Page - 208 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

208 6 SolvingNonlinearAlgebraicEquations whereˇ is related to importantbeamparameters through ˇ4 D!2%A EI ; where% is thedensityof thebeam,A is theareaof the cross section,E isYoung’s modulus,andI is themomentoftheinertiaofthecrosssection. Themostimportant parameterof interest is!,which is thefrequencyof thebeam.Wewant tocompute the frequencies of a vibrating steel beamwith a rectangular cross section having widthb D 25mmandheighthD 8mm. Thedensity of steel is7850kg/m3, and E D2 1011Pa.ThemomentofinertiaofarectangularcrosssectionisI Dbh3=12. a) Plot the equation tobe solved so that one can inspectwhere the zero crossings occur. Hint Whenwriting the equation asf.ˇ/ D 0, thef function increases its ampli- tude dramaticallywithˇ. It is thereforewise to look at an equationwith damped amplitude,g.ˇ/D e ˇf.ˇ/D0. Plotg instead. b) Compute thefirst three frequencies. Filename:beam_vib.py. OpenAccess This chapter is distributedunder the termsof theCreativeCommonsAttribution- NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, duplication, adaptation, distribution and reproduction in anymediumor format, as long as you give appropriate credit to the original author(s) and the source, a link isprovided to theCreativeCommons licenseandanychangesmadeare indicated. The images or other third partymaterial in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if suchmaterial is not included in thework’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce thematerial.