P

NASA Contractor Report 187632

ICASE Report No. 91-68

ICASE

RESONANT FREQUENCY CALCULATIONS USING A

HYBRID PERTURBATION-GALERKIN TECHNIQUE

James F. Geer

Carl M. Andersen

Contract No. NAS1-18605

September 1991

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

Nalional Aeron_ulic,_ and

Space Adminislralion

Langley Resesreh Center

Hampton, Virginia 23665-5225

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RESONANT FREQUENCY CALCULATIONS USING A

HYBRID PERTURBATION-GALERKIN TECHNIQUE

James F. Geer 1

Department of Systems Science

Watson School of Engineering, Applied Science and Technology

State University of New York

Binghamton, NY 13902

Carl M. Andersen

Department of Mathematics

College of William and Mary in Virginia

Williamsburg, VA 23185

ABSTRACT

A two step hybrid perturbation-Galerkin technique is applied to the problem of determining the resonant frequencies of one- or several-degree(s)-of-freedom nonlinear systems

involving a parameter. In step one, the Lindstedt-Poincar@ method is used to determine

perturbation solutions which are formally valid about one or more special values of the

parameter (e.g. for small or large values of the parameter). In step two, a subset of the

perturbation coordinate functions determined in step one is used in a Calerkin type approximation. The technique is illustrated for several one-degree-of-freedom systems, including

the Duffing and van der Pol oscillators, as well as for the compound pendulum. For all of the

examples considered, it is shown that the frequencies obtained by the hybrid technique using

only a few terms from the perturbation solutions are significantly more accurate than the

perturbation results on which they are based, and they compare very well with frequencies

obtained by purely numerical methods.

1This research was partially supported by the National Aeronautics and Space Administration under

NASA Contract No. NAS1-18605 while the first author was in residence at the Institute for Computer

Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

1. INTRODUCTION

In this paper we presentand discussa two-stephybrid perturbation-Galerkin technique

for the computationof the resonantfrequenciesof nonlinearoscillating systemswith a finite

number of degreesof freedominvolving a scalarparameter e. In previouspapers,we have

developedand applied different versionsof the hybrid techniqueto severalclassesof twopoint boundary-valueproblemsfor ordinary differential equations[2,6,7],to someboundary

value problems for elliptic partial differential equations[8], and to someintegral equations

of the first kind which arisein slenderbody theory [5]. Foreachof theseclassesof problems,

the method has yielded resultswhich aretypically far moreaccuratethan the perturbation

solutionson which they are based,and often provide at leastreasonablesolutionsevenfor

valuesof the expansionparameterfor which the perturbation solution is meaningless.

The ideaof exploiting perturbation expansionsin conjunctionwith Galerkin or variational

techniqueswas introduced by Noor and Petersin 1979[12]and developedby Noor and his

collaboratorsin a number of papers (seee.g. [11],as well as many other referencescited in

[5]). Noor's "reducedbasismethod" is acombinationof finite elementor other discretization

techniques,perturbation expansions,and Galerkin (or variational) techniques.

In generalterms, our two-stephybrid techniqueconsistsof computing a few terms in the

perturbation expansionof the solution about oneor morevaluesof the parametereand then

usinga subsetof thesefunctions, with new amplitudes,in a Galerkin type approximation.

This mannerof combiningthe perturbation and Galerkin approaches(which wewill describe

in more detail below) seemsto overcomesomeof the drawbacksassociatedwith eachof the

methodswhenthey areappliedby themselves,while preservingsomeof the goodfeaturesof

eachone. In particular, the perturbation method hasat leasttwo significantdrawbacks.The

first is that, for most practical problems,only a few terms in a perturbation expansioncan

becomputedbecauseof the rapidly increasingamountof "labor" that is requiredto compute

eachadditional term. Secondly,the expansionparametermustusually berestrictedto values

which lie closeto the point about which the expansionwas constructed,in order to obtain

approximationsof acceptableaccuracy.A drawbackof the Galerkin method is the problem,

from a practical point of view, of selectinga small numberof "good" basisfunctions. As we

shall demonstratebelow,the functions determinedby the perturbation method appearto be

very effectivebasisfunctions andhenceour methodovercomesthe main drawbackassociated

with the Galerkin method. Also, the new amplitudes determined by the method produce

approximationswhich aretypically muchmore accuratethan the perturbation solutionson

which they are based.

In the following sectionswe describeour method in the context of the problem of determining the resonantfrequenciesof nonlinear oscillating systems. For simplicity and in
order to illustrate explicitly some of the salient features of the method, we consider first

systems with only one degree of freedom. The method is described in detail for such systems and then applied to several model one-degree-of-freedom problems. This allows us to

obtain several explicit results and expressions, which provide insight into the application of

the method to more complicated systems. We then describe the method in the context of

a general (conservative) system with a finite number of degrees of freedom which oscillates

about a stable equilibrium state. The method is then applied to the classical problem of a

compound pendulum. Observations about the method are presented in the final section.

2. ONE-DEGREE-OF-FREEDOM SYSTEMS

We consider first the problem determining the resonant frequency u of a one-degree-offreedom nonlinear oscillator which we write in nondimensional form as

v2u"+ u + (e/a) f(au, avu', c) = O,

u(O)= 1, _'(0) = O,

u(x+2_')=u(x) for all x>O,

_o2"f(au, avu ', c)u' dx = O.

(la)

(l b)

(lc)

(2)

In Eq (la), a is the maximum amplitude of the response, c is a "small" parameter, and

f is a specified nonlinear function of its arguments. (Here the dimensional response y of the

oscillator has been nondimensionalized by defining u = y/a. Also, the primes in Eqs (1)-(2)

denote differentiation with respect to the nondimensional time x = wt, where t is time,

w = 2 _r/T is the (unknown) resonant frequency of the oscillator, and T is the (unknown)

period of the oscillation. Then v = w/wo, where w0 is the linear natural frequency of the

oscillator.) Conditions (lb) and (lc) insure that u has its maximum amplitude at x = 0 and

is 2_r periodic. Condition (2) is a necessary and sufficient condition for Eq (1) to have a 2_r

periodic solution. We note that in the special case when f does not depend explicitly on

u', i.e. f = f(a u, c), then Eq (la) represents a conservative system. Then condition (2) is

satisfied automatically and Eqs (1) can be used to express v as

(3)

F(u,e) = 1 - u 2 + -- f(ax, e)dx,

a

where c is defined by the requirements that c > 0 and F(-c, e) = 0. Thus, for this special

case, the computation of v reduces to a quadrature, which will be useful for comparison

purposes with our perturbation and hybrid results (see [9]).
3. THE HYBRID PERTURBATION-GALERKIN TECHNIQUE

We now apply a slightly modified version of a two step hybrid perturbation-Galerkin

technique which has been introduced and discussed in a series of papers [2,5-8]. In step one

of the method, we use the Lindstedt-Poincar6 method [10] to find approximate solutions for

u, u, and a in the form

N-1

= E 'UJ(X) £j + O(_'N)'

j=O

N-1

. = 1+ Z + (4)

S=I

N-1

a = E as ¢j + o(eN),

S=o

which are formally valid as _ ---. 0. In (4), the {uj(x)} are the unknown perturbation coordinate functions, while the constants {uj} and {as} , which determine the perturbation

approximations to the frequency and initial amplitude, respectively, are also unknown. To

determine these quantities, we substitute the expansions (4) into Eqs (1), expand the resulting equations in power series in _ about _ = 0, and then equate the coefficients of like powers

of e on each side of these expressions. In this manner, we obtain a sequence of problems to

solve for each of the unknowns. In particular, we find easily that

u0(x) = cos(x), (5)

while each of the functions us(x ) with j > 1 satisfies a problem of the form

u "s + u s=2u s cos(x) + gs, (6)

with us(0 ) = u_(0) = 0 and uj(x + 21r) = uj(x). Here gj depends on uk, uk, and ak with

k < j. Thus, in order for uj to be 21r periodic, the right side of (6) must be orthogonal to

both cos(x) and sin(x), since otherwise "secular" terms proportional to x sin(x) and x cos(x)

will appear in the solution for us(x ) . These conditions yield the relations

"S - 2_r gs cos(x)dx, ]0gj sin(x)dx = O, (7)

which are two equations for the two unknowns us and aj-1. In particular, for j = 1 Eqs (7)

become

1 fo 2'_ ul - 27r ao f(ao cos(x),-ao sin(x), O) cos(x)dx,

O _

fo2"f(ao cos(x),-ao sin(x), O) sin(x)dx.

(8)

3
The second equation in (8) is an equation for a0, while the first equation expresses vl as a

function of a0.

In the examples which follow, we shall exhibit several explicit expressions for the various terms in the perturbation expansions (4), as well as comment on the accuracy of the

approximations computed using them.

In step two of the hybrid method, we use a subset of the perturbation coordinate functions

{u j} determined in step one as both trial and test functions in a Galerkin type approximation.

In particular, we seek new approximate solutions _, _, and 5 for u, v, and a, respectively,

with

N-1

= u0(x)+ 6j (9)

j=l

In (9), the functions {uj} are the perturbation coordinate functions determined by the

Lindstedt-Poincar_ method in step one, while the {6j} represent new "amplitudes" of these

coordinate functions. (We note that (9) satisfies conditions (lb) and (lc) for any choice of

the {6j}.) To determine the amplitudes {hi}, as well as the quantities _ and 5, we substitute

(9) into (la) and require that the residual is orthogonal to each Uk, 0 < k < N - 1. Also,

we require that (2) is satisfied when u is replaced by _. Thus we obtain the conditions

2_['k2_" + _i + (e/5)f(au, a'_',e)]ukdx = O, O<_k< N-1,

(10)

2_f(a u, e) _t' dt =

a lJ _v_ O.

Equations (10) are a system of N+I equations to determine the N+I unknowns 51,..., 6N-l,

_, and _. Although these equations must, in general, be solved numerically, we note that

the solution to this system is a point in (N + 1)-dimensional space, where N is reasonably

small, while the solution to Eq (1) is a continuous function. Also, for small values of e, it is

reasonable to assume that the unknown quantities in (10) are "close to" the corresponding

values in the perturbation solution (4) (e.g., 5j _ eJ). Hence, beginning with small values

of e and then proceeding to larger values of e, good starting values are available for the

unknown quantities in (10), which can be used with an iterative method of solution, such as

Newton's method.

If we set N = 1 in (9), we find that _ = Uo(X) = cos(x), while Eqs (10) reduce to

£ _021r _2 = 1 + -- f(5 cos(x),-5 P sin(x), e) cos(x)dx,

7ra

f02"f(5 cos(x),-5 _ sin(x), e) sin(x)dx = O.

(11)
Equations (11) are a system of two nonlinear equations for the two unknowns _ and _.

In particular, if we examinethe solution to theseequationsfor small valuesof e and let

P = 1+ ffl ¢+ O(e 2) and _ = _0 + O(e), we find from (11) that the equations for Pl and _0

are identical to Eqs (8) for _1 and a0. Thus, for small values of e, our hybrid solution (with

N = 1) reproduces the perturbation approximations for _, and a, at least to within terms

which are O(e 2) and O(e), respectively.

In the special case when f is independent of u _, we see that the second equation in (11)

is satisfied for any choice of _ and hence _ is an arbitrary parameter. Then the first equation

in (11) yields the explicit expression

= [1 + _c_ f02 f(_ cos(x), e)cos(x)dx] 1/_ (12)

In the next section we shall comment on the precise form of Eqs (10) for several examples,

as well as comment on the accuracy of the hybrid approximations generated by their solution.

4. EXAMPLES - ONE-DEGREE-OF-FREEDOM SYSTEMS

We now illustrate some of the basic features of the hybrid solutions outlined above with

examples of several one-degree-of-freedom systems. The small parameter e which appears

in the nondimensional version of the oscillator Eq (la) can have several different physical

interpretations. The examples which follow illustrate some of these different interpretations

and hence indicate some of the classes of problems for which we feel the hybrid method

will be especially useful. We will discuss these classes of problems, as well as some possible

variations of the basic method, more fully in the discussion section at the end of the paper.

During Oscillator

As our first example, we consider the Duffing oscillator which, in dimensional form, can

be written as

/_+_0_y +_y3 = 0, y(0) = a, y(0) = 0, (13)

where w0 is the (linearized) natural frequency of the oscillator, a is a specified parameter,

and the dots denote differentiation with respect to time t. We let T be the (unknown) period

of the oscillation and then define w = 2 _r/T, _, = W/wo , x = w t, and u = y/a. With these

definitions, Eq (13) can be written in the form of Eq (la) with f = u 3, i.e.

v2u '' + u + cua=O, u(O)=l, u'(O)=O, e=c_a2/W2o . (14)

Thus, in this example, c can be interpreted as a measure of the amount of nonlinearity in

the restoring force in the system.

5
For this casecondition (2) is satisfied automatically. Thus, the initial amplitude a is

arbitrary and hence can be incorporated into our expansion parameter _. The LindstedtPoincar6 method yields

= cos(x),

u, = -[cos(x) - cos(3x)]/32,

u2 = [23 cos(x) - 24 cos(3x) + cos(5x)]/1024,

(15)

3 21 e2 + 81 e3 6549 e4

v = 1 + _e - 25----6 _ 262144

37737 936183

_5 _ + O(J).

+ 2097152 67108864

The hybrid approximation _ is obtained from Eqs (10) which, in general, will be cubically

nonlinear in the amplitudes {6j}. In particular, for N = 1, Eq (11) yields the solution

3 e ] 1/2

= [1 , (16)

while for N = 2, Eqs (10) yield the relations

= [1 + 3e/4 - 3e51/128 + 3e5_/2048] '/2,

(17)

61 -_ + 3_6,/4 - 9 c6_/512 + 23 _ 5_/16384 = 0.

The first equation in (17) expresses _ as a function of 61 and c, while the second equation

is a (cubic) equation for 61 as a function of e. For small values of e, we see that (16) agrees

with (15) to within terms which are O(e2), while the solution for _ from (17) agrees with

(15) to within terms which are O(e3).

In Figs 1 and 2, we have plotted several approximations to v as a function of the parameter

e. In particular, we have plotted perturbation approximations (denoted by P[N]), where

N, the number of terms in expansion (15), ranges from 1 to 6; hybrid approximations

(denoted by H[N]) determined from (16) and (17) with N = 1 and N = 2, respectively;

and numerical solutions obtained by evaluating the integral representation (3) for v. As the

figures illustrate, the perturbation results (15) are useless for values of c above 1 (the radius

of convergence of the perturbation series), while the one and two term hybrid results are in

excellent agreement with the numerical results, even for values of c as large as 500.
Simple Pendulum

The equation of motion of a simple pendulum canbeexpressedas

+ _02 sin(O)= 0, 0(0) = a, t_(0) = 0, (18)

with w_ = g/L, where g is the acceleration due to gravity and L is the length of the pendulum.

Here 0(t) is the angle the pendulum makes with the vertical. Using the same definitions of

T, w, u, and x as in the first example and letting u = 8/a, we find that (18) can be written

in the form of (la) as

u2u '' + uW(1/e)[sin(eu)-eu]=O, e-=a. (19)

Thus, in this case, e can be interpreted simply as the maximum amplitude of the oscillation.

(Condition (2) is again satisfied automatically and hence the initial amplitude is arbitrary.)

Also, since the nonlinear term in (19) can be expanded for small values of e as a power series

in e2, we see that the series in (4) involve only even powers of e. In particular, using the

symbolic manipulation system Mathematica [15], the Lindstedt-Poincar_ method yields the

results

UO = COS(X),

= [cos(x) - ¢os(ax)]/192,

= [17 cos(x)- 20 cos(3x)+ 3 cos(Sx)]/61440, (20)

1 e2 + 1 e4 23 e6 2519 e8+ O(el0).

v = 1- 1-6 _ 737280 1321205760

This expansion converges for [e[ < r.

The hybrid approximations _ and _ are obtained from Eqs (10) with (e/a) f replaced by

[sin(cu)- eu]/e. In particular, for N = 1, Eq (11) yields

" 1 r2_r

= [7-_r J0 sin(e cos(x))cos(x)dx] '/2

(21)

= 1-e2/16+e4/1536+O(e6), as e---+0,

which agrees with (20) to within terms which are O(ea). In a similar manner, when the

solutions to Eqs (10) with N = 2 are expanded for small values of e, we find that our

expression for _ agrees with (20) to within terms which are O(e6), and the solution for 62

is as = e2 + e4/16 + O(e6). Thus, our hybrid solution _ (with N = 2) agrees with the

perturbation solution for u to within terms which are O(e4).
In Fig 3 we have plotted various approximations to v as a function of e. Included are

perturbation solutions (20) (denoted by P[N]) of order O(_ 2N) where N = 1, 2,6; hybrid

approximations (10) (denoted by H[N]) where N = 1,2 based on the first g nonzero functions {uj}; and numerical solutions obtained by numerically evaluating Eq (3) for the case

of the pendulum equation. The figure illustrates that the perturbation method gives good

results for _ = 8r_x up to approximately ¢ = 31r/4. For _ near lr, the perturbation results

converge very slowly and give little hint that the frequency must go to zero as _ _ _r. In the

region near _ = 7_r/8 the H[2] solution seems definitely superior to the P[2] solution, but for

larger amplitudes the accuracy of the HI2] solution is not good, though it does hint that the

frequency is tending to zero at _ = _r. The HI3] solution (not shown) is little better than

the H[2] solution.

Large amplitude oscillations about an unstable equilibrium

We now consider the problem of determining approximations to the frequency of "large"

amplitude oscillations about an unstable equilibrium of a nonlinear system. As a model

problem in this area, we consider the (non-dimensional) problem

u 2u" - u + u 3=0, u(0)=a, u'(0) =0, (22)

where u is 2r periodic and a is now interpreted as the nondimensional initial amplitude.

The equilibrium state u -= 0 is unstable for the system described by (22), while the states

u ---_il are stable. However, stable oscillations about u = 0 exist for a 2 > 2.

To study the stable oscillations about u = 0, we consider the modified problem

u2u" + u -k- ¢(u 3 -- 2u) ----0, u(O) = a, u'(O) _- 0,

(23)

u(x+ = u(x),

where e is an unspecified parameter. We now make two observations. First, we note that Eq

(23) is of the form of Eqs (1) with f/a = u 3 - 2u and hence we can construct approximate

solutions which will be formally valid as E --+ 0. Secondly, we note that, by setting e = 1 in

(23), we recover our original Eq (22). Thus, our (formal) procedure will be to apply our two

step hybrid method, as outlined above, and then set c = 1 to obtain an approximation to the

solution to (22). In this sense, we can interpret e in this example as a type of "homotopy"

parameter. In particular, as ¢ varies from 0 to 1, we can think of our problem (23) as varying

from a trivial problem, corresponding to c = 0, which we can easily solve, to the problem we

really want to solve,: i.e. problem (22), corresponding to e = 1.

Following the general method outlined above, we first use the Lindstedt-Poincar6 method
on problem (23) to obtain

u0 = a cos(x),

'/21 _-- -- (a3/32) [cos(x) - cos(3x)],

u2 = (a3/1024)[(23a 2 - 64) cos(x)

-(24a 2 -64) cos(3x) -I- a 2 cos(5x)],

v = 1 + (3a 2-1) e - (21a \ 256 4 3a8 2 + 1) e2

(24)

(81a 6 63a 4 9a 2 _)

£3

+ k2- 256 + 16

6549 a s 405a 2048 6 ÷ 315a 512 4 15a16 2 _) -Jv _4 + O(e5).

The hybrid approximation _ is given by (9), where the amplitudes {Sj} and _2 are

determined by Eqs (10) with the term (e/5) f replaced by e (_3 _ 2 _). In particular, setting

N=1and e= l in (11) we find

= [3a2/4--1] 1/2

= (v_/2)lal+O(1/lal), as lal---,,oo.

(25)

From the integral representation (3) of u, we find for this case that

"'11"/_[_ (1 -t- sin2(x))- 2]-1/2 dx] -1

v - 23/2 kJO

7rIal[fo./2 -1 (26)

- 2 x/_ (1 + sin2(x)) -x/2 dx] + O(1/lal)

= 0.8472[a1 + O(1/lal), as a _ c_.

Thus, from (25) we see that _ = 0.866 lal + O(Ulal) as lal- which compares well with

the exact asymptotic result (26).

In Figs 4 and 5 we have plotted various approximations to u as functions of the amplitude

a. In Fig 4 it is seen from the numerical solution that the frequency goes rapidly to zei.o

as a _ v/2 from above. Also in Fig 4 the perturbation expansions (24) _P[N] for N =

1, 2, 4, 8, 16) indicate that the convergence of the perturbation solutions takes place only for
a very limited rangeof a values, perhaps v/2 < a < 1.75. The hybrid solutions (H[N] for

N = 1,2,3,4) show rapid convergence for a > 1.7 and slower convergence nearer a = V_.

Even so in the region just above a = x/_, H[4] seems much more accurate that P[16]. Figure

5 shows that the hybrid solution, H[2] continues to have good accuracy up to a = 50 and

that even H[1] is a fairly good approximation for a = 50.

Self excited oscillations - the van der Pol oscillator

As an example of a nonconservative system, we now consider the limit cycle of the van

der Pol oscillator, for which f(au, uau',e) = (a:u 2- 1)auu' in Eq (la), i.e.

v 2u" + u + e(a 2u 2-1)uu'--0, u(0)-- 1, u'(0)--0, (27)

u(x+27r)=u(x) for all x>0.

For this case, e is a "tuning" parameter and can be interpreted as a measure of the amount

of a special kind of nonlinear damping in the system.

Several terms in the perturbation expansions of u, u, and a have been reported [1,4]. In

particular, we have

u0 = 2 cos(x),

Ul -- [3 sin(x) -- sin(3x)]/4,

= [12 cos(x) - 18 cos(ax) + 5 cos(hx)]/96,

(28)

1 e2 + 17 e4 35 e6 + O(eS )

p = 1 - 1---6 _ + 88473-------6-

1 e2 1033 e4 + 1019689 e6 + O(eS).

a = 2 + _-_ 552960 55738368000

The radius of convergence for this expansion is known to be approximately e = 1.85 (see

Andersen and Geer [1]).

In Fig 6 we have plotted several approximations to u, including the approximations

P[N], obtained by including all of the terms in the perturbation expansion (28b) up to those

which are o(eg+l), the approximations H[N], which are the hybrid solutions based on the

first N perturbation coordinate functions {uj}, and the approximations of Zonnefeld [16],

obtained using purely numerical methods. The figure illustrates that the hybrid results are

superior to the perturbation results on which they are based, and appear to be converging

to the numerical approximations as N increases. However, the overall quality of the hybrid

10
approximation is not asgood asin the previousthreeexamples.In the next sectionweshall

discussa generaltechniqueto improvethe overallquality of the hybrid approximationsand

then apply it specificallyto the van der Pol oscillator.

5. COMBINING TWO OR MORE PERTURBATION EXPANSIONS

In manypractical applications,it maybepossibleto constructperturbation expansionsof

the solution u and the frequency u about more than one value of the perturbation parameter

e. That is, it may be possible to expand both u and u as formal asymptotic series of the

form

Np

it ---- E ttjp +

j=O

(29)

Np

P _ E Vj +

j=O

which will be formally valid as c _ %, for p = 1,2,...,P. Here {a_(_)} is an appropriate

p

asymptotic sequence of gauge functions and each uj can be determined completely by a

standard perturbation method (e.g. a composite expansion of inner and outer expansions).

For example, in the Lindstedt-Poincar@ method, we have constructed expansions of the form

of (29) with % = c_ = 0 and aJ(c) = ¢j. In addition, it may also be possible to construct

expansions of the form of (29) which will be valid as c -_ c2 = oc, where the gauge functions

{a_(¢) } might now include terms such as cq, where q may be a positive integer or a positive

rational number, and might also involve appropriate transcendental functions, such as log(c)

(see e.g., [13]).

p

A subset of all of the perturbation functions uj are now chosen as coordinate functions

for the hybrid technique and an approximation _ for u is sought in the form of (9), where

each uj is now one of the perturbation coordinate functions u_. To determine the unknown

amplitudes 6j, we again apply the conditions (10).

To illustrate the application of our method when more than one perturbation expansion

of the solution is available to us, we consider first the problem of determining the resonant

frequency of a simple mechanical system consisting of a mass m restrained by two identical

springs, each having natural length L and spring constant k, midway between two parallel

walls a distance 2d apart, as discussed by Arnold and Case [3]. Thus, if we let h be the

maximum displacement of the mass, its displacement along the centerline between the two

11
u2u" + u-b #u I1

planes is described by h u(x), where

(14-e21u2)l/2] = 0, u(0) = 1, u'(0) = 0,

u(x + = u(x).

(30)

Here e = h/d, # = A/(1 - A), A = Lid < 1, u = 2_r/Two, w0 = [2k(1 - A)/m] '/2, and

x = Wo v t, where t is time. Thus e can again be interpreted as a measure of the maximum

amplitude of the oscillation of the system.

For small values of e, u and v have expansions of the form (29), with aj1 = e2j. Using

the Lindstedt-Poincar$ method, we find

u = cos(x)+ (#/64)[cos(3x)-cos(x)le 2 + O(e4),

(31)

u = 1+(3#/8)e 2 + O(et), as c---*0.

2

For large values of e, u has a (composite) expansion in the general form of (29) with aj = e -J,

in which the formal outer expansion, valid for [eu[ > 1, must be supplemented by inner

expansions around x = _r/2 and x -- 3 _r/2, where u vanishes. In this case, it is easy to show

that u is still well approximated by the first term on the right side of (31a), while

u=l+_+O(c1), asc_oz. (32)

To apply our hybrid method to this problem, we look for an approximate solution _ for

u in the form

N

= __, uj(x) 5j, 50 = 1, (33)

j=0

where Uo(X) = cos(x) and the remaining {us(x)} can be selected from either the small or

large-e expansions of u. Substituting (33) into (10), we see that the orthogonality conditions

become

Fj(I]2,51,52,...,SN)=O, j = 0,1,...,N,

(34)

Fj= _2 fi" + _ + # _ (1 - (1 + e2 _2),/2 ujdx.

Equations (34) are a system of N + 1 equations for the N + 1 unknowns _2 and 5j, (j =

1,2,..., N). For the special case when N = 0, we can solve (34) explicitly and find

u 2 = 1 + _ Jo 1 - (1 + e2 cos2(x)) 1 _/2 cos (x) x.

If we can expand (35) for small values of c, we recover the expansion for u in (31b), while if

we expand it for large values of e we recover the expansion (32). In Fig 7 we have plotted

12
u determined by (35) as a function of e, and have also plotted some corresponding values

determined by the numerical evaluation of Eq (3). As the figure indicates, the agreement of

the hybrid results with the frequencies determined from Eq (3) is excellent.

As a second example, we again consider the van der Pol oscillator. For large values of e,

both u and u have expansions of the form of (29) (see e.g. [13]), where the gauge functions

now include e-l, e-7/3, ¢-3 log(e), .... In particular,

3--log(4) e- 1 Jr-O(e-7/3), as 6----+ oo. (36)

u- 2r

The leading term in the corresponding perturbation solution for u involves several inner and

outer expansions, which can be combined in a straightforward manner to form the leading

term in a composite expansion of the solution. We shall denote the leading term in this

expansion by u_(x, e).

We now use u_o(x, e) as one of our coordinate functions in our hybrid solution (9), which

we write in the form

N-1

K = 60u0+ y] t_jui + (1 - (_0)u_, (37)

j=l

where {uj, j = 0, 1,2,...} represent the perturbation coordinate functions determined from

the (regular) perturbation expansion about e = 0. Thus, our solution (37) combines N

terms from the smMl-e perturbation expansion of u with the leading term in the large-e

perturbation expansion of u. The N+2 unknowns 60,..., 6N-l, a and u are then determined

by substituting (37) into the N + 2 conditions (10), with N replaced by N + 1 and with UN

formally replaced by u_. In Fig 8 we have plotted two perturbation approximations to u: the

three-term small< approximation denoted by P013] and the one-term large-e approximation

(Eq (36)) denoted by P_[1]. We also show three hybrid approximations: H[3,0], which is

based on the three small-e perturbation functions; HI0, 1], which is based on the one large<

perturbation function; and H[3, 1], which combines the information from the three small-e

perturbation functions with the one large< perturbation function. As the figure illustrates,

the hybrid approximation H[0, 1] is a considerable improvement over P_, but it gives poor

results for e < 2. The hybrid approximation HI3, 0] is a better approximation than P[3], but

it gives poor results for e > 3. Finally, H[3, 1] gives good results for very small or very large

values of e and reasonable (but not accurate) results for "intermediate" (i.e. neither "small"

nor "large") values of e where neither perturbation expansion is accurate.

6. SYSTEMS WITH SEVERAL DEGREES OF FREEDOM

We now wish to study periodic solutions (_(t, e) to systems of nonlinear equations describing oscillating systems with several degrees of freedom, which can be expressed in the

13
form

•" 2, -7

AO + BO+ei(O,Q,Q,_)=o. (38)

Here e is a small, dimensionless parameter, the dots denote differentiation with respect to

time t, Q is an M-component vector of dependent variables (where M > 1 is the number

of degrees of freedom of the system), A and B are real, symmetric, positive definite, M by

M, constant matrices, and j7 is an M-component vector. (Here it is convenient to require

the components of A to be dimensionless, while the components of B and f are required to

have the units of t-2.) We assume that, for small values of e, the term e f can be expanded

in a series of the form

97=e97, + e2 972+... (39)

where each 97i is independent of _. We also assume that the eigenvalues Aj of the generalized

eigenvalue problem associated with the lineaxized version of Eqs (38), i.e.

[B - Aj A]_j =0, j= 1,2,...,M, (40)

are distinct and simple, and that the corresponding eigenvectors Kj have been normalized

so that (A cTj, _j) = 1 for each j. (Here ( , ) is the usual Euclidean vector inner product.)

Consequently, the set of eigenvectors {5j, 1 < j _< M} is complete and A-orthonormal.

Then, for e = 0, Eqs (38) have periodic solutions of the form

(2(t)= a cos(_pt- _)_p, _p = ..p.t1/2 , (41)

where a and fl axe arbitrary constants and 1 _< p _< M.

Let T be the (unknown) period of a periodic solution of (38) which reduces to the solution

(41) as e --_ 0. Then we define

2_ w

T' /]_--'

COp

x = cat = vwpt, {(x) = Q(t). (42)

In terms of these variables, Eqs (38) become

_2_.A,7"+ B _ + _97(_,_p _', .2 _p__", _)= o,

{(x + 2 _r) = {(x), (43)

where the primes denote differentiation with respect to x.

In the first step of our hybrid technique, we again use the Lindstedt-Poincax_ method

to construct perturbation solutions which are formally valid for small values of e. Thus, we

seek solutions to Eqs (43) in the form

_(_,_)= _0(_)+ _,(_) + d q:( -' X ) + ...,

14
q'o(X) = a cos(x - fl) 5p, (44)

v=l + ¢vl + ¢2v_ + ....

In Eqs (44), each _j is 27r periodic in x, and each vj is a constant.

Substituting (44) into (43) we find that the terms which are O(1) on the left side of (43)

sum to zero (from the definition of q0), while the terms which are O(¢ k) with k >_ 1 yield the

relations

2 A "_ll

wp._q k + B_k = 2vkw_a cos(x -/3)A_, + _k,k = 1,2,..., (45)

where _k involves _j and vj with j < k. In particular, _1 = -]l(_o, Wp_D, wp2 _6,). The

general solution for _k will be the sum of a particular solution to Eqs (45) and an arbitrary

multiple of q'0, which is.the solution to the homogeneous version of Eqs (45). To make our

solution unique, we impose the orthogonality condition

f02'_(A0'k, _0)dx = 0, for k _> 1. (46)

We suppose at this point that k is fixed and that _j and vj are known for j < k. If we

now take the inner product of Eqs (45) with cos(x -/3) 5p, integrate the resulting expression

with respect to x between 0 and 27r, and use the periodicity condition for _k, we find that

the left side vanishes and hence we can express vk as

1

f02_(g_, cTp) cos(x -/3)dx. (47) vk - 2 _rw_ a

In particular, for k = 1 in (47) we obtain

1 t2,_ .-. -. ) - ]0 ( -' -"' cos(x-/3) (4s) vl 2 _r w_ a fl (q0, wp q0, wp q0 ), (_p

With v_ determined by (47), we look for a solution for q'k in the form

_'k= y_cos(j(x-/3)) _,k,j,pc_,], (49)

j=0 p=l

where the positive integer dk and the constants 7k,j,p are to be determined. To determine

these quantities, we first express the right side of Eq (45) as a trigonometric polynomial in

the form

Jk

2 uk w_ a cos(x -/3) A _, + _k = _ _k,j cos(j (x -/3)), (50)

j=O

where the vectors {ffk,j} are independent of z and the inner product (g_,_, (Tp) = 0. This

follows from the definition of vk in (47). Substituting (49) and (50) into (45), using the

15
relations (40), and taking the inner product of the resultingexpression(40)with CTm, we find

that

(95,j, am)

- 0<j<Jk,

_,j,m _o_--j2co_' --- (51)

l_m_M, (j-l) 2+(m-p)2>0,

while the constant %,1,p is zero from condition (46). In deriving Eq (51), we have assumed

that co,,2 ¢ y"2 2 cop unlessj=l andm=p.

In the second step of our hybrid method, we seek new approximations _ and _ to _ and

v, respectively, where

N-1

_= _0(x)+ _ _j(_)_(x). (52)

j=l

In (52), the _i(x) are the perturbation coordinate functions which appear in the expansion

(44), while the constants {(hi} represent new "amplitudes", which must be determined. We

note that (is 2rr periodic for any choice of the {6j}, since each _j is 2re periodic. To determine

the amplitudes {6j }, as well as _, we substitute (52) into (43) and demand that the residual

is orthogonal to each of the perturbation coordinate functions _k , i.e.

2'_([_2_%2A("+ B_+ ef(_,Pwp(',_ 2wp2(',e)],_k)dx=0, k=0,1,...,N-1.

(ha)

Equations (53) are a system of N equations for the N unknowns _1,...,5N-1, and Y. In

particular, setting N = 1 in (52) we find that (= _o while (53) yields the expression

[ f0 ( ) ]1/2 2.. _ -, _2 2-,,. _ cos(x-/_) d_ (54)

= 1 + rew_-------_ f (q°' P cop q0, wp q0 ), ap

We note that, for small values of e, from (54) we can write _ = 1 + e vl + O(e2), where va

is given by (48). Thus, for small values of e, our hybrid approximation (with N = 1) agrees

with our perturbation result up to terms which are O(c2).

7. EXAMPLE - COMPOUND PENDULUM

As an application of the results of the previous section, we consider two pendula, each

of length L and each with attached mass m. They are attached in such a way as to form

a classical compound pendulum. Letting 0x(t) and 02(t) denote the angles the two pendula

make with the vertical, we find that 01 and 02 satisfy the relations

2//, +/i_ cos(01 - 0_) + b_ sin(01 - O_) + 2 w_ sin(O,) = 0

(55)

g2 cos(Ox - 02) + O=- t}_ sin(O1 - 02) + w_ sin(O=) = O.

16
In (55), w_ = g/L, where g is the acceleration due to gravity. We now let 01 = eQ1 and

02 = e Q2 to write (55) in the form of Eq (38) with ¢ replaced by e2, where

A= {21 1}1 ' 0 01 /'

and

Q_[eos(¢(Q1- Q2))- 1]+ eQ_sin(¢(Q1- Q2))

_-2_-1 _x)_[sin(¢Qa)-- eQ_]

(_1 [cos(¢(Q1 - Q2)) - 1] - ¢0_ sin(¢(Q1 - Q2))

+c -_ [sin(¢Q:) - ¢Q2]

=e: [ -(1/2)Q2(Q1 - Q2)2+ (_(Q1- Q2)-(1/3)w_Q_}

-(1/2)Ol(Q1 - Q:)2_ O_(Qi Q:) (1/6)w_Q_

For this problem we find

+ 0(¢4). (57)

C_ 1 ---- C 1 , C_2 = (32 _V/_ '

Cl = (1/2) (2 -- Vf2), c2 ----(1/2) (2 q- V/2).

Thus, for small amplitude oscillations, the periodic mode corresponding to p = 1 (i.e. the

mode with frequency Wl) represents an oscillation for which, at any instant of time, the

pendula are always on the same side of the vertical. We shall refer to this mode as the

"symmetric" mode of oscillation. In a similar manner, the periodic mode corresponding to

p = 2 (with frequency w2) represents an oscillation for which the pendula are on opposite

sides of the vertical, which we shall refer to as the "asymmetric" mode of oscillation. Then,

setting p = 1 in our formulae of the previous section (with a = 1 and fl = 0 in the definition of

q*0), we find that we can express the perturbation solutions for the frequency w of oscillation

of the symmetric mode, as well as the angular displacements 91 and 02, as

<_:/_ = (2-v_)_,_=2-_+

¢2(-102 + 71v_) + ¢4(470602- 332533v"-2)

16 14336 + (59)

¢6(-2676206382 + 1892439443v_)

12845056

+ 0(¢_),

17
0,(x) = _ql(x)=_ cos(x)+

e3[ (1 - v_)c°s(x)32 + 23(i 19 + +2688 24V/-2) c°s(3x) ]

es[ (233 + 1375vf2)28672cos(x)

(812575 - 535236v"-2) cos(3x)

+

802816

+

(60)

3(970497 -- 751760v/2)cos(5x)]

9748480 l

0_(x) = _q_(x)= _v_ cos(x)+

+ o(d),

dr(2- v_) cos(x) (576- 781v/2)cos(3x)] [ 32 + 2688 ]

+

es[ (-2750 -28672233v/2)cos(x)

(-1148584 + 756783V"-2) cos(3x)

+

802816 +

(61)

(-4066960 + 3111099v"-2) cos(5x) ]

O(C). 9748480 J +

It follows that the perturbation expansions for u and for the (total) energy rnw_ L 2 E for

the symmetric mode are given by

= l + e2 (-31 + 20vf2)

32

+ c4 (-61839 + 46888x/2)

28672

+ e6 (-284017367 + 197610484_/2)

E = 2 e2 + e4

25690112

3 (29 -I- 8v_)

32

+ O(d),

(62)

+ e6 (2738185 - 550288v/2)

401408 + o(eS)"

If O,(x) and 02(x) are computed to order O(£2N-1), then the energy E is conserved (in time)

to O(c2N).

The corresponding expressions for the asymmetric mode of oscillation are the v_ conjugates of these expressions, i.e. everywhere _ appears in Eqs (59)-(62) it is replaced by

-v_.

For both the symmetric and asymmetric modes u is a monotonic decreasing function of

E in the interval 0 < E < 6, the energy interval in which the pendulum exhibits a backand-forth motion. For any given energy in this interval the frequency of the asymmetric

18
modeis higher than the frequencyof the symmetricmode. Parametric plots of u(e) vs. E(e)

as computed to orders O(d), O(P), O(e6), O(eS), O(d°), and O(d2), are shown in Fig 9

for the symmetric case and in Fig 10 for the asymmetric case. These curves are labelled

P[1] through P[6]. The hybrid results are useful well past the radii of convergence for the

perturbation results. They appear to be converging to the correct values for the energy

interval shown.

8. DISCUSSION AND CONCLUSIONS

Each of the examples we have considered provides some insights into the hybrid method

which we now discuss briefly.

The Duffing oscillator illustrates the fact that the usefulness of a perturbation expansion

can often be limited because the expansion has a finite radius of convergence. In this case,

the perturbation series converges only for lel < 1, since the solution u, when regarded as a

function of both x and e, has a singularity at e = -1. (This follows from the fact that the

solution to (14) is non-oscillatory for e < -1.) Thus, the perturbation expansion fails to

converge over the range 1 < e < oc, which is most of the region of physical interest. The

hybrid method overcomes this limitation and provides meaningful and, in this case, at least,

very accurate approximations to the resonant frequency, even for e well beyond the radius

of convergence of the perturbation solution (see Figs 1 and 2).

The perturbation expansion of the frequency (and solution u) for the simple pendulum

converges over the entire interval of interest, i.e. -_r < e < 7r, and hence, in principle, could

be used to compute approximations to the resonant frequency for any value of e in this

range. However, the rate of convergence is very slow for values of e close to +Tr and a great

many terms in the perturbation series would be needed to provide a solution of acceptable

accuracy (see Fig 3). By contrast, the hybrid approximation, based on only two terms of

the perturbation expansion, provides an approximation to the resonant frequency which is

essentially as accurate as the perturbation solution based on six terms for e not close to in,

and is reasonably close to the numerically computed frequencies, even when e is close to -t-_-.

Thus, we see that the hybrid method has the effect of accelerating the convergence of an

otherwise slowly converging series.

In the example of oscillations about an unstable equilibrium, we saw that the perturbation

expansions depend on the parameter a (the maximum initial amplitude), as well as upon

e. Thus, the convergence of the perturbation solutions will also depend upon the parameter

a. In particular, for e = 1, which corresponds to the case of interest in this problem, we

conclude from Fig 4 that the series solution for the resonant frequency converges only for a

very limited range of values of a, this range being from a = v'_ to approximately a _ 1.75.

19
This is in contrast to the hybrid solutionswhich appearto convergeto the exact frequencies

for all valuesof a > v/2. However, the rate of convergence, even for the hybrid solutions, is

much slower for values of a close to x/_ (see Fig 4) than for larger values of a (see Fig 5).

The van der Pol oscillator is an example of a nonconservative system, for which both the

frequency v and the initial amplitude a of the motion appear as unknowns in the problem

formulation. For this case, we again note (see Fig 6) that the hybrid solutions appear to be

converging to the "exact" (numerically determined) frequencies as the number of terms in

the approximation is increased. This convergence appears to hold even for values of e greater

than the radius of convergence of the perturbation expansion, which is approximately 1.85.

However, the rate of convergence of the hybrid solutions for e > 1.85 is not as dramatic

as in the previous examples. Although we do not as yet have a good explanation for this

behavior, we feel (intuitively) that it is due to the rather complicated form of the response

u as e becomes large (see e.g. [13]). It simply appears that this behavior cannot be well

approximated by the perturbation coefficient functions which have been included so far in

our hybrid approximation.

When perturbation expansions of the solution about more than one value of the parameter

are available, it has been our experience that it is usually advantagous to use at least one

term from each of these expansions. This is illustrated in the springs-and-mass example as

well as for the van der Pol oscillator.

In the springs-and-mass example, the exact form of the first term in the small-e perturbation expansion, i.e. u0 = cos(x), was used in the hybrid solution, along with an

approximation to the exact form of the first term in the large-e perturbation expansion,

which is again cos(x). In particular, we did not include in our hybrid approximation the

various inner expansions which should be combined with the outer expansion cos(x) to form

a uniformly valid first approximation to the solution when ¢ is large. Nonetheless, the hybrid

results are very accurate for all values of the expansion parameter e, as well as for an entire

range of values of the parameter/_ (see Fig 7). This illustrates the fact that apparently one

does not need to determine the "complete" form of the perturbation coefficient functions

and that merely a good approximation to these functions may suffice for use in the hybrid

approximation.

For the van der Pol oscillator, the exact form of the large-¢ perturbation solution is very

complicated, although several terms in the various inner and outer expansions necessary

to construct this expansion have been computed [13]. However, in our approximation, we

used only the leading term from this expansion (which is "difficult" to compute), along with

several of the small-e perturbation coordinate functions (which are "easy" to compute) to

form our hybrid approximation. As illustrated in Fig 8, there is a definite improvemeIlt

2O
in the quality of the hybrid approximation, although the agreement with the numerically

determined frequencies is by no means entirely satisfactory. Presumably, the accuracy of our

approximations could be improved by including more terms from either the small- or large-e

perturbation expansions. However, we have not carried out these calculations.

The compound pendulum is an example of the application of our method to systems

with more than one degree of freedom. We note from Figs 9 and 10 that the perturbation expansions of the different resonant frequencies appear to have significantly different

radii of convergence. In particular, the perturbation expansion of the higher resonant frequency, corresponding to the asymmetric mode of oscillation, has a much smaller radius of

convergence (approximately c2 = 0.092) than the expansion of the lower resonant frequency,

corresponding to the symmetric mode of oscillation (approximately e2 = 1.2). Hence, the two

perturbation expansions will be useful for calculating approximations to the corresponding

resonant frequency over different ranges of values of energy E (approximately 0 < E < 0.17

for the asymmetric case vs. 0 < E < 2.3 for the symmetric case). In both cases, however,

the hybrid approximations based on only a few perturbation functions appear to converge to

the numerically determined solutions, even for values of _ well beyond the respective radius

of convergence. We are currently developing the hybrid method in the context of general

Hamiltonian or Lagrangian systems (and even more general systems whose solution can be

characterized in terms of a variational principle) and are applying the methods to several

other examples. The results of these investigations will be reported elsewhere.

We note that the perturbation parameters ¢ employed in the above-mentioned problems

have a variety of interpretations, but in each case _ = 0 corresponds to simple harmonic

motion.

While we have for reasons of space limitation only reported on frequency functions, the

hybrid technique appears to converge to correct mode shapes as well as to correct frequencies.

The hybrid method is more successful in some of the examples reported here than in

others. We think it is particularly effective for large values of _ in Fig 2, for large values

of a in Fig 5, and for all values of p in Fig 7. It seems to be less effective, as in the

simple pendulum problem, when "directly" approaching a singularity rather than merely

approaching a radius of convergence.

We also note that the equations, such as (10) and (53), which determine the hybrid

amplitudes {$j} and P for nonlinear systems will, in general, be nonlinear and have multiple

solutions. Some of these solutions may be ruled out on the basis that they involve complex

numbers. Of the remaining solutions, it appears that the only ones of interest are those for

which the _j(_) coincide with the gauge functions of the perturbation expansion in the limit

---* 0. Therefore following a solution path starting at ¢ = 0 seems to be an essential part of

21
the method. In our experiencethis leadsin eachcaseto a unique solution.

It isinteresting to notethat the hybrid method providesanice supplementto someof the

methodsand ideaspresentedby Van Dyke[14]for improving the usefulnessof a perturbation

expansion.In particular, VanDyke'sideasareapplicablewhenafairly largenumberof terms

in a perturbation expansioncanbecomputed(usually with the aid of numericalor symbolic

computation). The coefficientsin the seriesare then analyzedto help uncoversomeof the

analytical structure of the solution and then this information is used to recast the series

into a different form which, in general,will be valid for a wider range of parametervalues.

In contrast, the hybrid method requires only a few terms (often only one or two terms)

in the perturbation expansionto be computed and then usestheseterms to construct an

"improved" approximation to the solution.

In conclusion,it appearsthat the hybrid perturbation-Galerkin techniqueis a usefulway

to enhancethe usefulnessof perturbation solutions to resonantfrequencycalculation problems. In particular, the hybrid solutions _ provide useful approximations to the resonant

frequenciesfor the examples illustrated here, even for parameter values well beyond the

radius of convergenceof the perturbation solutionson which they are based. We are currently investigatingapplication of the method to moregeneralHamiltonian and Lagrangian

systemswith a finite number of degreesof freedom,as well asto systemswith an infinite

numberof degreesof freedom,i.e. to partial differentialequations.The initial resultsof these

investigationshavebeenvery promising.

While therearestill many theoreticalquestionsto beansweredabout the hybrid method,

it is basedon anintuitively plausibleidea,it isrelatively simpleto implement,and it appears

to provide reasonableand often very accurateapproximate solutions.

22
References

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

Andersen C M and Geer J F, "Power series expansions for the frequency and period of

the limit cycle of the van der Pol equation," SIAM J Appl Math 42, 678-693 (1982).

Andersen C M and Geer J F, "Investigating a hybrid perturbation-Galerkin technique

using computer algebra," ICASE Rep No 88-65 (1988), NASA Langley, Hampton VA,

and J Symb Comp (in press).

Arnold T W and Case W, "Nonlinear effects in a simple mechanical system," Amer J

Phys 50, 220-224 (1982).

Dadfar M B, Geer J F and Andersen C M, "Perturbation analysis of the limit cycle of

the free van der Pol equation," SIAM J Appl Math 44, 881-895 (1984).

Geer J F and Andersen C M, "A hybrid perturbation Galerkin technique with applications to slender body theory," SIAM J Appl Math 49,344-361 (1989).

Geer J F and Andersen C M, "A hybrid perturbation-Galerkin technique for differential

equations containing a parameter," Appl Mech Rev 42(11), 69-77 (1989).

Geer J F and Andersen C M, "A hybrid perturbation-Galerkin technique which combines

multiple expansions," SIAM J Appl Math 50, 1474-1495 (1990).

Geer J F and Andersen CM, "A hybrid perturbation-Galerkin technique for partial

differential equations," ICASE Rep No 90-57 (NASA-Langley) and Asymptotic Analysis

and the Numerical Solution of Partial Differential Equations, Lec Notes Pure Appl Math

130, Marcell Dekker (New York), 1991, 113-134.

Melvin P J, "On the construction of Poincar6-Lindstedt solutions: the nonlinear oscillator equation," SIAM J Appl Math 33, 161-194 (1977).

Nayfeh A H, Perturbation Methods, Wiley (New York), 1973.

Noor A K, "Hybrid analytical technique for nonlinear analysis of structures," AIAA

Jour 23, 938-946 (1985).

Noor A K and Peters J M, "Reduced basis technique for nonlinear analysis of structures," Proc. of the AIAA-ASME-ASCE-AHS 20th Structures, Structural Dynamics

and Materials Conference St. Louis, 1979, 116-126.

23
[13] Ponzo P J and Wax N, "On certain relaxation oscillations: asymptotic solutions," SIAM

J Appl Math 13, 740-766 (1965).

[14] Van Dyke M, "Analysis and Improvement of perturbation series," Quart J Mech Appl

Math 27, 423-450 (1974).

[15] Wolfram S, Mathematica - A System for doing Mathematics by Computer, AddisonWesley, Redwood City CA, 2nd ed, 1991.

[16] Zonnefeld J A, "Periodic solutions of the van der Pol equation," Inclag Math 28, 620-622

(1966).

24
/2

5

0 0

' I ! I I i I i I I

;P[5] , .-

! I "

! f

! I _._"

, ' . -- Hrl]

, ,,...--_ .HI2]

__ @4,

P[6]', ',P[4] "-..F[2]

I I I J I I • I I I

5 10

E

FIG 1. Comparison of hybrid (solid lines) frequency computations for the Duffing oscillator

with perturbation (dashed lines) and numerical (circles) computations for a small range of

values.

2O

/2

I0

0 i ! I i t f i I I

0 250 500

E

FIG 2. Comparison of hybrid (solid lines) frequency computations for the Duffing oscillator

with numerical (circles) computations for a large range of e values.

25
I I I t

V

FIG 3. Comparison of hybrid (solid lines) frequency computations for the simple pendulum with perturbation (dashed lines) and numerical (circles) computations. The amplitude

ranges from 0 to _'.

b"

2

,4,

HI2] . _...... _..... -.

_', P[16] P[161,, ',P[81 , P[4]

t l l

;_-- i l i t

1.8 2.4

FIG 4. Comparison of hybrid (solid lines) frequency computations with perturbation (dashed

lines) and numerical (dots) computations for a small range of e values.

26
4O

20

0 0

HI1] H[2]

50

a

FIG 5. Comparison of hybrid (solid lines) frequency computations with numerical (dots)

computations for a large range of e values.

v

1.5

I.Oc 4

!

P[9] ', ', P[3]

0.5

° , , ,, ,.

P[11] ],_P[7]

/,;,P[5]

o

HI2]

o

H[3]

o

H[5]

o

o

b

00 ' ' ' " 5 ..... 10

FIG 6. Comparison of hybrid (solid lines) frequency computations for the limit cycle of the

van der Pol oscillator with perturbation (dashed lines) and numerical (circles) computations

for a range of e values.

27
_._ Report Documentation Page

_oace -_oe_n6t r alO_

1. Report No, 2. Government Accession No. 3. Recipient's Catalog No.

NASA CR-t87632

[CASE Report No. 91-68

5. Report Date

September 1991

4. Titleand Subtitle

RESONANT FREQUENCY CALCUATIONS USING A HYBRID

P ERTURBAT ION-GALERKIN TECHNIQUE

7. Author(s)

James F. Geer

Carl M. Andersen

9, PerformingOrganization Name and Addre_

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, FA 23665-5225

12. SponsoringAgency Namea_ Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

6. Performing Organization Code

8. PerformingOrganization Re_rt No.

91-68

10. Work Unit No.

505-90-52-01

11. Contract or Grant No.

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring/_gency Code

15. Supplementaw Notes

Langley Technical Monitor:

Michael F. Card

To appear in Applied Mechanics

Review.

Final Report

16. Abstract

A two step hybrid perturbation-Galerkln technique is applied to the problem of

determining the resonant frequencies of one- or several-degree(s)-of-freedom nonlinear systems involving a parameter. In step one, the Lindstedt-Poincar_ method

is used to determine perturbation solutins which are formally valid about one or

more special values of the parameter (e.g. for small or large values of the parameter). In step two, a subset of the perturbation functions determined in step one

is used in a Gelerkin type approximation. The technique is illustrated for several

one-degree-of-freedom systems, including the Duffing and van der Pol oscillators,

as well as for the compound pendulum. For all of the examples considered, it is _

shown that the frequencies obtained by the hybrid technique using only a few terms

from the perturbation solutions are significantly more accurate than the perturbation results on which they are based, and they compare very well with frequencies

obtained by purely numerical methods.

17. K_ Words (Suggest_ by Au_or(s))

resonant frequency, Galerkin technique,

several degrees of freedom

Unclassified

18. Diatribut_n S_tement

59 - Mathematical and Computer Sciences

(General)

64 - Numerical Analysis

Uncalsslfled - Unlimited

21. No. of pages

Unclassified 31

22 Price

A03

19. Security Classif.(of this report) 120. Security Classif.(of this page)

NASA FORM 1(121OCTBS NASA-Lan_ey,1991