R3. (52)

By combining forms (3a) and (25), the first order approximation of the motion can bewritten as

1 = X 1() cos [ + 1()]X 1() cos 1(),

2 = X 2() cos [3 + 2()]X 2() cos 2(),

3 = X 3() cos [3 + 3()]X 3() cos 3(), (53)

where the X i and i , respectively, represent the i th amplitude and phase. Next, by followingthe procedure described in subsections 2.2 and 2.3, equation (53) can be applied toequation (52) to produce the governing bifurcation equations

2X 1 = −X 1[11 − r1( p11 + p13W 1)] + p13(C 1 cos − C 2 sin )X 21 X 2,

2X 2 = −X 2[22 − r2( p21 + p23W 2)] + 19 p23(D1 cos − D2 sin )X 31 ,

2 = − + 1

[331( p11 + p13W 1) − 13 32( p21 + p23W 2)]

−3 p13(C 2 cos + C 1 sin ) X 2X 1+ 13 p23(D2 cos + D1 sin ) X 1X 2X 21 ,2X 3 = −X 3[33 − r3( p31 + p33W 3)] and 3 = 3 −

3323 ( p

31 + p33W 3), (54)

where

= 2 − 31, =14 (1 + 2), =

13

[3(22 + 21 ) − 22 ],

C 1 = 3r2(231 − r21 2) − 2r13132, C 2 = −1

[32(231 − r21 2) − 6r1r2312],

D1 = r1(3231 − r21 2), D2 =31

(231 − 3r21 2). (55)

is the phase difference between X 2 and X 1, which are the amplitude components of theperiodic motion having frequency . On the other hand, W 1 and W 2 are described by

W i = W − (23i + r2i 2)X 2i (i = 1, 3),

W 2 = W − (232 + 9r22 2)X 22 ,

W = 2(232 + 9r22 2)X 22 + j =1,3 (23 j + r2 j 2 j )X 2 j . (56)


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The steady states which bifurcate from the IES can be found by setting X i = = 0,i = 1, 2, 3 in equation (54). Note, however, that the equation describing X 3 is decoupledfrom the first three equations so that only X 1, X 2 and need be solved simultaneously fromthe three equations given by X 1 = X 2 = =0. The resulting three, non-linear algebraicequations are coupled so that no general method exists to find analytical expressions and,

hence, numerical iterations are usually employed. A specific analytical procedure has beenproposed in references [9, 10], on the other hand, to find explicit solutions in order tosave substantial computational time. This saving is achieved by first deducing a polynomialthe coefficients of which are expressed explicitly in terms of known system parameters.

Then the X i and can be found from analytical expressions involving the roots of thepolynomial.

The particular polynomial can be found by eliminating two of the three variables, X 1,X 2 and , from the three equations obtained by setting X 1 = X 2 = = 0. This is achievedby first employing the first two equations (given by X 1 = X 2 = 0) to express cos and sin as

cos =

A

D(231 − 3r21 2)31X 21 + 3

B

D[23132 − (r132 − 4r231)r12]X 22 ,

sin = AD(3231 − r21 2)r1X 21 + 2B D[31(r132 − r231) + r21 r22]X 22 , (57)where

A = 11 + r1[(1−4)Y +2( 1−2)Z ],

B = 22 + 2r2[2(1−2)Y + ( 1 − 4)Z ], D = −(r132 + 3r231)Y 2X 1X 2. (58)

11 and

22 are detailed in Table 2. = 0 for periodic solutions and 1 for quasi-periodicsolutions. Consequently, the 1:3:0 resonance may have a family of periodic solutionsconsisting of X i 0, i =1, 2, X 3 = 0, and quasi-periodic motions lying on a twodimensional torus in which X i 0, i = 1, 2, 3.

Next, use sin2 +cos2 = 1 and the third equation, given by = 0, to yield the seconddegree polynomial

g3(Z ) − g6(Z )Z g3 − g6Z = 0 (59)

and the seventh degree polynomial

( g1 − g4)[( g3 g4 − g1 g6Z )2 − ( g3 g5 − g2 g6)( g2 g4 − g1 g5Z )]

+( g2 − g5Z )[( g2 − g5Z )( g3 g5 − g2 g6) − ( g3 − g6Z )( g3 g4 − g1 g6Z )]

+Z ( g3 − g6Z )[( g3 − g6Z )2 − 2( g1 − g4)( g3 g5 − g2 g6)] = 0, (60)

where

Y = (231 + r21 2)X 21 , Z = (232 + 9r22 2)X 22 . (61)


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T 2

The coefficients, Ai and B i , used for the 1 : 3 : 0 resonance

A0 = 1r1s2(1r1 + 2 prr2)2

A1 = r1s2(212r1 + 3 prr2)2 − s24 A2 = 2r1s2(2r1 + 1 prr2)2

+32r2[3(1r1 + 2 prr2)s3 +3r23(3r1 + 212 prr2)s3

−(3131 − 2 pr32)s4] −(3331 − 212 pr32)s4]

A3 = 31r2[3(2r1 + 1 prr2)s3 A4 = [1r1(211 + pr22/3)+ 2r2 pr11]s2

−(3231 − 1 pr32)s4]

A5 = [2r1(211 + pr22/3)+ 1r2 pr11]s2 A6 = 3[2r122 + 1r2(311 + 2 pr22)]s3

+3[1r122 + 2r2(311 + 2 pr22)]s3 −[41r22/ p13 + (3231 − 1 pr32)22]

+[42r22/ p13 + (3131 − 2 pr32)22]

A7 = 11(11 + pr22/3)s2 A8 = 22(311 + pr22)s3 − 422s4/3 p13

B 0 = −1r1 p23s2/12

B 1 = −2r1 p23s2/9 B 2 = 32r2 p13s1

(31r1 p13 − 2r2 p23)s3 +(32r1 p13 − 1r2 p23/3)s3

+(31 p1331 − 2 p2332/3)s4/ +(32 p1331 − 1 p2332/3)s4/

B 3 = 271r2 p13s1 B 4 = − p2311s2/9

B 5 = (3 p1311 − p2322/3)s3/ + s4 B 6 = 9 p1322s1

Terms used in the above expressions

1 = 1 − 4 2 =2(1−2) 3 = 21 +

22

s1 = 231 + r21 2 ti = −ri pi 3, i = 1, 2

s2 = 232 + 9r22 2 t3 = 31 p13/

s3 = 3132 − 3r1r22 t4 = 32 p23/2

s4 = −(r132 + 3r231) p4 = p23/3 p13

ii = [ii − ri ( pi 1 + 2 pi 333)]/ p13, i = 1, 2 = − + (3 p1131 − p2132/3)/33 = ( p31 − 33/r3)/ p33 +2(3 p1331 − p2332/3)33/

The gi (Z ), i = 1, 2, . . . , 6, are polynomials of Z given by

g1(Z ) = 1A0

(A1Z + A4),

gi (Z ) = 1

A0 (Ai Z 2

+ Ai + 3Z + Ai + 5), i = 2, 3,

gi (Z ) = 1

B 0(B i − 3Z + B i ), i = 4, 5, 6. (62)

The constant coefficients Ai , i = 0, 1, . . . , 8, and B i , i = 0, 1, . . . , 6, are presented moreconveniently in Table 2.


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Having found Z from equations (59) and (60) (by using, for example, the IMSL programRPOLY [16]), Y is determined from

Y =( g2 − g5Z )( g3 g5 − g2 g6) − ( g3 − g6Z )( g3 g4 − g1 g6Z )

( g3 − g6Z )2 − ( g1 − g4)( g3 g5 − g2 g6) . (63)

Both Y and Z must be positive. The corresponding X 3 is obtained from

X 23 =

233 + r23 23 1r3 p33 (33 − r3 p31) − 2 (Y + Z ). (64)The first order approximation of the periodic and quasi-periodic solutions is described by

i = X i cos ¯ i , i = 1, 2, 3, (65)

where X 3 is given by equation (64). The frequency, ¯ 3, for the quasi-periodic solution isgiven by the fifth equation of (54) as

¯ 3 = 3 − 3323

( p31 + p33W 3), (66)

whereas

3 ¯ 1 = ¯ 2 = 3[ − 1 − 31( p11 + p13W 1) + p13(C 2 cos + C 1 sin )X 1X 2] (67)

for both the periodic and quasi-periodic solutions.

3.3. 1:1:1 —

In this case the ratio of the natural frequencies, 1:2:3, is 1:1:1 and the KB methodcan still be adopted. However, it is very difficult, if not impossible, to find explicit steady

state solutions by using the KB method. Therefore, the EG method will be used for thiscase and the alternative form (4b) will be applied to simplify the solution procedure. Thisapproach requires equation (50) to be rewritten as

¨ i + (*)2i = Ri + (*2 − 2i )i , i = 1, 2, 3, (68)

where * is the unknown frequency of the assumed periodic motion. Next, the first orderapproximation is assumed to take the form

i = Ai () cos * + B i () sin *, i = 1, 2, 3, (69)

so that the averaged equations are given by

A i =−limT

1T

T

0

1*

[Ri + (*2 − 2i )i ] sin * d,

B i = limT

1T

T

0

1*

[Ri +(*2 − 2i )i ] cos * d, i = 1, 2, 3. (70)


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With the aid of equations (4)–(6), as well as equations (68) and (70), the following sixgoverning equations can be found:

2*A i = −* 3

j = 1

ij A j − (*2 − 2i )B i − F [ pi 1 + pi 3(E 2 + F 2)],

2*B i = −* 3

j = 1

ij B j + (*2 − 2i )Ai + E [ pi 1 + pi 3(E 2 + F 2)], i = 1, 2, 3, (71)

where

E = 3

j = 1

(3 j A j + *r j B j ), F = 3

j = 1

(3 j B j − *r j A j ). (72)

Periodic solutions can be derived from the non-linear equations obtained by settingA i = B i = 0, i = 1, 2, 3, in equations (71). The resulting six equations need to be solvedsimultaneously to find the periodic solutions. Frequency * is an appropriate candidate

for the variable of the polynomial which most reduces computational effort. Afteralgebraic manipulation, a fifth degree polynomial results, which takes the general form:

[r1*2 − (1131 + r121 )](J 2 + K 2) − 1232(L2 + G2) − 1333(M 2 + N 2)

+[r2*2 − (1132 + 1231 + r221 )](LJ + GK ) + [r3*2

−(1133 + 1331 + r321 )](NJ + MK ) − [*2(33 + r113 − r311)

−3321 ]

(GJ − LK )*

− [*2(32 + r112 − r211) − 3221 ] (MJ − NK )

*

−(1233 + 1332)(GM + LN ) − 1*

(r213 − r312)(GN − LM ) =0, (73)

where the G i , K i , J i and L i , as well as G , K , L , J , M and N , are all polynomials of *.Their detailed expressions are given in Table 3. Having found positive values of * fromequation (73), the amplitude components of the periodic motion, A2i + B 2i , i =1, 2, 3, aredetermined from

A21 + B 21 =(J 2 + K 2)2

P 2 + Q2 (E 2 + F 2), A22 + B 22 =L2 + G2J 2 + K 2 (A21 + B 21 ),

A23 + B 23 =M 2 + N 2J 2 + K 2 (A21 + B 21 ), (74)where E

2

+ F 2

, P and Q are given by

E 2 + F 2 = − p11 p13

+ 2*

p13Q [11(J 2 + K 2) − 12(GK + LJ ) − 13(MK + NJ )], (75)

P = 31(J 2 + K 2) − 32(LJ + GK ) − 33(NJ + MK )

−*[r2(GJ − LK ) + r3(MJ − NK )], (76)


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T 3

Polynomial functions used for the 1 :1 : 1 resonance

DegreePolynomial in * Expression

G(*) 4 G1J 2 − G3J 3 − K 1L2 + K 3L1K (*) 4 G2J 2 − G3J 2 − K 2L3 + K 3L2

M (*) 4 G2J 1 − G1J 2 − K 2L1 + K 1L2

L(*) 3 G1L3 − G3L1 − K 3J 1 + K 1J 3

J (*) 3 G2L3 − G3L2 + K 2J 3 − K 3J 2

N (*) 3 G2L1 − G1L2 + K 2J 2 − K 1J 1

Terms used in the above expressions

Gi (*), i = 1, 2, 3 1 *[( p132i − p231i ) + ri ( p11 p23 − p21 p13)]

J i (*), i = 1, 2, 3 1 *[( p133i − p331i ) + ri ( p11 p33 − p31 p13)]

K i (*) i = 1, 2 2 (−1)i (*2 − 2i ) p(3− i )3 + 3i ( p11 p23 − p21 p13)

K 3 0 33( p11 p23 − p21 p13)

Li (*), i = 1, 3 2 (−1)(i +1)/2(*2 − 2i ) p(4− i )3 + 3i ( p11 p33 − p31 p13)

L2 0 32( p11 p33 − p31 p13)

and

Q = *[r1(J 2 + K 2) − r2(LJ + GK ) − r3(NJ + MK )] + 32(GJ − LK ) + 33(MJ − NK ),

(77)

respectively.The first order approximation of the periodic solution can be obtained by assuming

B 1 = 0 because the system is autonomous. Therefore,

1 = A1 cos *, 2 = A2 cos * + B 2 sin *, 3 = A3 cos * + B 3 sin *, (78)

where

A1 = (J 2 + K 2) E 2 + F 2P 2 + Q21/2

,

A2 = −A1 GH + LJ J 2 + H 2 , B 2 = −A1 GJ − LH J 2 + H 2 ,A3 = −A1

MH + NJ

J 2 + H 2

, B 3 = −A1

MJ − NH

J 2 + H 2

. (79)

3.4.

Both periodic and quasi-periodic solutions are feasible for the 1 : 3 : 0 resonance, but

only a periodic solution is possible for the 1:1:1 resonance. Stability conditions can bedetermined straightforwardly from the Jacobian matrices of the time-averaged equations(54) and (71) for the 1:3:0 and 1:1:1 cases, respectively. A periodic or quasi-periodic


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Figure 3. (a) The angle section. (b) The D-section.

solution is stable if all the real parts of the non-zero eigenvalues of the Jacobian matrixare negative. It should be noted, however, that the 6 × 6 Jacobian matrix of the 1 :1: 1

resonance has one zero eigenvalue [17], because it is evaluated on the periodic solutionwhich actually has only five independent variables.

3.5.

3.5.1. Example 1: bluff angle section

The angle section, which is shown in Figure 3(a), is actually a 2 DOF, co-centric system.This particular example was studied analytically by Blevins and Iwan [12] for thenon-resonant case as well as for the 1:1 and 1:3 resonances. However, explicit solutions

and stability conditions were given only for the non-resonance. Dynamic responses in theresonances were obtained numerically from the time-averaged equations. Later, the 1:1resonance was re-examined and explicit solutions were derived by using the EG method[8]. Recently, a typical angle section model was experimentally considered in detail by

Modi et al . [18]. This model had 1 × 1 in (2·54 × 2·54 cm) and 3 × 3 in (7·62 × 7·62 cm)cross-sectional dimensions with uniform leg thicknesses of 1/16 in (1·6 cm) and 1/2 in(1·27 cm), respectively. The aerodynamic lift, drag and moment coefficients of the modelwere measured from wind tunnel tests and dynamic responses of the model were alsoobtained, especially for the case of / y = 2·92 (close to 1: 3 resonance). However, unlikereference [12] and this paper, in which we are particularly interested in the dynamicbehavior of the model with respect to the variation of the frequency ratio ( / y ) nearthe 1:3 resonance, Modi et al . [18] investigated the dynamics of the model with respectto the wind speed (U ) at a fixed frequency ratio (2·92). Moreover, the results presented

in reference [18] were obtained from a one-degree-of-freedom (either plunge or torsion)model under appropriate wind speeds. Therefore, the results are actually obtained fornon-resonant cases, and the word ‘‘resonance’’ used in reference [18] (e.g., plungingresonance or torsional resonance) means that the corresponding maximum plunge

(torsion) response is with respect to a particular wind speed, rather than the usualdefinition [8–10, 12] indicating the relationship between simultaneous plunge andtorsional motions. Although the main purpose of this paper is to compare the two


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approximation methods in resonant cases, and thus only the results regarding the variationof / y are presented, the behavior of the model under different wind speeds is veryimportant and should be studied. In fact, an interesting phenomenon can be observed fromthe results presented in reference [18]: galloping is only associated with certain values of wind speeds in resonant cases. This observation has indeed been revealed analytically in

our early results [9, 10].The form of the explicit solution for the 1:3:0 resonance given in the previous

subsections can also be applied for the 1:3 situation. The structural and aerodynamicproperties of the bluff angle section are well documented in reference [12], and they are

used to yield

[M ] = diag [1 1 1], [K ]= diag [1 1 2],

[C ] = 2 diag [0·004125 1 0·00513], = / y, (80)

and

{F } =

0·06024(−0·656 +7·833)0·0

0·45550(−0·105 +9·34

3

)

. (81)

It should be noted that the horizontal component of the aerodynamic forces is zero,which results, in turn, in a zero horizontal response so that the system really has 2 DOF.The normalized wind speed U y = U z / yd , on the other hand, equals 4·5.

The IES, {q} = {0}, was found to be stable for the specified parameters over the rangeof structural frequency ratios, 1·1114 1·2815. A dynamic analysis was performedfor the initially unstable regions. A stable periodic solution was obtained by using the

formulae derived for the 1:3:0 resonance over the range 2·8 3·2. The resultingvertical, A y , and torsional, A , amplitudes of the motion and that obtained by usingnumerical integration are shown in Figure 4, where the results given by Blevins and Iwan[12] are also indicated. It can be seen from this figure that the vertical amplitude is quitesmooth near = 3 but the torsional amplitude has a peak at = 3. Also, it can be noted

that, for the vertical motion, the present results and those of Blevins agree with thenumerically integrated data. For the torsional motion, however, our results are in excellentagreement with the numerically integrated solution for a large interval of / y(3·0 0·2), but the results of Blevins cover a very narrow interval (approximately for / y =3·0 0·015, while the results outside this region are given by the non-resonantanalysis).

3.5.2. Example 3: D-section

A D-section transmission line, which is depicted in Figure 3(b), has been analyzedpreviously in order to consider a 1:1:1 galloping motion having strong coupling betweenthe vertical, horizontal and torsional components [9, 10]. Although a detailed study hasbeen published, the dynamic responses were limited to fixed parameter values, althoughthe steady wind speed was changed. The main interest here, however, lies in the dynamic

behavior near resonance, so that this example will be reassessed.The dimensionless data are given by

[M ] = 1·000000·00000−0·00520 0·000001·03000−0·01780 −0·00520−0·017808·00000


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[K ] =1·000000·000000·00000 0·00008210·00000 0·000000·0000022 ,[C ] = 2 diag [0·00150 0·001501 0·544002], 1 = z / y , 2 = / y , (82)

and

{F } =0·00048(−0·600 + 0·3702 +0·2003)

0·00048(−0·330 − 0·7802 +0·1453)0·00006(−0·450 + 0·0962 +0·2103). (83)

The non-dimensional wind speed, U y , is chosen to be 25·8, which is equivalent to a windspeed, U z , of 9 m/s. The initial stability analysis predicted the D-section to be initiallyunstable, at least over the range 0·01, 2 2·0 of structural frequency ratios. A stableperiodic solution was obtained by using the formulae derived for the 1: 1: 1 resonance overthe subrange 0·91, 2 1·1.

Predictions obtained virtually instantaneously from the analytical formulae given earlier

are presented in the left column of Figure 5, where A y , Az and A represent the vertical,horizontal and torsional amplitudes, respectively. They are in excellent agreement with the

Figure 4. The dimensionless response for the 1 : 3 : 0 resonance; U y =4·5. , Blevins’ numerical integration[12]; , our numerical integration; - - - -, Blevins’ asymptotic solution [12]; ——, our asymptotic solution.


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Figure 5. The dimensionless response for the 1 : 1 : 1 resonance; U y = 25·8. (a) vertical amplitude; (b) horizontalamplitude; (c) torsional amplitude. The left subfigures are obtained from the analytical formulation; the right

subfigures represent the numerical results.

reference numerically integrated data, shown in the right column of this figure. However,

the computation of each integrated point consumed hours of CPU time on a SUN workstation.

4. CONCLUSIONS

Two seemingly different, time averaging procedures are shown analytically to produceidentical first order, approximate solutions and stability conditions for an n-DOFnon-linear, discrete system. It is suggested that, depending upon the non-resonant orresonant case under consideration, the application of one or the other procedure can be

more advantageous. The two procedures are used to analyze galloping for which periodicand quasi-periodic solutions are derived explicitly.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support of the Canadian ElectricalAssociation (CEA Project No. 321 T 672) through the auspices of Manitoba Hydro. Partial


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support was also received from the Natural Sciences and Engineering Research Councilof Canada.

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The Krylov–Bogoliubov and Galerkin Methods for Non-linear Oscillations