(************** Content-type: application/mathematica ************** 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). 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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[ 350312, 10642]*) (*NotebookOutlinePosition[ 350973, 10665]*) (* CellTagsIndexPosition[ 350929, 10661]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Calcolo di sollecitazioni e spostamenti in un sistema di travi rettilinee\ \>", "Title"], Cell["\<\ Anche se non sembra semplice assegnare i dati conviene leggere le istruzioni \ ed evitare adattamenti con conseguenze imprevedibili\ \>", "Subtitle", CellFrame->True, Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Left, FontSize->12, Background->GrayLevel[0.849989]], Cell[TextData[StyleBox["v. 2.02 (10/4/2003) \n\[Copyright] Amabile Tatone, \ Universit\[AGrave] dell'Aquila, L'Aquila, IT \ntatone@ing.univaq.it", FontSize->14, FontWeight->"Bold"]], "Subtitle", CellFrame->True, Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Left, FontSize->12, Background->GrayLevel[0.849989]], Cell[CellGroupData[{ Cell["Istruzioni", "Section", Evaluatable->False], Cell[TextData[{ "Sono da assegnare:\n- i vettori a1 e a2 della base adattata alla sezione \ [", StyleBox["D1", FontColor->RGBColor[0, 0, 1]], "]\n- la distribuzione di forza [", StyleBox["D2", FontColor->RGBColor[0, 0, 1]], "]\n- i vincoli e le basi adattate al bordo [", StyleBox["D3", FontColor->RGBColor[0, 0, 1]], "]\n- le forze e i momenti alle estremit\[AGrave] [", StyleBox["D4", FontColor->RGBColor[0, 0, 1]], "]\n- costanti (lunghezze, moduli, intensit\[AGrave] delle forze) [", StyleBox["D5", FontColor->RGBColor[0, 0, 1]], "]\n\nSono da adattare:\n- la funzione di semplificazione extraSimplify [", StyleBox["\[FilledCircle]", FontColor->RGBColor[0, 0, 1]], "]\n- la cornice per la visualizzazione della deformazione [", StyleBox["\[FilledCircle]", FontColor->RGBColor[0, 0, 1]], "]\n- i fattori di scala per i diagrammi tecnici N, Q, M [", StyleBox["\[FilledCircle]", FontColor->RGBColor[0, 0, 1]], "]\n\nSono da controllare:\n- alcune definizioni riguardanti \ semplificazioni" }], "SmallText", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Viene prima calcolata la soluzione bulk delle equazioni di bilancio in \ corrispondenza di una qualsiasi distribuzione di forze (integrabile). Vengono assegnati i vincoli. Esiste il problema di compatibilita' dei vincoli \ solo in forma banale. Non esiste certamente per gli atti di moto, essendo per \ questi i vincoli delle condizioni omogenee. Vengono poi costruite le equazioni di bilancio al bordo corrispondenti agli \ atti di moto vincolati, fornendo l'elenco delle forze attive da assegnare. Sostituendo in queste equazioni la soluzione bulk si generano delle equazioni \ algebriche nelle costanti di integrazione. Viene calcolata la soluzione che, nel caso di \"vincoli eccedenti\", lascia \ indeterminate alcune delle costanti. Si puo' dire che si determina lo spazio delle soluzioni in termini di \ tensione bilanciata al bordo. In caso di \"vincoli in difetto\" occorre verificare la compatibilit\[AGrave] \ dei dati al bordo sulle forze. Si prosegue calcolando, attraverso la funzione di risposta, lo spazio degli \ spostamenti corrispondente alla tensione, introducendo altre costanti di \ integrazione. Dalle equazioni di vincolo si generano le equazioni algebriche da cui si \ calcolano infine tutte le costanti. Vincoli \"eccedenti\" => equazioni di bilancio al bordo \"in difetto\" Vincoli \"in difetto\" => equazioni di bilancio al bordo \"eccedenti\" \ (occorre verificare la compatibilita' delle forze al bordo)\ \>", "SmallText", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Le lunghezze dei vari tratti possono essere assegnate utilizzando una \ lunghezza base (ad esempio ", StyleBox["\[ScriptCapitalL]", FontFamily->"Courier"], " ), in modo che non compaiano in tutte le espressioni ", StyleBox["L[1], L[2]", FontFamily->"Courier"], " ecc.; cos\[IGrave] pure gli angoli. Occorre poi assegnare i valori di \ tali parametri in datiO per poter realizzare le figure." }], "SmallText", CellFrame->True, Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell["Inizializzazione", "Section", Evaluatable->False], Cell[BoxData[ \(\(outputDir = "\";\)\ \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(SetDirectory[outputDir]\)], "Input"], Cell[BoxData[ \("C:\\Wrk\\Corsi\\Scost\\esercizi\\7-travi\\7-18\\outmath"\)], "Output"] }, Open ]], Cell["\<\ In fase di modifica del notebook riattivare gli \"spelling warning\"\ \>", "SmallText"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\[IndentingNewLine]", \(\(Off[General::"\"];\)\)}], "Input"], Cell[BoxData[{ \(\(Off[Solve::"\"];\)\), "\n", \(\(<< \ LinearAlgebra`MatrixManipulation`;\)\), "\[IndentingNewLine]", \(\(<< Graphics`Colors`;\)\), "\n", \(\(SetOptions[Plot, ImageSize \[Rule] 228];\)\), "\n", \(\(SetOptions[ParametricPlot, ImageSize \[Rule] {200, 200}];\)\), "\[IndentingNewLine]", \(\(SetOptions[Plot, PlotRange \[Rule] All];\)\), "\[IndentingNewLine]", \(\(SetOptions[ParametricPlot, PlotRange \[Rule] All];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Descrizione della configurazione originaria [", StyleBox["D1", FontColor->RGBColor[0, 0, 1]], "]" }], "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Definizione delle basi", "Subsection", CellFrame->False, Background->None], Cell["Base del sistema di coordinate (non modificare)", "SmallText", CellFrame->False, Background->None], Cell[BoxData[{ \(\(e\_1 = {1, 0};\)\), "\n", \(\(e\_2 = {0, 1};\)\)}], "Input", CellFrame->False, Background->None], Cell["\<\ Basi adattate alla sezione di ciascun tratto (non modificare)\ \>", "SmallText", CellFrame->False, Background->None], Cell[BoxData[{ \(\(a\_1[i_] := Cos[\[Alpha][i]]\ e\_1 + Sin[\[Alpha][i]]\ e\_2;\)\), "\n", \(\(a\_2[i_] := \(-Sin[\[Alpha][i]]\)\ e\_1 + Cos[\[Alpha][i]]\ e\_2;\)\)}], "Input", CellFrame->False, Background->None] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Dati [", StyleBox["D1", FontColor->RGBColor[0, 0, 1]], "]" }], "Subsection"], Cell["Numero di tratti di trave", "SmallText"], Cell[BoxData[ \(\(travi = 2;\)\)], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Angoli che definiscono le basi adattate (possono anche non essere \ assegnati; in tal caso se ne assegni il valore nella lista ", StyleBox["datiO", FontFamily->"Courier New", FontWeight->"Bold"], ")\n", "[ l'uso di caratteri script per i parametri rende tutto molto pi\[UGrave] \ leggibile]" }], "SmallText", FontFamily->"Arial"], Cell[BoxData[{ \(\(\[Alpha][1] = 0;\)\), "\n", \(\(\[Alpha][2] = \[Pi]\/2;\)\)}], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Lunghezze (possono anche non essere assegnate; in tal caso se ne assegni \ il valore nella lista successiva ", StyleBox["datiO", FontFamily->"Courier New", FontWeight->"Bold"], ")\n", "[ l'uso caratteri script per i parametri rende tutto molto pi\[UGrave] \ leggibile]" }], "SmallText"], Cell[BoxData[{ \(\(L[1] = \[ScriptCapitalL];\)\), "\[IndentingNewLine]", \(\(L[2] = \[ScriptCapitalL];\)\)}], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[{ \(YA[1] := \[ScriptCapitalY]\[ScriptCapitalA]\ \ \), \ "\[IndentingNewLine]", \(YA[2] := \[ScriptCapitalY]\[ScriptCapitalA]\), "\[IndentingNewLine]", \(YJ[1] := \[ScriptCapitalY]\[ScriptCapitalJ]\), "\[IndentingNewLine]", \(YJ[2] := \[ScriptCapitalY]\[ScriptCapitalJ]\)}], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Valori numerici (di angoli e lunghezze) necessari alla visualizzazione e \ utilizzati solo per questo\ \>", "SmallText"], Cell[BoxData[ \(\(datiO = {\[ScriptCapitalL] \[Rule] 1};\)\)], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Altri dati EVENTUALMENTE assegnati (anche per ottenere espressioni \ pi\[UGrave] semplici). \ \>", "SmallText"], Cell[BoxData[ \(\[ScriptCapitalY]\[ScriptCapitalA] := \ \[ScriptCapitalY]\[ScriptCapitalJ]\/\(\[Kappa]\ \[ScriptCapitalL]\^2\)\)], \ "Input", CellFrame->True, Background->GrayLevel[0.849989]] }, Closed]], Cell[CellGroupData[{ Cell["Definizioni per la visualizzazione", "Subsection"], Cell["lunghezza caratteristica", "SmallText"], Cell[BoxData[ \(\(maxL = Max[Table[ L[i] /. \[InvisibleSpace]datiO, {i, 1, travi}]];\)\)], "Input"], Cell["definizione dell'asse", "SmallText"], Cell[BoxData[ \(\(\(\(asseO[i_]\)[\[Zeta]_] := org[i] + a\_1[i]\ \[Zeta] /. datiO;\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(Clear[org]\)], "Input"], Cell["\<\ Coordinate dell'estremit\[AGrave] sinistra di ciascun tratto (utilizzate solo \ per la visualizzazione dei tratti separati). 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Cell[BoxData[ RowBox[{\(eqbilt[i_]\), ":=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[\(s[i]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "+", \(\(b[i]\)[\[Zeta]]\)}], ")"}], ".", \(a\_1[i]\)}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox[\(s[i]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "+", \(\(b[i]\)[\[Zeta]]\)}], ")"}], ".", \(a\_2[i]\)}], "==", "0"}], ",", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(sM[i]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "+", \(\(sQ[i]\)[\[Zeta]]\), "+", \(\(c[i]\)[\[Zeta]]\)}], "==", "0"}]}], "}"}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(svar = Flatten[Table[{sN[i], sQ[i], sM[i]}, {i, 1, travi}]]\)], "Input"], Cell[BoxData[ \({sN[1], sQ[1], sM[1], sN[2], sQ[2], sM[2]}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(eqbil = Flatten[Simplify[Table[eqbilt[i], {i, 1, travi}]]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(bulksolC = \(DSolve[eqbil, svar, \[Zeta], DSolveConstants \[Rule] \[ScriptCapitalC]]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket]\)], "Input"], Cell[BoxData[ \({sN[1] \[Rule] Function[{\[Zeta]}, \[ScriptCapitalC][1]], sQ[1] \[Rule] Function[{\[Zeta]}, \[ScriptCapitalC][2]], sM[1] \[Rule] Function[{\[Zeta]}, \(-\[ScriptC]\)\ \((\(-1\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\) - \[Zeta]\ \[ScriptCapitalC][ 2] + \[ScriptCapitalC][3]], sN[2] \[Rule] Function[{\[Zeta]}, \[ScriptCapitalC][4]], sQ[2] \[Rule] Function[{\[Zeta]}, \[ScriptCapitalC][5]], sM[2] \[Rule] Function[{\[Zeta]}, \[ScriptC]\ \((\(-1\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\) - \[Zeta]\ \[ScriptCapitalC][ 5] + \[ScriptCapitalC][6]]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Cambiamento delle costanti di integrazione", "Subsection"], Cell["\<\ Viene costruita la lista cNQMO delle costanti di integrazione delle equazioni \ di bilancio. La lista cNQM delle costanti di integrazione presenti nelle condizioni al \ bordo, costruita pi\[UGrave] avanti, \[EGrave] in generale contenuta in \ questa.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(cClist = Table[\[ScriptCapitalC][i], {i, 1, 3 travi}]\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalC][1], \[ScriptCapitalC][2], \[ScriptCapitalC][ 3], \[ScriptCapitalC][4], \[ScriptCapitalC][5], \[ScriptCapitalC][ 6]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cNQM = Table[{sNo[i], sQo[i], sMo[i]}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({sNo[1], sQo[1], sMo[1], sNo[2], sQo[2], sMo[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[{\(sN[i]\)[0] == sNo[i], \(sQ[i]\)[0] == sQo[i], \(sM[i]\)[0] == sMo[i]} /. bulksolC, {i, 1, travi}] // Simplify\) // Flatten\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalC][1] == sNo[1], \[ScriptCapitalC][2] == sQo[1], \[ScriptC] + \[ScriptCapitalC][3] == sMo[1], \[ScriptCapitalC][4] == sNo[2], \[ScriptCapitalC][5] == sQo[2], \[ScriptCapitalC][6] == \[ScriptC] + sMo[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fromCtoNQM = \(Solve[\(Table[{\(sN[i]\)[0] == sNo[i], \(sQ[i]\)[0] == \ sQo[i], \(sM[i]\)[0] == sMo[i]} /. bulksolC, {i, 1, travi}] // Simplify\) // \ Flatten, cClist]\)\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\)\)\ \)], "Input"], Cell[BoxData[ \({\[ScriptCapitalC][1] \[Rule] sNo[1], \[ScriptCapitalC][2] \[Rule] sQo[1], \[ScriptCapitalC][3] \[Rule] \(-\[ScriptC]\) + sMo[1], \[ScriptCapitalC][4] \[Rule] sNo[2], \[ScriptCapitalC][5] \[Rule] sQo[2], \[ScriptCapitalC][6] \[Rule] \[ScriptC] + sMo[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bulksol = bulksolC /. fromCtoNQM\)], "Input"], Cell[BoxData[ \({sN[1] \[Rule] Function[{\[Zeta]}, sNo[1]], sQ[1] \[Rule] Function[{\[Zeta]}, sQo[1]], sM[1] \[Rule] Function[{\[Zeta]}, \(-\[ScriptC]\)\ \((\(-1\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\) - \[Zeta]\ sQo[1] + \((\(-\[ScriptC]\) + sMo[1])\)], sN[2] \[Rule] Function[{\[Zeta]}, sNo[2]], sQ[2] \[Rule] Function[{\[Zeta]}, sQo[2]], sM[2] \[Rule] Function[{\[Zeta]}, \[ScriptC]\ \((\(-1\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\) - \[Zeta]\ sQo[2] + \((\[ScriptC] + sMo[2])\)]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Equazioni di bilancio e integrali (sintesi)", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Table[eqbilt[i], {i, 1, travi}] // Simplify\) // Flatten\) // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { RowBox[{ RowBox[{ SuperscriptBox[\(sN[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "==", "0"}]}, { RowBox[{ RowBox[{ SuperscriptBox[\(sQ[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "==", "0"}]}, { RowBox[{ RowBox[{\(2\ \[ScriptC]\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\), "+", \(\(sQ[1]\)[\[Zeta]]\), "+", RowBox[{ SuperscriptBox[\(sM[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], "==", "0"}]}, { RowBox[{ RowBox[{ SuperscriptBox[\(sN[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "==", "0"}]}, { RowBox[{ RowBox[{ SuperscriptBox[\(sQ[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}], "==", "0"}]}, { RowBox[{ RowBox[{\(\(-2\)\ \[ScriptC]\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\), "+", \(\(sQ[2]\)[\[Zeta]]\), "+", RowBox[{ SuperscriptBox[\(sM[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], "==", "0"}]} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ Derivative[ 1][ sN[ 1]][ \[Zeta]], 0], Equal[ Derivative[ 1][ sQ[ 1]][ \[Zeta]], 0], Equal[ Plus[ Times[ 2, \[ScriptC], DiracDelta[ Plus[ \[ScriptCapitalL], Times[ -2, \[Zeta]]]]], sQ[ 1][ \[Zeta]], Derivative[ 1][ sM[ 1]][ \[Zeta]]], 0], Equal[ Derivative[ 1][ sN[ 2]][ \[Zeta]], 0], Equal[ Derivative[ 1][ sQ[ 2]][ \[Zeta]], 0], Equal[ Plus[ Times[ -2, \[ScriptC], DiracDelta[ Plus[ \[ScriptCapitalL], Times[ -2, \[Zeta]]]]], sQ[ 2][ \[Zeta]], Derivative[ 1][ sM[ 2]][ \[Zeta]]], 0]}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Table[\(svar\[LeftDoubleBracket] i\[RightDoubleBracket]\)[\[Zeta]] == \((\(svar\ \[LeftDoubleBracket]i\[RightDoubleBracket]\)[\[Zeta]] /. bulksolC)\), {i, 1, Length[svar]}] // Simplify\) // Flatten\) // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(\(sN[1]\)[\[Zeta]] == \[ScriptCapitalC][1]\)}, {\(\(sQ[1]\)[\[Zeta]] == \[ScriptCapitalC][2]\)}, {\(\[ScriptC]\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + \[Zeta]\ \[ScriptCapitalC][2] + \(sM[ 1]\)[\[Zeta]] == \[ScriptC] + \[ScriptCapitalC][3]\)}, {\(\(sN[2]\)[\[Zeta]] == \[ScriptCapitalC][4]\)}, {\(\(sQ[2]\)[\[Zeta]] == \[ScriptCapitalC][5]\)}, {\(\[Zeta]\ \[ScriptCapitalC][5] + \(sM[ 2]\)[\[Zeta]] == \[ScriptC]\ \((\(-1\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\) + \[ScriptCapitalC][6]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], \[ScriptCapitalC][ 1]], Equal[ sQ[ 1][ \[Zeta]], \[ScriptCapitalC][ 2]], Equal[ Plus[ Times[ \[ScriptC], UnitStep[ Plus[ Times[ -1, \[ScriptCapitalL]], Times[ 2, \[Zeta]]]]], Times[ \[Zeta], \[ScriptCapitalC][ 2]], sM[ 1][ \[Zeta]]], Plus[ \[ScriptC], \[ScriptCapitalC][ 3]]], Equal[ sN[ 2][ \[Zeta]], \[ScriptCapitalC][ 4]], Equal[ sQ[ 2][ \[Zeta]], \[ScriptCapitalC][ 5]], Equal[ Plus[ Times[ \[Zeta], \[ScriptCapitalC][ 5]], sM[ 2][ \[Zeta]]], Plus[ Times[ \[ScriptC], Plus[ -1, UnitStep[ Plus[ Times[ -1, \[ScriptCapitalL]], Times[ 2, \[Zeta]]]]]], \[ScriptCapitalC][ 6]]]}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Table[\(svar\[LeftDoubleBracket] i\[RightDoubleBracket]\)[\[Zeta]] == \((\(svar\ \[LeftDoubleBracket]i\[RightDoubleBracket]\)[\[Zeta]] /. bulksol)\), {i, 1, Length[svar]}] // Simplify\) // Flatten\) // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(\(sN[1]\)[\[Zeta]] == sNo[1]\)}, {\(\(sQ[1]\)[\[Zeta]] == sQo[1]\)}, {\(\[Zeta]\ sQo[ 1] + \[ScriptC]\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + \(sM[1]\)[\[Zeta]] == sMo[1]\)}, {\(\(sN[2]\)[\[Zeta]] == sNo[2]\)}, {\(\(sQ[2]\)[\[Zeta]] == sQo[2]\)}, {\(\[Zeta]\ sQo[2] + \(sM[2]\)[\[Zeta]] == sMo[2] + \[ScriptC]\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], sNo[ 1]], Equal[ sQ[ 1][ \[Zeta]], sQo[ 1]], Equal[ Plus[ Times[ \[Zeta], sQo[ 1]], Times[ \[ScriptC], UnitStep[ Plus[ Times[ -1, \[ScriptCapitalL]], Times[ 2, \[Zeta]]]]], sM[ 1][ \[Zeta]]], sMo[ 1]], Equal[ sN[ 2][ \[Zeta]], sNo[ 2]], Equal[ sQ[ 2][ \[Zeta]], sQo[ 2]], Equal[ Plus[ Times[ \[Zeta], sQo[ 2]], sM[ 2][ \[Zeta]]], Plus[ sMo[ 2], Times[ \[ScriptC], UnitStep[ Plus[ Times[ -1, \[ScriptCapitalL]], Times[ 2, \[Zeta]]]]]]]}], Editable->False]], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Definizioni di spostamenti e forze al bordo", "Section"], Cell[BoxData[ \(meno = "\<-\>"; pi\[UGrave] = "\<+\>";\)], "Input"], Cell["\<\ Spostamento, atti di moto e forze al bordo come combinazioni lineari dei \ vettori delle basi adattate al bordo {d,n}\ \>", "SmallText"], Cell[BoxData[{ \(\(\(ub[i_]\)[ bd_] := \(ub\_d[i]\)[bd]\ \(d[i]\)[bd] + \(ub\_n[i]\)[bd]\ \(n[i]\)[ bd];\)\), "\n", \(\(\(wb[i_]\)[ bd_] := \(wb\_d[i]\)[bd]\ \(d[i]\)[bd] + \(wb\_n[i]\)[bd]\ \(n[i]\)[ bd];\)\), "\n", \(\(\(sb[i_]\)[ bd_] := \(sb\_d[i]\)[bd]\ \(d[i]\)[bd] + \(sb\_n[i]\)[bd]\ \(n[i]\)[ bd];\)\)}], "Input"], Cell["Lista delle componenti dello spostamento al bordo", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(spbd = Table[\({\(ub\_d[i]\)[#], \(ub\_n[i]\)[#], \(\[Theta]b[ i]\)[#]} &\)\ /@ \ {pi\[UGrave], meno}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({\(ub\_d[1]\)["+"], \(ub\_n[1]\)["+"], \(\[Theta]b[1]\)[ "+"], \(ub\_d[1]\)["-"], \(ub\_n[1]\)["-"], \(\[Theta]b[1]\)[ "-"], \(ub\_d[2]\)["+"], \(ub\_n[2]\)["+"], \(\[Theta]b[2]\)[ "+"], \(ub\_d[2]\)["-"], \(ub\_n[2]\)["-"], \(\[Theta]b[2]\)[ "-"]}\)], "Output"] }, Open ]], Cell["Lista delle componenti dell'atto di moto al bordo", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(ambd = Table[\({\(wb\_d[i]\)[#], \(wb\_n[i]\)[#], \(\[Omega]b[ i]\)[#]} &\)\ /@ \ {pi\[UGrave], meno}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({\(wb\_d[1]\)["+"], \(wb\_n[1]\)["+"], \(\[Omega]b[1]\)[ "+"], \(wb\_d[1]\)["-"], \(wb\_n[1]\)["-"], \(\[Omega]b[1]\)[ "-"], \(wb\_d[2]\)["+"], \(wb\_n[2]\)["+"], \(\[Omega]b[2]\)[ "+"], \(wb\_d[2]\)["-"], \(wb\_n[2]\)["-"], \(\[Omega]b[2]\)[ "-"]}\)], "Output"] }, Open ]], Cell["Lista delle componenti delle forze al bordo", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(fbd = Table[\({\(sb\_d[i]\)[#], \(sb\_n[i]\)[#], \(mb[ i]\)[#]} &\)\ /@ \ {pi\[UGrave], meno}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({\(sb\_d[1]\)["+"], \(sb\_n[1]\)["+"], \(mb[1]\)["+"], \(sb\_d[1]\)[ "-"], \(sb\_n[1]\)["-"], \(mb[1]\)["-"], \(sb\_d[2]\)[ "+"], \(sb\_n[2]\)["+"], \(mb[2]\)["+"], \(sb\_d[2]\)[ "-"], \(sb\_n[2]\)["-"], \(mb[2]\)["-"]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Basi adattate al bordo e vincoli [", StyleBox["D3", FontColor->RGBColor[0, 0, 1]], "]" }], "Section"], Cell[CellGroupData[{ Cell["Descrizioni di vincoli standard", "Subsection"], Cell[BoxData[ \(\(carrelloV[trv_]\)[bnd_] := \(ub[trv]\)[bnd] . \(n[trv]\)[bnd] == 0\)], "Input"], Cell[BoxData[ \(\(cernieraV[trv_]\)[ bnd_] := {\(ub[trv]\)[bnd] . a\_1[trv] == 0, \(ub[trv]\)[bnd] . a\_2[trv] == 0}\)], "Input"], Cell[BoxData[ \(\(pernoV[trv1_, trv2_]\)[bnd1_, bnd2_] := {\((\(ub[trv2]\)[bnd2] - \(ub[trv1]\)[bnd1])\) . a\_1[trv2] == 0, \((\(ub[trv2]\)[bnd2] - \(ub[trv1]\)[bnd1])\) . a\_2[trv2] == 0}\)], "Input"], Cell[BoxData[ \(\(saldaturaV[trv1_, trv2_]\)[bnd1_, bnd2_] := {\((\(ub[trv2]\)[bnd2] - \(ub[trv1]\)[bnd1])\) . a\_1[trv2] == 0, \((\(ub[trv2]\)[bnd2] - \(ub[trv1]\)[bnd1])\) . a\_2[trv2] == 0, \(\[Theta]b[trv2]\)[bnd2] - \(\[Theta]b[trv1]\)[bnd1] \[Equal] 0}\)], "Input"], Cell[BoxData[ \(\(incastroV[trv_]\)[ bnd_] := {\(ub[trv]\)[bnd] . a\_1[trv] == 0, \(ub[trv]\)[bnd] . a\_2[trv] == 0, \(\[Theta]b[trv]\)[bnd] == 0}\)], "Input"], Cell["\<\ Per ogni nuova definizione, anche occasionale, occorre dare la corrispondente \ definizione della figura\ \>", "SmallText"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Dati [", StyleBox["D3", FontColor->RGBColor[0, 0, 1]], "]" }], "Subsection"], Cell["\<\ n vettore normale al piano di scorrimento di un carrello; d vettore tangenziale; {d, n} base ortonormale orientata come {e1, e2}\ \>", "SmallText"], Cell[BoxData[ \(\(Clear[d, n];\)\)], "Input"], Cell[BoxData[{ \(\(\(d[i_]\)[bd_] := e\_1;\)\), "\n", \(\(\(n[i_]\)[bd_] := e\_2;\)\)}], "Input"], Cell["\<\ Si assume che {d,n} siano identici a {e1,e2} a meno di una esplicita diversa \ definizione\ \>", "SmallText"], Cell[BoxData[{ \(\(\(d[1]\)[meno] = \(-e\_2\);\)\), "\[IndentingNewLine]", \(\(\(n[1]\)[meno] = e\_1;\)\), "\[IndentingNewLine]", \(\(\(d[2]\)[meno] = e\_1;\)\), "\[IndentingNewLine]", \(\(\(n[2]\)[meno] = e\_2;\)\)}], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Vincoli in forma scalare. Non usare esplicitamente le componenti ! Si \ pregiudicherebbe il meccanismo di sostituzione utilizzato nel calcolo della \ soluzione in termini di spostamento dalle equazioni di vincolo, oltre che \ incorrere pi\[UGrave] facilmente in errore. Utilizzare SEMPRE vincoli \ definiti secondo il modello dei vincoli standard, anche per definizioni \ occasionali. Ricordare di dare una definizione anche della figura del vincolo \ per la visualizzazione.\ \>", "SmallText"], Cell[BoxData[ \(vincoliDef := {\(cerniera[1]\)[meno], \(perno[1, 2]\)[pi\[UGrave], pi\[UGrave]], \(cerniera[2]\)[meno]}\)], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[ \(vincoli := \(Block[{carrello = carrelloV, cerniera = cernieraV, perno = pernoV, incastro = incastroV, saldatura = saldaturaV}, vincoliDef] // Flatten\) // Simplify\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(vincoli\)], "Input"], Cell[BoxData[ \({\(ub\_n[1]\)["-"] == 0, \(ub\_d[1]\)["-"] == 0, \(ub\_n[2]\)["+"] == \(ub\_n[1]\)["+"], \(ub\_d[1]\)[ "+"] == \(ub\_d[2]\)["+"], \(ub\_n[2]\)["-"] == 0, \(ub\_d[2]\)["-"] == 0}\)], "Output"] }, Open ]], Cell["Condizioni di vincolo come regole di sostituzione", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(vsp = \(Solve[\ vincoli, spbd]\)\[LeftDoubleBracket]1\[RightDoubleBracket] // Sort\)], "Input"], Cell[BoxData[ 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sinistra indotta dalla \ presenza di vincoli di collegamento tra le tarvi\ \>", "SmallText"], Cell[BoxData[ \(Clear[org]\)], "Input"], Cell[BoxData[ \(\(org[1] = {0, 0};\)\)], "Input"], Cell[BoxData[ \(\(coll[i_, j_]\)[bi_, bj_] := Block[{p = Sort[{{i, bi}, {j, bj}}, #1\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\) < #2\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)\ &]}, Block[{ix = p\_\(\(\[LeftDoubleBracket]\)\(1, \ 1\)\(\[RightDoubleBracket]\)\), jx = p\_\(\(\[LeftDoubleBracket]\)\(2, 1\)\(\[RightDoubleBracket]\ \)\), bix = p\_\(\(\[LeftDoubleBracket]\)\(1, 2\)\(\[RightDoubleBracket]\)\), bjx = p\_\(\(\[LeftDoubleBracket]\)\(2, \ 2\)\(\[RightDoubleBracket]\)\)}, \[IndentingNewLine]Switch[{bix, bjx}, \[IndentingNewLine]{pi\[UGrave], meno}, {org[jx] = Evaluate[ org[ix] + a\_1[ix] L[ix] /. datiO]}, \[IndentingNewLine]{pi\[UGrave], pi\[UGrave]}, {org[jx] = Evaluate[ org[ix] + a\_1[ix] L[ix] - a\_1[jx] L[jx] /. datiO]}, \[IndentingNewLine]{meno, meno}, 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basi adattate ai vincoli, utilizzando le componenti nelle basi adattate ai \ vincoli {d,n} (vedi la definizione di ub, sopra).\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[\(\((Append[\(ub[i]\)[#], \(\[Theta]b[i]\)[#]] /. vsp)\) &\)\ \ /@ \ {meno, pi\[UGrave]}, {i, 1, travi}], TableSpacing -> {4, 2, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {"0"}, {"0"}, {\(\(\[Theta]b[1]\)["-"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(\(ub\_d[2]\)["+"]\)}, {\(\(ub\_n[2]\)["+"]\)}, {\(\(\[Theta]b[1]\)["+"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]}, {GridBox[{ {"0"}, {"0"}, {\(\(\[Theta]b[2]\)["-"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(\(ub\_d[2]\)["+"]\)}, {\(\(ub\_n[2]\)["+"]\)}, {\(\(\[Theta]b[2]\)["+"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->4, ColumnSpacings->2, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{{0, 0, \[Theta]b[ 1][ "-"]}, { Subscript[ ub, d][ 2][ "+"], Subscript[ ub, n][ 2][ "+"], \[Theta]b[ 1][ "+"]}}, {{0, 0, \[Theta]b[ 2][ "-"]}, { Subscript[ ub, d][ 2][ "+"], Subscript[ ub, n][ 2][ "+"], \[Theta]b[ 2][ "+"]}}}, TableSpacing -> {4, 2, 2}]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(vincoli // Simplify\) // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(\(ub\_n[1]\)["-"] == 0\)}, {\(\(ub\_d[1]\)["-"] == 0\)}, {\(\(ub\_n[2]\)["+"] == \(ub\_n[1]\)["+"]\)}, {\(\(ub\_d[1]\)["+"] == \(ub\_d[2]\)["+"]\)}, {\(\(ub\_n[2]\)["-"] == 0\)}, {\(\(ub\_d[2]\)["-"] == 0\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ Subscript[ ub, n][ 1][ "-"], 0], Equal[ Subscript[ ub, d][ 1][ "-"], 0], Equal[ Subscript[ ub, n][ 2][ "+"], Subscript[ ub, n][ 1][ "+"]], Equal[ Subscript[ ub, d][ 1][ "+"], Subscript[ ub, d][ 2][ "+"]], Equal[ Subscript[ ub, n][ 2][ "-"], 0], Equal[ Subscript[ ub, d][ 2][ "-"], 0]}], Editable->False]], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Generazione delle equazioni di bilancio al bordo", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Potenza residua al bordo", "Subsection", Evaluatable->False], Cell["\<\ Le forze al bordo sono da definire dopo la separazione tra forze attive e \ forze reattive\ \>", "SmallText"], Cell["\<\ Espressione della potenza totale residua per la soluzione bulk (soluzione \ generale delle equazioni differenziali di bilancio)\ \>", "SmallText"], Cell[BoxData[ \(pote := \[Sum]\+\(i = 1\)\%travi\((\((\(sb[i]\)[ pi\[UGrave]] . \(wb[i]\)[pi\[UGrave]])\) + \((\(sb[i]\)[ meno] . \(wb[i]\)[meno])\) + \(mb[i]\)[ pi\[UGrave]]\ \(\[Omega]b[i]\)[pi\[UGrave]] + \(mb[i]\)[ meno]\ \(\[Omega]b[i]\)[meno])\) // Simplify\)], "Input"], Cell[BoxData[ \(potbd := pote - \[Sum]\+\(i = 1\)\%travi\((\((\(s[i]\)[L[i]] . \(wb[i]\)[ pi\[UGrave]])\) - \((\(s[i]\)[0] . \(wb[i]\)[ meno])\) + \(m[i]\)[L[i]]\ \(\[Omega]b[i]\)[ pi\[UGrave]] - \(m[i]\)[0]\ \(\[Omega]b[i]\)[meno])\) // Simplify\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(pote\)], "Input"], Cell[BoxData[ \(\(mb[1]\)["-"]\ \(\[Omega]b[1]\)["-"] + \(mb[1]\)[ "+"]\ \(\[Omega]b[1]\)["+"] + \(mb[2]\)["-"]\ \(\[Omega]b[2]\)[ "-"] + \(mb[2]\)["+"]\ \(\[Omega]b[2]\)["+"] + \(sb\_d[1]\)[ "-"]\ \(wb\_d[1]\)["-"] + \(sb\_d[1]\)["+"]\ \(wb\_d[1]\)[ "+"] + \(sb\_d[2]\)["-"]\ \(wb\_d[2]\)["-"] + \(sb\_d[2]\)[ "+"]\ \(wb\_d[2]\)["+"] + \(sb\_n[1]\)["-"]\ \(wb\_n[1]\)[ "-"] + \(sb\_n[1]\)["+"]\ \(wb\_n[1]\)["+"] + \(sb\_n[2]\)[ "-"]\ \(wb\_n[2]\)["-"] + \(sb\_n[2]\)["+"]\ \(wb\_n[2]\)[ "+"]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Map[Factor, Collect[potbd, ambd], {2}]\)], "Input"], Cell[BoxData[ \(\((\(mb[1]\)["-"] + \(sM[1]\)[0])\)\ \(\[Omega]b[1]\)[ "-"] + \((\(mb[1]\)[ "+"] - 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Cell[CellGroupData[{ Cell[BoxData[ \(Map[\((# == 0)\) &, \(LinearEquationsToMatrices[vincoli, spbd]\)\[LeftDoubleBracket]1\[RightDoubleBracket] . spbd]\)], "Input"], Cell[BoxData[ \({\(ub\_n[1]\)["-"] == 0, \(ub\_d[1]\)["-"] == 0, \(-\(ub\_n[1]\)["+"]\) + \(ub\_n[2]\)["+"] == 0, \(ub\_d[1]\)["+"] - \(ub\_d[2]\)["+"] == 0, \(ub\_n[2]\)["-"] == 0, \(ub\_d[2]\)["-"] == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{ub = wb, \[Theta]b = \[Omega]b}, vincoli] // Simplify\)], "Input"], Cell[BoxData[ \({\(wb\_n[1]\)["-"] == 0, \(wb\_d[1]\)["-"] == 0, \(wb\_n[2]\)["+"] == \(wb\_n[1]\)["+"], \(wb\_d[1]\)[ "+"] == \(wb\_d[2]\)["+"], \(wb\_n[2]\)["-"] == 0, \(wb\_d[2]\)["-"] == 0}\)], "Output"] }, Open ]], Cell["Condizioni di vincolo sugli atti di moto", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(vam = \(Solve[\ Map[\((# == 0)\) &, \(LinearEquationsToMatrices[ Block[{ub = wb, \[Theta]b = \[Omega]b}, vincoli], ambd]\)\[LeftDoubleBracket]1\[RightDoubleBracket] . ambd], ambd]\)\[LeftDoubleBracket]1\[RightDoubleBracket] // Sort\)], "Input"], Cell[BoxData[ \({\(wb\_d[1]\)["-"] \[Rule] 0, \(wb\_d[1]\)["+"] \[Rule] \(wb\_d[2]\)["+"], \(wb\_d[2]\)[ "-"] \[Rule] 0, \(wb\_n[1]\)["-"] \[Rule] 0, \(wb\_n[1]\)["+"] \[Rule] \(wb\_n[2]\)["+"], \(wb\_n[2]\)[ "-"] \[Rule] 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ambdv = Complement[ambd /. vam, {0}]\)], "Input"], Cell[BoxData[ \({\(\[Omega]b[1]\)["-"], \(\[Omega]b[1]\)["+"], \(\[Omega]b[2]\)[ "-"], \(\[Omega]b[2]\)["+"], \(wb\_d[2]\)["+"], \(wb\_n[2]\)[ "+"]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Potenza al bordo per atti di moto vincolati", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(potbdv = Collect[potbd /. vam, ambdv]\)], "Input"], Cell[BoxData[ \(\((\(mb[1]\)["-"] + \(sM[1]\)[0])\)\ \(\[Omega]b[1]\)[ "-"] + \((\(mb[1]\)[ "+"] - \(sM[1]\)[\[ScriptCapitalL]])\)\ \(\[Omega]b[1]\)[ "+"] + \((\(mb[2]\)["-"] + \(sM[2]\)[0])\)\ \(\[Omega]b[2]\)[ "-"] + 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"]"}]}]}]}], ",", RowBox[{\(sNo[2]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(sMo[ 2] + \[ScriptC]\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]]\), "==", RowBox[{\(\[Zeta]\ sQo[2]\), "+", RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[2]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}]}]}]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(spsolDO = \(DSolve[eqnspO, Flatten[Table[{u\_1[i], u\_2[i]}, {i, 1, travi}]], \[Zeta], DSolveConstants \[Rule] \[ScriptCapitalD]]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket] // Simplify\)], "Input"], Cell[BoxData[ \({u\_1[1] \[Rule] Function[{\[Zeta]}, \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[1]\ \)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(\[Zeta]\^2\ sMo[1]\)\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[Zeta]\^3\ sQo[1]\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) + \(\[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + \ 2\ \[Zeta]]\)\/\(4\ \[ScriptCapitalY]\[ScriptCapitalJ]\) - \(\[ScriptC]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][3]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[2]\ \)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][4]], u\_2[2] \[Rule] Function[{\[Zeta]}, \(\[Zeta]\^2\ sMo[2]\)\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[Zeta]\^3\ sQo[2]\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + \ 2\ \[Zeta]]\)\/\(4\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \(\[ScriptC]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \[ScriptCapitalD][ 5] + \[Zeta]\ \[ScriptCapitalD][6]]}\)], "Output"] }, Open ]], Cell["\<\ Dopo la sostisuzione delle soluzioni delle equazioni di bilancio al bordo\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(eqnsp = \(\(\(\(\(Table[{\(sN[i]\)[\[Zeta]] == \(sNf[ i]\)[\[Zeta]], \(sM[i]\)[\[Zeta]] == \(sMf[ i]\)[\[Zeta]]}, {i, 1, travi}] /. bulksol\) /. cNQMval\) /. risp\) /. grad\) /. vinBer // Flatten\) // Simplify\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{\(\[ScriptC]\/\[ScriptCapitalL]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(\(\[ScriptC]\ \[Zeta]\)\/\[ScriptCapitalL]\), "==", RowBox[{\(\[ScriptC]\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]]\), "+", RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[1]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}]}]}], ",", RowBox[{\(\[ScriptC]\/\[ScriptCapitalL]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(\[ScriptC]\ \((\(-\(\[Zeta]\/\[ScriptCapitalL]\)\) + UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\)\), "==", RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[2]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}]}]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(spsolD = \(DSolve[eqnsp, Flatten[Table[{u\_1[i], u\_2[i]}, {i, 1, travi}]], \[Zeta], DSolveConstants \[Rule] \[ScriptCapitalD]]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket] // Simplify\)], "Input"], Cell[BoxData[ \({u\_1[1] \[Rule] Function[{\[Zeta]}, \(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(-\(\(1\/\(\[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-\(\(\[ScriptC]\ \[Zeta]\^3\)\/6\)\) - 1\/4\ \[ScriptC]\ \[ScriptCapitalL]\^2\ \((\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 1\/2\ \[ScriptC]\ \[ScriptCapitalL]\ \((\(-\(\ \[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\[ScriptCapitalL]\) \ + 2\ \[Zeta]])\)\)\) + \[ScriptCapitalD][2] + \[Zeta]\ \[ScriptCapitalD][3]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][4]], u\_2[2] \[Rule] Function[{\[Zeta]}, \(\(1\/\(\[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-\(\(\[ScriptC]\ \[Zeta]\^3\)\/6\)\) - 1\/4\ \[ScriptC]\ \[ScriptCapitalL]\^2\ \((\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 1\/2\ \[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]])\)\) + \[ScriptCapitalD][ 5] + \[Zeta]\ \[ScriptCapitalD][6]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(splist = Table[{\(u\_1[i]\)[\[Zeta]], \(u\_2[i]\)[\[Zeta]], \(\[Theta][ i]\)[\[Zeta]]}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({\(u\_1[1]\)[\[Zeta]], \(u\_2[1]\)[\[Zeta]], \(\[Theta][ 1]\)[\[Zeta]], \(u\_1[2]\)[\[Zeta]], \(u\_2[ 2]\)[\[Zeta]], \(\[Theta][2]\)[\[Zeta]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolDO // Simplify\)], "Input"], Cell[BoxData[ \({\(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[1]\)\/\[ScriptCapitalY]\ \[ScriptCapitalJ] + \[ScriptCapitalD][ 1], \(\(1\/\(24\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\((\(-3\)\ \ \[ScriptC]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\^2\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]] + 4\ \((3\ \[Zeta]\^2\ sMo[1] - \[Zeta]\^3\ sQo[1] + 6\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[ScriptCapitalD][2] + 6\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[Zeta]\ \ \[ScriptCapitalD][ 3])\))\)\), \(-\(\(\[ScriptC]\ \[ScriptCapitalL]\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\) + \(\[ScriptC]\ \((\[ScriptCapitalL]\ \^2 - 4\ \[Zeta]\^2)\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\)\/\(4\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \(\(1\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((2\ \[Zeta]\ sMo[1] - \[Zeta]\^2\ sQo[ 1] + \[ScriptC]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 2\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[ScriptCapitalD][ 3])\)\), \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ \ sNo[2]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][ 4], \(\(1\/\(24\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\((3\ \ \[ScriptC]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\^2\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]] + 4\ \((3\ \[Zeta]\^2\ sMo[2] - \[Zeta]\^3\ sQo[2] + 6\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[ScriptCapitalD][5] + 6\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[Zeta]\ \ \[ScriptCapitalD][ 6])\))\)\), \(\[ScriptC]\ \[ScriptCapitalL]\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\ \)\/\(2\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \(\[ScriptC]\ \((\(-\(\ \[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ DiracDelta[\[ScriptCapitalL] - 2\ \ \[Zeta]]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] - \(\(1\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\((\(-2\)\ \[Zeta]\ sMo[ 2] + \[Zeta]\^2\ sQo[ 2] + \[ScriptC]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] - 2\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[ScriptCapitalD][ 6])\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolD // Simplify\)], "Input"], Cell[BoxData[ \({\(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][ 1], \(\(1\/\(24\ \[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-3\)\ \[ScriptC]\ \[ScriptCapitalL]\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\^2\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 4\ \((\[ScriptC]\ \[Zeta]\^3 + 6\ \[ScriptCapitalL]\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \((\ \[ScriptCapitalD][2] + \[Zeta]\ \[ScriptCapitalD][ 3])\))\))\)\), \(\(1\/\(\[ScriptCapitalL]\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\((\(-\(1\/2\)\)\ \[ScriptC]\ \ \[ScriptCapitalL]\^2\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]] + 1\/4\ \[ScriptC]\ \[ScriptCapitalL]\ \((\[ScriptCapitalL]\^2 - 4\ \[Zeta]\^2)\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]] + 1\/2\ \[ScriptC]\ \((\[Zeta]\^2 + \[ScriptCapitalL]\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\))\)\) + \[ScriptCapitalD][ 3], \(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][ 4], \(\(1\/\(24\ \[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-4\)\ \[ScriptC]\ \[Zeta]\^3 + 3\ \[ScriptC]\ \[ScriptCapitalL]\ \((\[ScriptCapitalL] - 2\ \ \[Zeta])\)\^2\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 24\ \[ScriptCapitalL]\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \((\ \[ScriptCapitalD][ 5] + \[Zeta]\ \[ScriptCapitalD][ 6])\))\)\), \(\(1\/\(\[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((1\/2\ \[ScriptC]\ \[ScriptCapitalL]\^2\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]] + \[ScriptC]\ \[ScriptCapitalL]\ \((\(-\(\ \[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]] - 1\/2\ \[ScriptC]\ \((\[Zeta]\^2 + \[ScriptCapitalL]\ \((\ \[ScriptCapitalL] - 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]])\))\)\) + \[ScriptCapitalD][ 6]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Cambiamento delle costanti di integrazione", "Subsection"], Cell["\<\ Viene costruita la lista delle costanti di integrazione delle funzioni di \ risposta. La lista delle costanti di integrazione presenti nelle condizioni di vincolo \ in generale contiene la prima.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(cDlistO = Complement[ Map[If[FreeQ[\(splist /. vinBer\) /. spsolD, #], 0, #]\ &, Table[\[ScriptCapitalD][i], {i, 3\ travi}]], {0}]\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][1], \[ScriptCapitalD][2], \[ScriptCapitalD][ 3], \[ScriptCapitalD][4], \[ScriptCapitalD][5], \[ScriptCapitalD][ 6]}\)], "Output"] }, Open ]], Cell["\<\ Vengono elencate le costanti di integrazione presenti nelle espressioni \ calcolate\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(cDlist = Block[{splistV = \(splist /. vinBer\) /. spsolD}, Join[\n\tComplement[ Map[If[FreeQ[splistV, #], 0, #]\ &, cNQM], {0}], \n\t Complement[ Map[If[FreeQ[splistV, #], 0, #]\ &, cDlistO], {0}]\n]] // Union\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][1], \[ScriptCapitalD][2], \[ScriptCapitalD][ 3], \[ScriptCapitalD][4], \[ScriptCapitalD][5], \[ScriptCapitalD][ 6]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Table[\({\(u\_1[i]\)[0] \[Equal] uo\_1[i], \(u\_2[i]\)[0] \[Equal] uo\_2[i], \(\[Theta][i]\)[0] \[Equal] \[Theta]o[i]} /. vinBer\) /. spsolD, {i, 1, travi}] // Simplify\) // Flatten\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][1] == uo\_1[1], \[ScriptCapitalD][2] == uo\_2[1], \[ScriptCapitalD][3] == \[Theta]o[1], \[ScriptCapitalD][4] == uo\_1[2], \[ScriptCapitalD][5] == uo\_2[2], \[ScriptCapitalD][6] == \[Theta]o[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fromDtoU = \(Solve[%, cDlistO]\)\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][1] \[Rule] uo\_1[1], \[ScriptCapitalD][2] \[Rule] uo\_2[1], \[ScriptCapitalD][3] \[Rule] \[Theta]o[ 1], \[ScriptCapitalD][4] \[Rule] uo\_1[2], \[ScriptCapitalD][5] \[Rule] uo\_2[2], \[ScriptCapitalD][6] \[Rule] \[Theta]o[2]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRlist = cDlist /. fromDtoU\)], "Input"], Cell[BoxData[ \({uo\_1[1], uo\_2[1], \[Theta]o[1], uo\_1[2], uo\_2[2], \[Theta]o[2]}\)], "Output"] }, Open ]], Cell["\<\ Prima della sostituzione delle soluzioni delle equazioni di bilancio al bordo\ \ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(spsolO = spsolDO /. fromDtoU\)], "Input"], Cell[BoxData[ \({u\_1[1] \[Rule] Function[{\[Zeta]}, \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[1]\ \)\/\[ScriptCapitalY]\[ScriptCapitalJ] + uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(\[Zeta]\^2\ sMo[1]\)\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[Zeta]\^3\ sQo[1]\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) + \(\[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + \ 2\ \[Zeta]]\)\/\(4\ \[ScriptCapitalY]\[ScriptCapitalJ]\) - \(\[ScriptC]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + uo\_2[1] + \[Zeta]\ \[Theta]o[1]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[2]\ \)\/\[ScriptCapitalY]\[ScriptCapitalJ] + uo\_1[2]], u\_2[2] \[Rule] Function[{\[Zeta]}, \(\[Zeta]\^2\ sMo[2]\)\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[Zeta]\^3\ sQo[2]\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + \ 2\ \[Zeta]]\)\/\(4\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \(\[ScriptC]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + uo\_2[2] + \[Zeta]\ \[Theta]o[2]]}\)], "Output"] }, Open ]], Cell["\<\ Dopo la sostisuzione delle soluzioni delle equazioni di bilancio al bordo\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(spsol = spsolD /. fromDtoU\)], "Input"], Cell[BoxData[ \({u\_1[1] \[Rule] Function[{\[Zeta]}, \(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(-\(\(1\/\(\[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-\(\(\[ScriptC]\ \[Zeta]\^3\)\/6\)\) - 1\/4\ \[ScriptC]\ \[ScriptCapitalL]\^2\ \((\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 1\/2\ \[ScriptC]\ \[ScriptCapitalL]\ \((\(-\(\ \[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\[ScriptCapitalL]\) \ + 2\ \[Zeta]])\)\)\) + uo\_2[1] + \[Zeta]\ \[Theta]o[1]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(\[ScriptC]\ \[ScriptCapitalL]\ \[Zeta]\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ] + uo\_1[2]], u\_2[2] \[Rule] Function[{\[Zeta]}, \(\(1\/\(\[ScriptCapitalL]\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)\)\((\(-\(\(\[ScriptC]\ \[Zeta]\^3\)\/6\)\) - 1\/4\ \[ScriptC]\ \[ScriptCapitalL]\^2\ \((\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta])\)\ UnitStep[\(-\[ScriptCapitalL]\) + 2\ \[Zeta]] + 1\/2\ \[ScriptC]\ \[ScriptCapitalL]\ \ \((\(-\(\[ScriptCapitalL]\^2\/4\)\) + \[Zeta]\^2)\)\ UnitStep[\(-\ \[ScriptCapitalL]\) + 2\ \[Zeta]])\)\) + uo\_2[2] + \[Zeta]\ \[Theta]o[2]]}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Soluzione delle equazioni di vincolo ", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["Equazioni di vincolo", "Subsection", Evaluatable->False], Cell["\<\ Le variabili che hanno il significato di spostamenti al bordo vengono \ sostituite con i valori al bordo dello spostamento\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(eqvinO = Block[{\n\t\tub = \((Function[ j, \((Switch[j, meno, \(u[#]\)[0], pi\[UGrave], \(u[#]\)[ L[#]]])\)] &)\), \[Theta]b = \((Function[ j, \((Switch[j, meno, \(\[Theta][#]\)[0], pi\[UGrave], \(\[Theta][#]\)[L[#]]])\)] &)\)\n\t\t}, vincoli] // Simplify\)], "Input"], Cell[BoxData[ \({\(u\_1[1]\)[0] == 0, \(u\_2[1]\)[0] == 0, \(u\_1[2]\)[\[ScriptCapitalL]] == \(u\_2[ 1]\)[\[ScriptCapitalL]], \(u\_1[1]\)[\[ScriptCapitalL]] + \(u\_2[ 2]\)[\[ScriptCapitalL]] == 0, \(u\_1[2]\)[0] == 0, \(u\_2[2]\)[0] == 0}\)], "Output"] }, Open ]], Cell["\<\ Qui \[EGrave] essenziale che \"vincoli\" sia stata definita con \":=\" e \ utilizzando il prodotto scalare invece che i nomi delle componenti dello \ spostamento.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(eqvin = \(eqvinO /. vinBer\) /. spsol // Simplify\)], "Input"], Cell[BoxData[ \({uo\_1[1] == 0, uo\_2[1] == 0, \(\[ScriptC]\ \[ScriptCapitalL]\^2\ \[Kappa]\)\/\[ScriptCapitalY]\ \[ScriptCapitalJ] + uo\_1[2] == \(\[ScriptC]\ \[ScriptCapitalL]\^2\)\/\(24\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \[ScriptCapitalL]\ \[Theta]o[1] + uo\_2[1], \(\[ScriptC]\ \[ScriptCapitalL]\^2\ \((\(-1\) + 24\ \ \[Kappa])\)\)\/\(24\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \[ScriptCapitalL]\ \ \[Theta]o[2] + uo\_1[1] + uo\_2[2] == 0, uo\_1[2] == 0, uo\_2[2] == 0}\)], "Output"] }, 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scN\ \(\(sNQM[ i]\)[\[Zeta]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ \ a\_2[i]\)], "Input"], Cell["Valori al bordo", "SmallText"], Cell[BoxData[ \(diaNb[ i_] := {\(asseO[i]\)[0] + scN\ \(\(sNQM[i]\)[ 0]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ a\_2[ i]\ \[Xi], \(asseO[i]\)[L[i]] + scN\ \(\(sNQM[i]\)[ L[i]]\)\[LeftDoubleBracket]1\[RightDoubleBracket]\ a\_2[ i]\ \[Xi]}\)], "Input"], Cell["Segni dei valori al bordo", "SmallText"], Cell[BoxData[ \(diaNs[i_] := Block[{y1 = scN\ \(\(sNQM[i]\)[ 0]\)\[LeftDoubleBracket]1\[RightDoubleBracket] \ /. \[InvisibleSpace]datinum, y2 = scN\ \(\(sNQM[i]\)[ L[i]]\)\[LeftDoubleBracket]1\[RightDoubleBracket] \ /. \[InvisibleSpace]datinum, pt1 = \(asseO[i]\)[0] + 0.5\ y1\ a\_2[i] + 0.04\ a\_1[i], pt2 = \(asseO[i]\)[L[i]] + 0.5\ y2\ a\_2[i] - 0.04\ a\_1[i], dsh = 0.04}, Complement[{If[y1 \[NotEqual] 0, pt1 + dsh\ a\_1[i]\ \((\[Xi] - 0.5)\)], If[y1 > 0, pt1 + dsh\ a\_2[i]\ \((\[Xi] - 0.5)\)], If[y2 \[NotEqual] 0, pt2 + dsh\ a\_1[i]\ \((\[Xi] - 0.5)\)], If[y2 > 0, pt2 + dsh\ a\_2[ i]\ \((\[Xi] - 0.5)\)]}, {Null}]] /. \[InvisibleSpace]datinum\)], \ "Input"], Cell[BoxData[ \(\(figN := Table[\(diaN[i]\)[L[i] \[Xi]], {i, 1, travi}] /. datinum;\)\)], "Input"], Cell[BoxData[ \(\(figNb := Flatten[Table[diaNb[i], {i, 1, travi}], 1] /. datinum;\)\)], "Input"], Cell[BoxData[ \(\(figNs := Flatten[Table[diaNs[i], {i, 1, travi}], 1] /. datinum;\)\)], "Input"], Cell[BoxData[ \(\(pltN := ParametricPlot[Evaluate[Join[figN, figNb, figNs]], {\[Xi], 0, 1}, Axes \[Rule] False, AspectRatio \[Rule] Automatic, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] {{Hue[0.4]}}];\)\)], "Input"], Cell[BoxData[ \(\(diaQ[i_]\)[\[Zeta]_] := \(asseO[i]\)[\[Zeta]] - scQ\ \(\(sNQM[ i]\)[\[Zeta]]\)\[LeftDoubleBracket]2\[RightDoubleBracket]\ \ a\_2[i]\)], "Input"], Cell[BoxData[ \(diaQb[ i_] := {\(asseO[i]\)[0] - scQ\ \(\(sNQM[i]\)[ 0]\)\[LeftDoubleBracket]2\[RightDoubleBracket]\ a\_2[ i]\ \[Xi], \(asseO[i]\)[L[i]] - scQ\ \(\(sNQM[i]\)[ L[i]]\)\[LeftDoubleBracket]2\[RightDoubleBracket]\ a\_2[ i]\ \[Xi]}\)], "Input"], Cell[BoxData[ \(diaQs[i_] := Block[{y1 = scQ\ \(\(sNQM[i]\)[ 0]\)\[LeftDoubleBracket]2\[RightDoubleBracket] \ /. \[InvisibleSpace]datinum, y2 = scQ\ \(\(sNQM[i]\)[ L[i]]\)\[LeftDoubleBracket]2\[RightDoubleBracket] \ /. \[InvisibleSpace]datinum, pt1 = \(asseO[i]\)[0] - 0.5\ y1\ a\_2[i] + 0.04\ a\_1[i], pt2 = \(asseO[i]\)[L[i]] - 0.5\ y2\ a\_2[i] - 0.04\ a\_1[i], dsh = 0.04}, Complement[{If[y1 \[NotEqual] 0, pt1 + dsh\ a\_1[i]\ \((\[Xi] - 0.5)\)], If[y1 > 0, pt1 + dsh\ a\_2[i]\ \((\[Xi] - 0.5)\)], If[y2 \[NotEqual] 0, pt2 + dsh\ a\_1[i]\ \((\[Xi] - 0.5)\)], If[y2 > 0, pt2 + dsh\ a\_2[ i]\ \((\[Xi] - 0.5)\)]}, {Null}]] /. \[InvisibleSpace]datinum\)], \ "Input"], Cell[BoxData[ \(\(figQ := Table[\(diaQ[i]\)[L[i] \[Xi]], {i, 1, travi}] /. datinum;\)\)], "Input"], Cell[BoxData[ \(\(figQb := Flatten[Table[diaQb[i], {i, 1, travi}], 1] /. datinum;\)\)], "Input"], Cell[BoxData[ \(\(figQs := Flatten[Table[diaQs[i], 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