Nonlinear Dynamics

The Passage through Resonance of a Coupled Mechanical Oscillator: The Experiment(Resonance Capture, Escape and Quasiperiodicity)

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Full Title: The Passage through Resonance of a Coupled Mechanical Oscillator: The Experiment(Resonance Capture, Escape and Quasiperiodicity)

Article Type: Original research

Keywords: Couple Oscillators; Resonance Capture; Resonance Escape; Quasiperiodicity

Corresponding Author: Ghulam Mustafa, Ph. D.

San Jose, CA UNITED STATES

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Corresponding Author's SecondaryInstitution:

First Author: Ghulam Mustafa, Ph. D.

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Abstract: Coupled oscillators are ubiquitous in nature and man-made systems they range fromcircadian rhythms in biological systems to phase locked loops in electric circuits andmulti-link robots. The complexity of dynamics of individuals withstanding, whencoupled, these oscillators exhibit a myriad of intricate behaviors. The mechanicaloscillator presented consists of a large flexible column with a tip mass, attached to it, isa pendulum. This is a prototype of a vibration absorber from the flexible column to thependulum. In order for the energy to flow from the beam to the pendulum, the couplingmust satisfy certain resonance conditions which sets the stage for the complexbehavior to enter into the dynamics. The paper will focus on the experimentalobservations as the oscillator is forced through two resonance zones. Within eachzone, the oscillator displays distinct characteristics, starting from resonance captureand escape in the first zone (the principal parametric resonance), and enter into aregime that is quasiperiodic interrupted by windows of two-frequency motions in thesecond (the combination resonance). Motion in each zone finds its way into chaos,each following a distinct route, different from the other. The two scenarios, namely, thesaparatrix crossing in the first case and breakup of the torus in the second ispresented. Motivation for further investigating these scenarios are presented.

Suggested Reviewers: Ali H NayfehVirginia Ploy Techanayfeh@vt.edu

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The Passage through Resonance of a Coupled MechanicalOscillatorThe Experiment (Resonance Capture, Escape, and Quasiperiodicity)

Ghulam Mustafa

Received: date / Accepted: date

Abstract Coupled oscillators are ubiquitous in natureand man-made systems they range from circadian rhythms

in biological systems to phase locked loops in electric

circuits and multi-link robots. The complexity of dy-

namics of individuals withstanding, when coupled, these

oscillators exhibit a myriad of intricate behaviors. Themechanical oscillator presented consists of a large flex-

ible column with a tip mass, attached to it, is a pendu-

lum. This is a prototype of a vibration absorber from

the flexible column to the pendulum. In order for theenergy to flow from the beam to the pendulum, the cou-

pling must satisfy certain resonance conditions which

sets the stage for the complex behavior to enter into

the dynamics. The paper will focus on the experimental

observations as the oscillator is forced through two res-onance zones. Within each zone, the oscillator displays

distinct characteristics, starting from resonance capture

and escape in the first zone (the principal parametric

resonance), and enter into a regime that is quasiperi-odic interrupted by windows of two-frequency motions

in the second (the combination resonance). Motion in

each zone finds its way into chaos, each following a dis-

tinct route, different from the other. The two scenarios,

namely, the saparatrix crossing in the first case andbreakup of the torus in the second is presented. Moti-

vation for further investigating these scenarios are pre-

sented.

Keywords Couple Oscillators · Resonance Capture ·Resonance Escape · Quasiperiodicity

G. MustafaTel.: +1-408-910-9444E-mail: ghulam.mustafa@sbcglobal.net

1 Introduction

Coupled oscillators provide a handy paradigm for study-ing large complex systems with interacting components

and sub-systems. Even as the dynamics of the individ-

ual oscillators may be well understood (hardly ever the

case); when coupled, the aggregate system displays be-haviors that have little in common with the individual

oscillators in their un-coupled state. By far, the most

pre-dominant behavior of the coupled system is the syn-

chronicity, i.e., when connected they tend to oscillate in

harmony with one another, as first noted by ChristianHuygens in the winter of the year 1665 when he placed

two of his pendulum clocks next to each other. Huy-

gens further noticed that as he moved the clocks fur-

ther apart (thus weakening the coupling), they wouldfall out of phase and behaved as independent entities.

Overarching synchronicity might be in the coupled os-

cillators it is just the tip of the iceberg. Underneath

lie amplitude drifts, phase shifts, bifurcations of Hopf,

flip and saddle-node kinds. When multiple periodic mo-tions interact, they give rise to quasiperiodic behavior

this is fundamental to understanding the nature of tur-

bulence, the mother of everything that is nonlinear.

This paper proceeds as follows: Sect. 2 summarizesprevious experiments and some background. Sect. 3 de-

scribes the experimental setup and the instrumentation.

Sect. 4 briefly covers the sweep across the two resonance

zones. The details of the principal parametric resonance(resonance Zone 2) is covered in Sect. 5 followed by the

combination resonance (Zone 3) results in Sect. 6. A

summary of the qualitative dynamics is in Sect. 7.

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2 Ghulam Mustafa

2 Background

First experimental findings of the column-pendulum

were reported in [1]. and in more detail in [2]. The ex-periment reported in these references was limited to

one of the resonance zones, namely, the combination

resonance, (see below) where the response was found to

be pre-dominantly quasiperiodic in nature. Within thequasiperiodic responses, windows of periodic responses

were observed. The major findings of these experiments

can be summarized as follows:

1. The observed behavior (for the resonance consid-

ered) of the system provides ample evidence that

the underlying dynamics is that of a 2-torus to it-

self. This torus was re-constructed from the experi-mental data that consisted of the beam deformation

and the angular displacement of the pendulum.

2. The column-pendulum oscillator displayed rich dy-

namics consisting of quasiperiodic motions interrupted

by webs of periodic windows. The periodicity of eachof these windows can be characterized by two inte-

gers (m,n). Identification of the periodic responses

was suggested but not completed.

3. As the excitation (in frequency and amplitude) wasvaried, the coupling between the excitation and the

internal oscillator modes (the column and the pen-

dulum) resulted in resonance overlap when the pe-

riodic windows intersect one other, resulting in the

breakup of the torus.4. The aforementioned findings imply the breakup of

the experimental torus, a scenario similar to the one

proposed by [3]for transition to turbulence.

Employing the instrumentation reported in [4] and

the analysis methods proposed therein, [5] repeated the

experiment for various orientation of the beam with a

slight change in the natural frequencies of the beam(ω1 = 2.8 Hz) and the pendulum (ω2 = 0.5ω1 = 1.4

Hz). Their experiment consisted of sweeping the exci-

tation frequency through two ranges, one for each ori-

entation of the beam. For the horizontal orientation ofthe beam, the excitation frequency was swept in the

range 2.5-3.5Hz; while, for the vertical orientation, the

excitation frequency was in the range 3.9-4.65Hz. It is

interesting that of the two frequency ranges, the first

corresponds to the fundamental resonance of the beam,ω = ω1 (the principal parametric for the pendulum,

ω = 2ω2) while the second corresponds to combination

resonance (ω1 +ω2) (see [6] for details), respectively. It

is worthy of note that the pendulum mode, as used inthis application has a propensity of displaying what is

termed in literature as the auto-parametric resonance.

In simpler form, this oscillator resembles the spring-

mass-pendulum auto-parametric vibration absorber stud-

ied by [7] , [8], [9] and [10]. The most widely studied res-

onance condition involves 2 : 1 internal resonance and

1 : 1 external resonance with the primary spring-mass.

The motion of the primary mass acts as the paramet-ric excitation of the pendulum. Conversely, when ex-

cited, the pendulum is able to transfer energy to the

spring-mass. Auto-parametric systems display a myr-

iad of behaviors such as strong periodicity, quasiperi-odicity, chaos and saturation. [11] contains an excellent

exposition on auto-parametric resonance in mechani-

cal systems. [12] reported theoretical and experimental

finding of the column-pendulum. They propose a pen-

dulum type vibration absorber, where the pendulumfrequency is not commensurable with that of the beam.

It is shown that, in this case, the pendulum does not

act as an auto-parametric vibration absorber and that

the static friction at the pivot of the pendulum playsa dominant role in the suppression of parametric reso-

nance. Furthermore, it is also shown that under small

disturbances to the beam and pendulum, the amplitude

is stabilized at nearly zero, and the pendulum, whose

natural frequency is not commensurable with any nat-ural frequency of the beam, exhibits no reverse action.

[5] noted that for the vertical orientation (combination

resonance case), the response to be quasiperiodic with

windows of periodicity, however for the horizontal ori-entation (parametric resonance case), the response of

the system was strongly periodic. The objective of this

paper is to cover both the principal parametric, as well

as, the combination resonance behaviors of the column-

pendulum oscillator. Furthermore, additional data andanalysis is presented to gain more insight into the res-

onance behavior of this coupled oscillator.

3 Column-Pendulum Oscillator,

Instrumentation and Experimental Setup

The oscillator under test consists of a long slender steel

column with a tip-mass. Attached to the tip is a sim-

ple pendulum that is free to oscillate and rotate aboutthe pivot. The coupled oscillator is rigidly fixed to a

vibrating base that can impart vertical motions along

the un-deformed axis of the column. The two modes of

the oscillator (the column(ω1) and the pendulum(ω2))are resonantly coupled such that

ω1 ≈ 2× ω2 (1)

The beam and pendulum parameters are selected

such that ω1 = 2.264 Hz and ω2 = 1.086 Hz. As men-tioned, this is termed as the parametric coupling and

introduces time-dependent coefficients in the differen-

tial equation, as in the Mathieu equation. The so-called

The Passage through Resonance of a Coupled Mechanical Oscillator 3

Fig. 1 The experimental setup

principal parametric resonance occurs when the excita-

tion is twice the natural frequency implying that the

pendulum mode is tuned to this resonance mode. For

extensive coverage of principal parametric resonance,

see [6].

The experimental setup, shown in Figure 1, con-sists of an accelerometer (A) attached to the base that

provides a feedback via the signal conditioner (S/C)

to the vibration controller and amplifier (V/C) which

generates an input voltage and feeds it to the shaker(S). This arrangement provides a controlled input to

the oscillator. The outputs consists of two signals, one

generated by the encoder (E) housed inside of the tip

mass. The pulse train from the encoder is feed into the

encoder counter module that converts the digital countto an analog signal proportional to the angle of the

pendulum. The second output signal is generated by

the peizofilm strain gage (P) affixed to the base of the

steel column. The voltage generated by the peizofilm isproportional to the strain, which in turn is proportional

to the displacement of the column. Both output signals

are fed into the data acquisition system (DAC) for stor-

age and for display into the scope (O). The vibration

controller can be programmed to sweep through a fre-quency range at a desirable sweep rate at a constant

amplitude or step increments.

4 First Pass Through the Resonances

The first experiment conducted consisted of covering

both the principal parametric and the combination res-

onance zones, henceforth called Zone 2 and Zone 3, re-spectively. The parameters chosen for this experiment

were, frequency range (2-4 Hz) and a constant ampli-

fier gain of 375 mV. The data were collected on the

reverse sweep direction for Zone 3 and forward direc-

tion for Zone 2, indicated by arrows in Figure 2, which

shows the peak points of the data collected. This may be

thought of as a pseudo Poincare map for each increment

of the excitation frequency as the periodic motion of aperiod one will appear as a single point, an n-periodic

motion as n points. Quasiperiodic and chaotic motions

appear as a cloud of points.

Figure 2 clearly shows that the behavior in each ofthese two zones is distinctly different. In Zone 2, it is

pre-dominantly periodic (of period 2 for the column and

period 1 for the pendulum). Note that the motion be-

comes strongly periodic, albeit modulated, in the cen-

tral portion of Zone 2; this behavior is rather distinctlyobservable in the pendulum. This phenomenon, when

internal modes are locked to the excitation frequency

over an interval around the resonance, is termed as res-

onance capture and has been widely reported in lit-erature. The motion enters a transient region, as the

frequency is increased, and then escapes and the reso-

nance is lost. The anatomy of resonance capture and

routes of escape will be discussed in more detail in

the next section. Zone 3 on the other hand, consistsof quasiperiodic (possibly chaotic) motions as depicted

by the cloud of points within this resonance zone. This

observation is same as described in the previous exper-

iments. Also note periodic motions embedded in thecloud of quasiperiodicity. The two zones are separated

by a central region where no motion takes place; this

is the region where the upward position of the col-

umn and the downward position of the pendulum are

asymptotically stable and external perturbations haveno influence on the oscillator. Following sections will

be dedicated to unfolding the details of the oscillator

response in each of the two aforementioned resonance

zones, namely, Resonance Zone 2 (the principal para-metric resonance) and Resonance Zone 3 (the combina-

tion resonance).

5 Passage through Zone 2 – the Principal

Parametric Resonance (ω = ω1 = 2ω2)

Figure 3 shows the response characteristics to a low am-

plitude excitation (375 mV) during a frequency sweep

across the parametric resonance zone. The beam re-

sponse is shown on the left while the pendulum re-sponse is on the right. The top graph in this figure is

the maximum of the response amplitude plotted against

the excitation frequency. The middle graph is the spec-

trogram of the response and represents the short-timeFourier transform of the recorded signal. The dark re-

gions in this graph correspond to the peaks of the FFT

at any given instant of time. At the bottom are shown

4 Ghulam Mustafa

Fig. 2 The response of the column (top) and the pendulum(bottom) during the first full frequency sweep covering boththe principal parametric (Zone 2) and combination (Zone 3)resonances. The excitation amplitude is fixed at 375mV.

Fig. 3 Response in Zone 2. Excitation amplitude is fixed at375 mV. (Top) Amplitude versus excitation frequency. (Mid-dle) Response spectrum (Hz) versus time (s). The dark rep-resent peaks of the Fourier spectrum. (Bottom) Phase plotsat different excitation frequencies.

the phase plots drawn at the selected excitation fre-quency. As the frequency is swept, the static equilib-

rium position becomes unstable; both the column and

the pendulum undergo periodic motions. The column

natural frequency is set at twice that of the pendu-

lum, the excitation frequency is in the neighborhoodof twice that of the pendulum this corresponds to the

principal parametric resonance’ for the pendulum. The

points where resonance capture and escape occurs are

indicated. During capture, both the beam and the pen-dulum display strong periodicity as can be seen from

the phase diagrams. Prior to the escape, coupling is

lost, the response goes through a short transitory phase

and the motion dies out; at this point the resonance is

lost.

Figures 4 shows the response characteristics to a

higher amplitude excitation (575 mV). The main obser-

vation from these graphs is that whereas the response

Fig. 4 Response in Zone 2. Excitation amplitude is fixed at575 mV. (Top) Amplitude versus excitation frequency. (Mid-dle) Response spectrum (Hz) versus time (s). The dark rep-resent peaks of the Fourier spectrum. (Bottom) Phase plotsat different excitation frequencies.

frequency content remains qualitatively the same as

the low amplitude excitation, for the higher amplitude

of excitation, the escape route is longer. It must be

mentioned that whereas resonance capture phenomenaobserved in other situations is robust, in the column-

pendulum, it is quite delicate and depends on combi-

nation of judiciously selected excitation amplitude and

frequency sweep rate. For the excitation amplitude–sweep rate combinations where resonance capture is

not observed, the resonance zone can be swept with-

out any detectable motion. However, for the combina-

tions for which the resonance capture does occur, it

can be sustained for a very long time, particularly ifthe frequency is held in the center of the capture zone.

The dependence on excitation parameters and the ini-

tial conditions makes the experiments in this zone less

repeatable as small change can cause the resonance toevade capture.

The resonance capture phenomenon has been re-

ported in a variety of mechanical systems. [13] inves-tigated the dynamics of an unbalanced rotor spun by

a motor with limited torque supported on an elastic

foundation. He observed that depending on parame-

ters, when starting from rest, the system would eitherpass through the first resonant frequency of the system

or stall as the angular velocity approached the natu-

ral frequency. Sensitivity to initial conditions was also

noted, even when the excitation parameters where held

constant, small change in initial condition would eitherlet the system pass through or be captured. Similar be-

havior was observed during the passage through Zone

2 of the column-pendulum oscillator. For lower ampli-

tude input and depending on the frequency sweep rate,the motion would either initiate or the system would

pass through the resonance zone without responding

to the input. When captured, it would, just prior to

The Passage through Resonance of a Coupled Mechanical Oscillator 5

the first resonant frequency (ω1 ≈ 2 Hz), undergo sus-

tained oscillations, around the central portion of the

capture zone (see Figure 3), pass through a transient re-

gion where large amplitude and seemingly non-periodic

motion would occur (see Figure 4) and then die out.Sometimes, a small tap on the tip-mass was needed to

start the motion, at other times, depending in the input

parameters, the motion would start on its own. Inter-

estingly, the unbalanced rotor supported on a spring-mass represents a close discrete analog of the column-

pendulum oscillator, with pendulum acting as the un-

balanced rotor and beam with tip-mass as the primary

spring-mass system. In [14] it was noted that if the size

of the unbalance is small compared to the motor torque,the rotor will pass-through the resonance. Conversely,

if the unbalance is comparatively large, resonance cap-

ture will occur and the rotor will stall near the resonant

frequency. It was shown that there is a hole in the res-onance plane through which the passage through reso-

nance can be achieved by a motor with limited torque

capability. The proposed feedback control involved ad-

dition of negative damping to the pendulum-type equa-

tion representing the unbalance in the system near res-onance.

Resonance capture has been reported in spinup dy-

namics of dual spin spacecraft during the ’despin ma-

neuver’ of a dual spin spacecraft. [15] noted that asthe despin is initiated, the energy provided by the spin

motor can be channeled into modes other than relative

motion between the platform and the rotor, this mode,

referred to as the ’precession phase-lock’ is associatedwith resonance capture can send the craft tumbling

through the space when the rotation axis becomes syn-

chronized with the precession of the spacecraft about its

angular momentum axis. Subsequently, [16] proposed a

nonlinear control strategy to circumvent the precessionphase lock during despin and the ensuing instability of

the space craft.

During the past decade Vakakis and co-workers ([17]

and the reference therein) have published extensively onwhat they call the Targeted Energy Transfer or TET.

The main concept involves the transfer of vibration en-

ergy from the primary system to a strongly nonlinear

local attachment, the so-called Nonlinear Energy Sink

(NES). Presence of NES greatly affects the dynamics ofthe primary system to which it is appended. The idea

is that when properly designed, NES is capable of cap-

turing the vibration energy hence providing a means of

passive vibration control. They show that this passiveenergy transfer occurs over a broad range of frequency

during which a sustained transient resonance capture is

observed.

5.1 The Anatomy of Resonance (Capture and Escape)

The mechanism under which resonance capture occurs

is described by [18]. The scenario involves a separatrix(a homoclinic or a hetroclinic orbit) surrounding a cen-

ter, referred to as the frozen system. The frozen system

is structurally unstable; small perturbations (in param-

eters) break the separatrix loop thus creating an open-ing from which the nearby trajectories can enter the

central region and circulate around the center. These

trajectories are said to be ’captured’ and display the

periodic resonant behavior described in the previous

section. The trajectories that fail to enter the openingcreated by the split, escape the resonance they are lost

to infinity. When this situation (i.e. the opening cre-

ated by the splitting) is sustained, the motion is stable

and not susceptible to small changes. Alternatively, thestable and the unstable manifolds may evolve slowly

over time due to small and gradual change in parame-

ters (for example small change in excitation frequency

or amplitude). Under these conditions, the manifolds

exchange locations and the captured motions find away out. There are two possible outcomes, the first

corresponds to a smooth transition, where the stable

and the unstable manifolds exchange locations gradu-

ally and do not intersect. The other possibility is whenthe manifolds, at first become tangent to one another,

then later intersect, creating homoclinic tangles (the so-

called Melinikov scenario). This situation is described

in detail by [19] for two coupled homoclinic oscilla-

tors which have a four-dimensional phase space. Thetwo-dimensional (the simpler of the two) scenario is il-

lustrated in Figure 5, starting with the frozen system

(Figure 5a). Small perturbations break the saddle loop

(Figure 5b), creating an opening for the trajectories toenter the center and be captured. The escape routes

are shown in Figures 5c and 5d. The first of these is

related to the sustained criss-crossing of the stable and

the unstable manifolds resulting in chaotic motion. This

situation, however, is unstable and results in the man-ifolds exchanging locations permanently, as shown in

Figure 5d. In this situation, the trapped motions find

a way out and escape from resonance. Similar behavior

has been observed in the column-pendulum oscillator(the four dimensional case)

The resonance capture and escape behaviors of thecolumn-pendulum are shown in Figures 6–8. The input

frequency is varied between 2-3Hz and the excitation

amplitude is kept constant at 575mV. These graphs

show the resonance capture mode where the motionis pre-dominantly period, indicated by the sharp peaks

in the FFT spectra, shown in the bottom of Figure 10.

The escape takes through the homoclinic tangle where

6 Ghulam Mustafa

Fig. 5 The anatomy of resonance capture and escape.

large amplitude random-like motions occur. Betweenthe periodic and the non-periodic motions, there is a

transition zone. This behavior can be explained as the

stable and unstable first become tangent to one an-

other, followed by intersection and later, splitting and

exchanging locations. The phase space for the column-pendulum is four-dimensional and one might expect

that the capture and escape behaviors to be more com-

plicated than the simple two-dimensional prototype de-

scribed. The fact that experiments in this resonancezone could not be reliability repeated, suggests sensi-

tivity to small and imperceptible variations that are in-

evitable in an experimental setting. This would explain

why parametric resonance behavior for the column-pendulum

oscillator eluded repeatable experimental observationssince no two experimental runs are exactly the same

this certainly was the source of frustration for this ex-

perimenter.

Hilbert transform analysis performed on two fre-quency scan experiments is illustrative of the differences

in resonance capture and escape. The first panel on the

left in Figure 9 shows the instantaneous phase differ-

ence between the beam and the pendulum response. In

this figure the flat portions correspond to the periodicresponse associated with capture. For the 375mV, both

the beam and the pendulum oscillate to the natural

frequency of the pendulum (ω2), shown in the middle

panel of Figure 9. As the frequency is increased, thebeam jumps from oscillating about the pendulum fre-

quency to its own natural frequency (ω1) then jumps

back to oscillate about ω2 where it is captured. This

is indicated as repeated small steps in the phase dif-

ference plot. For the lower amplitude of excitation; theresonance capture persists over a longer period of time.

There is a subtle difference in the capture behavior for

the two experiments. Note that in the 575 mV case, the

beam jumps from the phase locked at ω2 to ω1, whilethe pendulum continues to oscillate at about its own

frequency ω2. For a while, both oscillate closer to their

resonant frequencies; this corresponds to the transition

Fig. 6 Resonance capture and escape for the column-pendulum. Excitation amplitude is fixed at 575 mV. (Top)The beam response versus time. (Bottom) The pendulum re-sponse versus time. The graphs show the regions of resonancecapture, the transition between the capture and escape andfinally, the escape through the homoclinic.

Fig. 7 Resonance capture for the column-pendulum. Excita-tion amplitude is fixed at 575 mV. (Left) The beam response.(Right) The pendulum response. The graphs show the regionof resonance capture is strongly periodic with sharp peaks inthe FFT shown in the bottom.

The Passage through Resonance of a Coupled Mechanical Oscillator 7

Fig. 8 Resonance escape for the column-pendulum. Excita-tion amplitude is fixed at 575 mV. (Left) The beam response.(Right) The pendulum response. The graphs show the regionof resonance escape is not periodic with multitude of frequen-cies in the FFT shown in the bottom.

Fig. 9 Hilbert Transform analysis for capture and escape.The phase difference (∆φ, on the left between the beam andthe pendulum for the two frequency scans at 375 mV and 575mV. The instantaneous frequency for the beam and pendulumfor 375 mV(middle) and 575 mV(right)

zone in Figure 9, the 575mV response goes through atransition that is not phase locked before the final es-

cape through the homoclinic as described above.

5.2 The Dynamics of the Captured (the Resonant

Behavior)

As mentioned, once captured, the resonant behavior is

predominantly of periodic nature. The periodicity de-

pends on the characteristic of the excitation (i.e. am-

plitude and frequency) and on the manner in whichthe amplitude and frequency is arrived, in particular

the rate of frequency sweep. Figure 10 shows the num-

ber of experiments conducted on the excitation fre-

quency versus the excitation amplitude plane. Circlesin this figure denote the general neighborhood where

the experiment was performed since the excitation fre-

quency/amplitude tend to drift during an experiment.

Fig. 10 Zone 2 Experimental Space: Excitation amplitude[A] versus excitation frequency [ω]. The circle indicated thegeneral neighborhood of the experiment identified by thenumber.

The general approach taken to capture the resonant

motion was to start at the lower end of the frequency

(around ≈ 2 Hz), wait for the motion to begin and sta-

bilize, then slowly sweep to the desired frequency point,while keeping the amplitude constant. After the de-

sired frequency was reached, the amplitude was changed

to arrive at the point where the data was collected.

The Lissajous plots (beam versus pendulum) for theseexperiments are shown in Figure 11. Based on their

approximate appearance, the responses are classified

into four general groups corresponding to experiments

(1,5,7), (2,4), (3,6) and (9-13). These are called Group

1–4, in the following discussion. The Lissajous plotswere also observed in real time on the scope during

the frequency scan experiments described earlier, one

could observe the morphing of the responses between

the classes.In order to characterize the observed responses, the

following methodology was adopted:

1. Trend in the data was removed and was smoothed

via a 5-point moving average. This insured that thedata oscillated about zero and sharp peaks were av-

eraged.

2. The post-processed data was subjected to Hilbert-

Huang Transformation (HHT) to extract the response

modes via the Empirical Mode Decomposition (EMD).3. The top 4 dominant modes were used to extract the

frequency spectrum via the Hilbert Transform.

Computationally, Hilbert Transform is sensitive tonoise and small drifts in frequency, therefore it is neces-

sary to pre-filter the data. Numerical difficulties asso-

ciated with the brute force application of the Hilbert

8 Ghulam Mustafa

Fig. 11 Dynamics of the captured responses in Region 2.The plots are the Lissajous curves of beam displacement ver-sus the pendulum angle. The numbered squares correspondto the experiments indicated in Figure 10.

Transform are discussed in [20]. The Hilbert–Huang

Transform (HHT) method (see [21]) circumvents the

aforementioned pitfalls by first, decomposing the re-sponse into intrinsic characteristics and extracting the

inherent modes within the data - referred to as the

Empirical Mode Decomposition (EMD). In that sense,

the EMD pulls the basis of the total response fromthe data itself. The second step in HHT involves sub-

jecting the EMD to Hilbert Transform to obtain the

time-frequency spectrum, the so-called Hilbert Spec-

trum which yields the frequency content of the EMD

in time.

One response from each of the four groups was cho-

sen as a representative of the group. Experiment 5 was

chosen to represent Group 1, Experiment 2 for Group 2,

Experiment 6 for Group 3 and Experiment 10 was cho-

sen to represent Group 4. These are shown in Figures12–15. The first panel on the left shows the result of

the EMD for the beam. The top graph is the recorded

beam time history (b(t)), followed by the four dom-

inant modes (b1(t) . . . b4(t)). The middle panel showsthe EMD results for the pendulum. The third panel on

the right is the Hilbert Spectrum of the top four em-

pirical modes.

Fig. 12 Characterization of the captured response for Exper-iment #5. The first panel on the left shows the result of theEMD for the beam. The top graph is the beam time history(b(t)), followed by the four dominant modes (b1(t, . . . b4(t)).The middle panel shows the results for the pendulum. Thethird panel on the right is the Hilbert Spectrum of the topfour empirical modes.

Fig. 13 Characterization of the captured response for Ex-periment #2.

Figure 12 shows the response for Group 1. The pre-

dominant frequency for the pendulum is its own natural

frequency (ω2). The beam responds with two frequen-cies, one coincident with ω2 and the other modulates

around the combination (ω1 + ω2) mode. A compari-

son with Figure 13, Group 2, shows that the combina-

tion frequency response for the beam is more dominant

(shows tighter modulations), this is also obvious in therespective Lissajous plots in Figure 11. Group 3, Figure

14, shows that both the beam and the pendulum display

modulated responses, one about ω2, while the other at

ω1. Whereas the pendulum modulates around ω2, thebeam modulations are centered on both, ω1 and ω2,

showing a preference for the pendulum seen as the tight

spread about ω2. Not apparent form its Lissajous plot,

Group 4 (Figure 15) has the most complex response,

compared with the other Groups. Both the beam andthe pendulum modulate about ω1) and ω2, with neither

showing any preference for either of the modes. This is

a quasiperiodic response where both the beam and the

pendulum phase planes resemble circular annuli (seehigher excitation frequency phase plots in Figure 4),

the motion of the combined system lies on a surface of

a two dimensional torus. This types of response it the

The Passage through Resonance of a Coupled Mechanical Oscillator 9

Fig. 14 Characterization of the captured response for Ex-periment #6.

Fig. 15 Characterization of the captured response for Ex-periment #10.

main topic covered in the response of Resonance Zone

3 (combination resonance case).

Based on group observations, one can summarize

the captured responses as being predominantly periodic

(or rather two-frequency quasiperiodic) at the lower end

of the excitation frequency. In the central region, the

responses show modulations about the two natural fre-quencies (ω1 and ω2, respectively) and the combination

frequency (ω1 + ω2). The nature of transition between

these behaviors is not clear due to the fewer number

of experiments performed. Also how these motions re-spond to the change in the excitation amplitude (at

fixed excitation frequency) cannot be inferred based on

the sparsely scattered experiments. At higher end of

the excitation frequency, the motion is quasiperiodic

indicated as annular Lissajous plots; HHT performedon this group shows that the responses are not phase

locked. The persistence of these responses for long pe-

riod of time, is however an indication that these are

indeed captured.

5.3 Summary of Observations for the Principal

Parametric Resonance

Two types of experiments were conducted in the prin-cipal parametric resonance zone (ω ≈ ω1). First exper-

iments consisted of a frequency sweep through the en-

tire zone. The main observations in these experiments

were the presence of resonance capture and escape phe-

nomena. During the resonance capture, the oscillator

response consisted of phase locked periodic motions or

modulated periodic (two-frequency) motions. As the

frequency was increased, the oscillator experienced theresonance escape. Two routes of escape were observed.

For lower amplitude of excitation, the escape was short

and the motion subsided quickly. At higher amplitude

of excitation, the oscillator first entered a transitionzone (tangency), followed by a passage through a ho-

moclinic tangle (intersection). The capture and escape

scenarios were explained using a two dimensional phase

space as a prototype consisting of a separatrix in the un-

perturbed system. Small perturbations split the stable-unstable manifold creating an opening for the trajecto-

ries to enter and be captured. Using this as an analogy,

the resonance escape phenomena was explained for the

four–dimensional mechanical oscillator. It was shownthat during the capture, the motion was either phase

locked periodic or modulated, depending on the excita-

tion frequency and excitation amplitude. The Fourier

spectrum for these responses consisted of sharp peaks.

Prior to escape, the response FFT showed multitudeof frequencies. The characteristics pertaining to reso-

nance capture and escape were confirmed with the help

of Hilbert transform. The captured responses were seen

as phase locked, whereas the escaped responses showeda phase drift. The instantaneous frequencies were calcu-

lated for the responses during the capture and escape.

The second type of experiments was used to charac-

terize the motion of captured responses. These exper-

iments consisted of first capturing the response, thanslowly increasing the frequency and amplitude to arrive

in the neighborhood of the desired excitation amplitude

and frequency. Thirteen experiments were conducted;

Lissajous plots of the steady state response were used

to classify the responses based on their appearance.The Hilbert-Huang transformation was the main tool

of analysis for the response characterization. Each clas-

sified response was subjected to Empirical Mode De-

composition (EMD) to extract the significant modes. Itwas discovered that for the responses captured at the

lower end of the excitation frequency (ω), the pendulum

response consists of motions that are predominantly pe-

riodic at its own natural frequency (ω2). The main con-

tributors to the beam response modulated about ω2 andabout the combination frequency (ω1+ω2). In the cen-

tral zone of excitation frequency, both the pendulum

and the beam display modulated response. The pendu-

lum modulates around ω2, whereas the beam modula-tions are about ω1 and ω2. At higher end of excitation

frequency, the motion is dominantly quasiperiodic. The

quasiperiodic nature of the response is confirmed by the

10 Ghulam Mustafa

phase plane plots for the beam and the pendulum each

occupying an annular region indicating the existence of

a two–dimensional torus.

5.4 Questions Concerning the Motion in Zone 2

Based on the experimental observations and the subse-

quent analysis of the observed dynamics, the following

questions have surfaced:

1. How is the excitation amplitude frequency (A,ω)

plane divided into the captured, un-captured sub-

sets? What is the nature of these sub-sets, are their

boundaries smooth or fractal or have holes?2. What is the nature of the captured responses and

how do these change, mutate and morph? What

types bifurcations occur and the parameters that

control them?3. What are the routes to capture and why do some

responses avoid capture? Is there a combination of

parameters and/or initial conditions that determine

the fate of these trajectories?

4. What are the escape routes? Slip through the backdoor (captured suddenly find and escape)? Climb

through the stairway (bifurcations leading to quasiperi-

odicity and escape)? Pass through the labyrinth (trapped

in the homoclinic tangle before the release)?5. Based on the myriad of responses over a wide range

of excitation frequencies, are there control applica-

tions for harvesting ambient energy for small devices

such as a network of sensors?

6 Passage through Zone 3 – the Combination

Resonance(ω = ω1 + ω2)

It is known that for parametrically excited nonlinear

systems, combination resonance is a distinct possibil-ity. According to [22], combination resonance can occur

as a response to a single harmonic excitation; the type

of combination will depend on the order of nonlinear-

ity. In Figure 16, the response amplitude for the beamand the pendulum are plotted as a function of the ex-

citation frequency (ω) in the neighborhood around the

combination resonance (ω1 + ω2 ≈ 3.3 Hz) for various

excitation amplitudes. The left panel corresponds to the

beam response; the right panel shows the response ofthe pendulum. For each experimental run, the ampli-

tude of excitation was kept constant while the frequency

was swept between 3–4 Hz, thus in Figure 16, b275 is

the beam response to the excitation amplitude of 275mV, correspondingly for the pendulum. These graphs

were obtained by plotting the maximum of the mea-

sured response against the excitation frequency step. As

Fig. 16 Response to the frequency scan in the ResonanceZone 3 as a function of amplitude of excitation. The left panelcorresponds to the beam response; the pendulum response isshown in the right.

Fig. 17 Response of frequency scan in Zone 3 to 475 mVexcitation. The top graphs show the response amplitude asa function of excitation. The middle graphs is response fre-quency as a function of time. The phase plots are shown inthe bottom for selected excitation frequencies.

evident from these graphs, the beam response for the

entire zone is predominantly quasiperiodic, whereas thependulum response consists of windows of apparent pe-

riodic motions. This point is further illustrated in Fig-

ure 17, where the phase plots, shown in the bottom of

the figure clearly show that whereas the beam response

is mostly quasiperiodic, the pendulum response on theother hand consists largely of periodic motions. Fur-

thermore, periodic motions for the pendulum change

with the frequency and amplitude of excitation, while

that of the beam remain quasiperiodic. This can beseen from Figure 18, which is the response to three dif-

ferent amplitudes of excitation while the frequency was

scanned from 3.3 to 3.65 Hz.

The Passage through Resonance of a Coupled Mechanical Oscillator 11

Fig. 18 Response of frequency scan in Zone 3 to 475 mV,350 mV and 275 mV excitations.

6.1 Re-Constructing the Torus

Quasiperiodicity is perhaps the most commonly ob-

served behavior in dissipative dynamical systems, qnd

is customary to represent the dynamics of coupled os-cillators that displays quasiperiodic motions by a torus

(T 2) with the two phase angles (θ1 and θ2) as its co-

ordinates. The construction of the torus in this man-

ner is topologically equivalent to map of a unit square

to itself. This is visualized as shown in Figure 19, thetrajectories of the coupled oscillators wrap around the

torus like a string around a doughnut. The dynamics is

further simplified by opening and flattening the surface

of the torus into a unit square with the phase anglesas the coordinates. The trajectories that wrap around

the torus in three dimensions appear as curves on the

unit square. A periodic orbit is represented by finite

number of curves. A quasiperiodic orbit fills the entire

surface of the torus and consequently so also the unitsquare. This geometric construction is quite illustrative

of the complex dynamics pertaining to quasiperiodic

motion in coupled oscillators. In case of the periodi-

cally forced oscillators, the forcing function introducesthe third phase; hence the aforementioned two dimen-

sional construction of the torus does not apply directly.

One can however work with the Poincare section taken

with respect to the forcing phase and reduce the dimen-

sion from five back to four. The problem is transformedfrom a five–dimensional dynamical system in continu-

ous time to a four–dimensional system in discrete time.

[22] proposed a method of re-constructing the dynamics

of two coupled periodically forced Van der Pol oscilla-tors

x1 − α(1 − x21)x1 + ω2

1x1 + ǫx31 = βx2 + γ sin(t) (2)

x2 − α(1 − x22)x2 + ω2

2x1 + ǫx32 = βx1 + γ sin(t+ φ)(3)

The method is adopted here for the re-construction

of the experimental dynamics of the column-pendulum

Fig. 19 The flow on the torus and its cover

oscillator. The sub-set of the method described here

pertains to the re-construction of the torus only; the

complete method in [22] covers detection of the breakup

of the torus and as such is not relevant to the scope

of the current experimental investigation. The adoptedmethod is as follows:

1. Obtain the solution of the coupled second order dif-ferential equations. Post-process the solution so that

the oscillations are about the origin.

2. Draw the phase plane (x, x) for each of the oscilla-

tors.

3. Superpose the phase planes with Poincare points,obtained by re-sampling the phase diagrams after

every forcing period. This results in two annuli of

points dispersed around the phase trajectories of

each of the oscillators.4. For each Poincare point on the annulus, determine

the phase angle (θ = arctan(xx) and its radial dis-

tance (r =√x2 + x2)from the origin.

5. Obtain the cover of the torus T 2 by plotting the twophases (mod(2π)).

6. Compute the orbit on the surface of (T 2) using the

transformation (x, y, z) → ((r2+r1 cos θ2) cos θ1, (r2+

r1 sin θ2) sin θ1, r1 sin θ2).

Figure 20 shows a three frequency quasiperiodic mo-

tions for ω1 = 2.016432454, ω2 = 1.016432454, α =

0.5, β = 0.5, ǫ = 0.5, γ = 0.5. Also shown in Figure21 is a two frequency quasiperiodic motion for ω1 = 0.5

and ω2 = 0.5. The two–frequency orbit is clearly visual-

ized as a closed curve on the surface of the torus. Figure

22 shows the re-construction of the periodic window for

excitation frequency of ω = 3.13 Hz and excitation am-plitude A = 650 mV. A comparison of these figures in-

dicates that whereas the three-frequency quasiperiodic

motion tends to fill the entire surface of the torus, a two-

frequency quasiperiodic motion on the other hand is aclosed curve. The nature of the two-frequency quasiperi-

odic motion is determined by the number of times the

orbit wraps around the torus.

12 Ghulam Mustafa

Fig. 20 Torus re-construction for a three frequencyquasiperiodic motion for the coupled Van der Pol oscillators.The Poincare points are shown as crosses on the phase plots.The flow on the cover and the re-constructed torus are shownon the right.

Fig. 21 Torus re-construction for a two-frequency quasiperi-odic motion for the coupled Van der Pol oscillators.

Fig. 22 Torus re-construction for the coupled column-pendulum oscillators for excitation frequency ω = 3.13 Hz,and excitation amplitude A=650 mV.

6.2 The Dynamics of the Quasiperiodic

This section covers the characterization of the quasiperi-

odic motions observed in the combination resonance

zone. Furthermore,an ad hoc method for de-construction

of the phase dynamics on the torus is proposed as anadditional tool to further understand the observed mo-

tions . The amplitude and frequency setting for the ex-

periments performed in this zone are shown in Figure

23. Most of the experiments were conducted around

3.13 Hz at different amplitudes of excitation. It is ob-

vious from Figure 16 that this is the region where the

pendulum shows an apparent strong periodicity even

though overall dynamics of the coupled system is quasiperi-odic. Figure 24 shows the dynamics for excitation fre-

quency of 3.13 Hz and the amplitude of excitation of 475

mV. The first panel on the right is the response ampli-

tude for the beam (top) and for the pendulum (bottom)in the neighborhood of the excitation frequency. The

second panel shows the phase plane for the beam and

the pendulum. The third panel is the re-constructed

torus, the first return maps (θk, θk+1) for the beam and

the pendulum are shown in the fourth panel. The nexttwo panels are the Hilbert-Huang Empirical Modal De-

compositions (EMD) for the top three modes for the

beam and the pendulum. The last panel on the right

is the instantaneous frequencies of the top modes ob-tained via the Hilbert transform of the EMD. The first

EMD for the beam modulates around beam frequency

(ω1); second EMD oscillates about the pendulum fre-

quency (ω2). The first EMD for the pendulum responds

with the pendulum natural frequency (ω2), while thesecond mode oscillates at approximately one-third of

the pendulum frequency (ω2

3). This behavior persists

as the amplitude of excitation is increased from 475

mV to 725 mV; becoming pronounced as the ampli-tude is increased, i.e, frequency of the top two EMD

become concentrated around the two respective cen-

ter frequencies (ω1 and ω2). The motion breaks down

as the amplitude of excitation is increased to 775 mV.

Shown in Figure 25, at 775 mV amplitude of excita-tion, the cover of the torus is full as opposed to the

distinct trajectories observed at the lower amplitude of

excitation. Similarly, the return maps are not smooth

and seem to develope some pathology not seen at lowerexcitation amplitudes. A comparison of the character-

istics observed at 725 mV and 775 mV suggests that

a qualitative change in the dynamics has occurred, a

bifurcation from a two-frequency quasiperiodic motion

to a three (or perhaps higher) frequency quasiperiodic-ity. A slow frequency scan around 3.13 Hz and slightly

lower amplitude of excitation of 750 mV reveals the de-

tail of this transition. Shown in Figure 26, it is clear

that the pendulum shows at least three different typesof motion. This point will be further illustrated while

de-constructing the return maps of the torus.

Slow frequency scans in the mid range of the com-

bination resonance zone show that the beam response

to be quasiperiodic with beam natural frequency (ω1)dominating the response with a small contribution from

the excitation frequency (ω). The pendulum response in

this range consists of three distinct frequencies, the ma-

The Passage through Resonance of a Coupled Mechanical Oscillator 13

Fig. 23 Zone 3 Experimental Space: Excitation amplitudeversus excitation frequency. The circle indicated the generalneighborhood of the experiment identified by the number.The inset shows the detail of close cluster of experiments.

Fig. 24 Zone 3 Motion Characterization: Excitation fre-quency=3.13 Hz, Excitation Amplitude=475 mV.

Fig. 25 Zone 3 Motion Characterization: Excitation fre-quency=3.13 Hz, Excitation Amplitude=775 mV.

Fig. 26 A slow frequency scan around 3.13 Hz, ExcitationAmplitude=750 mV.

Fig. 27 A slow frequency scan around 3.35 Hz, ExcitationAmplitude=475 mV.

Fig. 28 A slow frequency scan around 3.47 Hz, ExcitationAmplitude=1390.2 mV.

jor contributors are the natural frequency of the pendu-

lum (ω2) , the other being the beam natural frequency

(ω1), followed by the line around 1/3 of the pendulum

frequency. Figure 27 shows the low amplitude excita-tion response around 3.35 Hz, here again the aforemen-

tioned behavior is observed, albeit that the pendulum

response is remarkably periodic. The response around

3.47Hz is quite different. As in Figure 28, both the beamand the pendulum have two distinct frequency compo-

nents. The major contribution for the beam response

comes from (ω2), while the major contributor for the

pendulum response is around 1

2ω1 of the pendulum nat-

ural frequency. The beam response also contains a smallcontribution from the excitation frequency ω.

The re-construction of the torus as the Poincare sec-

tion of the forced coupled oscillator is a useful tool in

identifying the dynamics, particularly if the behavioris predominantly quasiperiodic. The re-construction al-

lows one to be able to qualitatively distinguish between

a three-frequency and two-frequency quasi-periodic re-

sponses. It must be mentioned that the assumption be-hind the re-construction of the torus is that the inter-

acting modes are weakly coupled; this results in the

amplitudes being slaved by the respective phases. The

14 Ghulam Mustafa

assumption while useful is not strictly valid when the

coupling is strong and amplitudes play an important

role in the dynamics as well. The shortcoming with-

standing, the following section is dedicated to an ad

hoc method of ’de-constructing’ the torus from the firstreturn maps of the individual phase angles of the re-

constructed torus. The notion of de-construction in the

present sense implies that the phase angles are non-

interacting and hence are regarded as independent en-tities and can be taken apart. This additional assump-

tion, as will be demonstrated in the following, is useful

in quantifying various two-frequency quasiperiodic re-

sponses observed in the experiment.

6.3 Torus De-Construction from First Return Maps

Systems with two or more competing phases, where

the coupling between the modes is weak, the ampli-tudes are slaved by the respective phase angles. The

dynamics for such systems is that of flow on a torus

of appropriate dimension. The parametrically forced

beam-pendulum consists of three coupled oscillators,

the beam and the pendulum being the first two; theexternal sinusoidal force acts as third oscillator. It was

shown that the Poincare cross-section taken with re-

spect to the external forcing period consists of points

that form two annular regions. The re-construction ofthe two-torus map consisted of calculating the phase

angle for each point in the two annular regions and plot-

ting them on the cover of the two-torus. Furthermore,

amplitudes and phases where used to visualize the flow

on the torus, for two-frequency quasiperiodic motions,the flow consisted of distinct trajectories, whereas for

three (or perhaps higher) frequency motions filled the

surface of the torus. The fact that the quasiperiodic mo-

tion is dense can be used to extract the first return mapfor individual phase first return maps since they form a

smooth curve. In this section, an ad hoc method for de-

constructing the torus from the first return maps of the

individual phases is described. In addition to the weak

coupling, this method further assumes that the phasesare de-coupled this clearly is not the case. The objec-

tive here is to find a means to quantify the observed

two-frequency quasiperiodic motions of the oscillator.

It will be demonstrated that the proposed method ade-quately captures the two-frequency motions on the sur-

face of the torus and provides a means of quantifying

them.

The map of family on two-tori is given by

θk+1 = Ω + ω ∗ θk + f(θk) (4)

Several versions of the torus map are reported in

the literature, see [24] for review and dynamics of torus

maps. In the above equation, the first two terms corre-

spond to linear translation and whereas f is a periodic

(or oscillating) function, dependent on the coupling be-

tween the modes and determines long term dynamical

behavior like phase drift,phase entrainment and modelocking (see [25]). Further note that the two phases are

in fact monotonically increasing functions of time. One

can use the definition of the map of the torus to split

it in a linear part and an oscillating part as

θk = α+ β ∗ tk + f(tk) (5)

where tk is the kth point where θk is calculated forthe re-construction. The method consists of fitting a

least square line to the linear part of the map and fit-

ting a periodic function (a sine or Fourier Series) to the

oscillating function f(tk). The method works as follows:

1. Start by first unwrapping the two phase angles so

that they are monotonically increasing or decreas-

ing, albeit oscillating about a mean.2. Fit a line through the mean this is the linear trans-

lation of the torus map, Equation (5).

3. De-trend the unwrapped phase angles. This will re-

move the linear trend from the data.4. Fit sine/Fourier series to the de-trended phase data.

This is the oscillating part of the map. The number

of terms in the series requires trial and error.

5. Re-generate the data and plot the first return (θk, θk+1)

map for each of the phases this should match closelywith the first returns extracted from the experimen-

tal data.

6. Plot the generated data on the cover of the torus

with the two phases as the coordinates.7. Count the number of times the curves cross θ1 (m)

and θ2 (n) axes to classify the orbit as (m,n)-periodic.

Figure 29 shows the de-construction of the torusmap for the two-frequency response for the coupled

Van der Pol oscillators of equations (2–3). The map

matches well with the data, also shown is the location

of the orbit on the torus. For this particular case, athree term sine series was used. Figure 30 shows the

de-construction of the two-frequency quasiperiodic mo-

tion of the beam-pendulum oscillator for excitation fre-

quency of 3.13 Hz and amplitude of excitation set at

650 mV. Again, the de-constructed map matches wellwith the data. One is able to identify this motion by

counting the number of times the orbit crosses θ1 and

θ2 axes, respectively, on the cover of the torus; this

particular motion is classified as a (4, 9) two-frequencymotion. Figures 31 and 32 show (5, 11) and (11, 10) or-

bits, respectively. Figure 33 shows the results of the

de-construction applied to a scan around 3.13 Hz for

The Passage through Resonance of a Coupled Mechanical Oscillator 15

Fig. 29 De-Construction of a two-frequency quasiperiodicorbit for the coupled Van der Pol oscillators.

Fig. 30 De-Construction of a two-frequency (4, 9) quasiperi-odic orbit for the coupled beam-pendulum for excitation fre-quency = 3.13 Hz and excitation amplitude =650 mV.

Fig. 31 De-Construction of a two-frequency (5, 11)quasiperiodic orbit for the coupled beam-pendulum forexcitation frequency = 3.13 Hz and excitation amplitude=725 mV.

Fig. 32 De-Construction of a two-frequency (11, 10)quasiperiodic orbit for the coupled beam-pendulum for exci-tation frequency = 3.355 Hz and excitation amplitude =475mV.

the excitation amplitude set at 750 mV (shown in Fig-

ure 26). Note that although first return maps for θ1 and

θ2 were matched by the de-construction, there were not

enough points to complete the entire orbit on the torus

and is one of the short-comings of the proposed method.

6.4 Summary of Observations for the CombinationResonance

The response in the combination resonance region for

the parametrically excited column-pendulum oscillator

consists of quasiperiodic motions. Two sets of exper-iments were performed in the combination resonance

zone. The first set consisted of frequency scan through

the entire resonance zone, between 3 and 4 Hz. The

Fig. 33 : De-Construction of a two-frequency quasiperiodicmotions for a scan around 3.13 Hz for amplitude of excitation= 750 mV.

experiments were repeated for different amplitudes of

excitation. The frequency scans revealed that as the

frequency of excitation changed, the system displayed

different types of two-frequency quasiperiodic motions.The pendulum locked on to the periodic motion, the

beam motion, however was observed to be quasiperi-

odic. Furthermore, periodic motions for the pendulum

change with the frequency and amplitude of excitation.

Some slow frequency scans were also run to capturedetails of the periodic motions of the pendulum.

The second set of experiments was conducted tocapture the details of various two-frequency motions

observed during the frequency scans. The motion be-

ing quasiperiodic, a torus was re-constructed from the

experimental observations where the phase angles wereused as the coordinates of the torus. The two-frequency

quasiperiodic motions were identified on the surface of

the torus. Data from single frequency experiments were

subjected to Hilbert-Huang empirical modal decompo-

sition (EMD) to extract the dominant modes and theirrespective frequencies. EMD revealed that the top beam

mode modulated around the beams own natural fre-

quency (ω1). The second beam EMD oscillates around

the pendulum frequency (ω2). The first pendulum EMDresponds with the pendulum natural frequency (ω2),

while the second mode oscillates at approximately one-

third of (ω2). The behavior persists as the amplitude of

excitation is increased; becoming pronounced at higher

amplitude with frequency of the top two EMD con-centrated around the two respective center frequencies.

This motion breaks down as the amplitude of excita-

tion is increased as both the beam and the pendulum

display quasiperiodicity. The transition is visualized onthe cover of the re-constructed torus becoming full as

opposed to the distinct trajectories. Similarly, the first

return maps are not smooth compared with those re-

16 Ghulam Mustafa

constructed at lower amplitudes of excitation. A slow

frequency scan at slightly lower amplitude of excitation

revealed the detail of transition; it is observed that the

pendulum displays at least three different types of mo-

tion prior to the breakdown. This observation suggestsa qualitative change in the dynamics has occurred, a bi-

furcation from a two-frequency quasiperiodic motion to

a three (or perhaps higher) frequency quasi-periodicity.

The two facts that the quasiperiodic motions aredense on the torus and, the phase angles increase mono-

tonically suggested an ad hoc method of characteriz-

ing the various types of two-frequency motions. The

proposed method consists of splitting the phase an-gles into a linear and an oscillating part and fitting a

least square line through the linear part and expressing

the oscillating part in sine/Fourier series. This so-called

de-construction was first used to demonstrate its util-

ity on a coupled Van der Pol oscillator to de-constructa two-frequency motion from the first return maps of

the torus. The method was applied to various two-

frequency responses recorded for the column-pendulum

oscillator. The de-constructed map matched well withthe data and one is able to identify motions by counting

the number of times the orbit crossed θ1 and θ2 axes on

the cover of the torus. Various types of two-frequency

motions were classified as a (4,9), (5,11) and (11,10)

orbits. The method was also applied to one scan dataand although first return maps were matched by the de-

construction, there were not enough points within the

return maps to complete the entire orbit on the torus.

6.5 Questions Concerning the Dynamics in Zone 3

The experimental observations have raised the follow-

ing questions related to the dynamics of the column-

pendulum oscillator in the combination resonance re-gion of the response:

1. Even though the amplitudes are not strictly slaved

by the phases, the re-construction of the torus seems

to work. Can this method be extended to extract themap of the torus? Can this map be used to identify

the various two-frequency motions?

2. How do the two-frequency responses change (i.e, the

bifurcations) with the amplitude and frequency of

excitation?3. What is the mechanism for the bifurcations from

two- to three-frequency motions?

4. What is the route to chaos (break–down of quasiperi-

odicity) for the column-pendulum?5. What is the efficacy of the column-pendulum as a

vibration absorbing device in the combination reso-

nance one?

7 Conclusions on the Dynamics of the

Column-Pendulum

The oscillator displays distinct and rich dynamics in

the principal parametric and the combination resonance

that are vastly different. In the former, the dynamics is

of resonance capture and escape. This type of behav-ior is typical of mechanical systems with rotating ec-

centric inertia under spin-up or spin-down conditions.

Under certain conditions, the motion is trapped inside

a resonance tunnel; one such tunnel for the pendulum

response at 375 mV excitation is shown in Figure 34.There is also the possibility of missing the resonance

tunnel altogether - in which case, no motion takes place,

hence a judicious choice of excitation parameters is es-

sential - these include excitation amplitude and fre-quency swept rate - and sometimes a small tap as the

initial condition. Depending on the excitation, the tun-

nel is wide and long or short and narrow. The emer-

gence from the tunnel can either be smooth, where the

motion dies out gracefully or it can enter a transientaperiodic section prior to the exit. This behavior is ex-

plained by the presence of a broken homoclinic orbit,

which at first, creates an opening (due to the splitting

of the stable and unstable manifolds) for the motion tobe trapped, then as the forcing parameters are varied,

the manifold exchange positions, at some point becom-

ing tangent to and/or criss-crossing each other. This

results in the transient aperiodic behavior prior to the

final exit from the resonance tunnel. Once inside thetunnel, a variety of motions are possible, mostly cen-

tered around the natural frequencies or a combination

of thereof, with the distinct possibility of quasiperiocity.

In the combination resonance, the dynamics is quasiperi-

odic and admits to reconstruction of a torus with thebeam and pendulum phase angles are the coordinates.

On the torus, the trajectories wrap around like a string

around a doughnut. The trajectories that close on each

other, constitute the two-frequency quasiperiodic or-bits, one orbit is shown in Figure 35. Depending on

excitation, various two-frequency motions are possible,

these are characterized by the number of time they go

around the torus before closing. As the excitation fre-

quency is swept across the combination resonance zone,various two-frequencymotions appear as windows, some-

times interrupted by a cloud of three- perhaps more fre-

quency quasiperidic motion with a distinct possibility

of breakdown resulting in chaos.

Acknowledgements The author gratefully acknowledges thefinancial support provided to him by the Department of Me-chanical Engineering of the Texas Tech University and toProfessor Atila Ertas for believing in author’s ability to doindependent research and in guiding him in developing the

The Passage through Resonance of a Coupled Mechanical Oscillator 17

Fig. 34 Resonance tunnel for the pendulum for amplitudeof excitation = 375 mV

Fig. 35 A two-frequency orbit around the torus for excita-tion = 650 mV @ ω = 3.13 Hz.

Dynamical Systems Laboratory where the experiments wereconducted.

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18 Ghulam Mustafa

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