an experimental investigation into the dynamics of a...

An experimental investigation into the dynamics of a stringTimothy C. MoltenoDepartment of Physics, University of Otago, Dunedin, New Zealand

Nicholas B. Tufillaroa)

P.O. Box 60276, Palo Alto, California 94306-0276

~Received 9 July 2002; accepted 30 April 2004!

We describe a detailed experimental investigation into the dynamics of a sinusoidally forced string.We find qualitative agreement with the predictions of the averaged equations of motion for a stringin the high damping regime. At low damping we observe more complex phenomena not present inthe averaged equations. ©2004 American Association of Physics Teachers.

@DOI: 10.1119/1.1764557#

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I. INTRODUCTION

Analysis of the motions of strings has motivated a surping amount of science: From ancient Greeks investigathe length rations which produce harmonious soundsd’Alembert who developed the theory of partial differentequations in order to model the motions of a string.1 Ourreasons for studying the motions of a string are capturedby the words of P.M. Morse2 who wrote in 1936: ‘‘The stringis the simplest case of a system with an infinite numberallowed frequencies, and it is best to discuss some ofproperties common to all such systems for as simple asystem as we can find, lest the mathematical complicatcompletely obscure the physical ideas.’’ More recent theoical and numerical investigations3–8 have predicted a widevariety of nonlinear phenomena in the motion of a sinusdally forced stretched string: for example, torus-doublingfurcations, boundary crises, and chaos. In this paper wescribe an experiment which attempts a more extensinvestigation than earlier efforts to look for chaotmotions7,9–11 and related nonlinear phenomena in the vibtion of strings and verifies, in the most part, these predtions.

We previously reported a simple exploratory experimwhich established the existence of chaotic motionsstrings.9 This previous experiment successfully observedtorus-doubling transition to chaos, but lacked the control aprecision necessary for a thorough investigation of thenamics of a sinusoidally forced string.

A more comprehensive experimental apparatus wassigned for this second more detailed study. This wprompted both by a desire to improve the initial string eperiment, and by further theoretical work, particularly thatBajaj and Johnson,5,6 which indicated that several other interesting nonlinear phenomena, such as boundary crshould occur in the string system.

This paper describes the results of this second investtion into a string’s dynamics and is organized as follows.order to make this paper self-contained, Sec. II outlinesderivation of the averaged equations of motion used to mothe dynamics of a string and provides an extensive overvof the nonlinear phenomena predicted by these equatiSection III and the Appendix describes the experimentalparatus. Section IV presents the results of our experimeinvestigation.

It is hoped that some of these results could be incorporainto standard mechanics courses that touch on the linearnonlinear motions of a string, as well as theoretical and

1157 Am. J. Phys.72 ~9!, September 2004 http://aapt.org

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perimental courses that are specifically focused on the snonlinear dynamics.12–17For a basic background on somethe concepts and terms used in nonlinear dynamics andpaper—such as Hopf bifurcation, Logistic map, or Loreattractor, and so on—see an introductory text such asbook by Strogatz.13

II. MODELS OF STRINGS

We consider a string stretched between two fixed mouWhen the string vibrates transverse to its resting configution, its length must also fluctuate. This coupling betweentransverse motions and the longitudinal motions is nonlineeven for small oscillation amplitudes. The simplest intereing model of a vibrating string, which takes account of theffect and allows for motion in only one transverse directioyields a forced damped Duffing equation.18

This section starts with the full equations of motion forforced, damped, taut string. Then, following the work of Bjaj and Johnson,5,6 these equations are reduced to the funmental mode, but also allow motion in two transverse dirtions. Using averaging theory this results in fosimultaneous ordinary differential equations, which we cthe averaged equations of motion. These equations arebasis for the predictions of basic nonlinear behavior achaos which we search for experimentally.

A. Narasimha’s equations of motion

The nonlinear equations of motion governing the behavof a string date back at least to Kirchhoff~1883!.7,8 In thissection these equations are presented as derivedNarasimha18 and the assumptions implicit in them are summarized.

Narasimha models the string as a long thin circular cylder composed of a linear elastic solid under tensionT0 , andsubject to an external forcingf(x,t) transverse to the restinaxis. The exact equations of motion for the transverse~v! andlongitudinal (u) amplitudes of oscillation of such a strinare, in units of the length of the string (l ), mass per unitlength (m0), and velocity of propagation of transverse wav(AT0 /m0)

S 12]u

]xD u5c12 ]L

]x, ~1!

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1

2 S ]v

]xD 2

1S ]u

]xD 2

21

8 S ]v

]xD 2S S ]v

]xD 2

24]u

]xD1¯ . ~5!

If we assume that the oscillation amplitudes (u and v! aresmall, then we can approximatel as

l']u

]x1

1

2 S ]v

]xD 2

. ~6!

If we approximate the apparent strain asl' ]u/]x, then thetransverse and longitudinal motions of the string arecoupled and the transverse amplitudes~v! obey a linear waveequation. The approximation in Eq.~6! is then the lowestorder inclusion of the longitudinal coupling to the transvemotions.

A simple geometric argument leads to the conclusion tfor small transverse amplitudes,uvu;d, the longitudinal am-plitude isu;d2. For forced motions of the string we use thamplitude of forcinge as a small parameter,f5e f 0(x,t).Expanding the longitudinal amplitudeu, and the transverseamplitudev, in e we get

v5e~v01e2v11e4v21...!, ~7!

u5e2~u01e2u11e4u21...!. ~8!

The equation for the transverse motions is found by negling all but the lowest powers ofe, and adding a phenomenological damping termb. The damping is assumeda priori tobe linearly related to the velocity, and is designed to mothe complex loss mechanisms such as air resistance, intfriction, and movements of the endpoints. This yieldsequation of motion

v01bv0v02d2v0

dx2 F11c1

2

2 E0

1U dv0

dx U2

dxG5f0~x,t !. ~9!

In standard units, the equations of motion for the traverse amplitudesr (x,t) are

r12bv0r2d2r

dx2 Fc021

c12

2l E0

lUdr

dxU2

dxG51

mf~x,t !, ~10!

whereb is the damping on the fundamental mode,v0 is thefundamental frequency of transverse vibration,c0 is thetransverse wave speed,c1 the longitudinal wave speed,m isthe mass per unit length, andf(x,t) is the external forcingper unit length.

1158 Am. J. Phys., Vol. 72, No. 9, September 2004

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If we expand the transverse amplitudesr (x,t) and theforcing f(x,t) in a Fourier series,5 then the boundary conditions force the cosine terms to be zero and we get

r ~x,t !5 (n51

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rn~ t !sinnpx

l, ~11!

f~x,t !5 (n51

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fn~ t !sinnpx

l. ~12!

Substituting into the equations of motion@Eq. ~10!# we getan equation for thenth mode

rn12bv0rn1vn2rnF11

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4s (j 51

` U j r jp

l U2G51

mfn~x,t !,

~13!

wherevn5nv0 ands5(c0 /c1)2.Now if we assume that the external forcing is along they

axis, excites only the fundamental modef (t)5f1(t)•ey , andrestrictsr (x,t) to lie only along they axis

S ey5S 10D D ,

i.e., let r 5r1 , then Eq.~13! becomes

r 12bv0r 1v02r @11jr 2#5

1

mf ~ t !, ~14!

wherej5 p2/4sl2. This is a forced damped Duffing’s equation governing motion confined to one transverse directio4

B. Averaged equations of motion

Next we turn our attention to getting modal equations ttake into account possible motions in both transverse ditions. The equations of motion for the transverse amplituof the string@Eq. ~10!#, take into account the coupling between the longitudinal motions of the string and its tranverse motions. Keeping only the lowest order nonlineterms, Miles,19 and Bajaj and Johnson5,6 reduce these equations to a set of four ordinary differential equations for tamplitudes of the motion averaged over the fast oscillatioThis section follows their derivation of these averaged eqtions of motion.

We start with the modal expansion of Narasimha’s eqtion of motion @Eq. ~13!#, and assume that the forcing is ithe vertical plane only and sinusoidal with frequencyv closeto the fundamental frequencyv1 . We write it as~using ascaling chosen with the benefit of hindsight!

f15emlv1

2

pcosvtey , ~15!

wheree is an amplitude of excitation and will be considere‘‘small.’’ We assume that the forcing frequency excites onthe fundamental mode, and that the higher order modesexcited only through their nonlinear coupling to the fundmental mode. TheN mode truncation of the modal equationcan now be written, explicitly including a ‘‘smallness’’ parametere, as

1158Timothy C. Molteno and Nicholas B. Tufillaro

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The dimensionless parametersa and b represent dampingand detuning from the natural frequency of the fundamemode. Bajaj and Johnson6 show that the energy of the unforced modes~to O( e)) decay exponentially to zero. Thijustifies taking a single mode truncation of these modal eqtions which gives us the equations of motion

z91z5 e@cost1ey22az822bz924uzu2z#. ~21!

While it is possible to directly solve this truncated modequation numerically, averaging over the fast oscillationsthe string produces a model which is more easily analyzBajaj and Johnson6 show that, in most of the parameter rgimes of interest here, predictions from the averaged eqtions are in close agreement with direct numerical solutioOur experiment described in Sec. III will hopefully shesome light on the behavior of an averaged quantity, in thregions of parameter space where the averaging assumpare violated.

Before we apply the method of averaging it is convenito express the equations of motion as a system of first oordinary differential equations

x85y, ~22!

y851

112be@2x1 e~2 cost1ey22ay24uxu2x!#. ~23!

Note that asx,y are two-dimensional vectors, this is a systeof four coupled ordinary differential equations.

The system of equations@Eqs. ~22! and ~23!# is trans-formed into a form suitable for application of the methodaveraging by the invertible van der Pol transformation20

S pqD5AS x

yD , ~24!

where

A5S cost1 2sint1

sint1 cost1D , ~25!

A215S cost1 sint1

2sint1 cost1D , ~26!


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A85S 2sint1 2cost1

cost1 2sint1D . ~27!

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S p8q8 D5A8S x

yD1AS x8y8 D , ~28!

which, substituting forA, A8, and x8 from Eq. ~22!, isequivalent to

p852~x1y8!sint1 , ~29!

q85~x1y8!cost1 . ~30!

Now substituting fory8 in terms ofx and y @Eq. ~23!# andthen forx,y in terms ofp and q using Eq.~24!, we get

p8~112be!52 e@~ p cost11q sint1!~2b

24~p2 cos2 t11q2 sin2 t11p"q sint1!!

12a~ p sint12q cost1!

12 cost1ey#sint1 , ~31!

q8~112be!51 e@~ p cost11q sint1!~2b

24~p2 cos2 t11q2 sin2 t11p"q sint1!!

12a~ p sint12q cost1!

12 cost1ey#cost1 . ~32!

We now have the equations of motion in a ‘‘standaform’’ suitable for the application of averaging techniques

p85 eg1~ p,q,t1 ,e !, ~33!

q85 eg2~ p,q,t1 ,e !, ~34!

where the functionsg1 andg2 are periodic~period 2p! in t1 .The associated autonomous averaged system is defined~see,for example, Guckenheimer and Holmes20! as

p85 e1

2p E0

2p

g1~ p,q,t1,0!dt11O~ e2!, ~35!

q85 e1

2p E0

2p

g2~ p,q,t1,0!dt11O~ e2!. ~36!

Carrying out this integration we arrive at theaveraged equa-tions of motion

p1852ap12~b21.5E!q11Mp2 , ~37!

p2852ap22~b21.5E!q22Mp1 , ~38!

q1852aq11~b21.5E!p11Mq211, ~39!

q2852aq21~b21.5E!p22Mq1 , ~40!

where

E5p121p2

21q121q2

2 ~41!

and

M5p1q22p2q1 . ~42!

These equations are identical to those obtained by Mile19

who used the method of multiple time scales. The comnents p1 and q1 are slowly varying averaged amplitudecalled theplanar components of motion because they liethe plane of forcing. The variablesp2 andq2 are called thenonplanar components. Both represent motion in th


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‘‘modulated’’ frequency well bellow the ‘‘carrier’’ frequencyset by the forcing frequency.

The averaging theorem states that fore small enough thefollowing results hold between the original and averagsystems: If there is a hyperbolic fixed point of the averagsystem, then the original system possesses a unique hbolic periodic orbit of the same stability type, and that peodic solutions of the averaged equations correspond tosiperiodic solutions of the original system of equations.particular if a Hopf or saddle-node bifurcation occurs in taveraged system, then the Poincare´ map of the original sys-tem also undergoes a Hopf or saddle-node bifurcation. Adtionally, if the averaged system has a hyperbolic perioorbit g, then the Poincare´ map has an invariant closed curvnear g and the original system has a hyperbolic invariatorus.

There are no results at present which make a formal cnection between chaotic solutions of the averaged equatand the original system. Additionally the above statemeonly apply for sufficiently smalle. However, extensive numerical solutions of Bajaj and Johnson between the origand averaged equations indicate that chaotic motionsrelated behavior also exist between both models. The nsection summarizes the results of these theoretical andmerical investigations by Bajaj and Johnson5,6 for the solu-tions to the averaged equations of motion of a string.

C. Predicted string dynamics

The averaged equations of motion were investigatedMiles20 and subsequently by Bajaj and Johnson.5,6 Miles dis-covered torus doubling bifurcation sequences, but nwhich resulted in a chaotic attractor. Johnson and Bajajamined higher damping regimes and discovered chaoticlutions connected to a new periodic branch of solutions. Tsolution they termed theIsolated Branchsince it is not con-nected to theHopf Branchin the parameter region exploreby Miles. Further exploration uncovered many interestphenomena including the formation of a homoclinic orbboundary crises, and intermittency. We summarize somthe predicted results in this concluding theory section. Fomore complete discussion and illustrations see the paperBajaj and Johnson.5,6

The averaged equations of motion for the string are invant under sign change of the nonplanar components ofmotion (p2 ,q2)→(2p2 ,2q2). This means that for everysolution found numerically, there is another mirror-imagelution which is found by making this sign change.

We start the discussion of the dynamics generated by thequations~37!–~40! with a summary of the behavior of thefixed points which are known, from the averaging theorem20

to correspond to periodic trajectories of the unaveraged etions of motion@Eqs.~31! and ~32!#.

1. Fixed points

The fixed points of the averaged equations can be demined as functions of the detuning parameterb, and thedamping parametera. The plot of the amplitude of the aveaged motionE, against detuningb for some fixed dampingais called aconstant amplitude response curve. These curvesare simply the familiar resonance curves, plotting amplituof oscillation versus frequency.


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The stability of the fixed points divides the motions of thstring into several classes catalogued by their dampingues. These are:

d a.0.991. All motions are confined to the plane of excittion (p25q250). These are the only stable constant sotions to the averaged equations in this regime, andamplitude is a single valued function of the detuning. This called theplanar solution branch.

d a,0.991. The motions on the planar solution branch cbecome multivalued over some frequency intervals. Tycal response curves in this regime, are confined toplane, but in some detuning frequency intervals denotedDb there are three possible steady state solutions. Inregime, the string exhibits hysteresis as the detuningvaried. That is constant amplitude response curves exhdifferent transition jumps depending on whether the fquency scan is slowing increasing or decreasing—acalled ‘‘upward’’ or ‘‘downward’’ scan. These transitionoccur when the current planar solution branch loses staity and the motion jumps to another stable branch whare typically created via saddle-node bifurcations.

d a,0.687. Orbits are no longer confined to the planeexcitation. These are the so-called ‘‘whirling’’ motionThere are at least two non-planar whirling solutions—clockwise and counterclockwise orbit—because of theflection symmetry in the (p25q250) plane. Thesebranches are single valued for damping values larger ta50.477. The transition from planar to non-planar motiis usually quite easy to detect experimentally sincestring literally ‘‘pops’’ out from a single plane of motion aa parameter is changed.

d a,0.477 The nonplanar whirling solutions can also bcome multivalued, so again the system can exhibit jumbetween different orbits and show complex hysteresisfects as the control parameters are slowly varied.

2. Periodic solutions

The plane defined byp25q250, i.e., the plane of forcing,is an invariant manifold of the averaged equations forstring @Eqs.~37!–~40!#. Bendixson’s criterion for the nonexistence of limit cycles in a two-dimensional flows says thathe expression

]p18

]p11

]q18

]q1

is not identically zero and does not change sign, then thare no closed orbits. For motions restricted to the planeforcing, this expression is equal to22a and therefore thereare no planar periodic solutions to the averaged equatiAny periodic solutions must arise as solutions of the coplete four-dimensional equations of motion.

3. Hopf branch

Hopf bifurcations occur in the nonplanar branch fora,0.577. These lead to limit-cycles~they occur in pairs dueto symmetry! which are stable for damping values near0.577. These motions are said to lie on the Hopf solutbranch. As the damping is lowered below this value, the limcycles undergo period-doubling bifurcations. Starting wthe Hopf bifurcation which creates the Hopf branch, there


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a sequence of period-doubling bifurcations followed byverse period-doubling bifurcations back to the point at whthe Hopf branch disappears in an inverse Hopf bifurcatiThese periodic solutions of the averaged equations of mocorrespond to quasiperiodic motions of the string.

The loci in thea–b plane for the onset and disappearanof quasiperiodicity form the Hopf bifurcation set. In the eperimental section we show data outlining a Hopf bifurcatset.

4. Isolated branch

While numerically investigating the Hopf solution brancJohnson and Bajaj5 discovered another disconnected but cexisting solution branch. They conjectured that it arisescause of a global saddle-node bifurcation which causestable and unstable limit-cycle to appear for low enoudamping. Thisisolated solution branch possesses motiowith characteristics that are qualitatively different from tHopf branch orbits. In particular, the isolated branch orbexhibit ‘‘sharp’’ points in their orbit plots, which allow themto be distingished from the Hopf branch orbits. For loenough damping, the isolated solution branch is connectethe Hopf branch by an unstable limit-cycle. Because thelated branch is created by a saddle-node bifurcation, wepect to see hysteresis in its connection to the Hopf brancthis regime.

5. Chaotic solutions

According to the model equations, the isolated branchdergoes a sequence of period-doubling bifurcations wheventually results in the appearance of a chaotic attractorthe damping is reduced below the point where the isolabranch merges with the Hopf branch, a series of isolabranches are created via saddle-node bifurcations and suquently merge with the main solution branch. This sequeof orbit creation followed by merger, followed by anothcreation, terminates in the formation of a homoclinic orbThe orbit originates from a saddle-type fixed point whoeigenvalues satisfy Shilnikov’s criterion. Bajaj and Johnsconjecture that the Shilnikov mechanism20 is responsible forthe stretching and folding which causes the chaos.Shilnikov mechanism involves a homoclinic orbit fromfixed point which has one real, and a complex conjugateof eigenvalues.21 If the magnitude of the real eigenvaluelarger than the real part of the complex pair, then Shilnikshowed that return maps defined near the homoclinic ocontain horseshoes22—stretching and folding maps whicproduce chaotic behavior.

For orbits near to the homoclinic orbit, there is a branchsolutions which encircle both nonplanar fixed points~again,there are two because of the symmetry under sign cha(p2 ,q2)↔(2p2 ,2q2)). According to simulations, thisbranch undergoes a series of bifurcations which leads toformation of Lorenz type chaotic attractors~roughly, chaoticattractors which encircle two fixed points!.22

6. Crises

For damping values below 0.495, the chaotic solutionssuddenly be destroyed by a boundary crisis.23 The chaoticattractor comes into contact with the stable manifold ofixed point and the chaos is extinguished. The pointswhich this crisis occurs move closer to the Hopf bifurcati


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points as the damping is reduced further. This reducesinterval of detuning in which the interesting periodic anchaotic motions can occur. Bajaj and Johnson concludeeventually for a50.25, ‘‘practically all initial conditionslead to the lower planar steady-state constant solution.’’

D. Summary of predictions

The early work of Miles19 and the more recent work oBajaj and Johnson5,6 provided a lot of information about interesting behavior to look for in the motions of a string. Thpredict that:

d For small enough damping, the nonplanar solutionscomes unstable and form, via a Hopf bifurcation, the Hobranch—a branch of limit-cycle solutions~quasiperiodicityof the string’s motion!. This branch exhibits perioddoubling bifurcations but these do not lead to a chaoattractor.

d At lower values of damping, there is another coexistibranch of limit-cycle solutions, the isolated branch, whiis created via a saddle-node bifurcation.

d The isolated branch has a period-doubling transitionchaos which results in a Ro¨ssler type chaotic attractor.22

d For lower damping still, a series of isolated branchespear ~via saddle-node bifurcations! and merge with theHopf branch. This sequence ends in the creation of amoclinic orbit, whose fixed point has eigenvalues whisatisfy Shilnikov’s inequality. Lorenz type chaotic attrators are found in the neighborhood of this homoclinic obit.

d Finally as the damping is reduced yet again, the chaattractors are progressively destroyed by boundary crieventually eliminating all interesting behavior complete

The rest of this paper describes experiments which weresigned to explore these predictions.

III. EXPERIMENTAL APPARATUS

Our string experiment is constructed from a thin wiwhich is subject to an external periodic forcing. The wiredriven by passing a sinusoidal current through it while reing in a permanent magnetic field. The force per unit lenalong the wire is a product of the the magnetic field strenand the current in the wire. The forcing method is usedcause most theoretical studies assume a constant excitper unit length when modeling the string. A disadvantagethis forcing technique is that the string must be made ononmagnetic material, otherwise the motion will be confinto a plane. A magnetic material can be used when puplanar oscillations are desired. Also, because of heatingthe wire, the system is prone to slow temperature driwhich effectively cause a very slow parameter variati~drift! in the wire’s length.

Several types of wires were tested. This experimentashowed that a thoriated tungsten wire, 0.15 mm in diameworked well. Tungsten is nonmagnetic so the nonplanar mtions are not damped. Further, under the conditions ofexperiment, the wire was well within its linear elastic rsponse regime. The nonlinear behavior observed here isnot an effect of the material nonlinearity, but arises from tfundamental geometry of the string deformation.

The wire is mounted between two clamps, held apart bsteel frame which also forms the yoke of the forcing magn


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Mounted along the yoke are five pairs of rare earth cera~RES 190! magnets~Philips catalog No. 4313 059 6601!which provide a field of 0.2 T along the forcing length of thwire. The clamps are electrically insulated from one anotby two pieces of polyvinyl chloride, one of which holds thamplitude detector. Figure 1 shows a schematic and ofmounting fixture.

The amplitude detector consists of an infrared photo di~OP 165D! coupled to a match infrared phototransistor~OP505D! mounted in a low-Q plastic block. Partial occlusion othe beam by the string causes the amount of light atphototransistor to vary with the position of the string.24

In addition to the rig for holding the wire, we also deveoped a digital electronic system for signal synthesis, procing, and visualization which is described in the Appendand in great detail in Refs. 10 and 25. The system allowsthe real-time control and real-time signal analysis ofstring’s vibrations. The string controller provides controver the forcing of the string as well as triggering outputsdata collection. Its basic functions are to take an input squwave and provide the following outputs:

d a forcing sine wave;d the sampling trigger at regular intervals 32 times per fo

ing period; andd a pulse once per forcing period for controlling the intens

of an oscilloscope beam~for analog Poincare´ sections!.12

We sample the motion of the string an integer numbertimes per forcing period. This sampling technique allowsto average the amplitude of the string over one forcingriod.

The current electronics could easily be replaced withperipheral component interconnect~PCI! plug-in board suchas National Instruments PCI-6036E for signal synthesis, ctrol, and data collection. Such a board can be programmusingMATLAB’S Data Acquisition Toolbox. For instance, wrecently built such a low-cost system to do stimuluresponse testing and model creation for nonlinear electrcircuits.26,27

IV. EXPERIMENTAL RESULTS

The results of the string experiment are presented in oof increasing complexity: starting with simple periodic mtions and ending with chaotic oscillations.

A. Periodic motion

After the constant solutions, the next simplest solutionsa sinusoidally forced pretensioned string are periodic, wperiod equal to the forcing signal. These periodic solutioare characterized by their rms amplitudeRi(t), which is re-lated to the slowly varying averaged amplitudes (pi ,qi), by

Fig. 1. Schematic of the string mounting system.


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R1(t) andR2(t) represent slowly varying amplitudes of thplanar and nonplanar components of the motion, respetively. When the motions are periodic, these amplitudesconstant.

Fixing the forcing amplitude to give the desired dampi~a!, and then measuring the rms amplitude of the strinresponse for a range of forcing frequencies, gives a consamplitude response~CAR! curve. The interesting range oforcing frequencies, for large damping, is typically less th25 Hz on either side of the resonance (f 0). Figure 2 shows aseries of experimental CAR curves which are in qualitatagreement with the predictions of the averaged modSimilar experimental results are presented by Hansonet al.11

The damping at which the rms amplitude becomes multivued ~onset of hysteresis! provides the first quantitative tesbetween the experiment and the averaged equations mThe model predicts that this should happen for values oa,0.991. In the experiment this is only observed to occurvalues ofa,0.65.

Fig. 2. Constant amplitude response curves from the string experimen~a!before the onset of nonplanar motion and~b! after nonplanar motion. Somehysteresis is evident between the ‘‘upscans’’~1! and ‘‘downscans’’~o!.


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As the forcing amplitude is increased, the amplitude ofnonplanar components starts to increase and the string beto whirl around its resting state. The onset of nonplanar mtion is predicted to occur at values ofa,0.687, and is ob-served experimentally at values ofa,0.45. Notice that theratio of the two values~experiment to theory! is approxi-mately the same ('0.65) in both instances. This might sugest that the difference may be due to errors in measufixed parameters of the experimental system, such asforcing length of the string or the magnetic field strengAlternatively, it could be that the model fails to account acurately for some basic phenomenon like damping, or tquantitative discrepancies could also arise due to some oassumptions made in deriving the averaged model equat

Because the model is symmetric about the vertical atwo nonplanar~whirling! motions coexist corresponding tdifferent directions of whirling~clockwise or counterclock-wise!. These two solutions can not be distinguished by thCAR curves, but can be distinguished by plotting the noplanar average amplitudesp2 versusq2 . The solutions ap-pear as fixed points in either the upper right, or lower lquadrant of thep–q plane, according to the direction of thwhirling. In the experiment one of these directions is pferred, thus apparently breaking the symmetry present inaveraged equations. Whirling motions of either directicould be induced by deliberate perturbation, but left toown devices the system would almost invariably jumpnonplanar motion in one particular direction. This asymmtry could be introduced by a nonuniformity in the siliconcoating applied to the wire for damping, or by asymmetrin the clamps holding the wire.

B. Quasiperiodic motion

As the amplitude of forcing is increased, the string cjump, via a Hopf bifurcation, from periodic orbits describein the previous section to more complex motions with ariodically modulated amplitude. These quasiperiodic motioare the first step in the torus doubling route to chaos pdicted by Bajaj and Johnson5 and described experimentallby Molteno and Tufillaro.9

The nonplanar periodic motions are roughly ellipticalcross section, and quasiperiodicity arises when this ellistarts to precess. Figure 3 shows an experimental recorof the variations in the rms amplitude of a quasiperiosignal from the string experiment. The period of the amptude modulation in this example is approximately 250 foing periods. Thus, the modulation frequency is slow copared to the forcing frequency. The use of averaging thein deriving the ‘‘averaged equations of motion’’ would thuappear to be appropriate in this regime. Additionally, tlong period of this modulation may be one reason why pvious experimenters have found amplitude modulated cotic motion difficult to observe. Most experiments have beperformed on strings with a frequency of free vibration mothan 1 order of magnitude lower than in this experiment~forexample O’Reilly and Holmes7 used 88 Hz!. Therefore, themodulation period is several seconds, during which timesystem is susceptible to low frequency noise, which is vdifficult to eliminate.

The Hopf bifurcation set is recorded by scanning the foing frequency~changingb! for a series of different forcingamplitudes~changinga! and noting the onset and disappeaance of quasiperiodicity. Figure 4 shows the experimen


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Hopf bifurcation set recorded from the damped string. Tlower branch of the plot corresponds to creation of theHopf branch and the upper branch corresponds ot its destion. For damping values lower than those shown on tplot, the onset of quasiperiodicity occura at approximatthe same detuning, but the Hopf branch is destroyed byapparent crisis. The detuning at which this crisis is very ssitive to small perturbations, the Hopf branch appears tolosing stability both as the damping is lowered and asdetuning is increased.

In contrast, the upper branch of the Hopf bifurcation spredicted from the averaged equations model extends ovgreater range of damping values. The bifurcation sets frthe experiment and the averaged model have many featin common despite the apparent instability of the Hobranch in the experiment, which leads to its early destrtion. The critical point found experimentally for the onsetquasiperiodicity isa50.38,b53.4, whereas analysis of thaveraged equations yieldsa50.577,b53. Once again, theratio of experimental to theoretical damping for this criticpoint is approximately 0.65.

Fig. 3. A time series of the planar rms ampltitude for an experimensolution after the onset of quasiperiodicity.

Fig. 4. The experimental Hopf bifurcation set after being transformed to~a,b! plane.


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C. Hopf branch: period doubling

For damping values just below the onset of quasiperiomotion, the rms amplitude of the string show a periodoubling bifurcation. A typical experimental sequenof motions observed on the Hopf branch when scannthe forcing frequency is shown in Fig. 5. This containssuccession of plots of the nonplanar average amplitu(p2 versusq2) which show the bifurcation sequenceperiodone→period two→period four→period two→period one.

Periods larger than period four were not observed instring experiment. This may be due to parameter drift whcould obscure the observation of delicate motions on smparameter intervals, or it may reflect the actual dynamThese period doubling bifurcations of the averaged motcorrespond to torus doubling bifurcations of the amplitudethe string. This behavior is consistent with that of the Hosolution branch of the averaged equations of motion. Inparameter regime, soon after the onset of quasiperiodithere seems to be excellent qualitative agreement betwexperiment and the averaged model.

D. Isolated branch and chaos

As the forcing amplitude increases beyond the valuewhich the Hopf branch appears, the string’s motion can juto a qualitatively different type of quasiperiodic solutioThis jump exhibits hysteresis, i.e., the new motions persisthe forcing amplitude is reduced immediately after the juoccurs. Thus, the new motion can coexist with the Hobranch solutions. An example of these new motions is sho

Fig. 5. A simple experimental recording of a bifurcation sequence showa period doubling bifurcation sequence followed by an inverse period dbling bifurcation in the Hopf branch. The forcing frequency is the bifurction parameter (I rms50.108 A, f 51300 Hz,a50.408).

Fig. 6. Two qualitatively different quasiperiodic~period-one nonplanar rmsamplitude! experimental plots. Both take with a forcing amplitude ofI rms

50.110 and at the same forcing frequency:~a! Hopf branch and~b! isolatedbranch.


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in Fig. 6~b! which also shows for comparison a solution frothe Hopf branch@Fig. 6~a!#. The distinguishing characteristiis a peak~sharply pointed section of the trajectory! where theorbit is attracted towards the upper planar fixed point whis unstable~leading to the nonplanar solutions!. All theseobservations are consistent with the conjecture that thnew orbits correspond to the isolated branch solutions ofaveraged equations.

On the isolated branch, an increase in the forcing amtude results in a period doubling cascade~of the rms signal!which can lead to the formation of a chaotic attractor. Thisdemonstrated in the bifurcation sequence shown in Figwhich starts on the Hopf branch, jumps to the isolatbranch, and then period doubles to form a chaotic attracThis is followed by an inverse sequence of bifurcationwhich leads back to the Hopf branch. These observationsonce again consistent with the predictions made fromaveraged model. An interesting feature of these bifurcat

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Fig. 7. A bifurcation sequence showing the merging of the Hopf branchthe isolated branch~going through chaotic motions and ending in crisis!. ~a!The period-one solution on the Hopf branch.~b! Period-two motion.~c!Period-four motion; these are stable over a small parameter range and chard to observe because of parameter drift.~d! Back to period two with nochaotic window. The period two shows ‘‘characteristics’’ of the isolatbranch solution before it merges with the period one isolated branch orb~e!Period-one isolated branch orbit.~f! Period-two isolated branch orbit.~g!Chaotic isolated branch orbit.~h! Back to period-two isolated branch orbi~i! Period-one isolated branch orbit at end of the bifurcation sequence wgoes through chaos.~j! Isolated branch is destroyed by a crisis.


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sequences is the presence of period-two solutions wshare characteristics of both the Hopf branch and the isolbranch. Such solutions are not described in the simulatof Bajaj and Johnson.5,6 However, until an exhaustive searchas been carried out, we don’t know whether or not thresults are consistent with the averaged model.

Some of the plots in Fig. 7 provide an example of a coplex sequence of bifurcations first termed ‘‘bubbles’’ bKnobloch and Weiss.28 They are also seen by Bajaj anJohnson in their simulations of the averaged equations. Tis significant because these bubble sequences are assowith homoclinic orbits whose fixed point has eigenvaluwhich satisfy Shilnikov’s inequality, i.e., the real eigenvalis larger than the real part of the complex eigenvalue20

Shilnikov showed that in this circumstance there are hoshoes present in return maps defined near the homocorbit. This provides anecdotal evidence that the Shilnikmechanism is causing chaos in this parameter regime.

Figure 8~a! shows a chaotic time series from the striexperiment in thep2–q2 plane. The power spectrum of thtime series@see Fig. 8~b!# shows the continuous structurconsistent with either chaotic motion or colored noise.

1. First return map

Averaging over the fast oscillations reduces the dimensof the trajectory by one. Similarly, if a Poincare´ section canbe defined on this reduced trajectory then the trajectorfurther reduced to the orbit of a mapxn115 f (xn) and thedimension is reduced by one again. This map,f , called thefirst return map, is conveniently displayed by plottingxn

againstxn11 .If we take xn to be thenth point @measured at the inter

section of the chaotic trajectory and the line indicated in F8~a!# then we get the first return map shown in Fig. 8~c!. Thesimplicity of this map is striking and illustrates the low dmensionality of chaos in a string. It is a unimodal map vesimilar to the Logistic map22 ~the canonical example ofsimple deterministic chaotic system! and indicates that thestretching and folding of a horseshoe is causing the sensdependence to initial conditions in these chaotic time serDespite the complexity of the partial differential equatimodel first proposed by Kirchhoff~formally an infinite-dimensional dynamical system!, our study provides direcevidence that for some parameter regions the chaotictions of a string can be modeled by a simple unimodal m~a one-dimensional dynamical system!. This conclusion issupported by dimension calculations using a box-counalgorithm29 which indicates that the dimension of the orignal attractor is about 2.3, as well as by a more sophistica‘‘topological’’ analysis of these data sets.22

2. Banded chaos

Not all the chaotic time series in this parameter regishow a continuous unimodal first return map. In moststances, if the attractor appears soon after a period-two lcycle, then the period-two solution seems to dominatechaos and the strange attractor looks like a band abouperiod-two orbit@see Fig. 9~a!#. The unstable period-one solutions are not in the closure of the invariant set formedthe chaotic attractor. This appearance of a banded chaattractor, which is dominated by the period two oribt is simlar to that seen in the numerical solutions of the averaequations of Miles for spherical pendulum.30


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During the transition to chaotic behavior, the doubltorus usually breakes into a chaotic band in the immedneighborhood of the torus. We call thisbanded chaos. Thechaotic bands then become wider until they finally join a

Fig. 8. Experimental chaotic time seriesI rms50.11, f 051.38 kHz.~a! Eachpoint represents thep2 , q2 variables sampled once per forcing period. Treturn time for this time series is approximately 250 forcing periods.~b! Thepower spectrum.~c! The first return map at the Poincare´ section. Thexi

values are the intersection of the chaotic trajectory at the line shown in~a!.


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occupy the neighborhood of the period one orbit~undoubledtorus!. This sequence of motions is seen in both simulatioof the model equations and the experiment.

E. Crisis

As the forcing amplitude is increased beyond the pointonset of chaos in the isolated branch, the chaotic atttracan suddenly be destroyed by a boundary crisis.23,31 This isperhaps not surprising as Grebogiet al.23 show that thesecrises are common in systems which are close to unimomaps. Figure 10 shows a time delay embedded experimetime series of the planar rms amplitude before and aftercrisis point.

F. Intermittency

Transitions to intermittent chaos are seen in the stringperiment. The intermittency transition to chaos begins witperiodic state, then burst to an intermittent state as a pareter is increased above a critical value. These bursts apat seemingly random times, but become frequent as a coparameter is increased. Figure 11 shows an experimetime series during which bursts are occurring at a relativ

Fig. 9. Chaotic orbit showing ‘‘banded’’ chaos.~a! Chaotic time series soonafter the period-two limit cycle.~b! First return map showing the absencea period-one orbit resulting in a ‘‘banded’’ structure.


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high rate. This figure is one of the more complex typesbehavior which are observed at low damping. This intermtent behavior does not seem to be predicted by the equafor the averaged model.

Fig. 10. Boundary crisis:~a! nonplanar rms amplitude~in arbitrary units!plot just before a boundary crisis and~b! after boundary crisis, the ‘‘ghost’’of the strange attractor in~a! is clearly visible.

Fig. 11. Intermittency of the nonplanar rms amplitude,I rms528 mA, a50.1. Time is in units of forcing periods.


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G. Very low damping

Bajaj and Johnson conclude, from their observations ofbehavior of the averaged equations of motion, that for daming values smaller thana50.25 all the interesting motions ote string system are destroyed by boundary crises: ‘‘ . . .as the damping is lowered, the frequency over whichnon-planar complex motions exist decrease, so much sofor a50.25 the critical frequencies at which crisis occuessentially coincide withb1* andb2* and practically all ini-tial conditions lead to the lower planar steady-state conssolution.’’ In the above quoteb1* and b2* are the Hopfbifurcation ~creation and merger! points. Our experimenfinds more exciting dynamics in this parameter region. Fdamping values neara50.05, many complex nonplanar motions are found.

In the low-damping regime, persistent chaotic motionsobserved over a wide range of forcing amplitudes and deings. We observe both Lorenz and Ro¨ssler22 like chaos~seeFig. 12!. Some typical parameters for which these motioare found areDt51 s, f 051.3375 kHz,f 51.45 kHz with

d I rms534.7 mA. Lorenz like chaos ata50.090, andd I rms531.5 mA. Rossler like chaos ata50.095.

To plot these results we use the ‘‘differential-integral’’ embedding described by Gilmore and Lefranc.22 Specifically, ifx1( i ) is the sampled data, then this embedding is definedy1( i )5x( i )1exp(21/t)x1( i 21), y2( i )5x( i ), and y3( i )5x( i )2x( i 21), wheret is chosen to cover several oscillations, but not long enough to show a systematic drift~vary-ing mean value! in the embedded data.

If the forcing is increased the motion gets far more coplicated with multiple, coexisting attractors—many of thechaotic. The motions of the string in this low-damping rgime appear to mirror those of the string with a higher daming, modulated by a third, much lower, frequency.

V. CONCLUSION

We have examined the dynamics of a forced taut strand found good qualitative agreement with predictions fr

Fig. 12. Lorenz like chaotic attractor in the non-planar component of mofor a50.1. This data have been embedded from a single nonplanarseries using the differential-integral embedding.


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the averaged equations of motion in the high dampinggime. These equations are derived by truncating to the fdamental mode and averaging out the fast forcing time scbut allow for motion in both transverse directions with implicit coupling via longitudinal motion. Quantitative agreement was not present since we found a systematic shift inparameters required to observe bifurcations predicted byaveraged equation model. In addition we found a rich secomplex motions occurring at low damping and large detings which do not appear to occur in the averaged mode

It is possible that the model could be improved by incluing a more realistic description of the damping. Air restance and internal friction are not, contrary to the assumtions of the model, proportional to the velocity of the strinover the entire decay envelope. Altering the damping momay yield a more comprehensive agreement with theseperimental results. We should note, however, that BajajJohnson5 also compared parameter values for various bifcations in the the averaged equation model with the unaaged string model and also found that bifurcation paramvalues where shifted between these different models.

We have tried to make these data sets widely availableftp download, and some of the data described in this pahave been investigated by other researchers exploringtopics as nonlinear noise reduction, chaotic synchronizattopological analysis of chaotic time series, andon.21,22,32–40There is still much to be learned from the souof one string vibrating.

APPENDIX: STRING CONTROLLER ANDPARAMETER CONVERSIONS

To collect data for the string experiment we needsample the motion an integer number of times per forcperiod. An appropriate sampling trigger can be generawith a phase locked loop~PLL!, locking a high frequencysquare wave to a square wave of the same frequency aforcing sine wave. Unfortunately, such PLL’s can be vedifficult to design without phase drift.

Therefore we used a different approach. Instead of obting the sampling trigger from the forcing signal, the forcinsignal is derived from a high frequency ('100 kHz) squarewave from which the sampling trigger is also derived. Thhas many advantages but poses several technical problNamely, producing a sine wave from a square wave keepthe amplitude of the sine wave independent of the frequeof the square wave. This is important since a commonperimental procedure is to scan the forcing frequency whobserving the resulting behavior. The solution lay in a filmodule based on a National Semiconductor LMF60 CIN-sixth-order, switch-capacitor, Butterworth low-pass filtchip. This chip gives 36 dB per octave suppression abovcritical frequency f c which is determined by an externaclock input divided by 50. If this external clock has frequency times the desired sine-wave output frequency tthe cutoff frequency is 1.28 times the input square wafrequency. The filter then attenuates thenth harmonic com-ponent of the input square wave by a factor of 1022.81n, andthe deviation from sinusoidal of the output of the filter cancalculated by evaluating the filter function for the Fouricomponents of the input and summing these together.filtered out function becomes 4/p (cos(vt)2108.9cos(3vt)1...). This gives a noise level~from filter imperfections! ofless than 1026%.

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Signals from the strings are sampled and averaged intime on a digital signal processor designed and construfor the task at hand. The ADSP 2105 Digital Signal Procsor ~DSP!, made by Analog Devices Ltd., is the main prcessing unit. It is a 16 bit device with a pipelined HarvaArchitecture, a peak performance of 60 million operationsand an instruction set optimized for digital signal processiThe data are sampled at 16 bit resolution and average onDSP computer over one cycle~32 samples! to producep andq components. This averaged data are then transmittedhost computer~Macintosh SE/30! via a small computer system interface port. A more detailed description of the haware and software used in this experiment is given in Refand the paper by Brundell and Molteno.25

The parameters of the averaged equations of motiondimensionless. In order to investigate phenomena in therameter ranges examined by Bajaj and Johnson5,6 and tomake a quantitative comparison to the theoretical modelsneed to express these parameters as functions of easiltered experimental quantities. The parameters of interest

d f 0 , the frequency of free vibration~small amplitude! of thefundamental mode, controlled by adjusting the tensionlength of the string;

d Dt, the 90%–10% decay time for the amplitude of tstring once forcing is switched off. This determines tnatural damping which is equal to the decay rate towaequilibrium behavior divided by 2p f 0 . Dt is controlled bythe application of a silicone coating to the string asO’Reilly and Holmes;7

d f , the frequency of excitation; andd I rms, the rms excitation current~a measure of the strengt

of the excitation!.

The expressions relating the parameters used in the aaged equations of motion~a,b! to the experimental parameters shown in Table I are

a5ka f 0

DtI rms2/3 , b5

kb~ f 22 f 02!

I rms2/3 . ~A1!

The constants of proportionality, also in terms of the expemental parameters~see Table I! are given by

ka54 log 9l 2m

~2B2Ylf2!1/3, kb5

8l 2~p2m2r!1/3

~2B2Ylf2!1/3 . ~A2!

The parameter ranges investigated by Bajaj and [email protected],1# andbP@0,5#. With Dt50.1 s, these theo

Table I. Parameters of the string experiment. These are used to detethe formulas for conversion between experimental and numerical paeters.

Symbol Quantity Value

l length 0.07 ml F forcing length 0.045 mm mass per unit length 3.3931024 kg m21

r density 2.13104 kg m23

Y Young’s modulus 197 7783106 N m22

B Magnetic field strength 0.260.05 TDt 90%–10% decay time 0.1 sf 0 free vibration frequency 1.385 kHz

I rms forcing current 20–600 mA


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retical parameters correspond to the experimental paramin the range I rmsP@20 mA,600 mA# and f P@ f 0 , f 0

180 Hz#.The phenomenological damping inserted into the eq

tions of motion implies an exponential decay towardsresting state of the string. In the experiment damping is mcomplex, and it does not depend linearly on the velocity othe entire decay envelope. However, as we are using a lidamping model, we take a linear fit of the logarithm of tdecaying amplitude as the damping term used in the eqtions of motion. This linear fit to a nonlinear function coube the cause of some discrepancies between the modeexperiment.

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39K. Judd and M. Small, ‘‘Towards long-term prediction,’’ Physica D136,31–44~2000!.

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manufac-Companybox ato

Steam Engine Half Model. Up to the middle of the 20th century, introductory textbooks had cutaway drawings of steam engines, and apparatusturers responded with half-models showing the working parts. This model of a locomotive engine can be found in the 1925 Chicago Apparatuscatalogue for $6.00. This is a fairly big device, 38 cm long and 18 cm high. Shifting the reversing gear back and forth moves the slider in the steampthe cylinder so that the steam will be admitted first to the front or the back of the piston. The model is in the Greenslade Collection.~Photograph and notesby Thomas B. Greenslade, Jr., Kenyon College!


an experimental investigation into the dynamics of a...

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