Page - 148 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

148 4 SolvingOrdinaryDifferentialEquations videdthesignumfunctionsign.x/ isdefinedtobezeroforxD0 (numpy.signhas thisproperty). Tocheck that thesigns in thedefinitionoff are right, recall that the actual physical force is f and this is positive (i.e.,f <0)when itworksagainst thebodymovingwithvelocityu0 <0. Thenonlinearspring force is takenas s.u/D k˛ 1 tanh.˛u/; which is approximately ku for smallu, but stabilizes at˙k=˛ for large˙˛u. Here is aplotwithkD1000andu2 Œ 0:1;0:1 for three˛values: If there is noexternal excitation force actingon thebody,wehave the equation ofmotion mu00C mgsign.u0/Ck˛ 1 tanh.˛u/D0: Let us simulate a situationwhere a bodyofmass 1 kg slides on a surfacewith D 0:4, while attached to a springwith stiffness k D 1000kg=s2. The initial displacement of the body is 10 cm, and the˛ parameter in s.u/ is set to 60 1/m. Using theEulerCromer function from theosc_EC_generalcode, we canwrite a functionsliding_frictionfor solving thisproblem: def sliding_friction(): from numpy import tanh, sign f = lambda v: mu*m*g*sign(v) alpha = 60.0 s = lambda u: k/alpha*tanh(alpha*u) F = lambda t: 0 g = 9.81 mu = 0.4 m = 1 k = 1000