(************** 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[ 199132, 6984]*) (*NotebookOutlinePosition[ 199793, 7007]*) (* CellTagsIndexPosition[ 199749, 7003]*) (*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[ \("C:\\Wrk\\Corsi\\Scost\\esercizi\\7-travi\\7-03\\outmath"\)], "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[ \(\(travi = 1;\)\)], "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;\)\)], "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];\)\)], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[BoxData[{ \(YA[1] := \[ScriptCapitalY]\[ScriptCapitalA]\ \ \), \ "\[IndentingNewLine]", \(YJ[1] := \[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)\), \(asseO[i]\)[\[Zeta] + maxL\/5]}} /. datiO\)], "Input"], Cell[BoxData[ \(\(vecOa2[ i_]\)[\[Zeta]_] := {{\(asseO[i]\)[\[Zeta]], \(asseO[i]\)[\[Zeta]] + maxL\/5\ \ a\_2[i]}, {\(asseO[i]\)[\[Zeta]] + 1\/5\ maxL\ a\_2[ i] + \(\(\(maxL\)\(\ \)\)\/15\) \((\(-\(1\/2\)\)\ a\_1[i] - 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\[Zeta]\ \[ScriptCapitalC][2] + \[ScriptCapitalC][3]]}\)], "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]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cNQM = Table[{sNo[i], sQo[i], sMo[i]}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({sNo[1], sQo[1], sMo[1]}\)], "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]}\)], "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]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bulksol = bulksolC /. fromCtoNQM\)], "Input"], Cell[BoxData[ \({sN[1] \[Rule] Function[{\[Zeta]}, sNo[1]], sQ[1] \[Rule] Function[{\[Zeta]}, \[ScriptB]\ \((\(-\(\[ScriptCapitalL]\/2\)\) + \ \[Zeta])\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]] + sQo[1]], sM[1] \[Rule] Function[{\[Zeta]}, \(-\[ScriptB]\)\ \[Zeta]\ \((\(-\(\ \[ScriptCapitalL]\/2\)\) + \[Zeta])\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) \ + \[Zeta]] + 1\/8\ \[ScriptB]\ \((\(-\[ScriptCapitalL]\^2\) + 4\ \[Zeta]\^2)\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \ \[Zeta]] - \[Zeta]\ sQo[1] + sMo[1]]}\)], "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]", "]"}], "==", \(\[ScriptB]\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \ \[Zeta]]\)}]}, { RowBox[{ RowBox[{\(\(sQ[1]\)[\[Zeta]]\), "+", RowBox[{ SuperscriptBox[\(sM[1]\), "\[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]], Times[ \[ScriptB], UnitStep[ Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]]]]], Equal[ Plus[ sQ[ 1][ \[Zeta]], Derivative[ 1][ sM[ 1]][ \[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]] == \[ScriptB]\ \((\(-\(\[ScriptCapitalL]\/2\)\ \) + \[Zeta])\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]] + \ \[ScriptCapitalC][2]\)}, {\(1\/8\ \[ScriptB]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\^2\ \ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]] + \[Zeta]\ \ \[ScriptCapitalC][2] + \(sM[1]\)[\[Zeta]] == \[ScriptCapitalC][3]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], \[ScriptCapitalC][ 1]], Equal[ sQ[ 1][ \[Zeta]], Plus[ Times[ \[ScriptB], Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]], UnitStep[ Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]]]], \[ScriptCapitalC][ 2]]], Equal[ Plus[ Times[ Rational[ 1, 8], \[ScriptB], Power[ Plus[ \[ScriptCapitalL], Times[ -2, \[Zeta]]], 2], UnitStep[ Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]]]], Times[ \[Zeta], \[ScriptCapitalC][ 2]], sM[ 1][ \[Zeta]]], \[ScriptCapitalC][ 3]]}], 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] + \[ScriptB]\ \((\(-\(\[ScriptCapitalL]\/2\)\) + \ \[Zeta])\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]]\)}, {\(\[Zeta]\ sQo[1] + 1\/8\ \[ScriptB]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\^2\ \ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]] + \(sM[1]\)[\[Zeta]] == sMo[1]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], sNo[ 1]], Equal[ sQ[ 1][ \[Zeta]], Plus[ sQo[ 1], Times[ \[ScriptB], Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]], UnitStep[ Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]]]]]], Equal[ Plus[ Times[ \[Zeta], sQo[ 1]], Times[ Rational[ 1, 8], \[ScriptB], Power[ Plus[ \[ScriptCapitalL], Times[ -2, \[Zeta]]], 2], UnitStep[ Plus[ Times[ Rational[ -1, 2], \[ScriptCapitalL]], \[Zeta]]]], sM[ 1][ \[Zeta]]], sMo[ 1]]}], 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]\)[ "-"]}\)], "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]\)[ "-"]}\)], "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[ \({\(sb\_d[1]\)["+"], \(sb\_n[1]\)["+"], \(mb[1]\)["+"], \(sb\_d[1]\)[ "-"], \(sb\_n[1]\)["-"], \(mb[1]\)["-"]}\)], "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[""], "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 := {\(incastro[1]\)[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[ \({\(ub\_d[1]\)["-"] == 0, \(ub\_n[1]\)["-"] == 0, \(\[Theta]b[1]\)["-"] == 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[ \({\(\[Theta]b[1]\)["-"] \[Rule] 0, \(ub\_d[1]\)["-"] \[Rule] 0, \(ub\_n[1]\)["-"] \[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[ \({\(incastro[1]\)["-"]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[ vincoliDef /. {carrello \[Rule] \((\((Null\ &)\)\ &)\), incastro \[Rule] \((\((Null\ &)\)\ &)\), cerniera \[Rule] \((\((Null\ &)\)\ &)\), perno \[Rule] coll, saldatura \[Rule] coll}, {Null}]\)], "Input"], Cell[BoxData[ \({}\)], "Output"] }, Open ]], Cell["\<\ Calcolo della posizione della estremit\[AGrave] 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] - 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\[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]\)["+"] + \(sb\_d[1]\)["-"]\ \(wb\_d[1]\)[ "-"] + \(sb\_d[1]\)["+"]\ \(wb\_d[1]\)["+"] + \(sb\_n[1]\)[ "-"]\ \(wb\_n[1]\)["-"] + \(sb\_n[1]\)["+"]\ \(wb\_n[1]\)[ "+"]\)], "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]\)[ "+"] + \((\(sN[1]\)[0] + \(sb\_d[1]\)["-"])\)\ \(wb\_d[1]\)[ "-"] + \((\(-\(sN[1]\)[\[ScriptCapitalL]]\) + \(sb\_d[1]\)[ "+"])\)\ \(wb\_d[1]\)[ "+"] + \((\(sQ[1]\)[0] + \(sb\_n[1]\)["-"])\)\ \(wb\_n[1]\)[ "-"] + \((\(-\(sQ[1]\)[\[ScriptCapitalL]]\) + \(sb\_n[1]\)[ "+"])\)\ \(wb\_n[1]\)["+"]\)], "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, \(\[Theta]b[1]\)["-"] == 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, \(\[Omega]b[1]\)["-"] == 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[ \({\(\[Omega]b[1]\)["-"] \[Rule] 0, \(wb\_d[1]\)["-"] \[Rule] 0, \(wb\_n[1]\)["-"] \[Rule] 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ambdv = Complement[ambd /. vam, {0}]\)], "Input"], Cell[BoxData[ \({\(\[Omega]b[1]\)["+"], \(wb\_d[1]\)["+"], \(wb\_n[1]\)[ "+"]}\)], "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[ \(\((\(mb[1]\)["+"] - 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Si tratta in genere della selezione \ di un sottoinsieme descritto da alcuni parametri, come f ad esempio, il cui \ valore verr\[AGrave] assegnato tra i dati numerici [ l'uso caratteri script \ per i parametri rende tutto molto pi\[UGrave] leggibile]. I DATI VANNO \ ASSEGNATI IN FORMA DI EQUAZIONI (per via delle condizioni di continuit\ \[AGrave])\ \>", "SmallText"], Cell[BoxData[ \(\(forze = {\ };\)\)], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Una assegnazione esplicita dei dati sulle forze \[EGrave] la lista \ seguente, data qui come esempio e non assegnata a ", StyleBox["forze", FontFamily->"Courier New"], ". 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\[Zeta]\^2\ sQo[1] + 2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\ \[ScriptCapitalD][3])\)\)\/\(48\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolD // Simplify\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][ 1], \(\(-\[ScriptB]\)\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\^4\ \ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]] + 8\ \((\[ScriptB]\ \ \[ScriptCapitalL]\ \[Zeta]\^2\ \((\(-9\)\ \[ScriptCapitalL] + 4\ \[Zeta])\) + \ 48\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \((\[ScriptCapitalD][2] + \[Zeta]\ \ \[ScriptCapitalD][3])\))\)\)\/\(384\ \[ScriptCapitalY]\[ScriptCapitalJ]\), \ \(-\(\(\[ScriptB]\ \[ScriptCapitalL]\^3\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\)\/\(48\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\ \)\) + \(\[ScriptB]\ \[ScriptCapitalL]\^2\ \((\[ScriptCapitalL]\^2 - 4\ \ \[Zeta]\^2)\)\ DiracDelta[\[ScriptCapitalL] - 2\ \[Zeta]]\)\/\(32\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) - 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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]}\)], "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]}\)], "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]}\)], "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]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRlist = cDlist /. fromDtoU\)], "Input"], Cell[BoxData[ \({uo\_1[1], uo\_2[1], \[Theta]o[1]}\)], "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] 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\[ScriptCapitalY]\[ScriptCapitalJ]\) + uo\_2[1] + \[Zeta]\ \[Theta]o[1]]}\)], "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]}, uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(-\(\(3\ \[ScriptB]\ \[ScriptCapitalL]\^2\ \ \[Zeta]\^2\)\/\(16\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\) + \(\[ScriptB]\ \ \[ScriptCapitalL]\ \[Zeta]\^3\)\/\(12\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \ \(\[ScriptB]\ \[ScriptCapitalL]\^3\ \((\(-\(\[ScriptCapitalL]\/2\)\) + \ \[Zeta])\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]]\)\/\(48\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) - \(\[ScriptB]\ \[ScriptCapitalL]\^2\ \ \((\(-\[ScriptCapitalL]\^2\) + 4\ \[Zeta]\^2)\)\ UnitStep[\(-\(\ \[ScriptCapitalL]\/2\)\) + \[Zeta]]\)\/\(64\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) + \(\[ScriptB]\ \[ScriptCapitalL]\ \ \((\(-\[ScriptCapitalL]\^3\) + 8\ \[Zeta]\^3)\)\ UnitStep[\(-\(\ \[ScriptCapitalL]\/2\)\) + \[Zeta]]\)\/\(96\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) - \(\[ScriptB]\ \((\(-\[ScriptCapitalL]\^4\) + 16\ \ \[Zeta]\^4)\)\ UnitStep[\(-\(\[ScriptCapitalL]\/2\)\) + \[Zeta]]\)\/\(384\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\) + uo\_2[1] + \[Zeta]\ \[Theta]o[1]]}\)], "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[#]]])\)] 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\[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[ \({0, 0, 0}\)], "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[ \({0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cRval = Table[cRlist\[LeftDoubleBracket]i\[RightDoubleBracket] \[Rule] 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