Page - 264 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

264 8 SolvingOrdinaryDifferentialEquations Fig. 8.33 Sketch of a one-dimensional, oscillating dynamic system subject to sliding friction and aspring force 8.4.11 Spring-MassSystemwithSlidingFriction Abodywithmassm is attached toa springwith stiffnesskwhile slidingonaplane surface.Thebodyisalsosubject toa frictionforcef(u′)dueto thecontactbetween the body and the plane. Figure 8.33 depicts the situation. The friction forcef(u′) canbemodeledbyCoulombfriction: f(u′)= ⎧ ⎨ ⎩ −μmg, u′<0, μmg, u′>0, 0, u′ =0 whereμ is the frictioncoefficient, andmg is the normal forceon the surfacewhere thebodyslides.Thisformulacanalsobewrittenasf(u′)=μmgsign(u′),provided the signum function sign(x) is defined to be zero forx = 0 (numpy.signhas this property). To check that the signs in the definition of f are right, recall that the actual physical force is−f and this is positive (i.e.,f < 0) when it works against thebodymovingwithvelocityu′<0. Thenonlinearspringforce is takenas s(u)=−kα−1 tanh(αu), whichisapproximately−ku forsmallu,butstabilizesat±k/α for large±αu.Here is aplotwithk=1000andu∈[−0.1,0.1] for threeαvalues: