Page - 145 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 145 Fig.4.27 Effect of lineardamping thatwe introducedimensionless independentanddependentvariables: Nt D t tc ; NuD u uc ; where tc anduc are characteristic sizes of time and displacement, respectively, such that Nt and Nu have their typical size around unity. In the present problem, we can chooseuc D U0 and tc D p m=k. This gives the following scaled (or dimensionless)problemfor thedimensionlessquantity Nu.Nt/: d2 Nu d Nt2 Cˇ d Nu d Nt C NuD0; Nu.0/D1; Nu 0.0/D0; ˇD bp mk : The striking fact is that there is only one physical parameter in this problem: the dimensionless numberˇ. Solving this problem corresponds to solving the original problem(with dimensions)with the parametersmD k DU0 D 1 and bDˇ. However,solvingthedimensionlessproblemismoregeneral: ifwehave asolution Nu.NtIˇ/,wecanfindthephysical solutionofa rangeofproblemssince u.t/DU0 Nu t p k=mIˇ : As long asˇ is fixed,we can findu for anyU0, k, andm from the above for- mula! In thisway, a timeconsumingsimulationcanbedoneonlyonce,but still providemanysolutions. Thisdemonstrates thepowerofworkingwith scaledor dimensionlessproblems.