Nonlinear Dyn (2007) 49:105–116DOI 10.1007/s11071-006-9116-y

O R I G I N A L A R T I C L E

Global dynamics of an autoparametricspring–mass–pendulum system

Khalid El Rifai · George Haller · Anil K. Bajaj

Received: 29 September 2004 / Accepted: 21 June 2006 / Published online: 22 September 2006C© Springer Science + Business Media B.V. 2006

Abstract In this paper, a study of the global dynam-ics of an autoparametric four degree-of-freedom (DOF)spring–mass–pendulum system with a rigid body modeis presented. Following a modal decoupling procedure,typical approximate periodic solutions are obtained forthe autoparametrically coupled modes in 1:2 internalresonance. A novel technique based on forward-timesolutions for finite-time Lyapunov exponent is used toestablish global convergence and domains of attrac-tion of different solutions. The results are compared tonumerically constructed domains of attraction in theplane of initial position and initial velocity for the pen-dulum. Simulations are also provided for a few interest-ing cases of interest near critical values of parameters.Results also shed some light on the role played by othermodes present in a multi-DOF system in shaping theoverall system response.

Keywords Autoparametric pendulum . Domains ofattraction . Finite-time Lyapunov exponents . Globaldynamics . Internal resonance

K. E. Rifai · G. HallerDepartment of Mechanical Engineering, MassachusettsInstitute of Technology, Cambridge, MA 02139, USA

A. K. Bajaj (�)School of Mechanical Engineering, Purdue University, 585Purdue Mall, West Lafayette, IN 47907-2088, USAe-mail: bajaj@ecn.purdue.edu

1 Introduction

Dynamical systems possessing both flexible and rigidbody modes arise in many practical applications, in-cluding robotics and hard disk drive systems. The vibra-tional response of harmonically excited linear mass–spring–damper systems exhibiting modal as well asrigid body dynamics is well understood. Due to thelinearity of system dynamics, the vibrational responsecan be projected onto the normal mode coordinatesand reduced to analyzing the response of only a flex-ible system or that of a rigid body system. For non-linear systems, however, this decoupling is not pos-sible since no general modal decoupling procedure isavailable. Harmonically excited spring–mass–dampersystems with nonlinear components are known to un-dergo complicated dynamics in their vibratory mo-tions near resonant conditions [1, 2]. More specifi-cally, linear spring–mass–damper systems when cou-pled to a pendulum possess quadratic nonlinearitiesdue to inertial coupling with the rotational motionof the pendulum even though the pendulum by itselfonly has geometric nonlinearities. These nonlineari-ties, even if assumed to be weak and usually approx-imated as quadratic for small excitation and responseamplitudes, lead to large vibration affects with manyscenarios.

The case of 1:2 internal resonance between a pen-dulum and another structural or flexible mode that isresonantly excited has received tremendous attentionin recent years. Many researchers have studied this

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106 Nonlinear Dyn (2007) 49:105–116

class of problems where 1:2 internal resonance leadsto nonlinear modal coupling in a natural manner. Seeextensive literature reviews in [1, 2]. Examples of non-linear modal coupling due to 1:2 internal resonance in-clude pitch and roll motion in ships and surface wavesin fluid containing cavities. Others [3–7] have studiedthese systems with a view to using the pendulum as avibration absorber or neutralizer. To enhance the effec-tiveness of the absorber, a computer-controlled masssliding over a collar on an oscillating rod was usedin [8] as a tunable vibration absorber, and active vi-bration control laws have been also developed on thebasis of this concept (e.g., see [9]). One important ad-vantage of the autoparametric vibration absorber overlinear passive vibration attenuation schemes based ondynamic vibration absorbers [10] is that no additionalresonances in the neighborhood of excitation are cre-ated when the secondary system is introduced into theexcited system. Banarjee et al. [11] studied a systemconsisting of an n degree-of-freedom (DOF) mass–spring–damper assembly with a pendulum attached tothe kth mass, and investigated periodic solutions andpossible bifurcation by using the asymptotic methodof averaging. More recently, Vyas and Bajaj [12] againconsidered a single spring–mass system, but now withmultiple pendulums tuned at slightly different frequen-cies, and showed significant increase in the effectivebandwidth of the autoparametric pendulum as a vibra-tion absorber.

Almost all the above studies were primarily localin that they were limited to finding small steady-stateperiodic or other more complex solutions near the equi-librium position of the unforced system, and includedtheir stability and bifurcation behavior. Only the workof Lee and Hsu [13] has investigated the domains of at-traction of a two DOF spring–pendulum system for dy-namics within a local region in the pendulum’s plane.To study the domains of attraction of periodic solu-tions, the authors used the cell-to-cell mapping tech-nique developed by Hsu [14]. The domain of attractionof an invariant set is the set of all initial conditionsfor whom the solutions converge to the invariant set.Thus, the domains of attraction for a system provide anunderstanding of the transient dynamics of the systemas it evolves or asymptotically moves toward a steadystate, and are known to be determined by the globalstable and unstable manifolds of hyperbolic invariantsets for the system. The basin boundaries of attractorshave received much attention in the context of sensitive

dependence on initial conditions [15]. Even when thesystem does not have a chaotic attractor, it can exhibitcomplex transient dynamics if the basin boundaries ofperiodic or other regular attractors are fractal in na-ture. We should also note that basin boundaries areinvariant manifolds along which sensitive dependenceto initial conditions must hold. Indeed, initial condi-tions selected on different sides of the basin boundaryapproach different invariant sets, and hence develop afinite distance, no matter how close they were initially.The converse is certainly not true: sensitive sets are notnecessarily basin boundaries.

The domains of attraction in a region of the state–space can also be constructed by direct numerical inte-gration of the equations of motion over a grid of initialconditions. This is a very numerically intensive andinefficient approach. A more efficient approach canbe based on the determination of stable and unstablemanifolds or hyperbolic behavior in time-dependentvelocity fields. The hyperbolic behavior is, in general,characterized by distribution of finite-time Lyapunovexponents in the flow. Haller [16] has recently derivedrigorous results that allow one to locate finite-time hy-perbolic sets and their local stable and unstable man-ifolds in time-dependent velocity fields. He also pro-vides a numerical algorithm for computing a first ap-proximation to the uniformly hyperbolic sets in two-dimensional velocity fields. It has also been shown[17, 18] that contours or level curves of finite-timeLyapunov exponents exhibit striking resemblance tostable and unstable manifolds which define the basinboundaries.

In this work, we consider a system consisting of anautoparametric pendulum that is coupled to a 3-DOFlinear spring–mass–damper system. The 3-DOF sys-tem possesses one rigid body mode and two elasticmodes. Following the problem setup, a modal decou-pling procedure is performed in Section 2 on the non-dimensional form of equations to study the dynamicsof the system. In Section 3, approximate steady-stateperiodic solutions of the system to a harmonic force ex-citation acting on the lowest mass are then developedby using the harmonic balance approach. The focusis on the dynamics of the system when the resonantlyexcited highest-frequency mode is in near 2:1 internalresonance with the pendulum linear frequency. For pa-rameter combinations with multiple periodic solutions,the domains of attraction of coexisting periodic solu-tions are then considered in Section 4. The domains of

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Nonlinear Dyn (2007) 49:105–116 107

attraction in a subset region of the state–space of thesystem are first obtained by direct numerical integrationover a grid of initial conditions. Here the initial condi-tions of importance are those for the pendulum, that is,its angular position and angular velocity. Subsequently,the maximal finite-time Lyapunov exponent is com-puted for a grid of initial conditions in pendulum sub-space, and level curves for the Lyapunov exponent arecontrasted with the domains of attraction determinedby direct numerical simulation. The work concludes inSection 5 with some remarks.

2 Problem formulation

Consider the 3-DOF linear mass–spring–damper as-sembly in Fig. 1, which has a pendulum attached tothe second mass, and is subjected to a single frequencyharmonic excitation applied to the bottom mass. Themasses, springs, and the dashpots in the 3-DOF sys-tem are all identical and the system is in vertical plane.Note that the system with pendulum in downward orupward vertical position can also translate as a rigidbody, which represents the first mode of the system.Clearly, the system shown may exhibit linear “locked-pendulum” vibrations (pendulum settled to the lower orupper equilibrium position) for the 4-DOF system withθ = 0, or nonlinear oscillations in a system in which allfour DOFs are active (nontrivial pendulum oscillationsin addition to vibrations of the spring–mass–dampersystem). Naturally, the linear forced oscillations withpendulum in upward equilibrium position are expectedto be unstable, while those with the pendulum in down-ward equilibrium position are expected to be stable atleast for some excitation parameters. The focus of thisstudy is on the case when a 1:2 internal resonance existsbetween the pendulum (rotational) mode and the thirdlocked-pendulum mode, and this third linear mode is innear-resonance with external excitation. It is assumedthat the coordinates x1, x2, and x3 are measured fromthe equilibrium position of the system which exists un-der the action of constant upward forces of magnitudesMg, (M + m)g, and Mg, applied, respectively, to thethree blocks. These forces, not shown in the figure,compensate for the gravitational forces on the blocks.The equations of motion governing the motion of thesystem are then given as

Mx1 + c1(x1 − x2) + k1(x1 − x2) = P0 cos ωt, (1)

Fig. 1 The three-degree-of-freedom spring–mass–damper sys-tem with a nonlinear pendulum attached to the middle block

(M + m)x2 + c1(2x2− x1− x3) + k1(2x2 − x1 − x3)

+ ml(θ sin θ + θ2 cos θ ) = 0, (2)

Mx3 + c1(x3 − x2) + k1(x3 − x2) = 0, (3)

ml2θ + ml(x2 + g) sin θ + c2θ + k2θ = 0. (4)

It is important to note that the 4-DOF system withpendulum has θ = 0 as a solution. The coupling in thependulum dynamics and the dynamics of the 3-DOFlinear system arises only through terms in the equa-tion for the second block. For small amplitude os-cillations, if only linear terms in generalized coor-dinates and velocities are retained in the model, thethree coordinates x1, x2, and x3 are uncoupled fromthe linear pendulum equation. Now, the natural fre-quencies of the 3-DOF linear undamped system (sys-tem without pendulum) are given by �1 = 0, �2 =√

k1/M , and �3 = √3k1/M . For the 4-DOF linear

system that includes pendulum, it is easy to show thatthe second natural frequency �2 = √

k1/M remainsthe same, ω2n = �2, while the third frequency �3

is modified to ω3n = √k1/M

√(3M + m)/(M + m) =

�2√

(3 + R)/(1 + R). The additional natural fre-

quency is that for the pendulum with ωp =√

ω2g + ω2

k

where ωg = √g/ l and ωk =

√k2/ml2. These two

frequencies, respectively, represent the linear grav-ity and stiffness controlled natural frequencies of thependulum about the bottom equilibrium position. In

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108 Nonlinear Dyn (2007) 49:105–116

the remaining work, we will only consider pendu-lum motions near θ = 0, that is, motions in which thependulum is near its bottom equilibrium position. Thecomplete (4-DOF) system then has four linear naturalmodes: three of them are the “locked-pendulum” ver-sions of the modes of the 3-DOF linear system, andthe fourth one is the additional pendulum mode of vi-bration. We now define the following non-dimensionalparameters:

τ = ωt, ηi = xi/ l, R = m/M, ε = P0/k1l,

ζ1 = c1/2M�2, ζ2 = c2/2ml2√

ω2g + ω2

k ,

α = ω/�2, βc =√

β22 + β2

2 ,

β1 = ωg/ω2n, β2 = ωk/ω2n. (5)

With these parameters, Equations (1)–(3) reduce to thefollowing non-dimensional form:

[M] {η}′′ + [C] {η}′ + [K ] {η} = {F} , (6)

where

[M] =

⎡⎢⎢⎣1

1+R 0 0

0 1 0

0 0 11+R

⎤⎥⎥⎦ ,

[C] = 2ζ1α

⎡⎢⎣ 1 −1 0

−1 2 −1

0 −1 1

⎤⎥⎦ ,

[K ] = α2

⎡⎢⎣ 1 −1 0

−1 2 −1

0 −1 1

⎤⎥⎦ ,

{F} =

⎧⎪⎨⎪⎩εα2 cos τ

−γ (θ ′′ sin θ + θ ′2 cos θ )

0

⎫⎪⎬⎪⎭ , {η} =

⎧⎪⎨⎪⎩η1

η2

η3

⎫⎪⎬⎪⎭ ,

and γ = R/(R + 1), ζ1 = ζ1/√

1 + R, α = 1/

(α√

1 + R). Note that there is also the pendulum equa-tion for the 4-DOF system. The influence of the pen-dulum on block motions is reflected in the vector F onthe right-hand side of Equations (6).

Since the analysis here is primarily concerned withthe response of the system when the third translationalmode of the system is resonantly excited (ω3n/ω ≈ 1),it is natural to perform a modal transformation to un-couple the modes of the translational system. It is note-worthy that the damping and stiffness matrices are pro-portional. This means that the damped linear 3-DOFsystem can be uncoupled on the basis of the undampedmodal matrix for the 3-DOF system. Let [P] be themodal matrix of the undamped system (6) correspond-ing to the translational coordinates ηi ’s with {F} = 0.Then, the transformation {η} = [P]{y} transforms thesystem (6) into the following form:

[P]T [M] [P] {y}′′ + [P]T [C] [P] {y}′

+ [P]T [K ] [P] {y} = [P]T {F} . (7)

Here the components of the vector {y}, that is, y1, y2,and y3, are the amplitudes of the three linear locked-pendulum modes of vibration of the system with thependulum at its lower equilibrium position. The equa-tions for the three modes are coupled due to the right-hand side in Equation (7) with the motion of the pendu-lum. Also, the pendulum motion is nonlinearly coupledonly to the response of the first and the third transla-tional modes; the second mode being uncoupled fromthe motion of the pendulum, as well as from motionsof the first and the third translational modes.

As already stated, the interest in this study is onsteady-state periodic solutions and their domains ofattractions resulting from the excitation acting on thebottom block. For small amplitude oscillations near thethird (highest) natural frequency of the 3-DOF system,it can be shown that the response of the system canbe well approximated by assuming negligible contri-butions to the third mode and the pendulum motionsfrom the first (rigid body) translational mode of thesystem. Such an analysis should follow standard argu-ments made when applying the method of averagingto weakly nonlinear resonant oscillations in systemswith internal resonance [2]. Thus, it is reasonable toneglect the contribution of the first mode (rigid bodymode) to the response of the third mode and the pen-dulum, that is, to let y1 ≈ 0 in the pendulum’s dynam-ical equation. Note that this does not suggest that thesemodes play no role in the full dynamic response ofthe system even at frequencies very close to the thirdmodal resonance. This assumption has been furthervalidated by comparing the later obtained solutions

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Nonlinear Dyn (2007) 49:105–116 109

with those from numerical integration. Also, assum-ing small oscillations of the pendulum near θ = 0, welet sin θ ∼= θ, cos θ ∼= 1. The resulting two equationsof motion for y3 and θ are with quadratic nonlinearitiesand are given by

y′′3 + 2ζ3 y′

3 + ω23 y3 + c3(θ ′′θ + θ ′2) = c f cos τ, (8)

θ ′′ + 2ζpθ′ + ω2

pθ + cp y′′3 θ = 0. (9)

Here, the coefficients in the model can be shown to be

ς3 = ς1(3 + R)/{α(1 + R)},ω3n =

√(3 + R)/{α2(1 + R)},

c3 = γ (−2)(R2+2R+3)/{(3+R)

(√2R2+4R+6

)},

c f = ε(R2 + 2R + 3)/{α2(3 + R)

(√2R2 + 4R + 6

)},

ςp = ς2βc/(α√

1 + R), ωp = βc/(α√

1 + R),

cp = −2/(√

2R2 + 4R + 6).

3 Local steady-state periodic solutions

An approximate steady-state periodic solution for thereduced system is easily obtained through the methodof harmonic balance. In this regard, one-term harmonicsolutions were assumed for the approximate solution,which is reasonable for small amplitude harmonic ex-citation. See [4] for a comprehensive comparison be-tween harmonic balance solutions and those obtainedby numerical integration. Therefore, the solutions areassumed in the following form:

y3 = Y3 cos(τ + φ3), (10)

θ = B cos(τ/2 + φB). (11)

Following some trigonometric manipulations andsolution of the resulting algebraic equations, explicitalgebraic expressions for the amplitudes Y3 and B wereobtained as a function of the system parameters. Theseare as follows:

Y3

=

⎧⎪⎪⎨⎪⎪⎩c2

f

/ {√(ω2

3n − 1)2 + (2ς3)2

}, when B = 0,

2√(

ω2p − 1/4

)2 + (ςp)2/cp, when B �= 0,

(12)

and the amplitude of the pendulum B governed by thefourth-order polynomial

B4 − (16q/c3cp)B2 + 64{dp

/c2

p − c2f

/4}/

c23 = 0,

(13)

where

q = (ω2

3n − 1)(

ω2p − 1/4

) − 2ς3ςp/c3p,

where c3p = f (R),

d = (ω2

3n − 1)2 + (2ς3)2, p = (

ω2p − 1/4

) + (ςp)2.

Shown in Fig. 2 is a typical frequency response am-plitude plot of the steady-state periodic solutions for thethird mode and the pendulum, plotted as a function ofα. The parameters were selected to yield zero internalmistuning (ω3n/ωp = 2), R = 0.2, βc = 0.895, smallprimary and secondary system (pendulum) dampings,ζ1 = 0.005, ζ2 = 0.015, and small forcing amplitude,ε = 0.02. The procedure and results obtained here arein accordance with those obtained by Savran [19], whocompared the performance of the autoparametric anddynamic absorbers for a similar system.

The periodic solutions in Fig. 2 are of two types: thesemi-trivial solution and the nontrivial solution. Thesemi-trivial solution corresponds to a trivial pendu-lum solution, that is, it is with B = 0 or θ = 0. There-fore, the semi-trivial solution, also termed the locked-pendulum solution, is the linear modal response of thethird mode. In the nontrivial solution, both Y3 and Bare nonzero. In the frequency interval (region A) be-tween the points denoted as “pitchfork bifurcations,” itis known that the semi-trivial solution has larger ampli-tude of response compared to the response amplitude ofthe third mode in the nontrivial solution [3, 4, 11]. How-ever, at larger mistunings (regions B) the amplitude ofthe semi-trivial solution is smaller than that in the non-trivial solution. This highlights the limited bandwidthover which the pendulum acts as a vibration absorber.

Figure 2 displays some typical phenomena. As theexcitation frequency is changed, there arise changesin stability in steady-state solution branches eitherthrough a pitchfork bifurcation or through a saddle–node bifurcation (turning point). These results are men-tioned only for completeness, and typical results for thisclass of systems, based on averaged equations, can befound in [4] as well as in [12]. For harmonic balance

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110 Nonlinear Dyn (2007) 49:105–116

Fig. 2 Amplitude responsesfor the third mode and thependulum as a function ofthe normalized frequency ofexcitation, for a nominal setof parameters. R = 0.2,βc = 0.895, ζ1 = 0.005,ζ2 = 0.015, ε = 0.02

solutions, Floquet multipliers of the system linearizedabout the periodic orbit may be used to indicate sta-bility. If all the Floquet multipliers lie inside the unitcircle, the corresponding periodic orbit is stable. Onthe other hand, an orbit is unstable if at least one Flo-quet multiplier is outside the unit circle. In case a Flo-quet multiplier is of modulus one, a bifurcation point iscreated. For details on the classification of bifurcationpoints based on such an approach, see [3] and [20]. Itis noteworthy that this system has a strong dependenceon its parameters R, ζ 2, ζ 1, βc, and ε. Many parametricstudies have been performed and details are availablein [4] and [11].

4 Global domains of attraction

Motivated by the existence of two very distinct sta-ble solutions and the change in their stabilities withthe frequency of excitation (or mistuning), we con-sider the domains of attraction for these solutions.These domains play an essential role in determiningthe transient response of the autoparametric vibra-tion absorber. For the following simulations, a dimen-sional set of parameters was chosen yielding the samenominal non-dimensional parameters as those used inFig. 2. For this set of parameters, the excitation frequen-cies ω = 14.55, 16.35 and ω = 14.95, 15.9 rad/s werefound to correspond, respectively, to the saddle–nodebifurcations and the pitchfork bifurcations. In regionB of Fig. 2, both a stable and an unstable nontrivialsolution exist in addition to a stable semi-trivial solu-tion. The domains of attraction then predict the conver-gence of a trajectory to one stable steady-state solution

compared to the other for different initial conditions.Note that contrary to steady-state behavior, the analy-sis here may not be reduced to two second-order dif-ferential equations (Equations (8) and (9)) for the thirdlocked-pendulum mode and the pendulum. So, evenif the periodic solutions may essentially be confinedto the four-dimensional subspace of (y3, y′

3, θ, θ ′) andthus well approximated by the harmonic balance pro-cedure outlined above, the transient motions may notbe limited to only this subspace.

It is known that the boundaries of domains of attrac-tion of attractive invariant sets in a finite-dimensionaldynamical system are characterized by the stable mani-folds of various hyperbolic structures, such as unstablefixed points, unstable periodic orbits, and other highercomplexity solutions. Finding domains of attraction ofsteady-state solutions of forced nonlinear systems isa very difficult problem and not much literature ex-ists on the subject. Tondl and colleagues have over theyears (e.g., see [21]) used a method, which involvesapplying a disturbance over a period and observing thesubsequent motion. However, such a technique, thoughinsightful, may not necessarily capture the true do-mains of attraction for a system. Another noteworthystudy is that of Lee and Hsu [13], who considered aharmonically excited 2-DOF planar spring–pendulumsystem, and used several techniques such as the cell-to-cell mapping and interpolated mapping to construct thedomains of attraction of stable periodic solutions. Thefirst technique partitions the state–space into cells andthe dynamics of the system is approximated by map-pings over a period of forcing. Iterates of the mappingthen correspond to evolution of the system in time. Es-cape of solutions from the cells then gives a measure

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Nonlinear Dyn (2007) 49:105–116 111

of the domains of attraction. The interpolated mappingmethod finds an image corresponding to the discretestate–space model using an interpolation-based proce-dure. We now give a short review of the finite-time Lya-punov exponents approach to approximating domainsof attraction.

4.1 Maximal Lyapunov exponent algorithm

Consider an n-dimensional dynamical system:

{x(t)} = { f ({x} , t)} . (14)

Let {x({x0}, t)} denote the solution to Equation (14)starting at an initial condition {x0}. Infinitesimal pertur-bations {ν(t)} about a specified trajectory {x(t)} satisfythe variational equation

{ν(t)} = ∂ { f ({x} , t)}∂ {x} {ν(t)} . (15)

These perturbations admit the unique solution

{ν(t)} = ∂ {x({x0} , t)}∂ {x0} {ν(t0)} , (16)

where ∂{x({x0}, t)}/∂{x0} is the differential transitionoperator (or state transition matrix for the linear system(15)). By taking norms in Equation (16) and dividingby the perturbation ‖{ν(t0)}‖ �= 0, the maximal stretchin the flow field at each time instant can be written as

sup‖{ν(t)}‖‖{ν(t0)}‖ = sup

∥∥∥ ∂{x}∂{x0} {ν(t0)}

∥∥∥‖{ν(t0)}‖

= σ (∂ {x} /∂{x0}), (17)

where ‖{·}‖ is the Euclidean norm and σ (·) is the maxi-mum singular value of a matrix. Then, the direct (finite-time) Lyapunov exponent [17, 18] at a trajectory start-ing at {x0} is defined as

δ({x0}, t) = ln σ (∂ {x} /∂{x0})(t − t0)

, (18)

where σ (∂{x}/∂{x0}) represents the maximum stretchin the flow field. Indeed, σ (∂{x}/∂{x0}) > 1 for a re-gion of stretch and conversely for contraction regions,whereas σ (∂{x}/∂{x0}) = 1 represents no stretch orcontraction. The aforementioned scenarios correspond

to positive, negative, and zero values for δ({x0}, t).Thus, a positive Lyapunov exponent represents a regionin state–space where trajectories are separating expo-nentially. In this discussion, properties of the Lyapunovexponent and the corresponding system behavior maybe either over a time interval t ∈ [t1, t2] or uniformlyin time ∀ t > t0. This notion of having a positiveLyapunov exponent has been commonly used as a mea-sure of chaos in dynamical systems [15, 20]. However,information about the sensitivity of system’s solutionsto initial conditions can be used to deduce many otherconvergence characteristics.

The algorithm to be used here involves solving forthe finite-time Lyapunov exponent over a grid of initialconditions for a large enough time such that all solu-tions settle to their final states. Therefore, the longerthe solution time is, the more refined the picture willbecome as nearby trajectories would separate (or comecloser) as the time traces of their corresponding Lya-punov exponents would separate (or merge). The valueof the maximal Lyapunov exponent, a measure of thesensitivity of a system to initial conditions, is expectedto be maximal on the boundaries between different do-mains of attraction. This is evident as these boundariesexhibit the largest sensitivity to initial conditions forthe system.

For the full eight-dimensional state–space of the sys-tem under study, the domains of attraction in the fre-quency range of interest are viewed through the sepa-ration between two physically different solutions bothon the stable manifold. Geometrically, this needs to berealized in a seven-dimensional phase space. However,since visualizing such an object even via taking slicesor projections into lower dimensional subspaces willnot necessarily yield much insight, we opt to focus onobserving the effect this object has on the domains ofattraction in the (θ0, θ0) plane. In this regard, the pen-dulum solutions would be either a fixed-point solutionat the origin, which corresponds to the semi-trivial so-lution for the system, or a periodic orbit (limit cycle)that corresponds to nontrivial solutions. These may besufficient to capture the targeted behavior of the system.

Observe that the problem of finding the finite-timeLyapunov exponent is, in essence, that of finding the op-erator norm or largest singular value of the differentialtransition operator (Equation (17)). Due to the fixingof initial conditions for the six translational states tozero, the problem reduces to that of finding the largestsingular value of an 8 × 2 matrix. It is not difficult to

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112 Nonlinear Dyn (2007) 49:105–116

show that this reduces to finding the largest eigenvalueof a 2 × 2 matrix, since the remaining six singular val-ues are zero. This means that this problem is mathe-matically equivalent to finding the finite-time invariantmanifolds in a 2D flow field. The algorithm here uses afourth-order fixed step Runge–Kutta approximation tothe solution of Equations (1)–(4), which translates to aglobal truncation error of order h4, where h is the stepsize for time-integration. Considering m different initialconditions for which the solution is to be sampled alongeach of the two axes yields m2 combinations. For thiseighth-order state–space model that translates to solv-ing 8m2 first-order differential equations. In addition,these equations need to be solved for long enough timesuch that steady-state behavior is reached.

4.2 Simulation results

The final algorithm used to produce the results pre-sented in the following uses 100 initial condition sam-ples (m) on each axis, a step size of 0.01 s (≈T/40)where T is the period of excitation, and 15,000 timesteps in integration. These parameters were selectedon the basis of experience from numerically obtainedsolutions. The domains of attraction at an excitationfrequency ω = 14.7 rad/s, which corresponds to 5%external mistuning from exact resonance condition(ω3 = 1), are shown in Fig. 3. The response curvesin Fig. 2 had indicated that the semi-trivial state anda nontrivial periodic solution are stable for this exci-tation frequency. The data in Fig. 3a was generatedvia direct numerical integration for a limited range ofinitial condition vector (θ0, θ0) near the lower pendu-lum equilibrium and using a relatively coarse grid. Thepoints “·” indicate initial conditions that converge tothe semi-trivial solution and the points “∗” representinitial conditions that converge to the nontrivial peri-odic response. Figure 3b is a plot of the level sets ofthe largest Lyapunov exponent for the same set of pa-rameters and excitation frequency. It is evident that thetwo techniques agree, up to the limited discretizationof the (θ0, θ0) plane used in direct numerical integra-tion. Knowing that at (θ0 = 0, θ0 = 0), the solution isthe semi-trivial one and that the boundaries betweendifferent domains correspond to the largest Lyapunovexponent, the picture becomes clear. As predicted bynumerical integration, the trajectories for initial condi-tions near the origin converge to the fixed-point solu-tion (semi-trivial periodic solution). This corresponds

Fig. 3 Domains of attraction in the (θ0, θ0) plane at ω =14.7 rad/s. (a) Result of direct numerical integration; “·” indi-cate initial conditions that converged to the semi-trivial solutionand “∗” represent initial conditions that converged to nontrivialresponse. (b) Level curves of the maximal Lyapunov exponent.R = 0.2, βc = 0.895, ζ1 = 0.005, ζ2 = 0.015, ε = 0.02

to a small Lyapunov exponent, that is, strongly attrac-tive solution. As the initial angle increases the sys-tem is more likely to be attracted to the nontrivial sta-ble periodic orbit. This is signified by the blue regionwhich is that of minimal stretch in the flow field orwith strongest attraction for the fixed-point semi-trivialsolution.

The plot also reveals variation in the rate of attrac-tion from different initial conditions, that is, the degreeof attraction within a domain. The simulation resultsstarting with different initial conditions (Fig. 4) showthat the regions with different degrees of attractioncould correspond to quite different global system be-havior. The three cases in Fig. 4a–c lie within the central

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Fig. 4 Time response (position x2 and velocity x2) of secondmass for different initial conditions in (θ0, θ0) plane for the case inFig. 3. (a) (θ0 = 0, θ0 = 1), (b) (θ0 = 0.2, θ0 = 1), (c) (θ0 = 0.5,

θ0 = −0.3), (d) (θ0 = 0.5, θ0 = −0.4). ω = 14.7 rad/s, R = 0.2,βc = 0.895, ζ1 = 0.005, ζ2 = 0.015, ε = 0.02

regions of Lyapunov exponent δ, (−0.01 < δ < 0.01),(0.01 < δ < 0.03), and (0.03 < δ < 0.04) and corre-spond to three different semi-trivial solutions based ondifferent contributions from the rigid body mode. Ob-serve also that the responses for the cases in Fig. 4c andd, which correspond to initial conditions on either sideof the boundary between different domains, have verysimilar time traces up to a critical time beyond whichsolutions’ paths deviate. It is noteworthy that for a verysmall change in initial conditions, the steady-state pen-dulum oscillations shift from an amplitude of 0 to about0.5 rad. Note that the results here in Fig. 5 were plottedfor x2 and x ′

2 rather than θ and θ ′ to show the role ofcoupling in the dynamics of the complete system. Fur-thermore, even though the drift in solutions arise forx1, x2, x3, these contribute a Lyapunov exponent equalto zero. Thus, as long as there is a positive Lyapunov

exponent, the rigid body translations do not show up inour calculations.

The next simulation results show the effect of exter-nal mistuning. Figure 5a shows the Lyapunov exponentlevel curves at the same frequency and parameters ofFig. 3a but over an extended range in the pendulum’sphase plane. As expected, the structure is quite com-plex further away from the fixed point at the origin.Figure 5b shows the level curves for the excitation fre-quency ω = 16.2 rad/s, which also corresponds to 5%external mistuning but now above the zero mistuningpoint, which is at ω = 15.45 rad/s. The behavior in thiscase is quite distinct even in regions somewhat close tothe origin. In fact, a much larger region of initial condi-tions converges to the fixed point at the origin, includ-ing some initial conditions much closer to the upperequilibrium position. Recall that the stable manifolds

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Fig. 5 Lyapunov exponent level curves in (θ0, θ0) plane. (a) ω =14.7 rad/s, (b) ω = 16.2 rad/s. R = 0.2, βc = 0.895, ζ1 = 0.005,ζ2 = 0.015, ε = 0.02

of the invariant repelling sets play a major role in shap-ing the domains of attraction. Reverting back to Fig.2, one notes the asymmetrical nature of the pendulumsolutions. This asymmetry leads to the two periodic or-bits being very close to each other in the phase space,for this mistuning. As a result, the unstable orbit repelsmost of the trajectories away from its nearby stableorbit, which then settle to the fixed point.

Next, the global behavior near bifurcation points isexplored. Let us consider the two important frequenciesof ω = 14.6 and 14.9 rad/s, which are very close to asaddle–node (to its right) and a pitchfork bifurcation (toits left), respectively. The corresponding level curvesfor the finite-time Lyapunov exponent are shown inFig. 6.

Observe that for ω = 14.6 rad/s, the domains of at-traction picture is similar to that for ω = 14.7 rad/s,

Fig. 6 Lyapunov exponent level curves in (θ0, θ0) plane. (a) ω =14.6 rad/s, (b) ω = 14.9 rad/s. R = 0.2, βc = 0.895, ζ1 = 0.005,ζ2 = 0.015, ε = 0.02

except that the regions of attraction for the semi-trivialsolution are appreciably stretched in size. This is quiteexpected as this parameter value is closer to the saddle–node bifurcation point (ω = 14.55 rad/s) below whichonly the semi-trivial solution exists. This bifurcation,which corresponds to hysteretic behavior or the com-monly referred to jump phenomenon, is one incentivefor pursing the domains of attraction. The other sig-nificant parameter values are those in the vicinity of apitchfork bifurcation, such as that for ω = 14.9 rad/s.Here, the semi-trivial solution is losing its stability. Thisis indicated by the fact that the origin in (θ0, θ0) planeitself is close to the boundary between solutions, thatis, a region of weakest attraction. Here, the unstablependulum orbit (see Fig. 2) is very small and surround-ing the trivial pendulum solution. Thus, on this scaleit looks like that the origin in (θ0, θ0) plane is unstable

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Fig. 7 Chaotic response of the system at large amplitude ofexcitation. (a) Level curves of the finite-time Lyapunov exponentin (θ0, θ0) plane. (b) Modal time responses with initial condition(θ0 = 1, θ0 = 0). Frequency ω = 15.45 rad/s and amplitude ε =1 of excitation

and repelling trajectories as represented by the red areaaround θ = 0. The plot in Fig. 6b then mostly con-sists of pairs of level curves settling onto a periodicorbit.

Finally, we illustrate the global characterization ofmotion of the system for sufficiently large amplitudeof excitation ε = 1 (see Fig. 7). This case correspondsto chaotic oscillations of the spring–mass–damper–pendulum system. In contrast to all previous cases, theLyapunov exponent values here are all positive over thependulum (θ0, θ0) plane though they are not necessar-ily large since no significant change in the final statetakes place for different trajectories. It is noteworthythat no significant changes in the structure are observedfor larger excitations, e.g., when ε = 10, and only the

intensity of chaotic mixing is just increased. Anotherinteresting observation (see Fig. 7b) here is that despitethe chaotic dynamics, the response of the second modeof the system is invariably that of a linear harmoni-cally driven system, as analytically predicted, with theexcitation amplitude linearly scaling the steady-stateamplitude. This is in contrast to the obviously chaoticbehavior of the third mode, a result which is driven bythe fact that the second mass, which is coupled to thependulum, does not contribute to the second mode’sdynamics.

5 Conclusions

In this work, we studied the global dynamics anddomains of attraction for different periodic solutionsof an autoparametric spring–mass–pendulum system.Through a modal decoupling procedure, the couplingbetween the pendulum and system modes was identi-fied. In addition to the autoparametric coupling with thethird flexible mode, coupling with a rigid body modewas also identified. It was shown that rigid body dy-namics of the system are excited even for zero ini-tial conditions for the rigid body mode and excitationnear the third mode, which depends on the pendulum’stransients. Though local steady-state weakly nonlin-ear behavior is well understood, the transient behav-ior and asymptotic convergence of solutions to a fixedpoint or a periodic orbit in a critical region of frequen-cies bounded by pitchfork and saddle–node bifurca-tions depends on domains of attraction. Furthermore,global and strongly nonlinear behavior is also of in-terest. A study of characterization of the domains ofattraction was performed through the development oflevel curves of the finite-time Lyapunov exponent in aglobal region in the pendulum’s plane. This approach,based on a dynamical performance index rather thanon a discretization or interpolation technique, provedto be very rewarding and insightful. It provided bothrelative and absolute measures of global convergenceto invariant sets. Furthermore, its conceptual robust-ness allows for global and strongly nonlinear condi-tions to be studied without concerns of adequacy ofan iteration size or other equivalent parameter. Usinglocal steady-state solutions as a guide, the behaviorof the three coupled modes for the key cases of in-terest was captured including strongly excited chaoticsolutions.

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