This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
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
 Young,T
 Equations

Click to edit the document details