Page - 124 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

124 4 SolvingOrdinaryDifferentialEquations Fig.4.14 Theeffect of avaccination campaign 4.3 OscillatingOne-DimensionalSystems Numerous engineering constructions and devices contain materials that act like springs. Such springsgive rise to oscillations, andcontrollingoscillations is a key engineering task.Weshall nowlearn to simulateoscillating systems. As always, we start with the simplest meaningfulmathematical model, which foroscillations is a second-orderdifferential equation: u00.t/C!2u.t/D0; (4.40) where! is a givenphysical parameter. Equation (4.40)models a one-dimensional system oscillating without damping (i.e., with negligible damping). One-dimen- sional heremeans that somemotion takesplace alongonedimensiononly in some coordinate system. Alongwith (4.40)weneed the two initial conditionsu.0/ and u0.0/. 4.3.1 DerivationofaSimpleModel Many engineering systems undergooscillations, and differential equations consti- tute the key tool to understand, predict, and control the oscillations. We startwith the simplest possiblemodel that captures the essential dynamics of an oscillating system. Some bodywithmassm is attached to a spring andmoves along a line without friction, see Fig. 4.15 for a sketch (rollingwheels indicate “no friction”). Whenthespring is stretched (orcompressed), thespringforcepulls (orpushes) the bodybackandwork “against” themotion.Moreprecisely, letx.t/be theposition