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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 24427, 906]*) (*NotebookOutlinePosition[ 25482, 942]*) (* CellTagsIndexPosition[ 25438, 938]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "(* This Mathematica program demonstrates the difference between the *)\n(* \ non-linear response of a system and the linearized approximation *)\n(* The \ example used is a draining cylindrical tank in which the *)\n(* outflow \ is proportional to the square root of the liquid level *)\n\n(* The \ parameters used are Fi [=] m^3/hr, A [=] m^2, beta [=] m^2.5/min *)\n\n(* The \ initial condition is h(0) = hs m. *)\n"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "(* Define a function which is the linear o.d.e. *)\n\nf[t_] = h'[t] + \ (beta/A) hs^0.5 + (beta/(2 A hs^0.5)) (h[t]-hs) - Fi/A "], "Input", AspectRatioFixed->True], Cell[TextData[ "General::spell1: \n Possible spelling error: new symbol name \"beta\" is \ similar to existing symbol \n \"Beta\"."], "Message", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData[ "\<\ -(Fi/A) + (beta*hs^0.5)/A + (beta*(-hs + h[t]))/(2*A*hs^0.5) + \ Derivative[1][h][t]\ \>", "\<\ 0.5 Fi beta hs beta (-hs + h[t]) -(--) + ---------- + ----------------- + h'[t] A A 0.5 2 A hs\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData[ "\n(* Solve the linear o.d.e. with the initial tank height *)\n(* set to hs \ m. *)\n\n sol = DSolve[{f[t]==0,h[0]==hs},h[t],t] "], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{h[t] -> ((2*Fi - beta*hs^0.5)*hs^0.5)/beta + ((-2.*Fi*hs^0.5)/beta + 1.*hs + \ 1.*hs^1.)/E^((beta*t)/(2*A*hs^0.5))}}\ \>", "\<\ 0.5 -2. 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