(************** 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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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[ \("C:\\Wrk\\Corsi\\Scost\\esercizi\\7-travi\\7-16\\outmath"\)], "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]\/6;\)\)}], "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). Quelle deivanti dai vincoli sono \ descritte a parte, pi\[UGrave] avanti.\ \>", "SmallText"], Cell[BoxData[ \(\(org[1] = {0, 0};\)\)], "Input"], Cell[BoxData[ \(org[i_] := org[i - 1] + {Max[\(\(asseO[i - \ 1]\)[0]\)\_\(\(\[LeftDoubleBracket]\)\(1\)\(\[RightDoubleBracket]\)\), \ \(\(asseO[i - 1]\)[L[i - 1]]\)\_\(\(\[LeftDoubleBracket]\)\(1\)\(\ \[RightDoubleBracket]\)\)], 0} + {maxL\/10, 0}\)], "Input"], Cell["definizione delle sezioni", "SmallText"], Cell[BoxData[ \(\(secO[ i_]\)[\[Zeta]_] := {\(asseO[i]\)[\[Zeta]] - maxL\/20\ a\_2[i]\ , \(asseO[i]\)[\[Zeta]] + maxL\/20\ a\_2[i]\ } /. datiO\)], "Input"], Cell["definizione della base adattata", "SmallText"], Cell[BoxData[ \(\(vecOa1[ i_]\)[\[Zeta]_] := {{\(asseO[i]\)[\[Zeta]], \(asseO[i]\)[\[Zeta]] + maxL\/5\ \ a\_1[i]}, {\(asseO[i]\)[\[Zeta] + maxL\/5] + maxL\/15\ \((\(-a\_1[i]\) + a\_2[i]\/2)\), \(asseO[ i]\)[\[Zeta] + maxL\/5]}, {\(asseO[i]\)[\[Zeta] + maxL\/5] + \(\(\(maxL\)\(\ \)\)\/15\) \((\(-a\_1[i]\) - a\_2[i]\/2)\), 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per 0 (zero)\n", "(si possono anche usare dei parametri; in tal caso se ne assegni il valore \ nella lista dei dati numerici ", StyleBox["datip(D5)", FontFamily->"Courier New", FontWeight->"Bold"], ")", "\n[ l'uso caratteri script per i parametri rende tutto molto pi\[UGrave] \ leggibile]" }], "SmallText"], Cell[BoxData[""], "Input", CellFrame->True, Background->GrayLevel[0.849989]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Propriet\[AGrave] di UnitStep nel contesto di questo calcolo (da \ controllare ogni volta)", " [", StyleBox["\[FilledCircle]", FontColor->RGBColor[0, 0, 1]], "]" }], "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Unprotect[UnitStep]\)], "Input"], Cell[BoxData[ \({"UnitStep"}\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(UnitStep[\(-\[ScriptCapitalL]\)] = 0;\)\), "\[IndentingNewLine]", \(\(UnitStep[\(-\(\[ScriptCapitalL]\/2\)\)] = 0;\)\), "\[IndentingNewLine]", \(\(UnitStep[\[ScriptCapitalL]\/2] = 1;\)\), "\[IndentingNewLine]", 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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] 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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], \[ScriptCapitalC][3] == sMo[1], \[ScriptCapitalC][4] == sNo[2], \[ScriptCapitalC][5] == sQo[2], \[ScriptCapitalC][6] == 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] sMo[1], \[ScriptCapitalC][4] \[Rule] sNo[2], \[ScriptCapitalC][5] \[Rule] sQo[2], \[ScriptCapitalC][6] \[Rule] 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]}, \(-\[Zeta]\)\ sQo[1] + sMo[1]], sN[2] \[Rule] Function[{\[Zeta]}, sNo[2]], sQ[2] \[Rule] Function[{\[Zeta]}, sQo[2]], sM[2] \[Rule] Function[{\[Zeta]}, \(-\[Zeta]\)\ sQo[2] + 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[{\(\(sQ[1]\)[\[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[{\(\(sQ[2]\)[\[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[ 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[ 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]\)}, {\(\[Zeta]\ \[ScriptCapitalC][2] + \(sM[ 1]\)[\[Zeta]] == \[ScriptCapitalC][3]\)}, {\(\(sN[2]\)[\[Zeta]] == \[ScriptCapitalC][4]\)}, {\(\(sQ[2]\)[\[Zeta]] == \[ScriptCapitalC][5]\)}, {\(\[Zeta]\ \[ScriptCapitalC][5] + \(sM[ 2]\)[\[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[ \[Zeta], \[ScriptCapitalC][ 2]], sM[ 1][ \[Zeta]]], \[ScriptCapitalC][ 3]], Equal[ sN[ 2][ \[Zeta]], \[ScriptCapitalC][ 4]], Equal[ sQ[ 2][ \[Zeta]], \[ScriptCapitalC][ 5]], Equal[ Plus[ Times[ \[Zeta], \[ScriptCapitalC][ 5]], sM[ 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] + \(sM[1]\)[\[Zeta]] == sMo[1]\)}, {\(\(sN[2]\)[\[Zeta]] == sNo[2]\)}, {\(\(sQ[2]\)[\[Zeta]] == sQo[2]\)}, {\(\[Zeta]\ sQo[2] + \(sM[2]\)[\[Zeta]] == sMo[2]\)} }, 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]], 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]]], sMo[ 2]]}], 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[{ \(\(\(d[2]\)[pi\[UGrave]] = \(-e\_2\);\)\), "\[IndentingNewLine]", \(\(\(n[2]\)[pi\[UGrave]] = \(-e\_1\);\)\)}], "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], meno], \(cerniera[2]\)[pi\[UGrave]]}\)], "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[ \({\(ub\_d[1]\)["-"] == 0, \(ub\_n[1]\)["-"] == 0, 1\/2\ \((\@3\ \((\(-\(ub\_d[1]\)["+"]\) + \(ub\_d[2]\)[ "-"])\) - \(ub\_n[1]\)["+"] + \(ub\_n[2]\)["-"])\) == 0, 1\/2\ \((\(ub\_d[1]\)["+"] - \(ub\_d[2]\)[ "-"] + \@3\ \((\(-\(ub\_n[1]\)["+"]\) + \(ub\_n[2]\)[ "-"])\))\) == 0, \(-\(1\/2\)\)\ \(ub\_d[2]\)["+"] - 1\/2\ \@3\ \(ub\_n[2]\)["+"] == 0, 1\/2\ \((\(-\@3\)\ \(ub\_d[2]\)["+"] + \(ub\_n[2]\)["+"])\) == 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[ \({\(ub\_d[1]\)["-"] \[Rule] 0, \(ub\_d[1]\)["+"] \[Rule] \(ub\_d[2]\)["-"], \(ub\_d[2]\)[ "+"] \[Rule] 0, \(ub\_n[1]\)["-"] \[Rule] 0, \(ub\_n[1]\)["+"] \[Rule] \(ub\_n[2]\)["-"], \(ub\_n[2]\)[ "+"] \[Rule] 0}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Definizioni per la visualizzazione", "Subsection"], Cell["Condizioni di vincolo sui collegamenti tra le travi", "SmallText"], Cell[BoxData[ \(Clear[coll]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(vincoliDef\)], "Input"], Cell[BoxData[ \({\(cerniera[1]\)["-"], \(perno[1, 2]\)["+", "-"], \(cerniera[2]\)[ "+"]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[ vincoliDef /. {carrello \[Rule] \((\((Null\ &)\)\ &)\), incastro \[Rule] \((\((Null\ &)\)\ &)\), cerniera 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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]\)[ "-"] + \((\(mb[1]\)[ "+"] - \(sM[1]\)[\[ScriptCapitalL]])\)\ \(\[Omega]b[1]\)[ "+"] + \((\(mb[2]\)["-"] + \(sM[2]\)[0])\)\ \(\[Omega]b[2]\)[ "-"] + \((\(mb[2]\)[ "+"] - \(sM[2]\)[\[ScriptCapitalL]])\)\ \(\[Omega]b[2]\)[ "+"] + \((\(sN[1]\)[0] + \(sb\_d[1]\)["-"])\)\ \(wb\_d[1]\)[ "-"] + \((\(-\(sN[1]\)[\[ScriptCapitalL]]\) + \(sb\_d[1]\)[ "+"])\)\ \(wb\_d[1]\)["+"] + 1\/2\ \((\@3\ \(sN[2]\)[0] - \(sQ[2]\)[0] + 2\ \(sb\_d[2]\)["-"])\)\ \(wb\_d[2]\)["-"] + 1\/2\ \((\(sN[ 2]\)[\[ScriptCapitalL]] + \@3\ \(sQ[2]\)[\[ScriptCapitalL]] + 2\ \(sb\_d[2]\)["+"])\)\ \(wb\_d[2]\)[ "+"] + \((\(sQ[1]\)[0] + \(sb\_n[1]\)["-"])\)\ \(wb\_n[1]\)[ "-"] + \((\(-\(sQ[1]\)[\[ScriptCapitalL]]\) + \(sb\_n[1]\)[ "+"])\)\ \(wb\_n[1]\)["+"] + 1\/2\ \((\(sN[2]\)[0] + \@3\ \(sQ[2]\)[0] + 2\ \(sb\_n[2]\)["-"])\)\ \(wb\_n[2]\)["-"] + 1\/2\ \((\@3\ \(sN[2]\)[\[ScriptCapitalL]] - \(sQ[ 2]\)[\[ScriptCapitalL]] + 2\ \(sb\_n[2]\)["+"])\)\ \(wb\_n[ 2]\)["+"]\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Vincoli sugli atti di moto al bordo", "Subsection"], Cell["\<\ Si generano le equazioni di vincolo omogenee per gli atti di moto\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[\((# == 0)\) &, \(LinearEquationsToMatrices[vincoli, spbd]\)\[LeftDoubleBracket]1\[RightDoubleBracket] . spbd]\)], "Input"], Cell[BoxData[ \({\(ub\_d[1]\)["-"] == 0, \(ub\_n[1]\)["-"] == 0, \(-\(1\/2\)\)\ \@3\ \(ub\_d[1]\)["+"] + 1\/2\ \@3\ \(ub\_d[2]\)["-"] - 1\/2\ \(ub\_n[1]\)["+"] + 1\/2\ \(ub\_n[2]\)["-"] == 0, 1\/2\ \(ub\_d[1]\)["+"] - 1\/2\ \(ub\_d[2]\)["-"] - 1\/2\ \@3\ \(ub\_n[1]\)["+"] + 1\/2\ \@3\ \(ub\_n[2]\)["-"] == 0, \(-\(1\/2\)\)\ \(ub\_d[2]\)["+"] - 1\/2\ \@3\ \(ub\_n[2]\)["+"] == 0, \(-\(1\/2\)\)\ \@3\ \(ub\_d[2]\)["+"] + 1\/2\ \(ub\_n[2]\)["+"] == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{ub = wb, \[Theta]b = \[Omega]b}, vincoli] // Simplify\)], "Input"], Cell[BoxData[ \({\(wb\_d[1]\)["-"] == 0, \(wb\_n[1]\)["-"] == 0, 1\/2\ \((\@3\ \((\(-\(wb\_d[1]\)["+"]\) + \(wb\_d[2]\)[ "-"])\) - \(wb\_n[1]\)["+"] + \(wb\_n[2]\)["-"])\) == 0, 1\/2\ \((\(wb\_d[1]\)["+"] - \(wb\_d[2]\)[ "-"] + \@3\ \((\(-\(wb\_n[1]\)["+"]\) + \(wb\_n[2]\)[ "-"])\))\) == 0, \(-\(1\/2\)\)\ \(wb\_d[2]\)["+"] - 1\/2\ \@3\ \(wb\_n[2]\)["+"] == 0, 1\/2\ \((\(-\@3\)\ \(wb\_d[2]\)["+"] + \(wb\_n[2]\)["+"])\) == 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]\)[ "-"] + \((\(mb[2]\)[ "+"] - \(sM[2]\)[\[ScriptCapitalL]])\)\ \(\[Omega]b[2]\)[ "+"] + \((\(-\(sN[1]\)[\[ScriptCapitalL]]\) + 1\/2\ \((\@3\ \(sN[2]\)[0] - \(sQ[2]\)[0])\) + \(sb\_d[1]\)[ "+"] + \(sb\_d[2]\)["-"])\)\ \(wb\_d[2]\)[ "-"] + \((\(-\(sQ[1]\)[\[ScriptCapitalL]]\) + 1\/2\ \((\(sN[2]\)[0] + \@3\ \(sQ[2]\)[0])\) + \(sb\_n[1]\)[ "+"] + \(sb\_n[2]\)["-"])\)\ \(wb\_n[2]\)["-"]\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Equazioni di bilancio al bordo (corrispondenti agli atti di moto vincolati)\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eqbilbd = \((#1 == 0 &)\) /@ Table[Coefficient[potbdv, ambdv\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, Length[ambdv]}]\)], "Input"], Cell[BoxData[ \({\(mb[1]\)["-"] + \(sM[1]\)[0] == 0, \(mb[1]\)["+"] - \(sM[1]\)[\[ScriptCapitalL]] == 0, \(mb[2]\)["-"] + \(sM[2]\)[0] == 0, \(mb[2]\)["+"] - \(sM[2]\)[\[ScriptCapitalL]] == 0, 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\(sNf[i]\)[\[Zeta]], \(sM[ i]\)[\[Zeta]] == \(sMf[i]\)[\[Zeta]]}, {i, 1, travi}] /. bulksol\) /. risp\) /. grad\) /. vinBer // Flatten\) // Simplify\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{\(sNo[1]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(sMo[1]\), "==", RowBox[{\(\[Zeta]\ sQo[1]\), "+", RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[1]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}]}]}], ",", RowBox[{\(sNo[2]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(sMo[2]\), "==", 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]}, \(-\(\(\(-\(1\/2\)\)\ \[Zeta]\^2\ sMo[1] + 1\/6\ \[Zeta]\^3\ sQo[ 1]\)\/\[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]}, \(-\(\(\(-\(1\/2\)\)\ \[Zeta]\^2\ sMo[2] + 1\/6\ \[Zeta]\^3\ sQo[ 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[{\(\(-\@3\)\ \[ScriptF]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[1]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], "==", "0"}], ",", RowBox[{\(\(-2\)\ \[ScriptF]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[2]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[2]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], "==", "0"}]}], "}"}]], "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]}, \(-\(\(\@3\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Zeta]\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\) + \ \[ScriptCapitalD][1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][3]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(-\(\(2\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Zeta]\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\) + \ \[ScriptCapitalD][4]], u\_2[2] \[Rule] Function[{\[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], \(-\(\(\[Zeta]\^2\ \((\(-3\)\ sMo[1] + \[Zeta]\ sQo[ 1])\)\)\/\(6\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\) + \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][ 3], \(\[Zeta]\ sMo[1] - 1\/2\ \[Zeta]\^2\ sQo[1]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][ 3], \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[2]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][ 4], \(-\(\(\[Zeta]\^2\ \((\(-3\)\ sMo[2] + \[Zeta]\ sQo[ 2])\)\)\/\(6\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\) + \[ScriptCapitalD][ 5] + \[Zeta]\ \[ScriptCapitalD][ 6], \(\[Zeta]\ sMo[2] - 1\/2\ \[Zeta]\^2\ sQo[2]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ] + \[ScriptCapitalD][6]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolD // Simplify\)], "Input"], Cell[BoxData[ \({\(-\(\(\@3\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\) + \[ScriptCapitalD][ 1], \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][3], \[ScriptCapitalD][ 3], \(-\(\(2\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\) + \[ScriptCapitalD][ 4], \[ScriptCapitalD][ 5] + \[Zeta]\ \[ScriptCapitalD][6], \[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]}, \(-\(\(\(-\(1\/2\)\)\ \[Zeta]\^2\ sMo[1] + 1\/6\ \[Zeta]\^3\ sQo[ 1]\)\/\[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]}, \(-\(\(\(-\(1\/2\)\)\ \[Zeta]\^2\ sMo[2] + 1\/6\ \[Zeta]\^3\ sQo[ 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]}, \(-\(\(\@3\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Zeta]\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\) + uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, uo\_2[1] + \[Zeta]\ \[Theta]o[1]], u\_1[2] \[Rule] Function[{\[Zeta]}, \(-\(\(2\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Zeta]\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\) + uo\_1[2]], u\_2[2] \[Rule] Function[{\[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, \(u\_1[2]\)[0] == 1\/2\ \((\@3\ \(u\_1[1]\)[\[ScriptCapitalL]] + \(u\_2[ 1]\)[\[ScriptCapitalL]])\), 1\/2\ \(u\_1[1]\)[\[ScriptCapitalL]] - 1\/2\ \@3\ \(u\_2[1]\)[\[ScriptCapitalL]] + \(u\_2[2]\)[0] == 0, \(u\_1[2]\)[\[ScriptCapitalL]] == 0, \(u\_2[2]\)[\[ScriptCapitalL]] == 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, uo\_1[2] == \(\(-3\)\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa] + \ \[ScriptCapitalY]\[ScriptCapitalJ]\ \((\[ScriptCapitalL]\ \[Theta]o[1] + \@3\ \ uo\_1[1] + uo\_2[1])\)\)\/\(2\ \[ScriptCapitalY]\[ScriptCapitalJ]\), 1\/2\ \((\(-\(\(\@3\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\) + uo\_1[1] - \@3\ \((\[ScriptCapitalL]\ \[Theta]o[1] + uo\_2[1])\) + 2\ uo\_2[2])\) == 0, uo\_1[2] == \(2\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ], \[ScriptCapitalL]\ \[Theta]o[2] + uo\_2[2] == 0}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Matrice delle equazioni di vincolo", "Subsection", Evaluatable->False], Cell[BoxData[ \(\(matvin = LinearEquationsToMatrices[eqvin, cRlist] // Simplify;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[matvin\[LeftDoubleBracket]1\[RightDoubleBracket]]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", "0", "0", "0", "0"}, {"0", "1", "0", "0", "0", "0"}, {\(-\(\@3\/2\)\), \(-\(1\/2\)\), \(-\(\[ScriptCapitalL]\/2\)\), "1", "0", "0"}, {\(1\/2\), \(-\(\@3\/2\)\), \(-\(\(\@3\ \ \[ScriptCapitalL]\)\/2\)\), "0", "1", "0"}, {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "\[ScriptCapitalL]"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ColumnForm[matvin\[LeftDoubleBracket]2\[RightDoubleBracket]]\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"0"}, {"0"}, {\(-\(\(3\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\)}, {\(\(\@3\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)}, {\(\(2\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ]\)}, {"0"} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ {0, 0, Times[ Rational[ -3, 2], \[ScriptF], Power[ \[ScriptCapitalL], 3], Power[ \[ScriptCapitalY]\[ScriptCapitalJ], -1], \[Kappa]], Times[ Rational[ 1, 2], Power[ 3, Rational[ 1, 2]], \[ScriptF], Power[ \[ScriptCapitalL], 3], Power[ \[ScriptCapitalY]\[ScriptCapitalJ], -1], \[Kappa]], Times[ 2, \[ScriptF], Power[ \[ScriptCapitalL], 3], Power[ \[ScriptCapitalY]\[ScriptCapitalJ], -1], \[Kappa]], 0}], Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Length[ Transpose[matvin\[LeftDoubleBracket]1\[RightDoubleBracket]]]\)], "Input"], Cell[BoxData[ \(6\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRnull = NullSpace[matvin\[LeftDoubleBracket]1\[RightDoubleBracket]]\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRlist\)], "Input"], Cell[BoxData[ \({uo\_1[1], uo\_2[1], \[Theta]o[1], uo\_1[2], uo\_2[2], \[Theta]o[2]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Propriet\[AGrave] della soluzione", "Subsection"], Cell[BoxData[ \(\(If[Length[cRnull] > 0, StylePrint["\", FontSlant \[Rule] "\", CellFrame \[Rule] True, Background \[Rule] Hue[0.17]]];\)\)], "Input"], Cell[BoxData[ \(\(If[nv > Length[cRlist], StylePrint["\", FontSlant \[Rule] "\", CellFrame \[Rule] True, Background \[Rule] Hue[0.17]]];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Soluzione delle equazioni di vincolo", "Subsection", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(cRsol0 = LinearSolve[matvin\[LeftDoubleBracket]1\[RightDoubleBracket], matvin\[LeftDoubleBracket]2\[RightDoubleBracket]]\)], "Input"], Cell[BoxData[ \({0, 0, \(7\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \[Kappa]\)\/\[ScriptCapitalY]\ \[ScriptCapitalJ], \(2\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ], \(4\ \@3\ \[ScriptF]\ \ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ], \ \(-\(\(4\ \@3\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(Clear[cA]\)], "Input"], Cell[BoxData[ \(\(cRsol1 = Array[cA[#] &, Length[cRnull]] . cRnull;\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(cRsol = If[Length[cRnull] > 0, cRsol0 + cRsol1, cRsol0]\)], "Input"], Cell[BoxData[ \({0, 0, \(7\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \[Kappa]\)\/\[ScriptCapitalY]\ \[ScriptCapitalJ], \(2\ \[ScriptF]\ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\ \[ScriptCapitalY]\[ScriptCapitalJ], \(4\ \@3\ \[ScriptF]\ \ \[ScriptCapitalL]\^3\ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ], \ \(-\(\(4\ \@3\ \[ScriptF]\ \[ScriptCapitalL]\^2\ \ \[Kappa]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRval = Table[cRlist\[LeftDoubleBracket]i\[RightDoubleBracket] \[Rule] cRsol\[LeftDoubleBracket]i\[RightDoubleBracket], {i, 1, 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