Page - 240 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

240 8 SolvingOrdinaryDifferentialEquations whereω is a given physical parameter. Equation (8.40) models a one-dimensional system oscillating without damping (i.e., with negligible damping). One- dimensionalhere means that some motion takesplace alongone dimensiononly in somecoordinatesystem. Alongwith (8.40)we need the two initial conditionsu(0) andu′(0). 8.4.1 DerivationofaSimpleModel Many engineering systems undergo oscillations, and differential equations consti- tute the key tool to understand, predict, and control the oscillations. We start with the simplest possible model that captures the essential dynamics of an oscillating system. Some body with mass m is attached to a spring and moves along a line without friction, see Fig. 8.18 for a sketch (rolling wheels indicate “no friction”). When thespring is stretched(orcompressed), thespring forcepulls (orpushes) the body back and work “against” the motion. More precisely, let x(t) be the position of the body on thex axis, along which the bodymoves.The spring is not stretched whenx=0, so theforce is zero,andx=0 ishence theequilibriumpositionof the body.Thespringforceis−kx,wherek isaconstanttobemeasured.Weassumethat therearenootherforces(e.g.,nofriction).Newton’ssecondlawofmotionF =ma thenhasF =−kx anda= x¨, −kx=mx¨, (8.41) whichcanbe rewrittenas x¨+ω2x=0, (8.42) by introducingω=√k/m (which isverycommon). Equation (8.42) is a second-order differential equation, and therefore we need two initial conditions,one on the positionx(0)and one on the velocityx′(0). Here Fig. 8.18 Sketch ofa one-dimensional, oscillating dynamic system (without friction)