(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 29683, 855] NotebookOptionsPosition[ 21902, 665] NotebookOutlinePosition[ 26446, 776] CellTagsIndexPosition[ 25319, 748] WindowTitle->Lecture 23 MIT 3.016 (Fall 2010) \251 W. Craig Carter \ 2003--2010 WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Resonance Phenomena", "Title"], Cell[CellGroupData[{ Cell["\<\ Resonance phenomena simulated by finite differences and biased noise inputs\ \>", "Subtitle"], Cell["\<\ In Lecture-21.nb, a second-order differencing scheme that iteratively solved \ y'' + \[Beta]y' + \[Gamma]y=0 with the specification of two initial values. Modify this to add a little random noise to y[i] at each step and observe how \ this behaves---this version will store the noise added at each iteration so \ that it can be visualized later....\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"GrowListGeneralNoise", "[", RowBox[{ "ValuesList_List", ",", "\[CapitalDelta]_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "randomamp_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Minus1", "=", RowBox[{"ValuesList", "\[LeftDoubleBracket]", RowBox[{"1", ",", RowBox[{"-", "1"}]}], "\[RightDoubleBracket]"}]}], ",", RowBox[{"Minus2", "=", RowBox[{"ValuesList", "\[LeftDoubleBracket]", RowBox[{"1", ",", RowBox[{"-", "2"}]}], "\[RightDoubleBracket]"}]}], ",", RowBox[{"noise", "=", RowBox[{"RandomReal", "[", RowBox[{"{", RowBox[{ RowBox[{"-", "randomamp"}], ",", "randomamp"}], "}"}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Append", "[", RowBox[{ RowBox[{ "ValuesList", "\[LeftDoubleBracket]", "1", 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"Text"], Cell[BoxData[ RowBox[{ RowBox[{"TheBiasedData", " ", "=", " ", RowBox[{"Nest", "[", RowBox[{"GrowListSpecificBiasedNoise", ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}]}], "}"}], ",", "20000"}], "]"}]}], ";"}]], "Input"], Cell[BoxData[{ RowBox[{"ListPlot", "[", RowBox[{"TheBiasedData", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "\[IndentingNewLine]", RowBox[{"ListPlot", "[", RowBox[{"TheBiasedData", "[", RowBox[{"[", "2", "]"}], "]"}], "]"}]}], "Input", CellTags->"mmtag:23:noise__periodic_bias"] }, Closed]], Cell[CellGroupData[{ Cell["Resonance Phenomena by Solution of the ODE", "Subtitle"], Cell[TextData[{ "The \"periodic forcing function\" can be any periodic function (i.e., a \ fourier series), but it is a bit simpler to analyze the effect of each mode \ separately. Below, the forcing function will be assumed to be ", Cell[BoxData[ FormBox[ SubscriptBox["F", "app"], TraditionalForm]]], "cos(", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "app"], TraditionalForm]]], "t)\nSolve problems in terms of the mass and natural frequency--eliminate \ the spring constant in equations by defining it in terms of the mass and \ natural frequency." }], "Subsection", CellTags->{"mmtag:23:resonance_phenomena", "mmtag:23:natural_frequency"}], Cell[BoxData[ RowBox[{"Kspring", " ", "=", " ", RowBox[{"M", " ", SuperscriptBox["\[Omega]char", "2"]}]}]], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can solve the nonhomogeneous ODE with a forcing function at with an \ applied frequency:" }], "Subsection"], Cell[BoxData[ RowBox[{"yGeneralSol", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"y", "[", "t", "]"}], "/.", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"M", " ", RowBox[{ RowBox[{"y", "''"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"\[Eta]", " ", RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"Kspring", " ", RowBox[{"y", "[", "t", "]"}]}]}], " ", "\[Equal]", " ", RowBox[{"Fapp", " ", RowBox[{"Cos", "[", RowBox[{"\[Omega]app", " ", "t"}], "]"}]}]}], ",", RowBox[{"y", "[", "t", "]"}], ",", "t"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}]}]], "Input", CellTags->"mmtag:23:harmonic_oscillator__general_solution_for_damped_forced"], Cell[TextData[StyleBox["Consider the form of the above solution: The \ homogeneous part of the solution (i..e, the part that depends on constants \ C[1] and C[2]) depends only on the characteristic frequency, while the \ heterogeneous part includes the forcing and the characteristic frequency. \ From a first glance at the denominator in the heterogeneous term, it appears \ the amplitudes are maximized as \[Omega]app \[LongRightArrow] \[Omega]char; \ and , it appears that, as long as the viscous term \[Eta] is non-zero, the \ solution is bounded. However, this is not the case as will be shown below.", FontVariations->{"CompatibilityType"->0}]], "Section"], Cell[TextData[{ "Consider the behavior of the general solution at time t=0. This will show \ that the homogeneous parts of the solution are needed to satisfy boundary \ conditions, even if the oscillator is initially at rest at zero displacement \ (i.e., y(0) = ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ OverscriptBox["y", "."], "(", "0", ")"}], " ", "=", " ", "0"}], ")"}], "."}], TraditionalForm]]] }], "Subsection"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"yGeneralSol", "/.", RowBox[{"t", "->", "0"}]}], "]"}]], "Input"], Cell["\<\ Therefore, the C[1] and C[2] in the general solution, must pick up terms \ that depend on \[Omega]app and these must be included in considerations of \ the limit \[Omega]app \[LongRightArrow] \[Omega]char.\ \>", "Section"], Cell["\<\ Consider the particular case of an oscillator at rest at zero displacement:\ \>", "Subsection", CellTags-> "mmtag:23:harmonic_oscillator__homogeneous_and_particular_solutions"], Cell[BoxData[ RowBox[{"yParticularSol", " ", "=", " ", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"y", "[", "t", "]"}], "/.", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"M", " ", RowBox[{ RowBox[{"y", "''"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"\[Eta]", " ", RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}]}], " ", "+", " ", RowBox[{"Kspring", " ", RowBox[{"y", "[", "t", "]"}]}]}], " ", "\[Equal]", " ", RowBox[{"Fapp", " ", RowBox[{"Cos", "[", RowBox[{"\[Omega]app", " ", "t"}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"y", "[", "0", "]"}], "==", "0"}], ",", " ", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "0", "]"}], "==", "0"}]}], "}"}], ",", RowBox[{"y", "[", "t", "]"}], ",", "t"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}]}]], "Input", CellTags->"mmtag:23:harmonic_oscillator__general_solution_for_damped_forced"], Cell["\<\ The resonant solution is the case: \[Omega]app \[Rule] \[Omega]char\ \>", "Subsection"], Cell[BoxData[ RowBox[{"ResonantSolution", " ", "=", RowBox[{"Simplify", "[", RowBox[{"yParticularSol", "/.", RowBox[{"\[Omega]app", " ", "\[Rule]", " ", "\[Omega]char"}]}], "]"}]}]], "Input"], Cell["\<\ To analyze the resonant behavior when the viscosity \[Eta] is small, the \ solution is expanded about small viscosity to second order: (Below, the function Map is used with a pure function to simplify each of the \ terms resulting from a series expansion (n.b. Normal converts the series form \ into a normal form). Each term in the expansion is operated on in the \ following manner: 1) Convert any Exp[] into trignometric functions with ExpToTrig 2) Remove fractional powers of containing powers of real numbers with \ PowerExpand 3) Simplify by using a pure function as the first argument to Map,\ \>", "Subsection"], Cell[BoxData[ RowBox[{"ResonantSolutionSmallViscosity", " ", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Simplify", "[", RowBox[{"PowerExpand", " ", "[", " ", RowBox[{"ExpToTrig", "[", "#", "]"}], "]"}], "]"}], "&"}], ",", RowBox[{"Normal", "[", RowBox[{"Series", "[", RowBox[{"ResonantSolution", ",", RowBox[{"{", RowBox[{"\[Eta]", ",", "0", ",", "2"}], "}"}]}], "]"}], "]"}]}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"ResonantSolutionSmallViscosityDetuned", " ", "=", RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"Simplify", "[", RowBox[{"PowerExpand", " ", "[", " ", RowBox[{"ExpToTrig", "[", "#", "]"}], "]"}], "]"}], "&"}], ",", RowBox[{"Normal", "[", RowBox[{"Series", "[", 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