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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[ 14327, 476]*) (*NotebookOutlinePosition[ 14957, 498]*) (* CellTagsIndexPosition[ 14913, 494]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Undamped, forced vibrations: Beats", "Section"], Cell[TextData[{ StyleBox["DE: ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`mu''\ + \ ku\ = F\_0\ cos\ \[Omega]\ t\)], FontSize->14], StyleBox["\nhas complimentary solution\n\t", FontSize->14], Cell[BoxData[ \(TraditionalForm\`c\_1\ cos\ \(\[Omega]\_0\) t\ \ + \ \ c\_2\ sin\ \(\[Omega]\_0\) t\)], FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell["Particular solution", "Subsection"], Cell[TextData[{ StyleBox["Propose particular solution: ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(A\ cos\ \[Omega]\ t\ + \ B\ sin\ \[Omega]\ t\)\)\)], FontSize->14] }], "Text"], Cell[BoxData[{ \(Clear[w0, F0, m]\), "\[IndentingNewLine]", \(uPart[t_] := a*Cos[w*t]\ + \ b*Sin[w*t]\), "\[IndentingNewLine]", \(Simplify[\(uPart''\)[t] + w0^2*uPart[t]]\)}], "Input"], Cell[BoxData[ \(Solve[{\((w0^2 - 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\[Delta]).\nIf we treat the value of \ \[CurlyPi] (the frequency) as a parameter in the forcing function which may \ be tweaked, we might wonder about how much amplitude we will see in the \ resulting solution at various values of \[Omega]. In particular, we might \ wonder if, by making \[Omega] approach some particular value, whether we \ might be able to force the system into a resonance-like state. So, let us \ define ", FontSize->14], StyleBox["R", FontSize->14, FontSlant->"Italic"], StyleBox[" as a function of \[Omega].", FontSize->14] }], "Text"], Cell[BoxData[ \(R[w_]\ := \ Sqrt[\((\(-\(\(F0\ \((k - m\ w\^2)\)\)\/\(\(-g\^2\)\ w\^2 - \((k - m\ w\^2)\)\ \^2\)\)\))\)^2\ + \ \((\(-\(\(F0\ g\ w\)\/\(\(-g\^2\)\ w\^2 - \((k - m\ \ w\^2)\)\^2\)\)\))\)^2]\)], "Input"], Cell[TextData[{ StyleBox["Without specifying specific values for the constants\n\t", FontSize->14], Cell[BoxData[ \(TraditionalForm\`F\_0, \ m, \ k, \ \[Gamma], \)], FontSize->14], StyleBox["\nit will be difficult to graph ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`R(\[Omega])\)], FontSize->14], StyleBox[". Neverthless, we can employ \"curve-sketching\" techniques from \ calculus. For instance, we may find the limits of ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`R\)], FontSize->14], StyleBox[" as \[Omega] goes to its most extreme values.", FontSize->14] }], "Text"], Cell[BoxData[{ \(Limit[R[w], \ w \[Rule] 0]\), "\[IndentingNewLine]", \(Limit[R[w], w \[Rule] Infinity]\)}], "Input"], Cell[TextData[{ StyleBox["We may also differentiate ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`R\)], FontSize->14], StyleBox[" to find its critical points.", FontSize->14] }], "Text"], Cell[BoxData[ \(Solve[\(R'\)[w]\ \[Equal] \ 0, \ w]\)], "Input"], Cell[TextData[{ StyleBox["Though three critical points are found, it is possible that only \ one of them is real (\[Omega]=0). If the others are non-real, then this \ \[Omega] is the frequency which will yield the largest amplitude. If the \ others are real as well, then one is negative (not a frequency that we are \ interested in), and the other one will yield the greatest ampllitude. ", FontSize->14], StyleBox["Your book has written this \[Omega] = ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[\(\[Omega]\_max\), FontSize->14], TraditionalForm]]], StyleBox["value in an equivalent form, in equation (12) on page 203. They \ point out that, when the damping coefficient \[Gamma] is small, the frequency \ ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[Omega]\_max\)], FontSize->14], StyleBox[" is quite close to what the natural frequency ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[Omega]\_0\)], FontSize->14], StyleBox[" would be if there were no damping at all.\n\nIt is interesting \ to compare the amplitude at ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[\(\[Omega]\_max\), FontSize->14], TraditionalForm]]], StyleBox["to the amplitude ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`F\_0\)], FontSize->14], StyleBox[" of the forcing function. 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See Figure 3.9.3 in your \ text, on page 204.", FontSize->14] }], "Text"], Cell[TextData[StyleBox["You should read the last couple of paragraphs in \ Section 3.9 (on pp. 204-205) to see the effect the choice of \[Omega] has on \ the phase angle \[Delta].", FontSize->14]], "Text"] }, Open ]] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{520, 600}, WindowMargins->{{172, Automatic}, {63, Automatic}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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