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International Journal of Bifurcation and Chaos, Vol. 12, No. 4 (2002) 671–686c© World Scientific Publishing Company

MECHANICAL MODELS OF CHUA’S CIRCUIT

J. AWREJCEWICZDepartment of Automatics and Biomechanics, 1/15 Stefanowskiego St.,

Technical University of Lodz, 90-924 Lodz, Poland

M. L. CALVISIApplied Science and Technology Graduate Group,

University of California, Berkeley, CA 94720, USA

Received March 15, 2001; Revised October 31, 2001

In this paper we present two different examples of electromechanical realization of Chua’s circuitand one of Chua’s unfolding circuit. In addition, a novel mechanism is proposed for realizingChua’s equations in a purely mechanical way using friction properties. All relevant equationsare derived and the mechanical realizations of the proposed devices are discussed.

Keywords: Chua’s circuit; Chua’s unfolding circuit; Chua’s diode; mechanical Chua’s circuit.

1. Introduction

Chua’s circuit (see Fig. 1) is one of the sim-plest physical models that has been widely inves-tigated by mathematical, numerical and experi-mental methods. One of the main attractions ofChua’s circuit is that it can be easily built withless than a dozen standard circuit components,and has often been referred to as the poor man’schaos generator. A mathematical analysis of theglobal unfolding behavior of Chua’s circuit is givenin [Chua, 1993]. Perhaps one of the most impor-tant observations is that by adding a linear resis-tor in series with the inductor in Chua’s circuit,the resulting unfolded Chua’s circuit is topologicallyequivalent to a 21-parameter family of continuousodd-symmetric, piecewise-linear differential equa-tions in R3. Any vector field belonging to the“unfolded” topologically conjugate family can betransformed (mapped) via a nonsingular lineartransformation to an unfolded Chua’s circuit withonly seven parameters. In addition, it extendsthe local concept of unfolding to a global one,where all results are valid for the whole space R3.In other words, any autonomous three-dimensionalsystem characterized by an odd-symmetric, three-

segment, continuous, piecewise-linear function canbe mapped to an unfolded Chua’s circuit havingidentical qualitative dynamics.

The following question was posed in [Chua,1993]: Since there are several different third-ordercircuits (which exhibit strange attractors) com-posed of a continuous, odd-symmetric, piecewise-linear vector field in R3, does a homeomorphicmapping of such circuits to an unfolded Chua’scircuit exist? If such a homeomorphism exists, thetwo circuits are said to be equivalent (or topo-logically conjugate). The unfolded Chua’s circuitis canonical in the sense that the governing equa-tions contain a minimum number of parameters forobserving the full generality of dynamical behav-iors. In this paper, we will present mechanical andelectromechanical device models of Chua’s circuit,as well as of the unfolded Chua’s circuit.

The Chua’s circuit is shown in Fig. 1. Thegoverning equations have the form

C1dv1

dt=

1

R(v2 − v1)− f(v1) ,

C2dv2

dt=

1

R(v1 − v2) + iL ,

LdiLdt

= −v2 ,

(1)

671

672 J. Awrejcewicz & M. L. Calvisi

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(a) (b)

Fig. 1. (a) Chua’s circuit is made of four standard linear circuit components and a nonlinear resistor; (b) vR–iR characteristicof the nonlinear resistor, which can be synthesized by two standard OP AMPS (operational amplifiers) and six linear resistors[Kennedy, 1992].

where f(·) is a piecewise-linear function repre-senting the vR–iR characteristic of Chua’s diode;namely, the nonlinear resistor, NR. In nondimen-sional form, (1) is described by the following two-parameter family of equations

dx

dτ= α[y − x− f(x)]

dy

dτ= x− y + z ,

dz

dτ= −βy .

(2)

The unfolded Chua’s circuit is shown in Fig. 2and examples of some typical continuous, piecewise-linear functions associated with the nonlinear resis-tor are shown in Fig. 3 [Chua, 1993]. The governingequations are

dv1

dt=

1

C1[G(v2 − v1)− f(v1)] ,

dv2

dt=

1

C2[G(v1 − v2) + iL] ,

diLdt

= − 1

L(v2 +R0iL) ,

(3)

where

G =1

R,

and

iR = f(v1)

= Gbv1 +1

2(Ga −Gb){|v1 +E| − |v1 −E|} .

(4)

In the nondimensional form, the unfoldedChua’s circuit can be reduced to the form

x = α[y − x− f(x)] ,

y = x− y + z ,

z = −βy − yz ,(5)

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Fig. 2. The unfolded Chua’s circuit.

Mechanical Models of Chua’s Circuit 673

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Fig. 3. A family of piecewise-linear resistor characteristics: (a) Gb < Ga < 0; (b) Ga < 0, Gb > 0; (c) Gb > Ga > 0;(d) Ga > 0, Gb < 0; (e) Ga > Gb > 0.


where the relations between the dimensional andnondimensional parameters are given below

x ≡ v1

E, y ≡ v2

E, z ≡ iL

EG,

τ ≡ tG

C2, m0 ≡

GaG, m1 ≡

GbG,

α ≡ C2

C1, β ≡ C2

LG2, γ ≡ C2R0

LG,

(6)

The piecewise-linear characteristics shown inFig. 3 can be realized in a mechanical model wherethe state variables represent, for example, verticaland/or horizontal positions.

2. Mechanical Models of Chua’sCircuit

It is well known that, in general, first-order nonlin-ear differential equations do not always have asso-ciated real mechanical models. In fact, the classicalapproach focuses on the application of known rules,laws and/or hypotheses to derive a set of differen-tial equations based on observed physical phenom-ena. Conversely, a recent trend in mathematics andcomputer sciences is to first generate the equationsand then find various physical realizations from dif-ferent fields to match their behavior. We use thislatter approach in this paper.

2.1. Geometrical constructionof the piecewise-linear function

In order to develop a mechanical model of Chua’scircuit, it is first necessary to replicate thepiecewise-linear behavior of the nonlinear resistorin Figs. 1 and 3 using mechanical means. Considerthe simple geometry shown in Fig. 4.

Assuming π/2 ≤ θ0 ≤ π, −x1 < x < x1, andx1 > 0, the equation of a straight line through theorigin is given by

f(x) = m0x , (7)

where m0 = tan θ0 < 0.The equation of a straight line through

(x1, f(x1)) for x ≥ x1 is given by

f(x) = m1(x− x1) + b ,

where m1 = − tan θ1 < 0. Since f(x1) = b = m0x1,the equation of the right straight-line segment for

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Fig. 4. Definition of slope parameters: m0 < tan θ0 < 0,m1 = − tan θ1 < 0, |m0| > |m1|.

x ≥ x1 is given by

f(x) = m0x1 +m1(x− x1) (8)

where |m0| > |m1|. Similarly, the equation of theleft straight-line segment for x < −x1 is given by

f(x) = −m0x1 +m1(x+ x1) . (9)

When Eqs. (7)–(9) are substituted into the firstequation in (2) above, the following equation isachieved after rearranging terms

dx

dτ=

α[y − (m1+1)x− (m0−m1)x1], x≥x1

α[y − (m0+1)x], |x|<x1

α[y − (m1+1)x+ (m0−m1)x1], x≤−x1

(10)

2.2. Electromechanical model ofChua’s circuit

Our first model of Chua’s circuit consists of threeseparate mechanical devices coupled together viaelectromechanical devices. Each mechanical systemhas a single degree of freedom representing either arotation or a translation. Therefore, the entire cou-pled system has three degrees of freedom denotedby the variables ϕ, y and z in analogy with the statevariables x, y and z, respectively, in (2).

Let us begin with our mechanical realization ofa response with negative slope, analogous to the v–icharacteristic of the nonlinear resistor in Chua’s cir-cuit (see Figs. 1 and 3). The first device we will con-sider for producing such behavior is shown in Figs. 5and 6. This mechanism is composed of a rotatingdisk of radius r whose center is fixed in space andwhose moment of inertia is negligible (I ≈ 0). Itsrotation is defined by the angle ϕ(t) and is positive


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Fig. 5. A mechanical device for producing negative stiffness.

in the counterclockwise direction. A dashpot withviscous damping coefficient c1 and a spring withstiffness coefficient k1 are attached to the disk. It isassumed that all springs and dashpots in Fig. 6 arelinear and massless, a classical mechanical assump-tion. Therefore, the damping force generated bythe dashpot is proportional to −c1rϕ, whereas theconservative force generated by the spring is pro-portional to −k(xA−xB), where xA and xB are thedisplacements of the spring terminals A and B, re-spectively. At the disk point D a bar perpendicularto the plane of the figure is attached. Observe thatthe construction is a mechanism, since the degreeof freedom of the mechanism is w = 3n− 2p [Paul,1979], where n denotes the number of rigid elements(n = 5 in our case), and p is the number of the firstclass kinematic pairs (p = 7 in our case). Therefore,w = 1 for our mechanism which is satisfied since itsdegree of freedom (ϕ) is equal to one.

When the bar at point D moves within the in-terval (−x1, x1) a force reaction is generated by thedashpot with damping coefficient c1, and by thespring with stiffness k1. The equations of motion

can be easily derived using Newton’s law. In or-der to get a piecewise-linear response, two linearsprings with stiffness k2 are positioned on either sideof point D with a small gap, x1. These springs areactivated when |ϕr| ≥ |x1|, i.e. when the point D(bar) contacts one of the springs k2 (neglecting anypossible impact effects and assuming the springs arecompressed linearly).

The lower structure in Fig. 6 consists of a rigidrectangular frame attached to the disk at pointE and at the other end to a rod upon which ismounted a slider at point C. The triangle OCBis comprised of two rigid linkages of equal length,OC = CB = a, that connect to the slider at pointC. Observe that point O at the end of the armOC is fixed in space, point B moves only hori-zontally, and points O, A and B lie in a straighthorizontal line for any movement, assuming smallrotations.

The connections at points A, B, C, E and F areassumed to be pin connections that allow rotationand yet have negligible friction. All forces acting onthe disk are assumed to act in the plane of its center


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Fig. 6. Notation for the mechanical device in Fig. 5.

of mass. While this is only approximately true forthe device considered here, it is valid if the disk issufficiently thin and the springs at point D are po-sitioned close to the disk. In a real device, placingtwo identical constructions of linkages, springs anddashpots on opposite sides of the disk can eliminateout-of-plane torques. Figures 5 and 6 show only onesuch construction for clarity.

In order to produce a negative stiffness, we haveconstructed the mechanism to cause a greater dis-placement at point B than at point A. In otherwords, the linkage satisfies the condition xB > xA.A counterclockwise rotation of the disk ϕ causespoint E and, therefore, the frame to move verti-cally upwards due to the coupling of the disk to thelinkage mechanism. The frame is coupled rigidly tothe slider such that an upward motion of the framecauses point C to move up, and, consequently, pointB to move to the right. Therefore, both pointsA and B move in the same direction although, ifthe linkage mechanism is designed properly, point Bmoves further than point A. This causes the springk1 to stretch and thus exert a force to the right

on the disk at point A. Therefore, a small counter-clockwise rotation ϕ of the disk creates a force atpoint A that tends to continue rotating the disk inthe counterclockwise direction. Likewise, a rotationϕ in the clockwise direction generates a clockwisemoment on the disk. This type of response is calleda “negative stiffness” and is manifested for the sys-tem in Fig. 6 whenever |xB| > |xA| for a givenrotation ϕ.

Consider now the kinematics related to thetriangle OCB in Fig. 7. Unless otherwise noted,all rotations of the disk are assumed to be smallsuch that sin ϕ ≈ ϕ and any rotation of the rigidframe about point E is negligible. As the linkageOC rotates in a clockwise direction, the point Cmoves to C′ and point B moves to B′. Observe thatCD ⊥ DC′ by construction and, for small rotationsof the linkages, the following approximations arevalid: CC′ ⊥ OC and ∆OKC ∼ ∆CDC′. Definingd ≡ CD and h ≡ CK, we obtain the relation

d

DC′=

OK

h, (11)


Fig. 7. Kinematics of the mechanical device in Fig. 5.

and hence

DC′ =hd

OK=

hd√a2 − h2

, (12)

where OK =√a2 − h2. The displacement of the

point B is defined as xB ≡ BB′. Using symme-try from Fig. 7, we deduce the following geometricrelation,

xB ≡ BB′ = 2OP− 2OK = 2(OP−OK) = 2DC′ .

(13)

Thus xB = 2DC′. Substituting (12) into (13) andusing d ≈ rϕ for small rotations we obtain

xB = 2hr√

a2 − h2ϕ . (14)

Therefore, the spring force FA acting at point Aon the disk for a small counterclockwise rotation ϕ,i.e. xA ≈ rϕ, is

FA = k1(xB − xA) = k1

(2h√a2 − h2

− 1

)rϕ .

(15)

It is easy to design a construction such that 2h(a2−h2)−1/2 > 1, thereby realizing the negative stiffnesscriterion. For example, if h = 2r then this criterionis satisfied as long as 4r > (a2 − h2)1/2, which canbe easily achieved.

There is a reaction force due to the frame actingon the disk at point E (see Fig. 6) that counteractsthe rotation of the disk. The derivation is omit-ted here for brevity but, neglecting the mass of thelinkages and assuming small rotations, this reactionforce FE can be approximated as

FE ≈ −k1(xB − xA)h√

a2 − h2

= −k1

(2h√a2 − h2

− 1

)h√

a2 − h2rϕ . (16)

This is identical to Eq. (15) save for the minus signand a geometric factor.

There is one additional force P1 acting on theperimeter of the disk at point F (see Fig. 6). Thisis generated by an electromechanical device and isgiven by

P1 = λ1y , (17)

where y is the displacement of a second mechanicaldevice coupled to the disk.

Summing moments about the center of the diskyields the equation of motion. A free-body diagramof the disk and the forces acting on it is shown inFig. 8. For the case where point D does not con-tact the springs of stiffness k1(−x1 < ϕr < x1), the


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Fig. 8. Free-body diagram of forces acting on the disk in Fig. 5.

equation of motion is derived by summing momentsabout the disk center

|ϕr|<x1 :∑

MO =Iϕ=−c1r2ϕ+FAr+FEr+P1r

Iϕ=−c1r2ϕ+k1δr2ϕ+λ1ry ,

(18)

where I is the moment of inertia of the disk. Theforces FA and FE from Eqs. (15) and (16), respec-tively, have been combined into a single term in (18)with the geometric factor δ defined as

δ ≡(

2h√a2 − h2

− 1

)(√a2 − h2 − h√a2 − h2

). (19)

To achieve negative stiffness, δ > 0 is required,which requires that both terms in parentheses in(19) be positive. This imposes the following boundson the allowable values of h

0.5 <h√

a2 − h2< 1.0 . (20)

For the case of slightly larger rotations whenpoint D of the disk contacts the springs of stiff-ness k2, i.e. |ϕr| ≥ x1, the equations of motion are

given by

ϕr ≥ x1 : Iϕ = −c1r2ϕ+ k1δr2ϕ

−k2r2(ϕ− ϕ∗) + P1r , (21)

ϕr ≤ −x1 : Iϕ = −c1r2ϕ+ k1δr2ϕ

−k2r2(ϕ+ ϕ∗) + P1r , (22)

where the critical angle of contact ϕ∗ is defined by

x1 = rϕ∗ .

Setting I = 0 and rearranging the terms in (18)yields the following first-order ordinary differentialequation (ODE) for ϕ,

ϕ =

(k1δ

c1

)ϕ+

(λ1

c1r

)y (23)

We now make the equation nondimensional byintroducing the following relations,

ϕ = ϕ∗ϕ, y = y∗y, z = z∗z, τ = ωt (24)

where ϕ, y, z, and τ are nondimensional quanti-ties and ϕ∗, y∗, z∗ and ω are constants of propor-tionality. Substituting (24) into (23) we obtain the


following nondimensional ODE,

|ϕr| < x1 : ϕ′ =(

λ1y∗

c1rωϕ∗

)[y +

(k1δrϕ

∗

λ1y∗

)ϕ

](25)

where ϕ = dϕ/dτ . Likewise, Eqs. (21) and (22) canbe expressed in nondimensional form as

ϕr≥x1 : ϕ′=(

λ1y∗

c1rωϕ∗

)[y+

(k1δrϕ

∗−k2rϕ∗

λ1y∗

)ϕ

+

(k2rϕ

∗

λ1y∗

)](26)

ϕr≤−x1 : ϕ′=(

λ1y∗

c1rωϕ∗

)[y+

(k1δrϕ

∗−k2rϕ∗

λ1y∗

)ϕ

−(k2rϕ

∗

λ1y∗

)](27)

By defining the following nondimensional quanti-ties,

α ≡(

λ1y∗

c1rωϕ∗

),

(m0 + 1) ≡ −(k1δrϕ

∗

λ1y∗

)and

(m0 −m1) ≡ −(k2rϕ

∗

λ1y∗

)Equations (25)–(27) transform into

ϕ′=

α[y − (m1+1)ϕ− (m0−m1)], ϕr≥x1

α[y − (m0+1)ϕ], |ϕr|<x1

α[y − (m1+1)ϕ+ (m0−m1)], ϕr≤−x1

(28)

which is identical in form to the first Chua’s equa-tion (10) with x1 set equal to unity.

To generate equations of motion analogous tothe other two Chua’s circuit equations in (2), it isnecessary to couple the mechanism in Fig. 6 to twoother mechanical devices whose displacements rep-resent the state variables y and z. The coupling isaccomplished using four electromechanical devicesand the entire system is shown in Fig. 9. One mech-anism consists of a dashpot with friction coefficientc2 and a spring with stiffness coefficient k3, bothbeing driven by the combined force P2 + P3, whereP2 and P3 are forces generated by electromechan-ical devices. Assuming the masses of all elements

in the device are negligible, the force balance in they-direction is∑

Fy = c2y + k3y = P2 + P3 , (29)

where P2 = λ2z, P3 = λ3rϕ and λ2 and λ3 are con-stants of proportionality. The forces P2 and P3 arein turn controlled by the linear position z and angu-lar position ϕ, respectively. Introducing the nondi-mensional quantities in (24), (29) is transformedinto the following nondimensional equation

y′ =(λ3rϕ

∗

c2ωy∗

)ϕ−

(k3

c2ω

)y +

(λ2z∗

c2ωy∗

)z , (30)

where y′ ≡ dy

dτ.

If we set ω = k3/c2, λ3rϕ∗ = y∗k, and λ2z

∗ =y∗k3, we get an equation identical in form to thesecond equation in (2),

y′ = ϕ− y + z . (31)

Summing forces in the z-direction for thedashpot system in the lower part of Fig. 9 whileneglecting all masses yields the following equationof motion ∑

Fz = c3z = P4 , (32)

where P4 = −λ4y represents the force generatedby a linear electromechanical device with constantof proportionality λ4. Using the relations in (24),Eq. (32) transforms in nondimensional form into

z′ = −βy , (33)

where z′ ≡ dz

dτand the nondimensional quantity β

is defined as

β ≡ λ4c2y∗

k3c3z∗(34)

using ω = k3/c2. Equations (28), (31) and (33)form a system of equations identical in form to theChua’s circuit equations in (2).

2.3. Electromechanical model ofunfolded Chua’s circuit

As previously stated, an unfolding of Chua’scircuit can realize much richer examples of bifurca-tion and chaotic dynamics. Comparing the dimen-sionless system of equations describing the Chua’s


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Fig. 9. Three mechanical components of a model for Chua’s circuit with their electromechanical couplings.

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Fig. 10. Spring-dashpot mechanism used for realizing the unfolded Chua’s circuit.


circuit (2) to that of the unfolded Chua’s circuit(5), it is clear that the first and second equationsare identical whereas the third equation in the un-folded case has an additional term. Building thecorresponding mechanical system for the unfoldedChua’s circuit requires a slight change to our previ-ous configuration in Fig. 9.

To realize the unfolded circuit in a mechan-ical model it is sufficient to replace the dashpotdevice shown in the lower part of Fig. 9 with thespring-dashpot combination shown in Fig. 10. Thecorresponding equation of motion for this device(neglecting masses) and its dimensionless versionare given respectively by∑

Fz = c3z + k4z = P4 ,

and

z′ = −βy − γz , (35)

where P4 = −λ4y as before, ω = k3/c2, β is givenin (34), and

γ ≡ k4c2k3c3

.

Note the differential equation for z (35) gains anadditional term and is now identical in form to thethird equation in (5). Equations (28), (31) and (35)combined represent the unfolded Chua’s circuit.

2.4. Alternative electromechanicalmodel of Chua’s circuit

We now propose a rough outline of a second elec-tromechanical concept for realizing Chua’s circuit,while not providing all the details. This alterna-tive device can be constructed by replacing the diskmechanism discussed earlier in Fig. 6 with the cammechanism shown in Fig. 11. This figure shows two

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Fig. 11. A simple cam device to realize cubic stiffness (note that two different curvatures ρ1 and ρ2 are applied).


cams mounted on a common rotating shaft withcables attached to each cam at one end and to aspring at the other. The springs, with negligiblemass, provide stiffness. While the mechanism inFig. 11 can be easily realized in the laboratory, elec-tromechanical coupling to the two mechanical de-vices presented in Fig. 9 is still required. By vary-ing the curvatures ρi(i = 1, 2) of the cams, it ispossible to achieve an approximate piecewise-linearresponse, similar to that shown in Fig. 3, as wellas both linear and nonlinear stiffness. In order torealize a negative rotational stiffness, some otherform of coupling of the cams is required, for exam-ple, to an additional electromechanical device thatprovides a moment in the direction of rotation.

The previous disk mechanism in Fig. 6 and thecam mechanism each have advantages and disad-vantages. The previous device utilizes both rota-tion and translation; however, relations betweenthe forces and corresponding displacements canbe derived relatively easily. The cam device pre-sented in Fig. 8 involves only rotational motionand can realize a broader class of behavior such

as the so-called Duffing (or cubic) stiffness with amaximum rotation of ϕmax = ±170◦. It is ratherdifficult, in general, to calculate analytically the re-quired cam curvatures a priori. In practice, thestiffness characteristics are defined using identifi-cation methods by measuring the moments andcorresponding displacements.

2.5. Purely mechanical Chua’scircuit mechanism

Our motivation in this paper is to develop a me-chanical analog of the well-known Chua’s circuit.Three examples of such mechanical realizationshave thus far been proposed. All of these used elec-tromechanical feedback devices to transmit signalsfrom one subdevice to another. The most interest-ing question, however, remains open: How do webuild Chua’s circuit using only purely mechanicalcomponents? We now address this problem.

To realize Chua’s circuit from purely mechani-cal means, we employ the following trick. Observethat we can eliminate the variable z from the two

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Fig. 12. Novel mechanism with friction.


Fig. 13. Notation for the mechanical device in Fig. 12.

last equations of system (2) and transform thesystem of three first-order ODE’s into an equiva-lent system of one first-order and one second-orderODE. Differentiating the second equation in system(2) with respect to time τ and substituting in thethird equation from (2) we get the following system

dx

dτ= α[y − x− f(x)] ,

dx

dτ=d2y

dτ2+dy

dτ+ βy .

(36)

We now construct a device whose equations ofmotion will match those of (36). Consider the sys-tem shown in Figs. 12 and 13. This mechanism iscomposed of an oscillating outer ring with radiusr0 and negligible inertial moment (I1 ≈ 0) whosecenter is fixed in space. Its rotation is defined bythe angle ϕ1(t) and is considered positive in thecounterclockwise direction. This ring contacts aninner rotor of radius ri. The center of the wheel isfixed in position and is driven by an external agentat an angular velocity ω. All linear springs and

viscous dampers shown in Fig. 13 are assumed tobe without mass.

The addition of a stiff beam with nonnegligi-ble mass, whose oscillations are denoted by ϕ2, in-troduces a second-order differential equation to thesystem. Note that ϕ2 is positive in the clockwisedirection. The key idea is that the beam movementtransmits a variable normal force to the outer ringvia a spring with stiffness kb and a dashpot withdamping coefficient cb. This normal force in turnalters the friction force between the ring and rotor.The beam also couples to the ring through the dash-pot of damping coefficient c1. (It is assumed theforces generated by kb and cb act at common pointson the beam and outer ring so as not to imparttorques, only normal forces. For illustration pur-poses, Figs. 12 and 13 show these two componentsseparated, connected at the top and bottom by blueand red linkages. The torque induced by this con-struction is negligible if the linkages are sufficientlyshort. In addition, as with the device in Fig. 5, itis assumed that all forces act in the plane of thering’s center of mass. While the device in Figs. 12


and 13 show some forces to act outside this plane,clever design can minimize or eliminate out-of-planetorques in a real system.)

Observe that a linkage mechanism is providedto produce negative stiffness identical to the devicein Fig. 6. The linkage mechanism OCB couples tothe rotation of the outer ring through the springwith stiffness coefficient k1 attached at point A.The outer ring in turn couples to the inner rotorthrough friction. In order to realize the piecewise-linear response of Fig. 3, springs of stiffness k2 areplaced on either side of point D in Fig. 13 with asmall gap, x1. At the disk point D, a bar perpen-dicular to the plane of the figure is attached sim-ilar to Fig. 5. These springs are activated when|ϕ1r| ≥ |x1|, i.e. when the point D (bar) contactsone of the springs (neglecting any possible impacteffects and assuming the springs are compressedlinearly).

The friction force between the rotor and outerring is assumed to be of the stick-slip variety. Suchfriction is commonly exhibited in everyday systemssuch as the squeak of door hinges or chalk on ablackboard, and in the oscillations of violin strings,to name a few. Stick-slip friction has the charac-teristic of decreasing locally as a function of therelative velocity between the two bodies in con-tact and can generate self-excited oscillations. Thecharacteristics of stick-slip friction versus relativevelocity have been discussed in various papers andbooks (see, e.g. [Awrejcewicz & Delfs, 1990a, 1990b;Awrejcewicz & Holicke, 1999]). A graph of thestick-slip friction coefficient µ versus relative veloc-ity w is shown in Fig. 14.

Now we are ready to derive the equationsof motion based on Newton’s laws and analyze

Fig. 14. Stick-slip friction coefficient versus relative velocity.

oscillations around the static equilibrium positions.Note that the mass m of the rigid beam causes aninitial deflection of the spring kb. When the pointD moves outside the interval (−x1, x1), the outerring experiences forces from the springs k1 and k2,the dashpot c1, and the friction caused by the ro-tating inner wheel. There is also a reaction forceat point E caused by the frame. Note that springkb and dashpot cb exert only a normal force on theouter ring; friction is neglected here. Likewise, thebeam is acted upon by the spring kb, the dashpotsc1 and cb, and gravity. The equations of motion ofthe beam and the outer ring for |ϕ1| ≥ ϕ∗, respec-tively, are given by

I2ϕ2 + kbl22ϕ2 + cbl

22ϕ2 + c1(ϕ2l1 − ϕ1r0)l1

= −mgl22,

I1ϕ1 − k1δr20ϕ1 + k2r

20(ϕ1 − ϕ∗)

+ c1(ϕ1r0 − ϕ2l1)r0

= riN(µ0 sgnw − α0w + β0w3) ,

(37)

where I1, I2 denote the moments of inertia of theouter ring and beam, respectively. The reactionforce of the frame acting at point E is includedthrough δ, the geometric factor defined in (19). Forthe case of |ϕ1| < ϕ∗ the first equation is iden-tical and so is the second equation save for thek1r

20(ϕ1 − ϕ∗) term. The right-hand side (RHS) of

the second equation in (37) represents the torquedue to friction where N is the normal force, wthe relative (slip) velocity between the two bodies,and µ0, α0 and β0 denote friction coefficients (seeFig. 14). The relations for w and N are

w = (ω − ϕ1)ri ,

N =

(mg

2− kbl2ϕ2 − cbl2ϕ2

).

Note the mg/2 term in the equation for N is due tostatic deflection of the spring kb. For this case, letus assume that w is in the regime where frictiondecreases linearly with increasing w, i.e. assumesimply w = 1, and β0 = 0.

Let us determine the static equilibriumpositions, ϕ10 and ϕ20, of the outer ring and beam,respectively. We assume the equilibrium position ofthe outer ring is such that point D does not contactthe two springs of stiffness k2, i.e. −ϕ∗ ≤ ϕ10 ≤ ϕ∗.We obtain the static equilibrium angles ϕ10 and ϕ20


from (37) assuming ϕ1 = ϕ2 = ϕ1 = ϕ2 = 0

ϕ10 =mgri(µ0 − α0riω)

k1δr20

,

ϕ20 = − mg

2kbl2.

(38)

We seek to determine oscillations about the equi-librium positions, for the case of |ϕ1| < ϕ∗ and|ϕ1| > ϕ∗. Therefore, we introduce new variablesϕ1 = ϕ1 + ϕ10, ϕ2 = ϕ2 + ϕ20 where the overbarsrepresent deviations from equilibrium. We also as-sume small displacements and velocities such thatwe may neglect nonlinear terms as being of second-order. Substituting these new variables into thesystem (37), assuming I1 = 0, and omitting over-bars we get

c1l1r0ϕ1 = I2ϕ2 + (cbl22 + c1l

21)ϕ2 + kbl

22ϕ2 ,

(c1r20 − 0.5mgα0r

2i )ϕ1

= [c1l1r0 − (µ0 − α0riω)cbl2ri]ϕ2

− [(µ0 − α0riω)kbl2ri]ϕ2 + k1δr20ϕ1

− k2r20(ϕ1 − ϕ∗0)

(39)

where ϕ∗0 ≡ ϕ∗ − ϕ10 in the last term on the RHSof the second equation.

To eliminate the ϕ2 term on the RHS of thesecond equation in (39), we impose the followingconstraint on the system parameters

c1l1r0 − (µ0 − α0riω)cbl2ri = 0 . (40)

Now the system of equations in (39) begins toresemble that of (36). All that remains is to nondi-mensionalize the system and impose certain con-straints on the system parameters (e.g. kb, c1, I2,etc.) to achieve the desired form of the equations.Such a transformation requires several steps, thedetails of which are omitted here for brevity.

First, we introduce the nondimensional time τand positions ϕ1, ϕ2

τ = ω∗t, ω∗ =cbl

22 + c1l

21

I2,

ϕ1 = ϕ∗0ϕ1, ϕ2 = ϕ∗0ϕ2 ,

where ω∗ and ϕ∗0 are constants of proportionality.Next, we impose the following constraint on thesystem parameters

I2ω∗ = c1l1r0 (41)

and, finally, make the following definitions

α ≡ (α0riω − µ0)kbl2ri(c1r2

0 − 0.5mgα0r2i )ω∗ ,

(m0 + 1) ≡ − k1δr20

(α0riω − µ0)kbl2ri,

(m0 −m1) ≡ k2r20

(α0riω − µ0)kbl2ri,

β ≡ kbl22

I2(ω∗)2.

(42)

Performing these steps transforms Eq. (39) into thefollowing nondimensional pair of equations

ϕ′1 = ϕ′′2 + ϕ′2 + βϕ2 ,

ϕ′1 = α[ϕ2 − ϕ1 − h(ϕ1)](43)

where ϕ′1 = dϕ1/dτ , ϕ′2 = dϕ2/dτ and ϕ′′2 =d2ϕ2/dτ

2. The function h(ϕ1) is the requiredpiecewise-linear function

h(ϕ1) =

m0ϕ

∗ +m1(ϕ1 − ϕ∗), ϕ > ϕ∗

m0ϕ1, |ϕ| < ϕ∗

−m0ϕ∗ +m1(ϕ1 + ϕ∗), ϕ < −ϕ∗ .

Note the system of equations in (43) has thesame form as the reduced Chua’s circuit system in(36). By varying the parameters α, β, m0, and m1

in (42) we should, in principle, be able to generatethe vast array of nonlinear behavior demonstratedby Chua’s circuit. This will not be trivial in prac-tice, however, as the parameters must be varied insuch a way as not to violate the required constraintsin (40) and (41). The large number of parame-ters present in the system should allow the prac-titioner flexibility to explore the desired parameterspace (α, β, m0, m1) while satisfying the assump-tions and constraints of this model.

3. Concluding Remarks

The unfolding of Chua’s circuit can exhibit variouschaotic and bifurcation behaviors which are well-documented in the literature [Madan, 1993]. Inthis paper, the idea of its potential equivalence andhomeomorphism in other engineering systems hasbeen discussed and illustrated. Several realizationshave been proposed: two electromechanical models


of Chua’s circuit and one of Chua’s unfolded circuit,and, finally, a purely mechanical model of Chua’scircuit. We employed various “tricks” to repro-duce the negative slope characteristics of the non-linear resistor in the Chua’s circuit. The electrome-chanical systems use electromechanical couplingsbetween three mechanical subsystems to achievethe required dynamic behavior of the Chua’s cir-cuit. Conversely, the purely mechanical device usesonly naturally observed mechanical properties andphenomena to satisfy this goal. Friction, often con-sidered a nuisance in industry, is exploited in thepurely mechanical system to produce some origi-nal couplings and synthesize the third-order Chua’sequations.

Acknowledgments

The first author acknowledges a discussion withDr. W. Wodzicki related to the construction of themechanism. This work is partially supported bythe Fulbright Foundation and by the ONR grantN000-14-98-0594.

ReferencesAwrejcewicz, J. & Delfs, J. [1990a] “Dynamics of a

self-excited stick-slip oscillator with two degrees offreedom. Part 1: Investigation of equilibrium andslip-stick, slip-slip and stick-slip transition,” Europ. J.Mech., A/Solids 9(4), 269–282.

Awrejcewicz, J. & Delfs, J. [1990b] “Dynamics of aself-excited stick-slip oscillator with two degreesof freedom. Part 2: Periodic and chaotic orbits,”Europ. J. Mech., A/Solids 9(5), 397–418.

Awrejcewicz, J. & Holicke, M. M. [1999] “Melnikov’smethod and stick-slip chaotic oscillations in veryweakly forced mechanical systems,” Int. J. Bifurca-tion and Chaos 9(3), 505–518.

Chua, L. O. [1992] “The genesis of Chua’s circuit,”Archiv fur Elektronik und Ubertragungstechnik 46(4),250–257.

Chua, L. O. [1993] “Global unfolding of Chua’s circuit,”IEICE Trans. Fundamentals E16-A(5), 704–734.

Kennedy, P. [1992] “Robust op amp realisation of Chua’scircuit,” Frequenz 46, 66–80.

Madan, R. A. [1993] Chua’s Circuit: A Paradigm forChaos (World Scientific, Singapore).

Paul, B. [1979] Kinematics and Dynamics of PlanarMachinery (Prentice-Hall, NJ).

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