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discrete & continuous nonlinear dynamics 303f l 1m 1θ 1l 2θ 2θIαmm 2Figure 12.5 Left: A pendulum of length l driven through air by an external sinusoidal torque.The strength of the torque is given by f and that of air resistance by α. Right: A doublependulum.the gravitational torque. The parameter ω 0 is the natural frequency of the systemarising from the restoring torque, α is a measure of the strength of friction,and f is a measure of the strength of the driving torque. In our standard ODEform, dy/dt = y (Chapter 9, “Differential Equation Applications”), we have twosimultaneous first-order equations:dy (0)dtdy (1)dt= y (1) , (12.31)= −ω 2 0 sin y (0) − αy (1) + f cos ωt,where y (0) = θ(t), y (1) = dθ(t) . (12.32)dt12.11.1 Free Pendulum OscillationsIf we ignore friction and external torques, (12.29) takes the simple formd 2 θdt 2 = −ω2 0 sin θ (12.33)If the displacements are small, we can approximate sin θ by θ and obtain the linearequation of simple harmonic motion with frequency ω 0 :d 2 θdt 2 ≃−ω2 0θ ⇒ θ(t)=θ 0 sin(ω 0 t + φ). (12.34)−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 303