(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 111027, 4125]*) (*NotebookOutlinePosition[ 111714, 4149]*) (* CellTagsIndexPosition[ 111670, 4145]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Coupled oscillators, jiggled from one end\ \>", "Title"], Cell[TextData[{ "A one-dimensional system of n masses connected by springs. The far end of \ the last spring, the spring connected to the right ", Cell[BoxData[ \(TraditionalForm\`m\_n\)]], ", is fixed. The first spring, the spring connected to the left of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], ", can be \"jiggled\" from the left with displacement ", Cell[BoxData[ \(TraditionalForm\`\(x\_0\)(t)\)]], ". The system of equations to solve is :\n", Cell[BoxData[ \(TraditionalForm\`d\^2\/dt\^2\)]], Cell[BoxData[ \(TraditionalForm\`X\ = \ A\ X\ + F\)]], "\nwhere, for an example of 3 masses,\n", Cell[BoxData[ FormBox[ RowBox[{"X", "=", RowBox[{"(", GridBox[{ {\(x\_1\)}, {\(x\_2\)}, {\(x\_3\)} }], ")"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {\(-\(\(k\_1 + k\_2\)\/m\_1\)\), \(k\_2\/m\_2\), "0"}, {\(k\_2\/m\_2\), \(-\(\(k\_2 + k\_3\)\/m\_2\)\), \(k\_3\/m\_3\)}, {"0", \(k\_2\/m\_3\), \(-\(\(k\_3 + k\_4\)\/m\_3\)\)} }], ")"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{"F", "=", RowBox[{"(", GridBox[{ {\(\(k\_1\/m\_1\) x\_0\)}, {"0"}, {"0"} }], ")"}]}], TraditionalForm]]], "\nWe can make this system dimensionless by using ", Cell[BoxData[ \(TraditionalForm\`\@\(m\_1\/k\_1\)\)]], "as the time scale: ", Cell[BoxData[ FormBox[ RowBox[{" ", FormBox[\(t = \[Tau] \@\( m\_1\/k\_1\)\), "TraditionalForm"]}], TraditionalForm]]], ". The dimensionless system will be written as\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(d\^2\/d\[Tau]\^2\), "TraditionalForm"], "X"}], " ", "=", " ", \(A\ X\ + X\_0\)}], TraditionalForm]]], ",\nwhere now all the ", Cell[BoxData[ \(TraditionalForm\`k\_j\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_j\)]], " should be interpreted as the ratio of the dimensional value to ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], ", respectively. For example, if all ", Cell[BoxData[ \(TraditionalForm\`k\_j = k\_1\)]], "and ", Cell[BoxData[ \(TraditionalForm\`m\_j = m\_1\)]], ", then\n", Cell[BoxData[ FormBox[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {\(-2\), "1", "1"}, {"1", \(-2\), "1"}, {"1", "1", \(-2\)} }], ")"}]}], TraditionalForm]]], "\nAlso,\n", Cell[BoxData[ FormBox[ RowBox[{\(X\_0\), "=", RowBox[{"(", GridBox[{ {\(x\_0\)}, {"0"}, {"0"} }], ")"}]}], TraditionalForm]]], "\nLet the eigenvectors of ", Cell[BoxData[ \(TraditionalForm\`A\)]], " be written as ", Cell[BoxData[ \(TraditionalForm\`K\_j\)]], " and the eigenvalues be ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_j\)]], ".\nConstruct a matrix ", Cell[BoxData[ \(TraditionalForm\`P\)]], " with the columns being the eigenvectors ", Cell[BoxData[ \(TraditionalForm\`K\_j\)]], ".\nLet ", Cell[BoxData[ \(TraditionalForm\`X = P\ Y\)]], ", which means Y is the amplitude of the eigenvectors that\nare summed to \ represent the displacment. Here is the clever way\nto solve the coupled \ problem: \n\n", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ RowBox[{ FormBox[\(d\^2\/d\[Tau]\^2\), "TraditionalForm"], "P", " ", "Y"}], " ", "=", " ", \(A\ P\ Y\ + X\_0\)}], "TraditionalForm"], ",", "\n"}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{Cell[""], \(P\^\(-1\)\)}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ RowBox[{ FormBox[\(P\ d\^2\/d\[Tau]\^2\), "TraditionalForm"], " ", "Y"}], " ", "=", " ", \(P\^\(-1\)\ A\ P\ Y\ + P\^\(-1\)\ X\_0\)}], "TraditionalForm"], ","}], TraditionalForm]]], "\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(d\^2\/d\[Tau]\^2\)\), "TraditionalForm"], " ", "Y"}], " ", "=", " ", \(D\ \ Y\ + P\^\(-1\)\ X\_0\)}], TraditionalForm]]], "\n\nwhere ", Cell[BoxData[ \(TraditionalForm\`D\)]], " is a diagonal matrix of the eigenvalues. Let \n", Cell[BoxData[ \(TraditionalForm\`G \[Congruent] \ \(P\^\(-1\)\) X\_0\)]], ". This is really cool: there are uncoupled O.D.E.s for the amplitudes of \ each of the eigenvectors:\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(d\^2\/d\[Tau]\^2\)\), "TraditionalForm"], " ", \(y\_j\)}], " ", "=", \(\[Lambda]\_j\ \ y\_j\ + \ g\_j\)}], TraditionalForm]]], "\nFinding a particular solution of the above equation is rather \ elementary. \n\nWe next present a technique for finding the homogeneous \ solution,\nand in particular an efficient way to satisfy the initial \ conditions\n", Cell[BoxData[ \(TraditionalForm\`\(\(\(X\_h\)(0)\)\(\ \)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(X\& . \_h\)(0)\)]], ". The general homogeneous solution is:\n", Cell[BoxData[ \(TraditionalForm\`\(X\_h\)(t) = \ Re {\ P\ \(\(Y\_h\)(t)\)\ }\)]], " \nwhere \n", Cell[BoxData[ \(TraditionalForm\`\(y\_hj\)(t) = \ \(c\_j\) e\^\(\(i\[Omega]\_j\) t\)\)]], "\nLet ", Cell[BoxData[ \(TraditionalForm\`c\_j = \ c\_jr + \ i\ \(\(c\_ji\)\(.\)\(\ \ \)\)\)]], "And let ", Cell[BoxData[ \(TraditionalForm\`C\)]], " be a column vector of the ", Cell[BoxData[ \(TraditionalForm\`c\_j\)]], ".\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\(X\_h\)(0)\), Cell[""]}], "=", " ", \(P\ C\_r\)}], TraditionalForm]]], "\n", Cell[BoxData[ \(TraditionalForm\`\(X\& . \_h\)(0)\ = \ \(-P\)\ D\ C\_i\)]], "\nAs a matrix problem, this gives\n", Cell[BoxData[ \(TraditionalForm\`C\_r = \ \(P\^\(-1\)\) \(\(X\_h\)(0)\)\)]], "\nand\n", Cell[BoxData[ \(TraditionalForm\`C\_i = \ \(-Q\^\(-1\)\) \(\(X\& . \)(0)\)\)]], "\nwhere ", Cell[BoxData[ \(TraditionalForm\`Q = P\ D\)]], "." }], "Text", FontSize->24], Cell["\<\ I like to turn off warnings about variable names being \ similar:\ \>", "Text"], Cell[BoxData[ \(Off[General::spell]\)], "Input"], Cell[BoxData[ \(\(\(Off[General::spell1]\)\(\ \)\)\)], "Input"], Cell["\<\ Create a matrix using \"Input\"-> \"Create Table/Matrix/Pallette\". Then check \"Matrix\". The matrix is stored (and can also be input) as a list of lists, with the inner lists being the rows of A. Comment one of these A matrices as text, \"uncomment\" as needed. Note the use of one decimal number forces the use of numerical computation, as opposed \ to symbolic.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {\(-2\), "1", "0"}, {"1", \(-2\), "1"}, {"0", "1", \(-2. \)} }], ")"}]}]], "Input"], Cell[BoxData[ \({{\(-2\), 1, 0}, {1, \(-2\), 1}, {0, 1, \(-2.`\)}}\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{"A", "=", RowBox[{"(", GridBox[{ {\(-2\), "1"}, {"1", \(-2\)} }], ")"}], " ", \( (*change\ this\ ' Format > Style'\ to\ ' Input'\ to\ use\ this\ one*) \)}]], "Text"], Cell[BoxData[ \({{\(-2\), 1}, {1, \(-2\)}}\)], "Output"], Cell[CellGroupData[{ Cell[BoxData[ \(n = Length[A]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["\<\ First, enjoy the exercise of finding eigenvalues without using the \ \"canned\" routine. 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ool06ooo0`00HOoo001?ool00`00ooooo`12ool20007ool00`00ooooo`0Lool00`00ooooo`1Qool0 04koo`03003ooooo04Coo`P001ooo`03003ooooo067oo`00C_oo00<00?ooool0Jooo00<00?ooool0 HOoo001>ool00`00ooooo`1[ool00`00ooooo`1Qool004koo`03003ooooo06_oo`03003ooooo067o o`00C_oo00<00?ooool0Jooo00<00?ooool0HOoo001=ool00`00ooooo`1/ool00`00ooooo`1Qool0 04goo`03003ooooo06coo`03003ooooo067oo`00COoo00<00?ooool0K?oo00<00?ooool0HOoo001= ool00`00ooooo`1/ool3001Qool004coo`03003ooooo06goo`03003ooooo067oo`00C?oo00<00?oo ool0KOoo00<00?ooool0HOoo001"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-6.23672, -4.56907, \ 0.0330085, 0.0516396}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ce \[Equal] 0, \[Lambda]]\)], "Input"], Cell[BoxData[ \({{\[Lambda] \[Rule] \(-3.414213562373095`\)}, {\[Lambda] \[Rule] \(-2.`\ \)}, {\[Lambda] \[Rule] \(-0.585786437626905`\)}}\)], "Output"] }, Open ]], Cell["Or use the canned routine:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(lam = Eigenvalues[A]\)], "Input"], Cell[BoxData[ \({\(-3.4142135623730954`\), \(-2.`\), \(-0.5857864376269049`\)}\)], \ "Output"] }, Open ]], Cell[TextData[{ "We will solve the homogeneous equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(\(\ \)\(d\^2\/d\[Tau]\^2\)\), "TraditionalForm"], " ", \(y\_j\)}], " ", "=", \(\[Lambda]\_j\ \ y\_j\)}], TraditionalForm]]], "by proposing ", Cell[BoxData[ \(TraditionalForm\`y\_j = \ e\^\(\(i\[Omega]\_j\) t\)\)]], ". Obviously, ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_j = \@\(-\[Lambda]\_j\)\)]], ". Let's make a list of these \"eigenfrequencies\":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ef = Sqrt[\(-lam\)]\)], "Input"], Cell[BoxData[ \({1.8477590650225737`, 1.4142135623730951`, 0.7653668647301795`}\)], "Output"] }, Open ]], Cell[TextData[{ "Let ", StyleBox["Mathematica", FontSlant->"Italic"], " find the eigenvectors:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(evect = Eigenvectors[A]\)], "Input"], Cell[BoxData[ \({{0.49999999999999994`, \(-0.7071067811865475`\), 0.5`}, {\(-0.7071067811865477`\), \(-7.577190362262357`*^-17\), 0.7071067811865474`}, {0.49999999999999994`, 0.7071067811865476`, 0.49999999999999994`}}\)], "Output"] }, Open ]], Cell[TextData[{ "Store the eigenvectors are columns in ", Cell[BoxData[ \(TraditionalForm\`P\)]], ":" }], "Text"], Cell[BoxData[ \(\(P = Transpose[evect];\)\)], "Input"], Cell[TextData[{ "Calculate ", Cell[BoxData[ \(TraditionalForm\`P\^\(-1\)\)]], ":" }], "Text"], Cell[BoxData[ \(\(Pinv = Inverse[P];\)\)], "Input"], Cell[TextData[{ "Have a look at ", Cell[BoxData[ \(TraditionalForm\`P\^\(-1\)\)]], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Pinv // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.5`", \(-0.7071067811865476`\), "0.5000000000000001`"}, {\(-0.7071067811865477`\), \(-1.9968293511751778`*^-16\), "0.7071067811865474`"}, {"0.4999999999999999`", "0.7071067811865476`", "0.5000000000000001`"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Have a look at ", Cell[BoxData[ \(TraditionalForm\`\(P\^\(-1\)\) A\ P\)]], ", just to see that it yields the expected diagonal matrix (except for \ round-off error):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Pinv . \((A . P)\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-3.4142135623730954`\), "3.0444397003392965`*^-16", "2.937239210534792`*^-16"}, {"8.651933336434325`*^-17", \(-2.`\), "1.8026216370287118`*^-16"}, { "9.074772183703672`*^-17", \(-1.0928757898653885`*^-16\), \ \(-0.585786437626905`\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(DiagonalMatrix[lam] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(-3.4142135623730954`\), "0", "0"}, {"0", \(-2.`\), "0"}, {"0", "0", \(-0.5857864376269049`\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "In this particular example, we set ", Cell[BoxData[ \(TraditionalForm\`x\_0 = \ sin\ \[CapitalOmega]\ \[Tau]\)]], ". In doing so, we have implicitly nondimensionalized displacement in \ terms of the amplitude of the \"jiggle\"." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(F = PadRight[{1}, n] Sin[\[CapitalOmega]\ t]\)], "Input"], Cell[BoxData[ \({Sin[t\ \[CapitalOmega]], 0, 0}\)], "Output"] }, Open ]], Cell["Project the forcing onto the eigenvectors:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(G = Pinv . F\)], "Input"], Cell[BoxData[ \({0.5`\ Sin[t\ \[CapitalOmega]], \(-0.7071067811865477`\)\ Sin[ t\ \[CapitalOmega]], 0.4999999999999999`\ Sin[t\ \[CapitalOmega]]}\)], "Output"] }, Open ]], Cell["\<\ A little \"pencil and paper math\" yields the amplitudes of the eigenvectors as caused by the jiggling:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Y = G/\((ef*ef - \[CapitalOmega]\^2)\)\)], "Input"], Cell[BoxData[ \({\(0.5`\ Sin[t\ \[CapitalOmega]]\)\/\(\(\(3.414213562373096`\)\(\ \[InvisibleSpace]\)\) - \[CapitalOmega]\^2\), \(-\(\(0.7071067811865477`\ Sin[ t\ \[CapitalOmega]]\)\/\(\(\(2.0000000000000004`\)\(\ \[InvisibleSpace]\)\) - \[CapitalOmega]\^2\)\)\), \(0.4999999999999999`\ \ Sin[t\ \[CapitalOmega]]\)\/\(\(\(0.5857864376269049`\)\(\[InvisibleSpace]\)\) \ - \[CapitalOmega]\^2\)}\)], "Output"] }, Open ]], Cell["\<\ Let's consider a particular value of \[CapitalOmega], so that we \ can have something to plot out: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Xp = P . Y /. \[CapitalOmega] \[Rule] 1/2\)], "Input"], Cell[BoxData[ \({1.1092436974789917`\ Sin[t\/2], 0.9411764705882355`\ Sin[t\/2], 0.5378151260504203`\ Sin[t\/2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Xpd = D[Xp, t]\)], "Input"], Cell[BoxData[ \({0.5546218487394958`\ Cos[t\/2], 0.47058823529411775`\ Cos[t\/2], 0.26890756302521013`\ Cos[t\/2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Xp0 = Xp /. t \[Rule] 0\)], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Xpd0 = Xpd /. t \[Rule] 0\)], "Input"], Cell[BoxData[ \({0.5546218487394958`, 0.47058823529411775`, 0.26890756302521013`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Q = P . DiagonalMatrix[ef]\)], "Input"], Cell[BoxData[ \({{0.9238795325112867`, \(-1.0000000000000002`\), 0.38268343236508967`}, {\(-1.3065629648763766`\), \ \(-1.0715765374994131`*^-16\), 0.541196100146197`}, {0.9238795325112868`, 0.9999999999999998`, 0.38268343236508967`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Qinv = Inverse[Q]\)], "Input"], Cell[BoxData[ \({{0.2705980500730985`, \(-0.38268343236508967`\), 0.27059805007309856`}, {\(-0.5`\), 0.`, 0.5`}, {0.6532814824381882`, 0.923879532511287`, 0.6532814824381885`}}\)], "Output"] }, Open ]], Cell["\<\ Now with all the masses assumed to have no displacement and no \ velocity at t=0, we have\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Xh0 = \(-Xp0\)\)], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Xhd0 = \(-Xpd0\)\)], "Input"], Cell[BoxData[ \({\(-0.5546218487394958`\), \(-0.47058823529411775`\), \ \(-0.26890756302521013`\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Cr = Pinv . Xh0\)], "Input"], Cell[BoxData[ \({0.`, 0.`, 0.`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ci = \(-Qinv . Xhd0\)\)], "Input"], Cell[BoxData[ \({0.04275913188839186`, \(-0.14285714285714285`\), 0.9727633537779373`}\)], "Output"] }, Open ]], Cell[TextData[{ "It is amazing what ", StyleBox["Mathematica", FontSlant->"Italic"], " can do with a list. Exp[list] makes a list. list*list makes a list of \ the product of the elements..." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Yh = \((Cr + I\ Ci)\)*Exp[I*ef*t]\)], "Input"], Cell[BoxData[ \({\((\(\(0.`\)\(\[InvisibleSpace]\)\) + 0.04275913188839186`\ \[ImaginaryI])\)\ \ \[ExponentialE]\^\(1.8477590650225737`\ \[ImaginaryI]\ t\), \((\(\(0.`\)\(\ \[InvisibleSpace]\)\) - 0.14285714285714285`\ \[ImaginaryI])\)\ \ \[ExponentialE]\^\(1.4142135623730951`\ \[ImaginaryI]\ t\), \((\(\(0.`\)\(\ \[InvisibleSpace]\)\) + 0.9727633537779373`\ \[ImaginaryI])\)\ \ \[ExponentialE]\^\(0.7653668647301795`\ \[ImaginaryI]\ t\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(Xh = P . Yh;\)\)], "Input"], Cell["Check that we really did satisfy initial conditions:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Re[Xh] /. t \[Rule] 0\)], "Input"], Cell[BoxData[ \({0.`, 0.`, 0.`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Re[D[Xh, t] /. t \[Rule] 0]\)], "Input"], Cell[BoxData[ \({\(-0.5546218487394959`\), \(-0.47058823529411775`\), \ \(-0.2689075630252102`\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Re[D[Xh + Xp, t] /. t \[Rule] 0]\)], "Input"], Cell[BoxData[ \({\(-1.1102230246251565`*^-16\), 0.`, \(-5.551115123125783`*^-17\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[Xp[\([1]\)], {t, 0, 30}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier 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