Tutorials and Reviews MECHANICAL MODELS OF CHUA'S CIRCUIT

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

MECHANICAL MODELS OF CHUA’S CIRCUIT J. AWREJCEWICZ Department of Automatics and Biomechanics, 1/15 Stefanowskiego St., Technical University of L´ od´z, 90-924 L´ od´z, Poland M. L. CALVISI Applied 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 circuit and one of Chua’s unfolding circuit. In addition, a novel mechanism is proposed for realizing Chua’s equations in a purely mechanical way using friction properties. All relevant equations are 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

segment, continuous, piecewise-linear function can be mapped to an unfolded Chua’s circuit having identical qualitative dynamics. The following question was posed in [Chua, 1993]: Since there are several different third-order circuits (which exhibit strange attractors) composed of a continuous, odd-symmetric, piecewiselinear vector field in R3 , does a homeomorphic mapping of such circuits to an unfolded Chua’s circuit exist? If such a homeomorphism exists, the two circuits are said to be equivalent (or topologically conjugate). The unfolded Chua’s circuit is canonical in the sense that the governing equations contain a minimum number of parameters for observing the full generality of dynamical behaviors. In this paper, we will present mechanical and electromechanical device models of Chua’s circuit, as well as of the unfolded Chua’s circuit. The Chua’s circuit is shown in Fig. 1. The governing equations have the form

Chua’s circuit (see Fig. 1) is one of the simplest physical models that has been widely investigated by mathematical, numerical and experimental methods. One of the main attractions of Chua’s circuit is that it can be easily built with less than a dozen standard circuit components, and has often been referred to as the poor man’s chaos generator. A mathematical analysis of the global unfolding behavior of Chua’s circuit is given in [Chua, 1993]. Perhaps one of the most important observations is that by adding a linear resistor in series with the inductor in Chua’s circuit, the resulting unfolded Chua’s circuit is topologically equivalent to a 21-parameter family of continuous odd-symmetric, piecewise-linear differential equations in R3 . Any vector field belonging to the “unfolded” topologically conjugate family can be transformed (mapped) via a nonsingular linear transformation to an unfolded Chua’s circuit with only seven parameters. In addition, it extends the local concept of unfolding to a global one, where all results are valid for the whole space R3 . In other words, any autonomous three-dimensional system characterized by an odd-symmetric, three-

1 dv1 = (v2 − v1 ) − f (v1 ) , dt R 1 dv2 = (v1 − v2 ) + iL , C2 dt R diL = −v2 , L dt C1

671

(1)

672 J. Awrejcewicz & M. L. Calvisi

R

1 G



;



5;

5



;



E

;5









/

 ;5 D

5

E

(a)

(b)

Fig. 1. (a) Chua’s circuit is made of four standard linear circuit components and a nonlinear resistor; (b) vR –iR characteristic of 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 representing the vR –iR characteristic of Chua’s diode; namely, the nonlinear resistor, NR . In nondimensional form, (1) is described by the following twoparameter family of equations dx = α[y − x − f (x)] dτ dy =x−y+z, dτ dz = −βy . dτ

and iR = f (v1 ) 1 = Gb v1 + (Ga − Gb ){|v1 + E| − |v1 − E|} . 2 (4) In the nondimensional form, the unfolded Chua’s circuit can be reduced to the form

(2) x˙ = α[y − x − f (x)] , y˙ = x − y + z , z˙ = −βy − yz ,

The unfolded Chua’s circuit is shown in Fig. 2 and examples of some typical continuous, piecewiselinear functions associated with the nonlinear resistor are shown in Fig. 3 [Chua, 1993]. The governing equations are 1 dv1 = [G(v2 − v1 ) − f (v1 )] , dt C1 1 dv2 = [G(v1 − v2 ) + iL ] , dt C2 1 diL = − (v2 + R0 iL ) , dt L

(3)

(5)

R



1 G

5





;



;



/

where G=

1 , R

Fig. 2.

The unfolded Chua’s circuit.

;5 5

Mechanical Models of Chua’s Circuit 673

#

-

# 





;#

,



-

-

;#

-

(b)

,





,

(a)

#





# ;#

, 

-



-

(c)

-

,



;#

(d)

#







;#

(e) 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.

674 J. Awrejcewicz & M. L. Calvisi



where the relations between the dimensional and nondimensional parameters are given below v2 iL v1 , y≡ , z≡ , E E EG tG Ga Gb τ≡ , m0 ≡ , m1 ≡ , C2 G G C2 C2 C2 R0 , α≡ , β≡ , γ≡ C1 LG2 LG

Û

x≡

(6)

Û





Û

The piecewise-linear characteristics shown in Fig. 3 can be realized in a mechanical model where the state variables represent, for example, vertical and/or horizontal positions.

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

x ≥ x1 is given by

2. Mechanical Models of Chua’s Circuit

f (x) = m0 x1 + m1 (x − x1 )

It is well known that, in general, first-order nonlinear differential equations do not always have associated real mechanical models. In fact, the classical approach focuses on the application of known rules, laws and/or hypotheses to derive a set of differential equations based on observed physical phenomena. Conversely, a recent trend in mathematics and computer sciences is to first generate the equations and then find various physical realizations from different fields to match their behavior. We use this latter approach in this paper.

(8)

where |m0 | > |m1 |. Similarly, the equation of the left straight-line segment for x < −x1 is given by f (x) = −m0 x1 + m1 (x + x1 ) .

(9)

When Eqs. (7)–(9) are substituted into the first equation in (2) above, the following equation is achieved after rearranging terms     α[y − (m1 +1)x − (m0 −m1 )x1 ],

dx = α[y − (m0 +1)x],  dτ   α[y − (m +1)x + (m −m )x ], 1 0 1 1

x ≥ x1 |x| < x1 x ≤ −x1 (10)

2.1. Geometrical construction of the piecewise-linear function In order to develop a mechanical model of Chua’s circuit, it is first necessary to replicate the piecewise-linear behavior of the nonlinear resistor in Figs. 1 and 3 using mechanical means. Consider the simple geometry shown in Fig. 4. Assuming π/2 ≤ θ0 ≤ π, −x1 < x < x1 , and x1 > 0, the equation of a straight line through the origin is given by f (x) = m0 x ,

(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 = m0 x1 , the equation of the right straight-line segment for

2.2. Electromechanical model of Chua’s circuit Our first model of Chua’s circuit consists of three separate mechanical devices coupled together via electromechanical devices. Each mechanical system has a single degree of freedom representing either a rotation or a translation. Therefore, the entire coupled system has three degrees of freedom denoted by the variables ϕ, y and z in analogy with the state variables x, y and z, respectively, in (2). Let us begin with our mechanical realization of a response with negative slope, analogous to the v–i characteristic of the nonlinear resistor in Chua’s circuit (see Figs. 1 and 3). The first device we will consider for producing such behavior is shown in Figs. 5 and 6. This mechanism is composed of a rotating disk of radius r whose center is fixed in space and whose moment of inertia is negligible (I ≈ 0). Its rotation is defined by the angle ϕ(t) and is positive

Mechanical Models of Chua’s Circuit 675










 






      




Fig. 5.

A mechanical device for producing negative stiffness.

in the counterclockwise direction. A dashpot with viscous damping coefficient c1 and a spring with stiffness coefficient k1 are attached to the disk. It is assumed that all springs and dashpots in Fig. 6 are linear and massless, a classical mechanical assumption. Therefore, the damping force generated by ˙ whereas the the dashpot is proportional to −c1 r ϕ, conservative force generated by the spring is proportional to −k(xA − xB ), where xA and xB are the displacements of the spring terminals A and B, respectively. At the disk point D a bar perpendicular to the plane of the figure is attached. Observe that the construction is a mechanism, since the degree of 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 first class kinematic pairs (p = 7 in our case). Therefore, w = 1 for our mechanism which is satisfied since its degree of freedom (ϕ) is equal to one. When the bar at point D moves within the interval (−x1 , x1 ) a force reaction is generated by the dashpot with damping coefficient c1 , and by the spring with stiffness k1 . The equations of motion

can be easily derived using Newton’s law. In order to get a piecewise-linear response, two linear springs with stiffness k2 are positioned on either side of point D with a small gap, x1 . These springs are activated when |ϕr| ≥ |x1 |, i.e. when the point D (bar) contacts one of the springs k2 (neglecting any possible impact effects and assuming the springs are compressed linearly). The lower structure in Fig. 6 consists of a rigid rectangular frame attached to the disk at point E and at the other end to a rod upon which is mounted a slider at point C. The triangle OCB is comprised of two rigid linkages of equal length, OC = CB = a, that connect to the slider at point C. Observe that point O at the end of the arm OC is fixed in space, point B moves only horizontally, and points O, A and B lie in a straight horizontal line for any movement, assuming small rotations. The connections at points A, B, C, E and F are assumed to be pin connections that allow rotation and yet have negligible friction. All forces acting on the disk are assumed to act in the plane of its center

676 J. Awrejcewicz & M. L. Calvisi





 




…

 









 

 

 Fig. 6.

Notation for the mechanical device in Fig. 5.

of mass. While this is only approximately true for the device considered here, it is valid if the disk is sufficiently thin and the springs at point D are positioned close to the disk. In a real device, placing two identical constructions of linkages, springs and dashpots on opposite sides of the disk can eliminate out-of-plane torques. Figures 5 and 6 show only one such construction for clarity. In order to produce a negative stiffness, we have constructed the mechanism to cause a greater displacement at point B than at point A. In other words, the linkage satisfies the condition xB > xA . A counterclockwise rotation of the disk ϕ causes point E and, therefore, the frame to move vertically upwards due to the coupling of the disk to the linkage mechanism. The frame is coupled rigidly to the slider such that an upward motion of the frame causes point C to move up, and, consequently, point B to move to the right. Therefore, both points A and B move in the same direction although, if the linkage mechanism is designed properly, point B moves further than point A. This causes the spring k1 to stretch and thus exert a force to the right

on the disk at point A. Therefore, a small counterclockwise rotation ϕ of the disk creates a force at point A that tends to continue rotating the disk in the counterclockwise direction. Likewise, a rotation ϕ in the clockwise direction generates a clockwise moment on the disk. This type of response is called a “negative stiffness” and is manifested for the system in Fig. 6 whenever |xB | > |xA | for a given rotation ϕ. Consider now the kinematics related to the triangle OCB in Fig. 7. Unless otherwise noted, all rotations of the disk are assumed to be small such that sin ϕ ≈ ϕ and any rotation of the rigid frame about point E is negligible. As the linkage OC rotates in a clockwise direction, the point C moves to C0 and point B moves to B0 . Observe that CD ⊥ DC0 by construction and, for small rotations of the linkages, the following approximations are valid: CC0 ⊥ OC and ∆OKC ∼ ∆CDC0 . Defining d ≡ CD and h ≡ CK, we obtain the relation OK d = , 0 DC h

(11)

Mechanical Models of Chua’s Circuit 677

Fig. 7.

Kinematics of the mechanical device in Fig. 5.

and hence DC0 =

hd hd =√ , 2 OK a − h2

(12)

√ where OK = a2 − h2 . The displacement of the point B is defined as xB ≡ BB0 . Using symmetry from Fig. 7, we deduce the following geometric relation, xB ≡ BB0 = 2OP − 2OK = 2(OP − OK) = 2DC0 . (13) 0

Thus xB = 2DC . Substituting (12) into (13) and using d ≈ rϕ for small rotations we obtain hr ϕ. xB = 2 √ 2 a − h2

(14)

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

FA = k1 (xB − xA ) = k1



2h √ − 1 rϕ . 2 a − h2 (15)

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

There is a reaction force due to the frame acting on the disk at point E (see Fig. 6) that counteracts the rotation of the disk. The derivation is omitted here for brevity but, neglecting the mass of the linkages and assuming small rotations, this reaction force FE can be approximated as h FE ≈ −k1 (xB − xA ) √ 2 a − h2 

= −k1



2h h √ −1 √ rϕ . 2 2 2 a −h a − h2

(16)

This is identical to Eq. (15) save for the minus sign and a geometric factor. There is one additional force P1 acting on the perimeter of the disk at point F (see Fig. 6). This is generated by an electromechanical device and is given by P1 = λ1 y ,

(17)

where y is the displacement of a second mechanical device coupled to the disk. Summing moments about the center of the disk yields the equation of motion. A free-body diagram of the disk and the forces acting on it is shown in Fig. 8. For the case where point D does not contact the springs of stiffness k1 (−x1 < ϕr < x1 ), the

678 J. Awrejcewicz & M. L. Calvisi

 
 …  



…


  





  
 

Fig. 8.


   


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

equation of motion is derived by summing moments about the disk center |ϕr| < x1 :

X

MO =I ϕ¨ = −c1 r ϕ+F ˙ A r+FE r+P1 r 2

2 ˙ I ϕ¨ = −c1 r 2 ϕ+k 1 δr ϕ+λ1 ry , (18)

given by ϕr ≥ x1 : I ϕ¨ = −c1 r 2 ϕ˙ + k1 δr 2 ϕ −k2 r 2 (ϕ − ϕ∗ ) + P1 r , (21) ϕr ≤ −x1 : I ϕ¨ = −c1 r 2 ϕ˙ + k1 δr 2 ϕ −k2 r 2 (ϕ + ϕ∗ ) + P1 r , (22)

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

where the critical angle of contact ϕ∗ is defined by

! √ a2 − h2 − h √ . (19) a2 − h2

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



δ≡



2h √ −1 2 a − h2

To achieve negative stiffness, δ > 0 is required, which requires that both terms in parentheses in (19) be positive. This imposes the following bounds on the allowable values of h h < 1.0 . 0.5 < √ a2 − h2

(20)

For the case of slightly larger rotations when point D of the disk contacts the springs of stiffness k2 , i.e. |ϕr| ≥ x1 , the equations of motion are

x1 = rϕ∗ .



ϕ˙ =







k1 δ λ1 ϕ+ y c1 c1 r

(23)

We now make the equation nondimensional by introducing the following relations, ˜ y = y ∗ y˜, z = z ∗ z˜, τ = ωt ϕ = ϕ∗ ϕ,

(24)

where ϕ, ˜ y˜, z˜, and τ are nondimensional quantities and ϕ∗ , y ∗ , z ∗ and ω are constants of proportionality. Substituting (24) into (23) we obtain the

Mechanical Models of Chua’s Circuit 679

following nondimensional ODE, 

0

|ϕr| < x1 : ϕ˜ =

λ1 y ∗ c1 rωϕ∗





k1 δrϕ∗ y˜ + λ1 y ∗

 

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

λ1 y ∗ ϕr ≥ x1 : ϕ˜ = c1 rωϕ∗ 

+ ϕr ≤ −x1 : ϕ˜0 =



−





k1 δrϕ∗ −k2 rϕ∗ y˜+ ϕ ˜ λ1 y ∗

k2 rϕ∗ λ1 y ∗

λ1 y ∗ c1 rωϕ∗ 



(26) 

y˜+



k1 δrϕ∗ −k2 rϕ∗ ϕ ˜ λ1 y ∗



(27)

By defining the following nondimensional quantities, 

α≡

λ1 y ∗ c1 rωϕ∗ 

(m0 + 1) ≡ − 



k2 rϕ∗ (m0 − m1 ) ≡ − λ1 y ∗



and

ϕ ˜=

α[y − (m0 +1)ϕ], ˜

   α[y − (m +1)ϕ ˜ + (m0 −m1 )], 1

y˜ =













λ3 rϕ∗ k3 λ2 z ∗ ϕ ˜ − y ˜ + z˜ , (30) c2 ωy ∗ c2 ω c2 ωy ∗

d˜ y . dτ If we set ω = k3 /c2 , λ3 rϕ∗ = y ∗ k, and λ2 z ∗ = ∗ y k3 , we get an equation identical in form to the second equation in (2), where y˜0 ≡

y˜0 = ϕ˜ − y˜ + z˜ .

(31)

Fz = c3 z˙ = P4 ,

(32)

where P4 = −λ4 y represents the force generated by a linear electromechanical device with constant of proportionality λ4 . Using the relations in (24), Eq. (32) transforms in nondimensional form into

Equations (25)–(27) transform into 0

0

X



  ˜ − (m0 −m1 )],   α[y − (m1 +1)ϕ

(29)

Summing forces in the z-direction for the dashpot system in the lower part of Fig. 9 while neglecting all masses yields the following equation of motion

,

k1 δrϕ∗ λ1 y ∗

Fy = c2 y˙ + k3 y = P2 + P3 ,

where P2 = λ2 z, P3 = λ3 rϕ and λ2 and λ3 are constants of proportionality. The forces P2 and P3 are in turn controlled by the linear position z and angular position ϕ, respectively. Introducing the nondimensional quantities in (24), (29) is transformed into the following nondimensional equation





k2 rϕ∗ λ1 y ∗

X

ϕ ˜ (25)

0

in the device are negligible, the force balance in the y-direction is

ϕr ≥ x1 |ϕr| < x1 ϕr ≤ −x1 (28)

which is identical in form to the first Chua’s equation (10) with x1 set equal to unity. To generate equations of motion analogous to the other two Chua’s circuit equations in (2), it is necessary to couple the mechanism in Fig. 6 to two other mechanical devices whose displacements represent the state variables y and z. The coupling is accomplished using four electromechanical devices and the entire system is shown in Fig. 9. One mechanism consists of a dashpot with friction coefficient c2 and a spring with stiffness coefficient k3 , both being driven by the combined force P2 + P3 , where P2 and P3 are forces generated by electromechanical devices. Assuming the masses of all elements

z˜0 = −β y˜ ,

(33)

d˜ z and the nondimensional quantity β where z˜0 ≡ dτ is defined as β≡

λ4 c2 y ∗ k3 c3 z ∗

(34)

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

2.3. Electromechanical model of unfolded Chua’s circuit As previously stated, an unfolding of Chua’s circuit can realize much richer examples of bifurcation and chaotic dynamics. Comparing the dimensionless system of equations describing the Chua’s

680 J. Awrejcewicz & M. L. Calvisi

  



 …



 





   

 



Fig. 9.

 …







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





Fig. 10.






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

Mechanical Models of Chua’s Circuit 681

circuit (2) to that of the unfolded Chua’s circuit (5), it is clear that the first and second equations are identical whereas the third equation in the unfolded case has an additional term. Building the corresponding mechanical system for the unfolded Chua’s circuit requires a slight change to our previous configuration in Fig. 9. To realize the unfolded circuit in a mechanical model it is sufficient to replace the dashpot device shown in the lower part of Fig. 9 with the spring-dashpot combination shown in Fig. 10. The corresponding equation of motion for this device (neglecting masses) and its dimensionless version are given respectively by X

Fz = c3 z˙ + k4 z = P4 ,

and z˜0 = −β y˜ − γ z˜ ,

(35)

where P4 = −λ4 y as before, ω = k3 /c2 , β is given in (34), and γ≡

k4 c2 . k3 c3

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

2.4. Alternative electromechanical model of Chua’s circuit We now propose a rough outline of a second electromechanical concept for realizing Chua’s circuit, while not providing all the details. This alternative device can be constructed by replacing the disk mechanism discussed earlier in Fig. 6 with the cam mechanism shown in Fig. 11. This figure shows two

 





  

Fig. 11.

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

682 J. Awrejcewicz & M. L. Calvisi

cams mounted on a common rotating shaft with cables attached to each cam at one end and to a spring at the other. The springs, with negligible mass, provide stiffness. While the mechanism in Fig. 11 can be easily realized in the laboratory, electromechanical coupling to the two mechanical devices presented in Fig. 9 is still required. By varying the curvatures ρi (i = 1, 2) of the cams, it is possible to achieve an approximate piecewise-linear response, similar to that shown in Fig. 3, as well as both linear and nonlinear stiffness. In order to realize a negative rotational stiffness, some other form of coupling of the cams is required, for example, to an additional electromechanical device that provides a moment in the direction of rotation. The previous disk mechanism in Fig. 6 and the cam mechanism each have advantages and disadvantages. The previous device utilizes both rotation and translation; however, relations between the forces and corresponding displacements can be derived relatively easily. The cam device presented in Fig. 8 involves only rotational motion and can realize a broader class of behavior such

as the so-called Duffing (or cubic) stiffness with a maximum rotation of ϕmax = ±170◦ . It is rather difficult, in general, to calculate analytically the required cam curvatures a priori. In practice, the stiffness characteristics are defined using identification methods by measuring the moments and corresponding displacements.

2.5. Purely mechanical Chua’s circuit mechanism Our motivation in this paper is to develop a mechanical analog of the well-known Chua’s circuit. Three examples of such mechanical realizations have thus far been proposed. All of these used electromechanical feedback devices to transmit signals from one subdevice to another. The most interesting question, however, remains open: How do we build Chua’s circuit using only purely mechanical components? We now address this problem. To realize Chua’s circuit from purely mechanical means, we employ the following trick. Observe that we can eliminate the variable z from the two

  



   

 


Fig. 12.

Novel mechanism with friction.

Mechanical Models of Chua’s Circuit 683

Fig. 13.

Notation for the mechanical device in Fig. 12.

last equations of system (2) and transform the system of three first-order ODE’s into an equivalent system of one first-order and one second-order ODE. Differentiating the second equation in system (2) with respect to time τ and substituting in the third equation from (2) we get the following system dx = α[y − x − f (x)] , dτ d2 y dy dx = 2+ + βy . dτ dτ dτ

(36)

We now construct a device whose equations of motion will match those of (36). Consider the system shown in Figs. 12 and 13. This mechanism is composed of an oscillating outer ring with radius r0 and negligible inertial moment (I1 ≈ 0) whose center is fixed in space. Its rotation is defined by the angle ϕ1 (t) and is considered positive in the counterclockwise direction. This ring contacts an inner rotor of radius ri . The center of the wheel is fixed in position and is driven by an external agent at an angular velocity ω. All linear springs and

viscous dampers shown in Fig. 13 are assumed to be without mass. The addition of a stiff beam with nonnegligible mass, whose oscillations are denoted by ϕ2 , introduces a second-order differential equation to the system. Note that ϕ2 is positive in the clockwise direction. The key idea is that the beam movement transmits a variable normal force to the outer ring via a spring with stiffness kb and a dashpot with damping coefficient cb . This normal force in turn alters the friction force between the ring and rotor. The beam also couples to the ring through the dashpot of damping coefficient c1 . (It is assumed the forces generated by kb and cb act at common points on the beam and outer ring so as not to impart torques, only normal forces. For illustration purposes, Figs. 12 and 13 show these two components separated, connected at the top and bottom by blue and red linkages. The torque induced by this construction is negligible if the linkages are sufficiently short. In addition, as with the device in Fig. 5, it is assumed that all forces act in the plane of the ring’s center of mass. While the device in Figs. 12

684 J. Awrejcewicz & M. L. Calvisi

and 13 show some forces to act outside this plane, clever design can minimize or eliminate out-of-plane torques in a real system.) Observe that a linkage mechanism is provided to produce negative stiffness identical to the device in Fig. 6. The linkage mechanism OCB couples to the rotation of the outer ring through the spring with stiffness coefficient k1 attached at point A. The outer ring in turn couples to the inner rotor through friction. In order to realize the piecewiselinear response of Fig. 3, springs of stiffness k2 are placed on either side of point D in Fig. 13 with a small gap, x1 . At the disk point D, a bar perpendicular to the plane of the figure is attached similar to Fig. 5. These springs are activated when |ϕ1 r| ≥ |x1 |, i.e. when the point D (bar) contacts one of the springs (neglecting any possible impact effects and assuming the springs are compressed linearly). The friction force between the rotor and outer ring is assumed to be of the stick-slip variety. Such friction is commonly exhibited in everyday systems such as the squeak of door hinges or chalk on a blackboard, and in the oscillations of violin strings, to name a few. Stick-slip friction has the characteristic of decreasing locally as a function of the relative velocity between the two bodies in contact and can generate self-excited oscillations. The characteristics of stick-slip friction versus relative velocity have been discussed in various papers and books (see, e.g. [Awrejcewicz & Delfs, 1990a, 1990b; Awrejcewicz & Holicke, 1999]). A graph of the stick-slip friction coefficient µ versus relative velocity w is shown in Fig. 14. Now we are ready to derive the equations of motion based on Newton’s laws and analyze

oscillations around the static equilibrium positions. Note that the mass m of the rigid beam causes an initial deflection of the spring kb . When the point D moves outside the interval (−x1 , x1 ), the outer ring experiences forces from the springs k1 and k2 , the dashpot c1 , and the friction caused by the rotating inner wheel. There is also a reaction force at point E caused by the frame. Note that spring kb and dashpot cb exert only a normal force on the outer ring; friction is neglected here. Likewise, the beam is acted upon by the spring kb , the dashpots c1 and cb , and gravity. The equations of motion of the beam and the outer ring for |ϕ1 | ≥ ϕ∗ , respectively, are given by I2 ϕ¨2 + kb l22 ϕ2 + cb l22 ϕ˙ 2 + c1 (ϕ˙ 2 l1 − ϕ˙ 1 r0 )l1 l2 = −mg , 2 (37) I1 ϕ¨1 − k1 δr02 ϕ1 + k2 r02 (ϕ1 − ϕ∗ ) + c1 (ϕ˙ 1 r0 − ϕ˙ 2 l1 )r0 = ri N (µ0 sgn w − α0 w + β0 w3 ) , where I1 , I2 denote the moments of inertia of the outer ring and beam, respectively. The reaction force of the frame acting at point E is included through δ, the geometric factor defined in (19). For the case of |ϕ1 | < ϕ∗ the first equation is identical and so is the second equation save for the k1 r02 (ϕ1 − ϕ∗ ) term. The right-hand side (RHS) of the second equation in (37) represents the torque due to friction where N is the normal force, w the relative (slip) velocity between the two bodies, and µ0 , α0 and β0 denote friction coefficients (see Fig. 14). The relations for w and N are w = (ω − ϕ˙ 1 )ri , 

N=

Fig. 14.

Stick-slip friction coefficient versus relative velocity.

mg − kb l2 ϕ2 − cb l2 ϕ˙ 2 2



.

Note the mg/2 term in the equation for N is due to static deflection of the spring kb . For this case, let us assume that w is in the regime where friction decreases linearly with increasing w, i.e. assume simply w = 1, and β0 = 0. Let us determine the static equilibrium positions, ϕ10 and ϕ20 , of the outer ring and beam, respectively. We assume the equilibrium position of the outer ring is such that point D does not contact the two springs of stiffness k2 , i.e. −ϕ∗ ≤ ϕ10 ≤ ϕ∗ . We obtain the static equilibrium angles ϕ10 and ϕ20

Mechanical Models of Chua’s Circuit 685

from (37) assuming ϕ ¨1 = ϕ¨2 = ϕ˙ 1 = ϕ˙ 2 = 0 mgri (µ0 − α0 ri ω) , k1 δr02 mg =− . 2kb l2

and, finally, make the following definitions α≡

ϕ10 = ϕ20

(38)

(m0 + 1) ≡ −

We seek to determine oscillations about the equilibrium positions, for the case of |ϕ1 | < ϕ∗ and |ϕ1 | > ϕ∗ . Therefore, we introduce new variables ϕ1 = ϕ1 + ϕ10 , ϕ2 = ϕ2 + ϕ20 where the overbars represent deviations from equilibrium. We also assume small displacements and velocities such that we may neglect nonlinear terms as being of secondorder. Substituting these new variables into the system (37), assuming I1 = 0, and omitting overbars we get c1 l1 r0 ϕ˙ 1 =

I2 ϕ ¨2 + (cb l22

+

c1 l12 )ϕ˙ 2

+

= [c1 l1 r0 − (µ0 − α0 ri ω)cb l2 ri ]ϕ˙ 2

(39)

− [(µ0 − α0 ri ω)kb l2 ri ]ϕ2 + k1 δr02 ϕ1

β≡

(40)

Now the system of equations in (39) begins to resemble that of (36). All that remains is to nondimensionalize the system and impose certain constraints on the system parameters (e.g. kb , c1 , I2 , etc.) to achieve the desired form of the equations. Such a transformation requires several steps, the details of which are omitted here for brevity. First, we introduce the nondimensional time τ and positions ϕ ˜1 , ϕ˜2 cb l22 + c1 l12 , I2

˜002 + ϕ ˜02 + β ϕ˜2 , ϕ˜01 = ϕ

(43)

ϕ˜01 = α[ϕ˜2 − ϕ˜1 − h(ϕ˜1 )]

where ϕ ˜01 = dϕ˜1 /dτ , ϕ˜02 = dϕ˜2 /dτ and ϕ˜002 = ˜2 /dτ 2 . The function h(ϕ˜1 ) is the required d2 ϕ piecewise-linear function

h(ϕ˜1 ) =

m ϕ˜ ,

0 1    −m ϕ ∗ ˜1 + ϕ∗ ), 0 ˜ + m1 (ϕ

ϕ > ϕ∗ |ϕ| < ϕ∗ ϕ < −ϕ∗ .

Note the system of equations in (43) has the same 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 generate the vast array of nonlinear behavior demonstrated by Chua’s circuit. This will not be trivial in practice, however, as the parameters must be varied in such a way as not to violate the required constraints in (40) and (41). The large number of parameters present in the system should allow the practitioner flexibility to explore the desired parameter space (α, β, m0 , m1 ) while satisfying the assumptions and constraints of this model.

3. Concluding Remarks

ϕ1 = ϕ∗0 ϕ ˜1 , ϕ2 = ϕ∗0 ϕ ˜2 , where ω ∗ and ϕ∗0 are constants of proportionality. Next, we impose the following constraint on the system parameters I2 ω ∗ = c1 l1 r0

kb l22 . I2 (ω ∗ )2

  ˜∗ + m1 (ϕ˜1 − ϕ∗ ),   m0 ϕ

where ϕ∗0 ≡ ϕ∗ − ϕ10 in the last term on the RHS of the second equation. To eliminate the ϕ˙ 2 term on the RHS of the second equation in (39), we impose the following constraint on the system parameters

(42)

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

− k2 r02 (ϕ1 − ϕ∗0 )

c1 l1 r0 − (µ0 − α0 ri ω)cb l2 ri = 0 .

k1 δr02 , (α0 ri ω − µ0 )kb l2 ri

k2 r02 (m0 − m1 ) ≡ , (α0 ri ω − µ0 )kb l2 ri

kb l22 ϕ2 ,

(c1 r02 − 0.5mgα0 ri2 )ϕ˙ 1

τ = ω ∗ t, ω ∗ =

(α0 ri ω − µ0 )kb l2 ri , (c1 r02 − 0.5mgα0 ri2 )ω ∗

(41)

The unfolding of Chua’s circuit can exhibit various chaotic and bifurcation behaviors which are welldocumented in the literature [Madan, 1993]. In this paper, the idea of its potential equivalence and homeomorphism in other engineering systems has been discussed and illustrated. Several realizations have been proposed: two electromechanical models

686 J. Awrejcewicz & M. L. Calvisi

of Chua’s circuit and one of Chua’s unfolded circuit, and, finally, a purely mechanical model of Chua’s circuit. We employed various “tricks” to reproduce the negative slope characteristics of the nonlinear resistor in the Chua’s circuit. The electromechanical systems use electromechanical couplings between three mechanical subsystems to achieve the required dynamic behavior of the Chua’s circuit. Conversely, the purely mechanical device uses only naturally observed mechanical properties and phenomena to satisfy this goal. Friction, often considered a nuisance in industry, is exploited in the purely mechanical system to produce some original couplings and synthesize the third-order Chua’s equations.

Acknowledgments The first author acknowledges a discussion with Dr. W. Wodzicki related to the construction of the mechanism. This work is partially supported by the Fulbright Foundation and by the ONR grant N000-14-98-0594.

References Awrejcewicz, J. & Delfs, J. [1990a] “Dynamics of a self-excited stick-slip oscillator with two degrees of freedom. Part 1: Investigation of equilibrium and slip-stick, slip-slip and stick-slip transition,” Europ. J. Mech., A/Solids 9(4), 269–282. Awrejcewicz, J. & Delfs, J. [1990b] “Dynamics of a self-excited stick-slip oscillator with two degrees of freedom. Part 2: Periodic and chaotic orbits,” Europ. J. Mech., A/Solids 9(5), 397–418. Awrejcewicz, J. & Holicke, M. M. [1999] “Melnikov’s method and stick-slip chaotic oscillations in very weakly forced mechanical systems,” Int. J. Bifurcation 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’s circuit,” Frequenz 46, 66–80. Madan, R. A. [1993] Chua’s Circuit: A Paradigm for Chaos (World Scientific, Singapore). Paul, B. [1979] Kinematics and Dynamics of Planar Machinery (Prentice-Hall, NJ).