(************** 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[ 216370, 7066]*) (*NotebookOutlinePosition[ 217031, 7089]*) (* CellTagsIndexPosition[ 216987, 7085]*) (*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[ \("C:\\Wrk\\Corsi\\Scost\\esercizi\\7-travi\\7-01a\\outmath"\)], "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] = \[Pi];\)\)], "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). 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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]", \(\(UnitStep[\[ScriptCapitalL]] = 1;\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Protect[UnitStep]\)], "Input"], Cell[BoxData[ \({"UnitStep"}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Soluzione generale delle equazioni differenziali di bilancio (bulk)\ \>", "Section", Evaluatable->False], Cell[CellGroupData[{ Cell["\<\ Descrittori della tensione (forza normale, taglio e momento) e integrali \ delle equazioni di bilancio\ \>", "Subsection"], Cell[BoxData[ \(\(s[ i_]\)[\[Zeta]_] := \(sN[i]\)[\[Zeta]]\ a\_1[ i] + \(sQ[i]\)[\[Zeta]]\ a\_2[i]\)], "Input"], Cell[BoxData[ \(\(m[i_]\)[\[Zeta]_] := \(sM[i]\)[\[Zeta]]\)], "Input"], 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[ \({sN[1], sQ[1], sM[1]}\)], "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]}, \(-\[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]}, sQo[1]], sM[1] \[Rule] Function[{\[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]", "]"}], "==", "0"}]}, { 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]], 0], 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]] == \[ScriptCapitalC][2]\)}, {\(\[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]], \[ScriptCapitalC][ 2]], Equal[ Plus[ 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]\)}, {\(\[Zeta]\ sQo[1] + \(sM[1]\)[\[Zeta]] == sMo[1]\)} }, 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]]}], 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 := {\(incastro[1]\)[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, \(\[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 ]] }, Open ]], Cell[CellGroupData[{ Cell["Definizioni 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\(\[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 ]] }, 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\(eqbilbd = \((#1 == 0 &)\) /@ Table[Coefficient[potbdv, ambdv\[LeftDoubleBracket]j\[RightDoubleBracket]], {j, 1, Length[ambdv]}]\)], "Input"], Cell[BoxData[ \({\(mb[1]\)["-"] + \(sM[1]\)[0] == 0, \(-\(sN[1]\)[0]\) + \(sb\_d[1]\)["-"] == 0, \(-\(sQ[1]\)[0]\) + \(sb\_n[1]\)["-"] == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(eqbilbd /. bulksol // Simplify\)], "Input"], Cell[BoxData[ \({sMo[1] + \(mb[1]\)["-"] == 0, \(sb\_d[1]\)["-"] == sNo[1], \(sb\_n[1]\)["-"] == sQo[1]}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Matrice delle equazioni di bilancio al bordo", "Subsection", Evaluatable->False], Cell["\<\ Vengono elencate le costanti di integrazione presenti nelle espressioni \ calcolate (per sicurezza vengono utilizzate le espressioni con le costanti C)\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(cNQM\)], "Input"], Cell[BoxData[ \({sNo[1], sQo[1], sMo[1]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cNQMb = 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forze al \ bordo reattive)\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(nv = Length[vincoli]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["\<\ numero di descrittori degli atti di moto vincolati (o numero descrittori \ delle forze al bordo attive)\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(nf = Length[ambdv]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell["controlli", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \({nf == Length[matbilbd\[LeftDoubleBracket]1\[RightDoubleBracket]], nc == no, nf == 2 no - nv}\)], "Input"], Cell[BoxData[ \({True, True, True}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(rango = nc - Length[ If[Length[matbilbd\[LeftDoubleBracket]1\[RightDoubleBracket]] > 0, NullSpace[matbilbd\[LeftDoubleBracket]1\[RightDoubleBracket]], 0]]\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Propriet\[AGrave] dei vincoli e delle forze attive", 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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 = {\(sb[1]\)[meno] . e\_2 \[Equal] \(-\[ScriptF]\)\ };\)\)], "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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della sostituzione delle soluzioni delle equazioni di bilancio al bordo\ \ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(eqnspO = \(\(\(\(Table[{\(sN[i]\)[\[Zeta]] == \(sNf[i]\)[\[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]", "]"}]}]}]}]}], "}"}]], "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]]}\)], "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[{ FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ 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i]\)[\[Zeta]]}, {i, 1, travi}] // Flatten\)], "Input"], Cell[BoxData[ \({\(u\_1[1]\)[\[Zeta]], \(u\_2[1]\)[\[Zeta]], \(\[Theta][ 1]\)[\[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]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolD // Simplify\)], "Input"], Cell[BoxData[ \({\[ScriptCapitalD][ 1], \(\[ScriptF]\ \[Zeta]\^3\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) + \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][ 3], \(\[ScriptF]\ \[Zeta]\^2\)\/\(2\ 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\(u\_2[i]\)[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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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[ \({\(u\_1[1]\)[\[ScriptCapitalL]] == 0, \(u\_2[1]\)[\[ScriptCapitalL]] == 0, \(\[Theta][1]\)[\[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, \(\[ScriptF]\ \[ScriptCapitalL]\^3\)\/\(6\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\) + 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\[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]\ 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