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Unformatted text preview: Lecture 29 Reduction of Higher Order Equations to Systems The motion of a pendulum Consider the motion of an ideal pendulum that consists of a mass m attached to an arm of length ℓ . If we ignore friction, then Newton’s laws of motion tell us: m ¨ θ =- mg ℓ sin θ, where θ is the angle of displacement. a116 a65 a65 a65 a65 a65 a65 a65 a65 a119 ℓ θ Figure 29.1: A pendulum. If we also incorporate moving friction and sinusoidal forcing then the equation takes the form: m ¨ θ + γ ˙ θ + mg ℓ sin θ = A sinΩ t. Here γ is the coefficient of friction and A and Ω are the amplitude and frequency of the forcing. Usually, this equation would be rewritten by dividing through by m to produce: ¨ θ + c ˙ θ + ω sin θ = a sinΩ t, (29.1) where c = γ/m . ω = g/ℓ and a = A/m . This is a second order ODE because the second derivative with respect to time t is the highest derivative. It is nonlinear because it has the term sin θ and which is a nonlinear function of the dependent variable θ . A solution of the equation would be a function θ ( t ). To get a specific solution we need side conditions. Because it is second order, 2 conditions are needed, and the usual conditions are initial conditions: θ (0) = θ and ˙ θ (0) = v . (29.2) 106 107 Converting a general higher order equation...
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.
- Fall '08