Physica D 240 (2011) 814–824

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Physica D

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Experimental versus theoretical robustness of rotating solutions in aparametrically excited pendulum: A dynamical integrity perspectiveStefano Lenci a,∗, Giuseppe Rega b

a Department of Architecture, Buildings and Structures, Polytechnic University of Marche, via Brecce Bianche, 60131 Ancona, Italyb Department of Structural and Geotechnical Engineering, Sapienza University of Rome, via A. Gramsci 53, 00197 Rome, Italy

a r t i c l e i n f o

Article history:Received 21 August 2010Received in revised form21 December 2010Accepted 22 December 2010Available online 4 January 2011Communicated by G. Stepan

Keywords:Parametric pendulumRotating solutionsExperiments vs theoryRobustnessDynamical integrityPractical stability

a b s t r a c t

The objective of this paper is showing how global safety arguments can be fruitfully used to interpretexperimental results of a pendulum parametrically excited by wave motion. In fact, the results ofan experimental campaign developed with the aim of simulating sea-waves energy production by aparametric pendulum show that rotations exist in a region which is smaller than the theoretical one.This discrepancy can be partially attributed to the experimental approximations and constraints, but ithas a deeper theoretical motivation. By comparing the experimental results with the dynamical integrityprofiles we have found that experimental rotations exist only where a measure of dynamical integrityaccounting for both attractor robustness and basin compactness is large enough, so that they can supportexperimental imperfections leading to changes in initial conditions.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Among various applications of nonlinear dynamics to advancedtechnologies is the extraction of energy from sea waves by meansof a parametric pendulum, which was proposed by Prof. MarianWiercigroch at the beginning of the present millennium [1].The basic idea consists of having a pendulum on a pontoon.The sea waves generate a vertical motion of the buoy at agiven (average) frequency. In addition to the rest position, thisparametric excitation produces non-equilibrium solutions, namelyoscillations and rotations, which compete with each other. If onesucceeds in getting robust rotations, a generator applied to thepivot of the pendulum may provide electrical energy.

As with all clever ideas, while being conceptually simple, thisis very difficult to realize in practice (‘‘Genius is one percentinspiration, ninety-nine percent perspiration’’ Thomas A. Edison,Harper’s Monthly, 1932), so that to date it is still at an early stage.The aim of this work is to perform a step ahead toward its practicalimplementation.

The study of the nonlinear dynamics of a mathematicalpendulum is one of the oldest scientific topics, and dates back at

∗ Corresponding author.E-mail addresses: Lenci@univpm.it (S. Lenci), Giuseppe.Rega@uniroma1.it

(G. Rega).

0167-2789/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2010.12.014

least to Galileo, Huygens and Foucault, and nowadays its interestextends from science to philosophy and education [2]. Practicallyalmost all cases have been studied in the literature, includingcoupled pendulums [3] and related phenomena [4,5], which couldbe useful in producing a larger amount of energy. Since it is notpossible to report on all contributions, we focus only on the caseof interest for our purposes, i.e. the pendulum rotations underparametric excitation, i.e. vertical displacement of the pendulumpivot. We do not consider different types of motion (nonlinearoscillations, quasi-periodic orbits, chaos, etc.), and different typesof excitation.

Initially, the investigations were directed toward full under-standing of the nonlinear dynamics of a parametrically excitedpendulum, by adding to generic studies on the subject [6–9]researches with the energy extraction hidden in the back-ground [10–14]. Both purely numerical [8–10,13,14] and analyt-ical [11,12] studies have been carried out. For a more detailedreview of the literature we refer to the introduction of [12].

Side by side with theoretical investigations, the experimentalapproach has been developed, both for single [15,16] andmultiple [17,18] generic pendulums. Restricting attention to theexperiments developed within the energy production project, wenote that initially a shaker has been used to provide verticalmotion of the pivot [19]; these authors faced the problem ofpendulum–shaker dynamical interaction, which was importantin the experiments, and which is somewhat propaedeutical forfurther experimental investigations.

S. Lenci, G. Rega / Physica D 240 (2011) 814–824 815

Successively, the first pendulum-buoy prototype was put ina wave flume at the Aberdeen Laboratories [20], and wave-induced rotations and tumbling chaos were observed [20]. Then,a systematic experiment was performed at the PolytechnicUniversity of Marche, Ancona.

An extensive experimental campaign has been performed byvarying the frequency and the amplitude of thewaterwaves,whichcan be easily done with the laboratory facilities. The specific goalwas to detect rotating solutions – those necessary to generateenergy by the considered mechanism –, so that in the experimentwe did not pay attention to nonlinear oscillations and other typesof motion. This goal has been challenging, since rotations havea small basin of attraction with respect to competing attractors(at least for ‘small’ excitation frequencies), and so they are verydifficult to be realized and maintained; but eventually we weresuccessful with it. The details of the experimental work will bepublished separately [21].

In the excitation frequency–amplitude parameter plane, rota-tions have been found within a strip strictly contained in the re-gion of theoretical existence of rotations. In spite of many efforts,we were unable to enlarge this strip up to the whole theoreticalregion, and this looked unpleasant at the first instance.

This paper aims at finding a theoretical justification of thisexperimental evidence, and at elucidating the apparent drawback.By using dynamical integrity arguments, in fact, we are ableto show not only that the experimental observations are fullyexplained, but also that we could not expect much better resultseven if improving the experimental precision and control.

Dynamical integrity consists of studying the robustness ofattractors and the safety of their basins in phase space. Thebasic idea, introduced earlier by Thompson [22,23], recentlyreconsidered by Lenci and Rega [14,24,25] and applied with acertain success to a multiple d.o.f. system in [26], is that stabilityis not enough for observing attractors in real cases. In fact, if thebasin of attraction is ‘‘small’’ (in an appropriate sense), there is nohope to observe experimentally the associated attractor, since evensmall (and unavoidable) experimental uncertainties will certainlylead the response out of the basin of attraction. As a consequence,the dynamics could settle onto a different, more robust, attractor,or a chaotic transient leading to the same attractor,with an ensuingescape from the basin of the reference one, which could trigger aloop leading to jumps onto different attractors or to chaotic beatingphenomena on the same attractor. All these situations are usuallyunwanted and therefore they must be carefully detected.

From an operative point of view the dynamical integritybasically consists of studying the properties of the safe basin (to beproperly defined andmeasured) and their evolutionwith a varyingdriving parameter. The whole matter requires some attention, andwe refer to [24,25] for a detailed discussion. In this paper the safebasin is the basin of attraction of the clockwise rotating solution,and the considered measure of integrity is the Integrity Factor(IF ), which is the normalized radius of the largest circle entirelybelonging to the safe basin, and which has been shown to beappropriate for rotations [14]. This measure balances the oppositerequirements of being computationally simple, and of ruling thefractal part of the safe basin out of the integrity evaluation [25].

Several integrity profiles of rotating solutions are determined,extending those reported in [14], and the whole dynamicalintegrity scenario is clarified. It is shown that the IF is large enoughonly close to the stripwhere rotations are detected experimentally,a fact that shows why rotations cannot be observed in otherparameter regions, where the dynamical integrity is not sufficientto guarantee their practical stability.

The paper is organized as follows. After briefly presentingthe pendulum governing equation (Section 2), in Section 3 theexperimental set-up is illustrated together with some properties

Fig. 1. The pendulum.

and constraints of the experiment. The experimental results aresummarized in Section 4,while in Section 5 the dynamical integrityof the parametric pendulum is discussed in detail and used tojustify the experimental observations. The paper ends with someconclusions (Section 6).

2. The mechanical model

The planar equation of motion of a stumpy pendulum (Fig. 1)subjected to a vertical displacement Y0(T ) of its axis of rotations Ois

I0d2θ

dT 2+ C

dθdT

+

g +

d2Y0

dT 2

Sx0 sin θ − Sy0 cos θ

= 0, (1)

where I0, Sx0 and Sy0 are the polar moment of inertia and thestatic moments of the pendulum with respect to O, C is the lineardamping constant, g the acceleration of gravity, and θ the angulardisplacement, which is zero in the downward g direction (Fig. 1)and positive anti-clockwise.

Without loss of generality (a rotation of reference frame ispossibly required) it is possible to assume that the axis Y passesthrough the centre of mass of the pendulum, so that θ = 0corresponds to the rest position. In this case Sy0 = 0, and Eq. (1)simplifies accordingly. By defining

ω0 =

gl, l =

I0Sx0

, t = ω0T ,

y0 = Y0Sx0I0

, h =C

√gI0Sx0

,

(2)

where ω0 is the natural pulsation (the natural frequency is f0 =

ω0/2π ), l is the nominal length of the pendulum, i.e. the distancefrom the pivot at which to concentrate the mass of the pendulumto have the same dynamics, t, y0 and h are dimensionless time,vertical displacement of O and damping coefficient, respectively,we obtain

θ + hθ + (1 + y0) sin θ = 0, (3)

where the dotmeans the derivativewith respect to the dimension-less time t .

For a point-mass mathematical pendulum we have I0 = ML2,Sx0 = ML and l = L, so that the well known expressionω0 =

√g/L

is obtained. For a rigid bar with uniformly distributed mass wehave I0 = ML2/3 and Sx0 = ML/2, which give l = (2/3)L andω0 =

√1.5

√g/L.

When the imposed vertical motion of the pivot point O is har-monic, Y0(T ) = −A cos(2π fT ) or y0(t) = −(ASx0/I0) cos[(2π f /ω0)t], which is the case for regularmonochromaticwaves, we have

θ + hθ + [1 + p cos(ωt)] sin θ = 0, (4)

816 S. Lenci, G. Rega / Physica D 240 (2011) 814–824

Fig. 2. Design drawing (a) and picture (b) of the pendulum.

Fig. 3. The buoy cross-section. The buoy length, perpendicular to the picture, isb = 950 mm.

where

ω =2π fω0

=ff0

, p =Aω2

l, (5)

are the dimensionless ratio between the external and naturalfrequencies, and the dimensionless excitation amplitude, respec-tively.

3. The experimental set-up

In this section we summarize the experimental rig we havebuilt. More details, and the motivations which are behind somechoices, can be found in [21]. The experiments were conducted inthe wave flume of the Hydraulics Laboratory of the PolytechnicUniversity of Marche, Ancona, Italy. It has length 50 (m), width1 (m) and height 1.3 (m). The maximum level of the water is1 (m). The wave generator is a HR Wallingford, and it is ableto produce monochromatic and coloured waves by an actuatormoving a vertical aluminium plate in the horizontal direction.

After some preliminary tests, we chose a PVC pendulum bar,having mass of 0.23 kg, with an added steel mass of 0.3 kg at theend of the bar (Fig. 2). This pendulumhas L = 653mm (Fig. 2), I0 =

164 114 kgmm2 and Sx0 = 280 kgmm, so that l = 586mm, whichprovides ω0 = 4.090 and f0 = 0.651 Hz. The flume characteristicspermit us to perform experiments for frequencies above and belowthe parametric resonance f = 2f0 = 1.302 Hz.

The buoy is made of polyurethane (density ρ = 35 kg m−3).The length is b = 950 mm, while the height is h = 200 mm, avalue which permits us to have no waves overcoming the buoy,

Fig. 4. The experimental rig on the flume.

and to reduce as much as possible the buoy weight. The width isd = 160mm, allowing us to have a sufficient floating force but nota too heavy buoy, and not to disturb the wave regularity. Finally, acut inclined at 20° is done to increment the floating force (Fig. 3).

Based on the previous considerations, the pendulum was firstdesigned (Fig. 2(a)) and then realized (Fig. 2(b)). Two verticalsupporting rigid bars, made of steel and with a tubular rectangularcross section, connect the axis of rotation with the buoy; on theexternal sides of these bars there are (i) eight wheels which movealong two vertical runways (one per side), to facilitate the verticalmovement and to eliminate the unwanted rotation around theaxis perpendicular to the pictures in Fig. 2, and (ii) four pivots,which run into apposite vertical guides located in the supportingframe (Fig. 5) and needed to have vertical motion only. The axis ofrotation of the pendulumwas connected by a fan belt to an encoderwhich measures the angular position.

The pendulum described above was inserted in an aluminiumframe built over the flume (Fig. 4), whose major elements are thetwo vertical bars with guides for the pivots and runways for thewheels. A lot of attention was paid to the details of the wholerig in order to guarantee that the pendulum axis can move onlyvertically.

The data we needed to measure were the angular positionof the pendulum bar, which is the system response, and thevertical position of the pendulum pivot O (Fig. 1), whichactually constitutes the system excitation. We did not measurewave amplitudes and frequencies, since they were imposedindependently by the wave generator. The rotation was measuredby the encoder (Figs. 2 and 5), while the vertical displacement was

S. Lenci, G. Rega / Physica D 240 (2011) 814–824 817

Fig. 5. The measurement system.

measured by a wire potentiometer mounted on one of the fixedvertical guides (Fig. 5).

4. Experimental results

A summary of the experimental results needed for the aim ofthis paper is reported in this section. Before proceeding with theirillustration, various comments are required.

A first remark follows from the experimental observation,which is also theoretically supported, that pendulum rotationsrequired harmonic excitation. This had practical consequences. Infact, when the waves reflected from the end of the flume (whichare small due to the anti-reflection apparatus, but not negligible)arrived at the pendulum buoy, they interacted with the principalwave so that the sumwas no longermonochromatic and destroyedrotations. To limit this unwanted effect we put the experimentalrig as close as possible to the wave generator. With this trick thereflectedwave arrived after about 40 s,which is thus themaximumreliable time interval measured in the experiments. Fortunatelyenough, this time is sufficient for the onset, observation andmeasurement of the searched steady state rotations.

Then, we note that the initial conditions θ0 and θ0 of thependulum motion were imposed manually by the operator, whoactually also fixed the initial time t0, i.e., the initial phase shiftbetween the pivot and the pendulum motions, which plays acertain role. After a preliminary training, the operator got anexpertisewhich permitted him to optimize the starting phasewithrespect to the wavelength. The initial conditions θ0 and t0 werethen measured by the data acquisition system, so that their valueswere known, although they could not be imposed exactly. In fact,this would have required much more complicated and expensiveexperimental tools, which are out of the scope and of the budget ofthe present work.

To compare the experimental results with the numerical sim-ulation we need the damping coefficient, which takes all dissi-pating phenomena into account, on average. It was determinedexperimentally bymeasuring the decay of free oscillations startingfrom an angle θ0 = −90°. The maximum amplitude reached wasθmax = 87.7°. Using these values and Eq. (3) with y0(t) = 0 wefound that the value of hwhichminimizes the differences betweenexperimental and theoretical results is h = 0.015. This value willbe used in all forthcoming numerical simulations. Note that it ismarkedly different from the value h = 0.1 used in [8] and in [14],whose results cannot be then directly used in this work, at leastfrom a quantitative point of view.

4.1. Time histories

The whole experimental campaign was based on the construc-tion of several time histories, obtained by measuring the pivotdisplacement and the pendulum rotation. Thus, they need to bepreliminarily discussed before going on.

A representative example of the time histories is reported inFig. 6, which corresponds to ω = 1.84, i.e. to the imposed wavefrequency f = 1.2 Hz, and to a nominal amplitude of the wavesimposed at the generator of 60 mm.We can clearly distinguish thefollowing different phases and behaviours.

(1) At T ∼= 12 s the first travellingwave produced by the generatorarrives at the buoy.

(2) From T ∼= 12 s to T ∼= 25 s the transient behaviour develops,with an increasing amplitude, until, at T ∼= 25 s, a steady statewave supports the buoy.

(3) During the transient the operator manually brings thependulum from the rest position to the chosen initial position(first isolated peak in the time history of Fig. 6(b)).

(4) At T ∼= 33 s, after the steady state waves have been set, theoperator launches the pendulum.

(5) From T ∼= 33 s to T ∼= 75 s we have ‘regular’ pivot motion(Fig. 6(a)) and pendulum rotation (Fig. 6(b)). This is the goodpart of the time histories, and it is actually the sole one usedfor interpreting the pendulum dynamics.The motion of the pivot looks non-regular, but its fast Fouriertransform (FFT) (Fig. 7(a)) confirms that in fact it is basicallya harmonic oscillation at f = 1.2 Hz, i.e. exactly at theimposed wave frequency. There is a small superharmonic atf = 2.4 Hz, but its amplitude is very small and can beneglected, as confirmed by the fact that it does not influencethe outcome. There is also an irregular, low frequency (about1/7 to 1/8 of the main frequency and thus out of the scale ofFig. 7(a)) amplitude modulation, likely due to the buoy-waverelative floating motion, which is clearly seen to occur in thevideoclips recorded during the experiments.The measured average pivot amplitude is about A = 48 mm,corresponding to p = 0.262, to be compared with the nominalimposed wave amplitude of 60 mm. This A has been obtainedby measuring (Amax − Amin)/2 in each cycle, and by averagingon the ∆ time/period = ∆ time× f cycles of ‘good’ behaviour(in the case of Fig. 6 we have (75 − 33) × 1.2 ∼= 50 cycles).The pendulum angular velocity has an f = 1.2 Hz oscillation ofamplitude about 1.1 rad s−1 around the average value of about7.5 rad s−1, as confirmed by the enlargement of Fig. 8 whichshows that the velocity oscillates from a minimum of about6.4 rad s−1 to a maximum of about 8.6 rad s−1. This agreeswell with the theoretical predictions [11] which prescribe thatfor very small damping (as in the considered case, being h =

0.015) and for large frequencies (which actually is not exactlythe present case, but can be used as a first guess) the velocitycan be approximated by (β0 is an unessential phase shift)

θ (t) = ω +1ω

cos(ωt + β0) + · · · , (6)

which in the present dimensional case becomes (l =

586 (mm))

dθdT

= 7.54 + 2.66 cos(7.54T + β0) + · · · , (7)

and the harmonic oscillation around the average velocity hasjust frequency f . The difference between 1.1 (experimental)and 2.66 (theoretical) is due to the fact that (6) is valid onlyfor large frequencies, which does not correspond exactly to theconsidered case. Furthermore, the experimental amplitude 1.1varies quite a lot from test to test.

818 S. Lenci, G. Rega / Physica D 240 (2011) 814–824

Fig. 6. Time histories (a) of the pendulum pivot Y0(T ) and (b) of the angular velocity.

Fig. 7. Fast Fourier Transform of the signal of Fig. 6(a) (upper) for (a) T = 25 → 70 (s) and (b) T = 70 → 120 (s).

Fig. 8. A three period enlargement of the time history of Fig. 6(b).

Also for the pendulum velocity the oscillations around theaverage value have a low frequency amplitude modulation,which again can be at least partially attributed to the buoyfloating motion.

(6) At T ∼= 75 s the reflected wave arrives at the buoy,and suddenly destroys the harmonicity of the pivot motion,as confirmed by the FFT reported in Fig. 7(b) where otherfrequencies are observed close to the principal one, likelydue to a beating phenomenon between the dominant and thereflected wave possibly ensuing from a small frequency shiftat reflection.

4.2. Experimental behaviour chart

In this subsection we investigate the effects of the excitation(wave) frequency and amplitude on the rotational behaviourof the pendulum. More precisely, we aim at detecting inthe (ω, p) parameter space the region where rotations canbe found experimentally, irrespective of their robustness, i.e.even if they exist for just one single initial condition (in fact,

experimental robustness with respect to initial conditions isconsidered in [21]). With this objective, we spanned the frequencyrange ω ∈ [1.15; 2.30], and for each considered frequency weapplied different increasing amplitudes, paying special attentionto determining the lowest and the highest amplitudes whererotations can be found experimentally.

The 34 considered couples of parameters are reported in Fig. 9,where the region of existence of experimental rotations is clearlyseen to be a strip of finitemagnitude, shrinking for low frequencies.Fig. 9 is the most important result of this section and of the wholeexperimental campaign.

Before proceeding, it should be noted that each rotation pointwas repeated at least twice to assess its reliability. Furthermore,as regards the oscillation points, we tried several initial conditionsbefore deciding that rotations do not exist. Thus, strictly speaking,we cannot say that in the oscillation points rotations do not exist,but only that we were not able to find them, in spite of repeatedattempts. They could exist in principle, but evidently they are notso robust as in the case of rotation points.

S. Lenci, G. Rega / Physica D 240 (2011) 814–824 819

Fig. 9. The behaviour chart, i.e. the map where rotations can be foundexperimentally in the (ω, p) parameter plane.

Fig. 10. Thebifurcationdiagramof the rotating solution forω = 1.3 andh = 0.015.

Several repetitions were made also to assess the reliability ofthe two detected PD points. Since after each single test we hadto stop the wave generator and wait for the water to recoverthe flat rest position (which required a certain time, more than20 min), we see how the whole experimental campaign was verytime consuming, even without considering the initial pendulumconstruction and set-up. This showswhywewere not able to checkmore points.

To better understand the experimental scenario it is usefulto report the theoretical behaviour of the rotating solutions asprescribed by (4). A numerical bifurcation diagram for a fixed ωand for increasing p is reported in Fig. 10, wherewe see that period1 rotations appear by a saddle-node (SN) bifurcation at a certainamplitude threshold pSN . Then, they have an interval of existenceand stability, which ends up with a period-doubling (PD) localbifurcation at pPD. After pPD the rotation still exists, but it is nolonger stable (and thus it is not reported in Fig. 10). The stabilityis ‘‘captured’’ by another rotation with a double period, born atthe PD bifurcation, which successively undergoes a cascade of PDbifurcations accumulating onto a boundary crisis global bifurcationat pBC , where the rotations definitely disappear [14].

For pSN we have the following analytical expression, whichcomes from the Melnikov theory for subharmonic orbits [6](see [27] for the case with a more general excitation). For a givenω solve the equation

π

ω=

1kK1k

(8)

with respect to k = k(ω) ∈]1, ∞[. In (8) K(.) is the completeelliptic integral of the first type, while H = 2k2 − 1 is the energy(Hamiltonian) of the undamped unforced system. The critical

threshold is then

pSN(ω) = h4k

πω2E1k

sinh

ω

kK

√k2 − 1k

, (9)

which is seen to agree very well with the numerical results. In(9) E(.) is the complete elliptic integral of the second type. Ananalytical approximation of pPD was obtained in [12].

The previous scenario is remarkably captured by the experi-mental data of Fig. 9. In fact, for each value of the frequency therotation is seen to appear at a certain amplitude threshold. Further-more, for the first two frequencies, also the first period doublingbifurcation is experimentally detected, a fact that was particularlyencouraging.

By building several bifurcations diagrams for different valuesof ω it was possible to determine the curves pSN(ω), pPD(ω) andpBC (ω), which are also reported in Fig. 9.

The first observation which follows from Fig. 9 is that thetheoretical ptheSN and the experimental pexpSN , which can be approx-imately obtained by joining the red square dots, share the samequalitative behaviour, but the latter is significantly higher than theformer. This has variousmotivations. In fact, as previously said, thereal pexpSN would be certainly lower, and closer to ptheSN , if we wereable to test many more initial conditions. Furthermore, the actualdamping could be (slightly) larger than that measured experimen-tally, likely as a consequence of aerodynamic dissipation during ro-tations, and this would rise ptheSN . Then, of course, there are alwaysexperimental uncertainties which play a role. But these are not themain motivations. In fact, in Section 5 it will be shown by globaldynamics arguments [14] that the theoretical threshold will neverbe obtained even in extremely careful and controlled experiments,since it is not robust enough.

Concerning the upper threshold of stability pPD, we see thattheoretically it is an increasing function of ω. We were able toexperimentally detect this threshold for low excitation frequencies(an upper bound approximation of pexpPD is obtained by joining theblack circle dots of Fig. 9). As the wave frequency ω increased, wereached the limit due to the wave flume characteristics, which didnot allow us to have amplitudes larger than 70 mm, this being anexperimental constraint. But, as in the case of the lower bound,this is not enough to completely justify the experimental results. Infact,wewere not able to find rotations even for amplitudes allowedby the flume. Once more, this fact will be theoretically justified inSection 5 in terms of dynamical integrity.

For low excitation frequencies, both the theoretical and theexperimental results agree very well in showing that the ranges ofexistence pBC–pSN and stability pPD–pSN of simple rotations shrinkto zero for a certain frequency. On the contrary, for large excitationfrequencies we have that ptheSN is a nearly constant function of ω (inthe considered frequency range, this is no longer true for higherω, see Eq. (9)), while pthePD is an increasing function of ω, so thatpthePD –ptheSN is increasing.

Thus, the theoretical results, based on the classical concept ofstability, suggest that it would be better to use large excitationfrequencies to have simple rotations. This general indication,however, has to be complemented with the modern dynamicalintegrity point of view (Section 5), which agrees with theexperimental results in showing a smaller excitation amplituderegion of practical existence of rotations, for whatever frequency.

The main conclusions which can be drawn from Fig. 9 is that,above a certain frequency threshold, experimental rotations havea well defined region of existence in the parameter plane. This issmaller than the theoretical one but in any case large and robustenough since it spreads over a ‘large’ frequency range, thus beingpossibly able to account, up to a certain extent, also for wavefrequency modifications as they occur, e.g., in transients or in seawaves.

820 S. Lenci, G. Rega / Physica D 240 (2011) 814–824

Fig. 11. The integrity profile of period 1 rotation for h = 0.015 and ω = 1.3.

5. Theoretical justification of experimental results

The main result of the previously described tests is Fig. 9,which shows the experimental robustness of the rotating solutionsproduced by wave motion in parameter space. While this pictureis very satisfactory from an experimental point of view, it deservesmore investigation from a theoretical/numerical point of view. Infact, it is apparent that the experimental region of existence andstability is a subset of the theoretical one, which is delimited by pSN(below) and pPD (above). Although this difference is partially dueto the experimental uncertainties, and in particular to the fact thatwe were not able to impose exactly the initial conditions and thatreal damping could be slightly larger than that used in simulations,there is a deeper theoretical reason lurking in the background,whose explanation is the subject of this section and the main goalof the paper.

5.1. Dynamical integrity and integrity profiles

The key tool for understanding why the rotating solutions canbe practically observed only in a subset of the stability domain isthe integrity profile, i.e. a curve which reports how the IF varies forincreasing excitation amplitude p. We recall that in this paper thesafe basin [24,25] is the basin of attraction of the clockwise rotatingsolution, and the considered measure of attractor robustnessand basin integrity is the Integrity Factor (IF ), which is thenormalized radius of the largest circle entirely belonging to the safebasin [14,24,25] (examples are given forward).

Various integrity profiles for rotations of pendulum have beenobtained and carefully discussed in [14] for the damping coefficienth = 0.1. In this work we have experimentally measured h =

0.015, so we need to rebuild them, since the smaller dampingentails quantitative differences as well as some minor qualitativedifferences (roughly speaking, the dynamical behaviour is richerin phenomena with less damping). Furthermore, while in [14]the integrity of rotations is confronted with the integrity ofoscillations, here we focus only on rotations, and do not usedifferent integrity measures, as done in [14].

A representative integrity profile for ω = 1.3 is reported inFig. 11, where IF has been normalizedwith respect to itsmaximumvalue (accordingly, themaximum value of the curve is IF = 1). Theassociated bifurcation diagram is reported in Fig. 10.

For increasing amplitude p, we can clearly identify the followingsuccession of events and properties.

• At p = pSN ∼= 0.0476 the period 1 rotating solution (R1) appearsthrough a saddle–node (SN) bifurcation. This is the startingpoint of the integrity profile.

• Just after the attractor is born, its basin of attraction enlargesaround it; accordingly, the integrity profile increases.

• At p = p6 ∼= 0.080 a rotation of period 6 (R6—the velocityhas a period which is 6 times that of the excitation) appears, bymeans of a SN bifurcation, inside the basin of attraction of themain rotation. Although the basin of R6 is very small (Fig. 12(a)),the fact that it is inside the basin of R1 (even though it is closeto the relevant boundary) entails an instantaneous decrementof the compact part of the safe basin of R1, and the associatedsudden reduction of its IF .

• Soon after being born, R6 undergoes a classical period doublingcascade, and disappears at about p ∼= 0.090, likely by aboundary crisis (numerically it is difficult to detect thisphenomenon since it occurs in a very small part of the phasespace). The former basin of attraction of the disappearingsolution is (re)captured by the main rotation, which in factincreases its robustness and recovers (and slightly increases)the former IF .

• At p = p5 ∼= 0.100 a period 5 rotation (R5) appears, again bya SN bifurcation and again inside the basin of attraction of themain rotation (Fig. 12(b)). Since R5 ismore robust than R6, thereis a larger fall down of the integrity profile.

• Like R6, also R5 undergoes a classical period doubling cascadeand disappears at about p ∼= 0.110 by a boundary crisis.The former basin of attraction of the disappearing solutionis partially recaptured by the main rotation, which slightlyincreases its IF up to a value which is however well belowthe value before the sudden fall. This is due to the fact that afractalization from out of the basin is proceeding for increasingp, which erodes from outside the compact part of the basin ofthe main rotation, thus smoothly decreasing its IF .

• In the p-range 0.130 ÷ 0.140 there is another phenomenonlike the two just described, which entails a sudden reductionof IF and a subsequent recovery. In spite of careful numericalsimulations, we were not able to detect the secondary attractorlikely responsible for this phenomenon.

• At p = p3 ∼= 0.190 a period 3 rotation (R3) appears by a SNbifurcation inside the basin of attraction of themain rotation R1.It is more robust than the rotations R6 and R5 previously seen,and accordingly there is a large decrement of IF . The suddendecrease due to R3 was also observed in [14] for h = 0.1, whilethe other minor attractors were not present in that case.

• The R3 has an interval of existence and stability which is largerthan those of R6 and R5, but it is in any case small. Thus, itsuddenly disappears and leaves themain rotation R1, which hashowever a merely residual IF , since the disappeared attractorhadpreviously tangledwith the surrounding fractal part, so thatintegrity is definitely lost.

• After p ∼= 0.200 the dynamical integrity is residual. The basinof attraction is definitely small and almost completely eroded.

• At p = pPD ∼= 0.298 the rotating solution R1 loses stabilitythrough a period doubling bifurcation (Fig. 10). A period 2rotating solution appears and captures the stability of theformer rotation, which still exists but is no longer stable. Thereare no effects on the integrity profile, since the integrity is justresidual in this range.

• At p = pBC ∼= 0.369 the path ensuing from the rotationsolution, which previously underwent a period doublingcascade, disappears by a boundary crisis. This is the last pointof the integrity profile.

For the purpose of interpreting Fig. 9 the main property ofFig. 11 is that the IF of the period 1 rotation is high only inthe central part, where the attractor is relatively robust and itsbasin is substantially uneroded. More precisely, we can distinguishbetween two different zones, reported in dark and light grey,respectively, where the integrity is close to its maximum value

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Fig. 12. Basins of attraction for h = 0.015, ω = 1.3 and (a) p = 0.085 (b) p = 0.105. In addition to the principal attractors R (rest), P2 (period 2 oscillation) and R1 (period1 rotation), it can be seen in (a) the (very small) basin of the secondary R6 attractor, and in (b) the (slightly larger) basin of the secondary R5 attractor.

Fig. 13. Integrity profiles of period 1 rotation for h = 0.015 and (a) ω = 1.20, (b) ω = 1.50, (c) ω = 1.80 and (d) ω = 2.00.

IF = 1, or it is on a ‘‘plateau’’ around the average value IF = 0.7.It is clear that out of this central part the dynamical integrity ismarginal (small and eroded basin), and thus the attractor cannotbe observed in practice. This provides the theoretical justificationfor experimentally observing rotations only in the central strip ofthe excitation amplitude range.

This justification is not limited to the case ω = 1.3, sinceintegrity profiles built for different values of the excitationfrequency share the same qualitative properties, as shown inFig. 13. Note that the maximum of each curve is obtained at theend of, or just after, the initial steep part, in proximity to the pointwhere p5 is born.

It is worthwhile to remark that, looking at the dynamicalintegrity, not only do we understand why we have not observedrotations for large excitation amplitudes, which is somewhatexpected (and also partially related to the fact that in experimentswe have a constraint on the maximumwave amplitude supportedby the generator), but also whywe have not observed rotations forexcitation amplitudes just above the SN bifurcation, which ismuchless intuitive.

To close this subsection it is useful to stress the importanceof secondary attractors, which variously affect the reduction of

robustness of the main attractor and/or of integrity of its basin.They are completely lost by a local analysis, also of path-followingor brute force bifurcation type (see, e.g., Fig. 10), and this furtherstresses the importance of a global analysis for a modern andreliable approach to system dynamics.

5.2. Comparison of numerical and experimental results

In the previous subsection we have justified by dynamicalintegrity arguments the experimental evidence that rotations areobserved only in the central part of their amplitude range ofexistence. In this subsection we proceed with a more accuratecomparison between numerical and experimental data.

We start by noticing that the integrity profiles of Figs. 11 and13 have been normalizedwith respect to theirmaximumvalues (infact, all curves havemaximum IF = 1), and thus they can be barelycompared. The comparison is possible only from a qualitative pointof view, i.e. only as far as the ‘shapes’ of the curves are considered.That a quantitative comparison is not possible can be foreseen alsoby the different scales of the abscissa axis.

Actually, the basins of attraction of rotations for low excitationfrequencies are smaller than those for large frequencies. An

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Fig. 14. The basins of attraction of period 1 rotation for h = 0.015 and (a) ω = 1.2, p = 0.07, (b) ω = 1.8, p = 0.23, and (c) ω = 2.2, p = 0.30. R1 = anti-clockwiserotation, R2 = clockwise rotation, P2 = period 2 oscillation, R = rest position.

Fig. 15. The surface IF (p, ω). Contrary to Figs. 11 and 13, here the largest circle used in the construction of IF has been normalized with respect to a unique value for allfrequencies. (a) and (b) are just different views of the same surface.

example is reported in Fig. 14, where the basins and the associatedcircles used in the IF are reported. In all three cases the amplitudep is that corresponding to the largest circle, i.e., to the largestdynamical integrity for that excitation frequency ω (and thus theycorrespond to IF = 1 according to Figs. 11 and 13).

The difference between Fig. 14(a)–(c) is remarkable, both inabsolute value (the circle of Fig. 14(c) is 4.4 times larger thanthat of Fig. 14(a)) and with respect to the competing attractors.This suggests that for a quantitative comparison the circles fordifferent frequencies must be normalized with respect to a uniquevalue. This has been done, and the results are reported in Fig. 15,where the surface IF (p, ω) is reported and the increment of IF forincreasing frequency is clear. More precisely, we can say that in theconsidered range IF increases almost linearlywith respect toω, seeFig. 15(a). The variation of IF for increasing amplitude ismuchmorecomplex and irregular: the corresponding ‘‘three-ridge’’ behaviourof Fig. 15 can be appreciated also in Fig. 13(c)–(d).

Themain conclusionwhich can be drawn from Figs. 14 and 15 isthat if one is interested in rotating solutions, as in the present case,large excitation frequencies are preferable. This prescription isfurther supported by the fact that the SN thresholdwhere rotationsappear does not increase somuch formoderately large frequencies(but not for very large frequencies, see Eq. (9)).

Having understood the overall dynamical integrity behaviour,we can proceed with the justification of the experimentalresults by also getting some hints on more involved quantitativecomparisons with numerical data. We report in Fig. 16 the contourplot of IF (p, ω) together with the experimental points of Section 4.The contour curves are partially smoothed for graphical reasons.

Fig. 16 shows that for ‘low’ excitation frequencies, say ω <1.6, the experimental points are on the ‘plateau’ of high IF (seethe light grey region of Fig. 11 or Fig. 13(a)). It is particularlyremarkable that the PD points are just after the sudden fall ofIF , which therefore can be considered, up to the experimentalapproximation, as the upper practical threshold for the existenceof rotation. Note that it corresponds to the birth of the period3 rotations, p3, and only global safety arguments show why itconstitutes an upper limit formain rotations: above, the dynamicalintegrity is marginal, and there is no hope for the relevant regionto be observed experimentally.

In this range, the bottom curve pexpSN seems to follow the minorfall after the first peak of the integrity profile, i.e. it approximatelyfollows the boundary between the dark and the light grey of Fig. 11.The fact that it is not below, at least in correspondence of theprincipal peak, is likely due to experimental approximations andto the imperfect control of the initial conditions.

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Fig. 16. The contour plot of IF (p, ω) and the experimental data. Red square = os-cillations, blue triangles = rotations and black circles = rotations of period 2. Thecrosses are the points corresponding to the basins of attraction of Figs. 12 and 14.The value increases from dark to light green.

For ‘large’ values of excitation frequencies, say ω > 1.6, the ex-perimental points are clearly around the main ridge of IF , which isa definitive confirmation that only rotations with large dynamicalintegrity can be practically observed. The fact that for ‘very large’frequencies, say ω > 2.2, the points no longer follow the ridge is aconsequence of the fact that in the present experiment the ampli-tudes have a technical upper bound (see Section 4.2) which cannotbe overcome.

The bottom curve pexpSN now approximately stands on a contourlevel of IF , showing the minimal dynamical integrity necessary forthe onset of experimental rotations.

6. Conclusions

An experimental apparatus to simulate the production ofenergy from seawaves has been built at the Polytechnic Universityof Marche, Ancona, Italy. An extensive experimental campaign hasbeen developed, showing that main rotations of the pendulumare possible only in a strict subset of their theoretical region ofexistence and stability.

With the aim of justifying this experimental evidence, thedynamical integrity of the pendulum has been systematicallyinvestigated, extending previous work in the literature. It has beenshown that the generic integrity profile suddenly increases afterthe appearance of the associated attractor. Then, the birth of asecondary, highly periodic rotation within the basin of attractionof the main rotation suddenly reduces its safe region, so thatthe integrity profile suddenly falls down. When the secondaryattractor disappears and the main one regains its robustness, theintegrity profile grows again, partially recovering the former IFvalue. However, the involved fractalization induced by the onsetof further secondary attractors entails a non-recoverable loss ofbasin compactness, which is properly registered by the IF measure.The sequence of such mechanisms observed for fixed excitationfrequency and increasing excitation amplitude ends up with theappearance of a period 3 secondary attractor, which (i) is the mostimportant one, (ii) leads to a deep fall down of the integrity, and(iii) commonly reduces it to a marginal value (this is true for lowexcitation frequencies, while for increasing frequency the residualintegrity is less and less marginal).

Overlapping the integrity profiles with the experimental datawe have found that the latter occur in ranges of high integrity, thuspermitting us to understandwhy rotations have not been observed

elsewhere, namely where the dynamical integrity is not enough tosustain the experimental imperfections.

The main conclusion is that the experimental results aretheoretically fully justified, up to the experimental uncertainties,of course. Furthermore, this paper constitutes an experimentalproof of Thompson’s and the authors’ idea that local stability is notenough for practical use, and thus it implicitly encourages furtherdevelopments of the dynamical integrity issue.

The study ofmulti-d.o.f. systems is felt to be the new frontier forthis topic, with plenty of practical consequences, since the largerthe system dimension, the larger the complexity of its nonlineardynamics, at least in principle, and the stronger the expectedreduction of dynamical integrity.

Acknowledgements

The authors wish to thank the undergraduate students (nowEngineers) Enrico Venturi and Williams Luzi for building theexperimental pendulum and for the experimental tests. Theexperimental part of this work was done at the HydraulicLaboratory of the Polytechnic University of Marche, Ancona, Italy.We wish to thank Prof. A. Mancinelli, Prof. M. Brocchini andProf. C. Lorenzoni for permission to use this facility, and Prof.M. Brocchini and Prof. C. Lorenzoni for their contributions to thehydraulic part of the experiment. Thanks are also due to the Ph.D.studentM. Postacchini for helpwith thewave flume.We also thankProf. P. Castellini, of theDepartment ofMechanics, for helpwith themeasurement set-up and tools.

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