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302 chapter 1212.10 Unit II. Pendulums Become Chaotic (Continuous)C DIn Unit I on bugs we discovered that a simple nonlinear difference equation yieldssolutions that may be simple, complicated, or chaotic. Unit III will extend that modelto the differential form, which also exhibits complex behaviors. Now we search forsimilar nonlinear, complex behaviors in the differential equation describing a realisticpendulum. Because chaotic behavior may resemble noise, it is important to be confidentthat the unusual behaviors arise from physics and not numerics. Before we explorethe solutions, we provide some theoretical background on the use of phase space plotsfor revealing the beauty and simplicity underlying complicated behaviors. We alsoprovide two chaotic pendulum applets (Figure 12.5) for assistance in understandingthe new concepts. Our study is based on the description in [Rash 90], on the analyticdiscussion of the parametric oscillator in [L&L,M 76], and on a similar study of thevibrating pivot pendulum in [G,T&C 06].Consider the pendulum on the left in Figure 12.5. We see a pendulum of length ldriven by an external sinusoidal torque f through air with a coefficient of drag α.Because there is no restriction that the angular displacement θ be small, we callthis a realistic pendulum. Your problem is to describe the motion of this pendulum,first when the driving torque is turned off but the initial velocity is largeenough to send the pendulum over the top, and then when the driving torque isturned on.12.11 Chaotic Pendulum ODEWhat we call a chaotic pendulum is just a pendulum with friction and a driving torque(Figure 12.5 left) but with no small-deflection-angle approximation. Newton’s lawsof rotational motion tell us that the sum of the gravitational torque −mgl sin θ, thefrictional torque −β ˙θ, and the external torque τ 0 cos ωt equals the moment of inertiaof the pendulum times its angular acceleration [Rash 90]:I d2 θdθ= −mgl sin θ − βdt2 dt + τ 0 cos ωt, (12.28)⇒whered2 θdt 2 = −ω2 0 sin θ − α dθ + f cos ωt, (12.29)dtω 0 = mglI , α= β I , f = τ 0I . (12.30)Equation (12.29) is a second-order time-dependent nonlinear differential equation.Its nonlinearity arises from the sin θ, as opposed to the θ, dependence of−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 302