Page - 144 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

144 4 SolvingOrdinaryDifferentialEquations # Step equations forward in time for n in range(N_t): v[n+1] = v[n] + dt*(1./m)*(F(t[n]) - f(v[n]) - s(u[n])) u[n+1] = u[n] + dt*v[n+1] return u, v, t The4-th orderRunge-Kuttamethod TheRK4method just evaluates the right- handsideof theODEsystem, . 1 m .F.t/ s.u/ f.v//;v/ for knownvalues ofu,v, and t, so themethod is very simple to use regardless of howthe functionss.u/andf.v/arechosen. 4.3.9 IllustrationofLinearDamping Weconsider anengineering systemwith a linear spring, s.u/Dkx, andaviscous damper, where the damping force is proportional to u0, f.u0/ D bu0, for some constant b > 0. This choicemaymodel the vertical spring system in a car (but engineers often like to illustrate such a system by a horizontalmovingmass like the one depicted in Fig. 4.25). Wemay choose simple values for the constants to illustratebasiceffectsofdamping(and later excitations). Choosing theoscillations to be the simpleu.t/ D cost function in the undampedcase,wemay setm D 1, kD1,bD0:3,U0 D1,V0 D0. The followingfunction implements this case: def linear_damping(): b = 0.3 f = lambda v: b*v s = lambda u: k*u F = lambda t: 0 m = 1 k = 1 U_0 = 1 V_0 = 0 T = 12*pi dt = T/5000. u, v, t = EulerCromer(f=f, s=s, F=F, m=m, T=T, U_0=U_0, V_0=V_0, dt=dt) plot_u(u, t) Theplot_u function is a collection of plot statements for plottingu.t/, or a part of it. Figure 4.27 shows the effect of the bu0 term: we have oscillationswith (an approximate)period2 , as expected,but theamplitude is efficientlydamped. Remarkaboutworkingwithascaledproblem Instead of settingb D 0:3 andm D k D U0 D 1 as fairly “unlikely” physical values, itwouldbebetter toscale theequationmu00Cbu0CkuD0. Thismeans