Rotating Orbits of a Parametrically-excited Pendulum

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  • orbits. Assuming no damping and small angle oscillations, this system can be simplied to the Mathieu equation in

    which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques

    0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved.

    * Corresponding author.

    Chaos, Solitons and Fractals 23 (2005) 15371548

    www.elsevier.com/locate/chaosare employed to explore the dynamic responses to dierent parameters and initial conditions. Particularly, the param-

    eter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational

    orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter

    space plots. Dierent kinds of rotations, together with oscillations and chaos, are found to be located in regions within

    the resonance tongues.

    2004 Elsevier Ltd. All rights reserved.

    1. Introduction

    The parametrically excited pendulum has been of scientic interest for a considerable amount of time. It has been

    found that this system can generate various types of motion, from simple periodic oscillation to complex chaos. The

    system motion depends on the initial condition, the excitation frequency and amplitude, and the parameters of the pen-

    dulor such as the length of the pendulor rod.

    The parametric pendula, as a kind of parametric vibration system, can be analysed by a means of various methods.

    Application of perturbation techniques, originally emanating from orbited mechanics, to problems in parametric and

    nonlinear vibration [1] have been popularised by fevered key researchers, in particular Nayfeh [2]. A good example of

    using the perturbation method of multiple scales for analysis of the externally loaded parametric oscillator is in the

    work by Watt and Cartmell [3], in which numerical and experimental investigations were also discussed for a system

    with power take-o capable of doing measurable work. Analytical studies of the parametrically-excited pendulum

    can be traced back to 1981 and the work of Koch and Leven [4], in which chaotic behaviour of the pendulum system

    was identied. In their later work [5], subharmonic and homoclinic bifurcations were analysed by using the Melnikov

    and averaging methods. More recently Bishop and Cliord [6,7,8,9,10,11,12] have been heavily involved in research intoRotating orbits of a parametrically-excited pendulum

    Xu Xu a, M. Wiercigroch a,*, M.P. Cartmell b

    a Centre for Applied Dynamics Research, Department of Engineering, Fraser Noble Building, Kings College, University of Aberdeen,

    Aberdeen, AB24 3UE, Scotland, UKb Department of Mechanical Engineering, James Watt Building, University of Glasgow, Glasgow, G12 8QQ, Scotland, UK

    Accepted 22 June 2004

    Abstract

    The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotationaldoi:10.1016/j.chaos.2004.06.053

  • Yc

    g

    x independent variable obtained by non-dimensionalising real time with respect to real exciting frequency,

    X

    1538 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548this system. Analytical approaches to the analysis include those of symbolic dynamics, invariant manifolds and the

    a non-dimensionalised natural frequency with respect to real exciting frequency, Xc non-dimensionalised damping ratio with respect to natural frequency, x0 non-dimensionalised exciting amplitude with respect to real exciting frequency, Xh; _h angle displacement and velocityh0,h1,h2 asymptotic expansion of hh0; _h0 initial conditions of angle displacement and velocitys independent variable obtained by non-dimensionalising real time with respect to natural frequency, x0/ dened by / = (1 2a)x + 2uu phase angle of the systemx non-dimensionalised exciting frequency with respect to natural frequency, x0x0 natural frequency of the systemX real exciting frequency real length of the pendulor rodm mass of the pendulor

    p non-dimensionalised exciting amplitude with respect to natural frequency, x0t real timemetho

    imate

    nized

    forced

    tions,

    and T

    of att

    exami

    the sa

    within

    Th

    dition

    nonlin

    cated

    2. Sm

    Th

    lumpe

    where

    eratio

    Noreal excitation amplitude

    damping coecient

    acceleration due to gravityNomenclature

    A amplitude of the systemd of braids, as well as the potential energy theory [9]. Oscillatory orbits have also been identied [10]. The approx-

    boundary of the escape zone was calculated analytically in the work of [6], rotating periodic orbits were scruti-

    in [8] and a detailed description of the rotational orbits was given. Tumbling chaos was evident in [7] and the

    inverted pendulum was discussed in [11]. The inverted oscillation of a driven pendulum, as well as its bifurca-

    was investigated using the method of manifolds and braids by Sudor and Bishop [12] and Szemplinska-Stupnicka

    yrkiel [13] identied the features of oscillation-rotational attractors by means of bifurcation diagrams and basins

    raction using the numerical integration method. Utilising the same method, chaotic motions were subsequently

    ned and categorized into three types, oscillating chaos, tumbling chaos and rotating chaos in another paper by

    me authors [14]. In particular, tumbling chaos was precisely described and then shown to exist in a broad area

    the parameter space.

    is paper is aimed at deepening the understanding of rotational orbits under various excitation and loading con-

    s and the pendulum system is initially investigated as a linear case with no damping and for small angles. The full

    ear model is later examined numerically using the software package, Dynamics [15] and the rotational orbits lo-

    in dierent resonance zones are discussed.

    all angles: analytical studies

    e mathematical model of the parametric pendulum, as shown in Fig. 1, assumes the form of a massless rod with a

    d mass at the end and viscous damping at the pivot and has the following form:

    h cm

    _h g Y

    X2 cosXt

    sin h 0; 1

    h is the angular displacement, t is time, c is the viscous damping coecient, m is the mass, g is gravitational accel-n, l is the length of the rod, Y is the excitation amplitude and X is the excitation frequency.n-dimensionalisation with respect to the system natural frequency, x0

    g=

    p, gives,

    h c _h 1 p cosxs sin h 0: 2

  • the fo

    veloci

    lar dis

    x

    y

    y

    l

    X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548 1539orbit in which the stable manifold merges with the unstable manifold at the saddles A and B, thus forming a closedorbit.

    and th

    be fou

    rotatio

    In

    damprm of circles 1 and 2, are oscillating orbits. The angular displacement is always smaller than p and the angularty changes its sign twice per period. The trajectory of wave form 4, depicts a typical rotational motion. The angu-

    placement is growing with time and the angular velocity does not change its sign. Trajectory 3 is a heteroclinicThe parameters in Eq. (2) are dened as follows

    s x0t; c cmx0 ;p YX2

    x20

    ; x Xx0 :

    In order to understand the fundamental behaviour of the parametrically-excited pendulum the case assuming no

    damping and no external excitation is considered using the phase diagram shown in Fig. 2. The phase trajectories in

    Fig. 1. Physical model of parametrically-excited pendulum.mIt is necessary to pass through the heteroclinic orbit in order to travel from oscillatory to rotational motion,

    is means that a heteroclinic bifurcation (global bifurcation) has to occur. As a result, rotational motions cannot

    nd by following oscillations using local bifurcation methods, such as the continuation method for example. The

    ns can develop from chaotic motion, as the heteroclinic orbits are being crossed.

    this section it is considered that the pendulum oscillates with small angular displacements, sin h h, and theing ratio is small enough to be neglected, c ! 0. Thus the equation of motion, Eq. (2), becomesh 1 p cosxsh 0: 3

    Fig. 2. Phase trajectories diagram for free motion of the undamped pendulum.

  • 0 2

    . . .

    where

    and r

    Th

    Fo

    Eq

    tion,

    In

    1540 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548higher value of jj. The instabilities manifested by large amplitude oscillations and rotations must obviously occurwithin the areas of the resonance zones.the case of the Mathieu equation with a damping term, it is known that the resonance zones will start from aFig. 3).

    Once the solution of h0 has been constructed it can be substituted into Eq. (7) and then solved. By substituting therst two terms of the general solution (Eq. (5)) into the Mathieu equation (Eq. (4)), the second set of critical curves can

    be obtained by applying the same approach as for the rst set of critical curves.dx dxj1 2aj = jj/2a, produces two boundary curves, the so-called rst set of critical curves (see the dashed curves inwhere a = 1 2a and b 2a. For the condition of a

    2 < b2, i.e. j1 2aj < jj/2a, when x!1, / tends to a constantvalue, as the amplitude A, noting that it is derived from Eq. (13).

    As a result, the critical resonance curves are determined for the stability condition, i.e. d/ ! 0 and dA ! 0. The equa-. (12) is solved by rearrangement and by direct integration to give

    tan/2

    tan 12

    a2 b2

    px

    aba2b2

    p ; a2 > b2

    eb2a2

    px1

    ab

    eb2a2

    px1

    b2a2

    p ; a2 < b2

    8>>>: ;1

    AdAdx

    4a

    sin/: 13A dx

    4asin1 2ax 2u: 11

    r / = (1 2a)x + 2u, Eqs. (10) and (11) becomed/dx

    1 2a 2a

    cos/; 121 dA 2cos1 2ax 2u cosax u

    2sin1 2ax 2u sinax u

    2cosa 1x u: 9

    e slowly varying phase and amplitude can be calculated as

    dudx

    4a

    cos1 2ax 2u; 10A A A0

    A and u are treated as slowly varying amplitude and phase. Neglecting second order derivatives which emergeearranging with respect to cos(ax-u) and sin(ax-u), the resultant equation is

    d2Adx2

    2Aa dudx

    A dudx

    2" #cosax u A d

    2udx2

    2adAdx

    2 dAdx

    dudx

    sinax uThe fundamental harmonic equation is obtained by substituting the solution of Eq. (6), h = A cos (ax-u), into Eq. (4), : h0 a h0 0; 61 : h1 a2h1 cosxh0; 72 : h2 a2h2 cosxh1; 8.. .. ..Eq. (3) can be re-scaled with respect to x to obtain a standard Mathieu equation,

    h a2 cos xh 0; 4where x = xs = Xt, a = 1/x and = p/x2 = Y/ is a small number.

    Fundamental parametric resonances can be determined by using a number of asymptotic methods, all of which

    assume the general solution of Eq. (4) in the form of a converging series:

    h h0 h1 2h2 : 5Then by substituting the general solution into Eq. (4), a set of dierential equations arranged accordingly to order of

    the power of is obtained

  • Fig. 3. First and second set of critical curves for Mathieu equation.

    Fig. 4. Parameter space plot for the parametric pendulum for dierent settings of initial conditions and damping, keeping p 2 (0.0,2.0)and x 2 (0.2,3.2); (a) h(0) = 0.01p, _h0 0, c = 0.1; (b) h(0) = 0.01p, _h0 0, c = 0.3; (c) h(0) = 0.5p, _h0 0, c = 0.1; (d) h(0) = 0.5p,_h0 0, c = 0.3.

    X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548 1541

  • 3. Large angles: numerical studies

    Numerical studies have been carried out using Dynamics [15], which allows one to investigate the global and local

    dynamics by considering bifurcation diagrams, Poincare maps, phase trajectories, time histories and parameter plots. In

    this paper the numerical integrations are carried out with a time step of 1/300 of the external excitation period to guar-

    antee that the steady state is reached, also with consideration being paid to computing time. This was decided on the

    basis of tests using dierent time steps. The rst 300 excitation periods were excluded and the analysis presented in this

    section has been based on the construction of parameter space plots which have been used to determine the regions of

    1542 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548Fig. 5. Parameter space plot for h(0) = 0.01p, _h0 0, c = 0.1 in the range of x 2 (0.1,3.5), p 2 (0,4.5). Bifurcation diagrams shownabove were constructed by varying p for (a) x = 1.8, (b) x = 0.87, (c) x = 0.575. F stands for xed point; O1 for period-one oscillation;O2 for period-two oscillation; R1 for period-one rotation; R2 for period-two rotation; Or1 for period-one oscillation-rotation; Or2 forperiod-two oscillation-rotation; C for chaotic motion; PDB for period-doubling bifurcation; SNB for saddle-node bifurcation.

  • the rotating solutions. Bifurcation diagrams were plotted to investigate the eects of varying the selected system

    parameter for typical ranges. For each value of the varied parameter, the same initial conditions were used so that a

    X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548 1543Fig. 6. Time histories and phase trajectories from the primary resonance zone, x = 2. (a) c = 0.1, p = 0.5, x = 1.8; (b) c = 0.1, p = 0.97,x = 1.8; (c) c = 0.1, p = 1.15, x = 1.8; (d) c = 0.1, p = 1.5, x = 1.8; (e) c = 0.1, p = 1.8, x = 1.8; (f) c = 0.1, p = 2.8, x = 1.8; (g) c = 0.1,

    p = 3.3, x = 1.3.

  • comparison could be made between dierent parameter values for the same initial conditions. They were obtained by

    direct numerical integration of the nonlinear equation, Eq. (2).

    3.1. Parameter space plots

    A parameter space plot is a diagram which graphically depicts the distribution of dierent types of steady state mo-

    tions when two or more parameters of the system are varied. Such plots are obtained by dividing the parameter space

    into a grid for which Lyapunov exponents are calculated keeping the same initial conditions for each element of the

    grid. Then by adding a colour code, i.e. allocating dierent colours for dierent values of the Lyapunov exponents,

    a parameter space is obtained. The meaning depends on the colour settings as summarised in Dynamics [15]. In this

    study, blue marks the xed points and period-one motion (oscillation, rotation or oscillation-rotation); cyan denes

    period-two motion (oscillation, rotation or oscillation-rotation); green represents chaotic motion; red depicts the area

    around period-doubling bifurcations and pink illustrates the region around saddle-node bifurcations.

    Four parameter space plots are shown in Fig. 4 and a series of resonance tongues is clearly visible, located at the

    frequency ratios from right to left, x = X/x0 = 2,1,2/3,1/2,. . .. The position of the main resonance zones coincides withthat calculated by the asymptotic method. When p is large, the resonance zones become wide and overlap each other, so

    it can be seen that the behavior in a particular resonance zone can be inuenced by neighbouring zones within the over-

    1544 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548lapped region.

    Comparisons between the parameter space plots of small and large damping ratios, c, and between that of small andlarge initial angle values h(0), are made in Fig. 4a and b, a and c. Large viscous damping requires more energy input,which is manifested by a larger excitation amplitude to initiate unstable motion in particular rotations. As a result, the

    various regions are moved up the amplitude axis, as shown in Fig. 4b and d. Furthermore, the heavily damped system is

    not so sensitive to parameter changes as for the case with a small damping ratio. Thus the boundaries between the dif-

    ferent regions are clearer than for the case with small damping (compare Fig. 4a and b). When the motion is initiated

    with a large angle, as in Fig. 4c, the pendulum can be made to oscillate, or rotate, even with the parameters outside the

    resonance tongue zone. In Fig. 4d, both c and h(0) are large.All parameter space plots have been carefully examined by plotting time histories and phase trajectories for points

    located close to the boundaries of dierent regions. The specic characteristics of the dynamic responses for dierently

    colour regions is shown in Fig. 5. The bifurcation scenarios for varying p within the rst three resonance zones of x = 2,x = 1 and x = 2/3 are shown in Fig. 5a, b and c. Clockwise and anti-clockwise pure rotations lie mainly in the principalresonance zone, coexisting with oscillations for small x. When increasing p, period-one rotation experiences a cascadeof period doubling bifurcations which leads to chaotic motion, and back to rotations again after a chaotic interval.

    3.2. Rotating solution

    A cross-examination of the system dynamics determined by the space parameters, in particular with respect to the

    rotating orbits is carried out in this section. In the work of Bishop and Cliord [8], rotations were dened to be the

    motions which exceed h = p. Two kinds of rotations can occur in a parametrically-excited pendulum. Those that ro-

    Fig. 7. Basins of attractions for (a) c = 0.1, p = 0.5, x = 1.8; (b) c = 0.1, p = 3.3, x = 1.3.nMarks period-two oscillational attractors or

    Period-two oscillation-rotational attractors; marks Period-one pure rotational attractors; marks xed points.

  • tate always in one direction, or in other words, when the angle, h, undergoes mono changes, are termed pure rotations,and those which change the direction of rotation are called oscillations-rotations [13].

    X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548 1545Fig. 8. Time histories and phase trajectories from the second resonance zone, x = 1. (a) c = 0.1, p = 0.88, x = 0.87; (b) c = 0.1,p = 1.03, x = 0.87; (c) c = 0.1, p = 1.15, x = 0.87; (d) c = 0.1, p = 1.7, x = 0.87; (e) c = 0.1, p = 1.95, x = 0.87; (f) c = 0.1, p = 2.3,

    x = 0.73.

  • As shown in the parameter space plot and the bifurcation diagram within the primary resonance zone (Fig. 5a), by

    increasing the excitation amplitude, p, the dynamic responses of the system change in terms of the sequence of equilib-

    rium, oscillations, rotations, chaos and rotations. The details of the motions can be checked with the time histories,

    phase planes and Poincare maps in Fig. 6. Period-two oscillations with egg-shaped phase trajectories are generated

    1546 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548Fig. 9. Time histories and phase trajectories from the third resonance zone, x = 2/3. (a) c = 0.1, p = 1, x = 0.575; (b) c = 0.1, p = 1.15,

    x = 0.575; (c) c = 0.1, p = 1.3, x = 0.575; (d) c = 0.1, p = 1.5, x = 0.575; (e) c = 0.1, p = 1.59, x = 0.575; (f) c = 0.1, p = 1.7, x = 0.575.

  • X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548 1547Fig. 10. Time histories and phase trajectories from the forth resonance zone, x = 1/2. (a) c = 0.1, p = 1, x = 0.44; (b) c = 0.1, p = 1.2,via a period-doubling bifurcation, as shown in Fig. 5. An example is shown in Fig. 6a, which co-exists with period-one

    rotations as shown in the basin of attraction, as in Fig. 7a. Increasing the amplitude, p, and maintaining the same initial

    conditions, period-one pure rotations (Fig. 6b) are obtained, and period-two pure rotations occur (Fig. 6c) after a per-

    iod doubling bifurcation. A series of period-doubling bifurcations brings on tumbling chaos which exists in a wide re-

    gion of parameter space, and for which one point is chosen for examination, shown in Fig. 6d. Within the region of

    chaos, narrow zones of periodic motions are found, for example, period-eight oscillations-rotations in Fig. 6e. After-

    wards, period-one pure rotations return, but with superharmonics as shown in the phase trajectory of Fig. 6f. As inu-

    enced by the neighbouring zone, period-two oscillations-rotations (Fig. 6g) are found near the resonance zone of x = 1and co-exist with the period-one pure rotations with superharmonics, as shown in Fig. 7b.

    For the second resonance zone, at x = 1 (Fig. 5b), the saddle node bifurcation gives birth to the period-one oscil-lations shown in Fig. 8a. After a narrow interval of chaos (Fig. 8b), the system develops oscillations-rotations, Fig. 8c.

    Then a series of period-doubling bifurcations bring chaos (Fig. 8d) existing in a large range of the parameter space. By

    increasing p further, oscillations-rotations (Fig. 8e) with superharmonics are generated. Both kinds of oscillations-rota-

    tions are kept within the range of 0 to 4p. The oscillations-rotations with large angular displacements (Fig. 8f) can beobtained for higher values of p.

    The time histories, phase planes and Poincare maps for the points in the resonance zones at x = 2/3 and x = 1/2 areplotted in Figs. 9 and 10. The orbits becomemore complex than those from the resonance zones ofx = 2 andx = 1, as thefrequency ratio, x Xx0, decreases and superharmonics are added in the motions. Pure rotations can be easily obtained inthe main resonance zone but can rarely be found in the higher order resonance zones. Also, despite the region inuenced

    by the neighbouring resonance zones, the motions with increasing angular displacement are born from period doubling

    bifurcations whereas motions emanating from saddle node bifurcations are limited to certain angular displacements.

    4. Closing remarks and future work

    This paper focusses on the numerical investigations of rotational orbits of a parametrically-excited pendulum.

    The dynamic responses of the system have been examined by constructing parameter plots to determine the resonance

    x = 0.44; (c) c = 0.1, p = 1.5, x = 0.44.

  • structure (resonance zones) within which the various types of motions are colour coded. Time histories, phase trajec-

    tories and Poincare maps have been used to provide cross-checks. Pure rotations are found in the main resonance zone;

    while oscillations-rotations are found in two other examined resonance zones.

    Although the method of numerical simulation for investigating the system dynamics is powerful, it has proved to be

    time consuming; for example a typical parameter space plot needs 72h of computing time on a standard PC. The work

    done for small angular oscillations with no damping has been shown to be a good beginning for an analytical analysis,

    principally because of its simple critical curve expression and its good correspondence with the numerical results. There-

    fore, more eort is being directed into devising robust analytical and approximate solutions.

    References

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    [2] Nayfeh AH. Introduction to perturbation techniques. John Wiley & Sons, Inc.; 1981.

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    [4] Leven RW, Koch BP. Chaotic behaviour of a parametrically excited damped pendulum. Phys Lett 1981;86A:714.

    [5] Koch BP, Leven RW. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica D 1985;16:113.

    [6] Cliord MJ, Bishop SR. Approximating the escape zone for the parametrically excited pendulum. J Sound Vib 1994;172(4):5726.

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    [11] Cliord MJ, Bishop SR. Inverted oscillations of a driven pendulum. Proc R Soc Lond Series A 1998;454:28117.

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    1548 X. Xu et al. / Chaos, Solitons and Fractals 23 (2005) 15371548[13] Szemplinska-Stupnicka W, Tyrkiel E. The oscillation-rotation attractors in the forced pendulum and their peculiar properties. Int

    J Bifurcation and Chaos 2002;12:15968.

    [14] Szemplinska-Stupnicka W, Tyrkiel E. Common features of the onset of the persistent chaos in nonlinear oscillators: a

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    Rotating orbits of a parametrically-excited pendulumIntroductionSmall angles: analytical studiesLarge angles: numerical studiesParameter space plotsRotating solution

    Closing remarks and future workReferences