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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[ 18236, 536]*) (*NotebookOutlinePosition[ 18874, 559]*) (* CellTagsIndexPosition[ 18830, 555]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["3.9 Forced Mechanical Vibrations", "Subtitle"], Cell[BoxData[{ RowBox[{\(Sometimes\ it\ is\ useful\ to\ model\ a\ damped\ mechanical\), " "}], RowBox[{ RowBox[{"vibration", " ", "with", " ", "an", " ", StyleBox["inhomogeneous", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "differential"}], " "}], RowBox[{"equation", "\n"}], RowBox[{" ", StyleBox[\(\(mu'\)'\ + \ \[Gamma]u'\ + \ ku\ = \ g\), FontWeight->"Bold"], "\n"}], RowBox[{\(where\ the\ function\ g\ accounts\ for\ some\ extra\ force\), " "}], \(applied\ to\ the\ vibrating\ object . \ \ A\ good\ example\ is \), RowBox[{\(that\ of\ a\ car\ \((the\ object)\) moving\ on\ top\ of\ its\), " "}], RowBox[{ \(shock\ absorbers\ \((the\ damped\ spring)\)\ on\ a\ bumpy\ road\), " "}], \(\(( the\ bumps\ compress\ the\ shock\ absorbers\ an\ extra\namount\ and\ therefore\ exert\ an\ extra\ force\ on\ the\ car)\) . \)}], "Input",\ FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(\nHere\ we\ consider\ two\ \(examples : \ one\ with\ no\ damping\)\ \), \(\((\[Gamma]\ = \ 0)\) and\ another\ with \((under)\) damping\ \((\[Gamma]\ = \ 1)\) . \ \ \ \nThe\ first\ example\ is\ designed\ to\ illustrate\ \), \(phenomena\ such\ as\ beat\ and\ resonance . The\ example\ \), \(revisits\ resonance\ but\ also\ \ highlights\ the\ role\ of\ \), \(the\ particular\ solution\ \((versus\ the\ solution\ of\ the\nhomogeneous\ equation)\)\ in\ the\ general\ solution\ of\ an\ \), \(inhomogeneous\ linear\ \(equation . \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(In\ both\ examples, \ we\ will\ assume\ that\ m = 1\ and\ that\), \(the\ system\ is\ subject\ to\ a\ periodic\ \((i . e . \ of\ the\ form\n`cos[\[Omega]t]\ ')\)\ external\ force . \ \ We\ will\ also\ set\ the\ \), \(initial\ conditions\ u[0]\ = \ 3, \ \(u'\)[0]\ = \ 0. \)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ StyleBox[\(1. \ No\ \(damping . \)\), "Subsubtitle"]], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ RowBox[{ RowBox[{"We", " ", "will", " ", "work", " ", "with", " ", "the", " ", RowBox[{"equation", ":", "\n", "\t\t", "\n", StyleBox[ RowBox[{" ", StyleBox[" ", FontWeight->"Bold"]}]], StyleBox[\(\(u'\)'\ + \ 25\ u\ == \ a\ Cos[\[Omega]t]\), FontWeight->"Bold"]}]}], StyleBox[ RowBox[{ StyleBox["\t", FontWeight->"Bold"], "\t"}]], "\n"}], RowBox[{ \(where\ a\ and\ \[Omega]\ will\ take\ on\ values\ specified\ below . \ \ \nLet\ us\ first\ compute\ and\ plot\ the\ solution\ when \), " "}], \(the\ right\ side\ is\ identically\ \(zero . \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ \(Clear[u]\)], "Input"], Cell[BoxData[ \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ 0, \tu[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)], "Input"], Cell[BoxData[ \(u1\ = \ u[t] /. First[soln]\)], "Input"], Cell[BoxData[ \(Plot[u1, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ \(Now\ we\ introduce\ a\ \((relatively)\)\ slowly\ varying\ \), \(external\ force, again\ solving\ the\ differential\ \), \(equation\ and\ plotting\ the\ \(solution . \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ Cos[t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u2\ = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u2, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ \(\( ... not\ much\ difference . \ \ Let' s\ try\ increasing\)\ \), \(the\ magnitude\ of\ the\ \(force : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ 5\ Cos[t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u3\ = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u3, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ \(\( ... still\ not\ all\ that\ different\) - just\ a\ bit\ of\ \), \(wavering\ in\ the\ original\ solution . \ \ Suppose\ we\ \), \(make\ the\ external\ force\ vary\ \((relatively)\)\), \(rapidly\ rather\ than\ slowly . \ \ We' ll\ increase\), \(the\ magnitude\ again\ for\ the\ sake\ of\ \(drama : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ 500 Cos[25 t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u4 = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u4, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ \(This\ introduces\ a\ bit\ of\ a\ wiggle\ into\ the\ solution\ \), \(\((as\ you\ mightexpect)\), \ but\ still\ we' re\ not\ getting\ \), \(our\ effort' s\ worth\ out\ of\ this\ external\ force . \ \ The\ \), \(real\ trick\ is\ to\ make\ the\ period\ of\ the\ force\ nearly\ \), \(equal\ to\ the\ period\ of\ the\ unforced\ \(solution . \)\)}], "Input",\ FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ 5\ Cos[4.5 t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u5 = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u5, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ \(That' s\ a\ bit\ more\ impressive . \ \ Let' s\ try\ making\ \), \(the\ periods\ even\ close\ and\ plot\ the\ solution\ for\ a\ \), \(longer\ \(time : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ 5\ Cos[4.9 t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u6 = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u6, \ {t, 0, \ 100}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[{ RowBox[{\(This\ is\ really\ suggestive . \ \ Our\ solution\ has\), " "}], RowBox[{\(developed\ a\ `beat' that\ varies\ much\ more\ slowly\), " "}], \(than\ either\ the\ unforced\ oscillation\ or\ the\ external\), \(force . \ \ \ The\ mathematical\ reason\ for\ the\ beat\ is\ the\), RowBox[{"formula", "\n"}], RowBox[{ StyleBox[ RowBox[{" ", StyleBox[" ", FontWeight->"Bold"]}]], RowBox[{ StyleBox[\(2\ sin \(\(B - A\)\/2\) sin \(B + A\)\/2\), FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{ StyleBox[\(cos[A]\), FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox["+", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{ RowBox[{ StyleBox[\(cos[B]\), FontWeight->"Bold"], ".", "\n", "\n", "\t\t\t", "This"}], " ", "formula", " ", "applies", " ", "directly", " ", "to", " ", "the", " ", "solution", " ", "u6"}]}]}], " "}], \(in\ our\ last\ example\ *after*\ we\ eliminate\ the\), RowBox[{ RowBox[{ \(effects\ of\ the\ unforced\ solution . \ \ That\ is\), ",", "\n", " ", "\n", StyleBox[ RowBox[{" ", StyleBox[" ", FontWeight->"Bold"]}]], RowBox[{ StyleBox["u6", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{ StyleBox[\(u1\ - \ 4.101010 ... \), FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], StyleBox[\(sin[ .05 t]\), FontWeight->"Bold"], RowBox[{ StyleBox[\(sin[4.95\ t]\), FontWeight->"Bold"], ".", "\n", "\n", "Note"}], " ", "that", " ", "the", " ", "second", " ", "term", " ", "contains", " ", "a", " ", "slowly", " ", "varying"}]}]}], " "}], RowBox[{ \(factor\ \((the\ beat)\) and\ a\ factor\ which\ varies\ at\ about\), " "}], RowBox[{ \(the\ same\ rate\ as\ the\ external\ force . \ \ Note\ also\ that\), " "}], RowBox[{\(the\ overall\ amplitude\ of\ the\ solution\ has\ grown\), " "}], RowBox[{ \(from\ about\ 3\ to\ over\ 6. \ \ Before\ moving\ on\ to\ damped\), " "}], RowBox[{\(examples, we\ consider\ the\ case\ where\ the\ external\), " "}], RowBox[{\(force\ has\ a\ period\ exactly\ matching\ that\ of\ the\), " "}], \(unforced\ \(solution : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ 25\ u[t]\ == \ \ \ 5 Cos[5 t], \t u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \)}], "Input"], Cell[BoxData[ \(u7 = \ Expand[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u7, \ {t, 0, \ 100}, \ PlotPoints\ -> \ 1000]\)], "Input"], Cell[BoxData[ \(\(This\ is\ called\ the\ resonant\ case . \ \ \(Why?\)\n\t\t\t\n\n\n\n\n \)\)], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"2", StyleBox[".", "Subsubtitle"]}]], StyleBox[" ", "Subsubtitle"], StyleBox["An", "Subsubtitle"], StyleBox[" ", "Subsubtitle"], StyleBox["Underdamped", "Subsubtitle"], StyleBox[" ", "Subsubtitle"], StyleBox["System", "Subsubtitle"], StyleBox[" ", "Subsubtitle"], StyleBox["with", "Subsubtitle"], StyleBox[" ", "Subsubtitle"], StyleBox[\(forcing . \), "Subsubtitle"]}]], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{\(Now\ we\ will\ modify\ the\ first\ equation, \ adding\ a\), RowBox[{ RowBox[{"damping", " ", RowBox[{"term", ":", "\n", "\t\t", "\n", StyleBox[ RowBox[{" ", StyleBox[" ", FontWeight->"Bold"]}]], StyleBox[\(\(u'\)'\ + \ 5 u'\ + 25\ u\ == \ a\ Cos[\[Omega]t]\), FontWeight->"Bold"]}]}], StyleBox[ RowBox[{ StyleBox["\t", FontWeight->"Bold"], "\t"}]], "\n"}], RowBox[{\(where, \ as\ before, \ we\ will\ experiment\ with\), " "}], RowBox[{ \(different\ values\ of\ a\ and\ \[Omega] . \ \ \ First\ we\ compute\ and\), " "}], RowBox[{\(plot\ the\ solution\ when\ the\ right\ side\ is\ identically\), " "}], \(zero . \)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ \(1\/2\) \(u'\)[t] + 25\ u[t]\ == \ 0, \tu[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \n \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[t], \ t] \n\)}], "Input"], Cell[BoxData[ \(u1\ = \ u[t] /. First[soln]\)], "Input"], Cell[BoxData[ \(Plot[u1, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000, \ \ PlotRange\ -> \ {\(-3\), 3}]\)], "Input"], Cell[BoxData[{ \(\nNow\ we\ apply\ a\ periodic\ external\ force\ that\ is\ pretty\), \(much\ out\ of\ sync\ with\ the\ period\ of\ the\ unforced\ \), \(\(system : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ \(1\/2\) \(u'\)[t] + 25\ u[t]\ == \ 2 Cos[3 t], \t\n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[0]\ == \ 3, \ \(u'\)[0]\ == \ 0}, \ u[t], \ t]\n\)}], "Input"], Cell[BoxData[ \(u2\ = \ Simplify[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u2, \ {t, 0, \ 20}, \ PlotPoints\ -> \ 1000, \ \ PlotRange\ -> \ {\(-3\), 3}]\)], "Input"], Cell[BoxData[ \(It\ might\ help\ to\ increase\ the\ domain\ a\ bit\ \(here : \)\)], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ \(Plot[u2, \ {t, 0, \ 40}, \ PlotPoints\ -> \ 1000, \ \ PlotRange\ -> \ {\(-3\), 3}]\)], "Input"], Cell[BoxData[{ RowBox[{ StyleBox[\(So, \ as\ you\ might\ expect, \ the\ solution\ doesn' t\ die\), FontSize->14, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}]}], \(off\ completely . \ \ After\ the\ initial\ vibration\ settles\), RowBox[{\(down, there\ remains\ vibration\ caused\ by\ the\), " "}], \(forcing . \ \ Perhaps\ more\ surprisingly, \ this\ `long\), \(term'\ vibration\ is\ completely\ independent\ of\ the\), \(initial\ condition . \ \ That\ is, if\ we\ try\ a\ different\), RowBox[{\(initial\ \(condition : \)\), " "}]}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ \(1\/2\) \(u'\)[t] + 25\ u[t]\ == \ 2 Cos[3 t], \t\n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[0]\ == \ \(-2\), \ \(u'\)[0]\ == \ 1}, \ u[t], \ t]\n\)}], "Input"], Cell[BoxData[ \(u3\ = \ Simplify[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[{ \(and\ compare\ with\ the\ solution\ corresponding\ to\), \(the\ first\ set\ of\ initial\ \(conditions : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ \(Simplify[u3\ - \ u2]\)], "Input"], Cell[BoxData[{ \(we\ see\ that\ the\ difference\ goes\ to\ zero\ as\ t\ \), \(increases . \ \ Mathematically\ speaking, \ this\ merely\), \(serves\ to\ illustrate\ the\ principle\ that\ any\ two\), \(solutions\ of\ a\ linear, \ inhomogeneous\ equation\ \), \(differ\ by\ a\ solution\ of\ the\ corresponding\), \(homogeneous\ \(equation . \)\n\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(A\ final\ point\ worth\ making\ is\ that\ damped\ systems\ have\), \(resonances\ just\ like\ undamped\ systems . \ \ However, \n the\ eventual\ solution\ of\ a\ resonant\ damped\ system\), \(will\ not\ grow\ \(forever : \)\)}], "Input", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(Clear[u]\), \(soln\ = \ DSolve[{\ \(\(u'\)'\)[t]\ + \ \(1\/2\) \(u'\)[t] + 25\ u[t]\ == \ 2 Cos[\(\(\@399\) t\)\/4], \t\n \t\t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u[0]\ == \ \(-2\), \ \(u'\)[0]\ == \ 1}, \ u[t], \ t]\n\)}], "Input"], Cell[BoxData[ \(u4\ = \ Simplify[u[t] /. First[soln]]\)], "Input"], Cell[BoxData[ \(Plot[u4, \ {t, 0, \ 40}, \ PlotPoints\ -> \ 1000, \ \ PlotRange\ -> \ {\(-3\), 3}]\)], "Input"] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{520, 600}, WindowMargins->{{146, Automatic}, {Automatic, 153}} ] (*********************************************************************** 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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