(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 299145, 5865] NotebookOptionsPosition[ 292146, 5630] NotebookOutlinePosition[ 292516, 5646] CellTagsIndexPosition[ 292473, 5643] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "The Duffing Equation with ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Title", CellChangeTimes->{{3.4808890228177824`*^9, 3.4808890334047823`*^9}, { 3.4817126807974434`*^9, 3.481712691784443*^9}}], Cell["\<\ Dr. Gustavo A. P\[EAcute]rez M. Honduras National Academy of Sciences gus@pepper.org\ \>", "Subsubtitle", CellChangeTimes->{{3.4808890465237827`*^9, 3.480889074484782*^9}, { 3.481306310510768*^9, 3.481306338977768*^9}, {3.4817109107034435`*^9, 3.4817110110144434`*^9}, {3.4817120401804433`*^9, 3.481712044176443*^9}}], Cell["\<\ Perception is everything \ \>", "Subsection", CellChangeTimes->{{3.4814009273190002`*^9, 3.481400944682*^9}, 3.4814010007980003`*^9}], Cell[CellGroupData[{ Cell[TextData[{ "Abstract :\nWe use ", StyleBox["Mathematica", FontSlant->"Italic"], " to obtain the analytical and numerical solutions of Duffing equation." }], "Subsubtitle", CellChangeTimes->{{3.4817111321914434`*^9, 3.481711197308443*^9}}], Cell["Introduction", "Section", CellChangeTimes->{{3.480889196380782*^9, 3.480889200557782*^9}, 3.4814011056210003`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "We consider as Duffing problem the differential equation:\nu''+u+\[Epsilon] \ ", Cell[BoxData[ FormBox[ SuperscriptBox["u", "3"], TraditionalForm]], "None"], "=0, as initial conditions, we take u(0)=", Cell[BoxData[ FormBox[ SubscriptBox["x", "0"], TraditionalForm]], "None"], " and u'(0)=", Cell[BoxData[ FormBox[ SubscriptBox["v", "0"], TraditionalForm]], "None"], ".\nThe solution u of our problem is a function of the independent variable \ t and the parameter \[Epsilon] that is a dimensionless quantity, and its a \ measure of the strength of the non linearity. Then, an exact solution has a \ frequency \[Omega] that is a function of \[Epsilon], the non linearity.\n", StyleBox["Mathematica", FontSlant->"Italic"], " solves numerically the Duffing problem." }], "Subsubtitle", CellChangeTimes->{{3.4808891644527826`*^9, 3.4808891825947824`*^9}, { 3.4808892125967827`*^9, 3.4808897564077826`*^9}, {3.4808899035027823`*^9, 3.4808901265697823`*^9}, 3.4817110669034433`*^9, 3.4817394152288423`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqdu", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"u", "''"}], "[", "t", "]"}], "+", " ", RowBox[{"u", "[", "t", "]"}], "+", RowBox[{"\[Epsilon]", " ", RowBox[{ RowBox[{"u", "[", "t", "]"}], "^", "3"}]}]}], "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.476474192783375*^9, 3.476474261965375*^9}, 3.481317743839768*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"u", "[", "t", "]"}], "+", RowBox[{"\[Epsilon]", " ", SuperscriptBox[ RowBox[{"u", "[", "t", "]"}], "3"]}], "+", RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.481307205263768*^9, 3.481307825435768*^9, 3.481307922936768*^9, 3.4813081382907677`*^9, 3.481310345140768*^9, 3.481311423297768*^9, 3.4813133003497677`*^9, 3.481314661708768*^9, 3.4813179891777678`*^9, 3.481318637766768*^9, 3.4813188767827682`*^9, 3.4813189348237677`*^9, 3.4813190960197678`*^9, 3.4813993872980003`*^9, 3.4817122221464434`*^9, 3.48191759073837*^9, 3.4819184692633696`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eqdv", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "''"}], "[", "t", "]"}], "+", " ", RowBox[{"v", "[", "t", "]"}], "+", RowBox[{"\[Epsilon]", " ", RowBox[{ RowBox[{"v", "[", "t", "]"}], "^", "3"}]}]}], "\[Equal]", "0"}]}]], "Input", CellChangeTimes->{{3.481317750763768*^9, 3.481317779470768*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"v", "[", "t", "]"}], "+", RowBox[{"\[Epsilon]", " ", SuperscriptBox[ RowBox[{"v", "[", "t", "]"}], "3"]}], "+", RowBox[{ SuperscriptBox["v", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "\[Equal]", "0"}]], "Output", CellChangeTimes->{3.481317783154768*^9, 3.481317989298768*^9, 3.481318637798768*^9, 3.481318876873768*^9, 3.481318934854768*^9, 3.4813190960517683`*^9, 3.481399387322*^9, 3.481712222185443*^9, 3.48191759077137*^9, 3.4819184692823696`*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We write the same equation two times in order to solve one for the amplitude \ eqdu and the other for the velocity eqdv with diferent initial conditions.\ \>", "Subsubtitle", CellChangeTimes->{{3.48148765284276*^9, 3.48148773184776*^9}, { 3.481712491319443*^9, 3.481712528982443*^9}}], Cell["The potential energy and the solutions", "Section", CellChangeTimes->{{3.4808902764357824`*^9, 3.4808902966717825`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The Duffing problem, as defined, is periodic and conservative, we keep the \ parameter \[Epsilon] in the necessary range as to satisfy these conditions. 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When the trajectory is closed and periodic the initial conditions are in some \ way limited by the energy conservation and a problem of initial velocity is \ equivalent after some time to a initial amplitude problem.\ \>", "Subsubtitle", CellChangeTimes->{{3.4808907858187823`*^9, 3.4808908168047824`*^9}, { 3.4808911909087825`*^9, 3.4808913919907827`*^9}, {3.480891429300782*^9, 3.480891644076782*^9}, {3.481401257785*^9, 3.48140126204*^9}}], Cell[TextData[{ "The equation of energy conservation eqce could be written as:\n", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["1", "2"], SuperscriptBox["v", "2"]}], TraditionalForm]], "None"], "=h-", Cell[BoxData[ RowBox[{ RowBox[{ FractionBox["1", "2"], SuperscriptBox["u", "2"]}], "-", " ", RowBox[{ FractionBox["1", "4"], " ", "\[Epsilon]", " ", SuperscriptBox["u", "4"]}]}]], CellChangeTimes->{{3.4764849246359997`*^9, 3.4764850467860003`*^9}}], "=h-F(u,\[Epsilon])" }], "Subsubtitle", CellChangeTimes->{{3.480964368525958*^9, 3.4809644964809585`*^9}, 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StyleBox["Mathematica", FontSlant->"Italic"], " could be used to bring the students of the level of the Nayfeh book to use \ the Jacobi Elliptic functions without effort." }], "Subsubtitle", CellChangeTimes->{{3.48148814899576*^9, 3.48148844245576*^9}, { 3.4817108233914433`*^9, 3.481710827520443*^9}, {3.481710861732443*^9, 3.4817108711984434`*^9}, 3.4817121392534432`*^9, 3.4817400301507225`*^9, { 3.48191723823837*^9, 3.4819173356833696`*^9}, {3.4819173744983697`*^9, 3.48191746621937*^9}, {3.4819178272963696`*^9, 3.48191783772437*^9}}], Cell["References", "Section", CellChangeTimes->{{3.4814002237200003`*^9, 3.481400228193*^9}}] }, Open ]], Cell["\<\ Nayfeh, Ali Hasan. 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