Page - 201 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

7.7 Exercises 201 When the program is executed, the original equation should be solved both withNewton’smethodandviafixedpoint iterations(asjustdescribed).Compare theoutput fromthe twomethods. Filename:fixed_point_iteration.py. Exercise7.8:SolveNonlinearEquationforaVibratingBeam Animportantengineeringproblemthatarisesinalotofapplicationsis thevibrations of a clamped beam where the other end is free. This problem can be analyzed analytically, but the calculations boil down to solving the following nonlinear algebraicequation: coshβcosβ=−1, whereβ is related to importantbeamparameters through β4 =ω2 A EI , where is the density of the beam,A is the area of the cross section,E is Young’s modulus,andI is themomentof theinertiaof thecrosssection.Themost important parameterof interest isω,which is the frequencyof thebeam.We want tocompute the frequencies of a vibrating steel beam with a rectangular cross section having widthb = 25mm and heighth = 8mm. The density of steel is 7850 kg/m3, and E = 2 × 1011Pa. The moment of inertia of a rectangular cross section is I = bh3/12. a) Plot the equation to be solved so that one can inspect where the zero crossings occur. Hint When writing the equation as f(β) = 0, the f function increases its amplitude dramatically with β. It is therefore wise to look at an equation with dampedamplitude,g(β)= e−βf(β)=0.Plotg instead. b) Compute the first three frequencies. Filename:beam_vib.py. Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,distributionandreproduction inanymediumor format, as longasyougiveappropriate credit to the original author(s) and thesource, provide a link to theCreative Commons licenceand indicate ifchanges were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons licence and your intended use is not permitted by statutory regulation orexceeds thepermitted use, you willneed toobtain permission directly from thecopyright holder.