(************** 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[ 280755, 8163]*) (*NotebookOutlinePosition[ 281416, 8186]*) (* CellTagsIndexPosition[ 281372, 8182]*) (*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. 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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]}\)], "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]}, \[ScriptB]\ \[Zeta]\ Sin[\[ScriptA]] + sNo[1]], sQ[1] \[Rule] Function[{\[Zeta]}, \[ScriptB]\ \[Zeta]\ Cos[\[ScriptA]] + sQo[1]], sM[1] \[Rule] Function[{\[Zeta]}, \(-\(1\/2\)\)\ \[ScriptB]\ \[Zeta]\^2\ Cos[\ \[ScriptA]] - \[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[ 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\[LeftDoubleBracket]i\[RightDoubleBracket]\)[\[Zeta]] /. bulksolC)\), {i, 1, Length[svar]}] // Simplify\) // Flatten\) // ColumnForm\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {\(\(sN[ 1]\)[\[Zeta]] == \[ScriptB]\ \[Zeta]\ Sin[\[ScriptA]] + \ \[ScriptCapitalC][1]\)}, {\(\(sQ[ 1]\)[\[Zeta]] == \[ScriptB]\ \[Zeta]\ Cos[\[ScriptA]] + \ \[ScriptCapitalC][2]\)}, {\(1\/2\ \[ScriptB]\ \[Zeta]\^2\ Cos[\[ScriptA]] + \[Zeta]\ \ \[ScriptCapitalC][2] + \(sM[1]\)[\[Zeta]] == \[ScriptCapitalC][3]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], Plus[ Times[ \[ScriptB], \[Zeta], Sin[ \[ScriptA]]], \[ScriptCapitalC][ 1]]], Equal[ sQ[ 1][ \[Zeta]], Plus[ Times[ \[ScriptB], \[Zeta], Cos[ \[ScriptA]]], \[ScriptCapitalC][ 2]]], Equal[ Plus[ Times[ Rational[ 1, 2], \[ScriptB], Power[ \[Zeta], 2], Cos[ \[ScriptA]]], 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]] == \[ScriptB]\ \[Zeta]\ Sin[\[ScriptA]] + sNo[1]\)}, {\(\(sQ[1]\)[\[Zeta]] == \[ScriptB]\ \[Zeta]\ Cos[\[ScriptA]] + sQo[1]\)}, {\(1\/2\ \[ScriptB]\ \[Zeta]\^2\ Cos[\[ScriptA]] + \[Zeta]\ sQo[ 1] + \(sM[1]\)[\[Zeta]] == sMo[1]\)} }, GridBaseline->{Baseline, {1, 1}}, ColumnAlignments->{Left}], ColumnForm[ { Equal[ sN[ 1][ \[Zeta]], Plus[ Times[ \[ScriptB], \[Zeta], Sin[ \[ScriptA]]], sNo[ 1]]], Equal[ sQ[ 1][ \[Zeta]], Plus[ Times[ \[ScriptB], \[Zeta], Cos[ \[ScriptA]]], sQo[ 1]]], Equal[ Plus[ Times[ Rational[ 1, 2], \[ScriptB], Power[ \[Zeta], 2], Cos[ \[ScriptA]]], 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]\)[ "+"], 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Cell["\<\ Per ogni nuova definizione, anche occasionale, occorre dare la corrispondente \ definizione della figura\ \>", "SmallText"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Dati [", StyleBox["D3", FontColor->RGBColor[0, 0, 1]], "]" }], "Subsection"], Cell["\<\ n vettore normale al piano di scorrimento di un carrello; d vettore tangenziale; {d, n} base ortonormale orientata come {e1, e2}\ \>", "SmallText"], Cell[BoxData[ \(\(Clear[d, n];\)\)], "Input"], Cell[BoxData[{ \(\(\(d[i_]\)[bd_] := e\_1;\)\), "\n", \(\(\(n[i_]\)[bd_] := e\_2;\)\)}], "Input"], Cell["\<\ Si assume che {d,n} siano identici a {e1,e2} a meno di una esplicita diversa \ definizione\ \>", "SmallText"], Cell[BoxData[{ \(\(\(d[1]\)[pi\[UGrave]] = \(-e\_1\);\)\), "\[IndentingNewLine]", \(\(\(n[1]\)[pi\[UGrave]] = \(-e\_2\);\)\)}], "Input", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Vincoli in forma scalare. Non usare esplicitamente le componenti ! Si \ pregiudicherebbe il meccanismo di sostituzione utilizzato nel calcolo della \ soluzione in termini di spostamento dalle equazioni di vincolo, oltre che \ incorrere pi\[UGrave] facilmente in errore. Utilizzare SEMPRE vincoli \ definiti secondo il modello dei vincoli standard, anche per definizioni \ occasionali. Ricordare di dare una definizione anche della figura del vincolo \ per la visualizzazione.\ \>", "SmallText"], Cell[BoxData[ \(vincoliDef := {\(cerniera[1]\)[meno], \(carrello[1]\)[ 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[ \({Cos[\[ScriptA]]\ \(ub\_d[1]\)["-"] + Sin[\[ScriptA]]\ \(ub\_n[1]\)["-"] == 0, Cos[\[ScriptA]]\ \(ub\_n[1]\)["-"] == Sin[\[ScriptA]]\ \(ub\_d[1]\)["-"], \(ub\_n[1]\)["+"] == 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[ \({\(ub\_d[1]\)["-"] \[Rule] 0, \(ub\_n[1]\)["-"] \[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[ \({\(cerniera[1]\)["-"], \(carrello[1]\)["+"]}\)], "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[ 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(sinistra, destra)", "Subsection"], Cell["\<\ Gli spostamenti al bordo ub sono descritti nella base {e1, e2}, non nelle \ basi adattate ai vincoli, utilizzando le componenti nelle basi adattate ai \ vincoli {d,n} (vedi la definizione di ub, sopra).\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[ Table[\(\((Append[\(ub[i]\)[#], \(\[Theta]b[i]\)[#]] /. vsp)\) &\)\ \ /@ \ {meno, pi\[UGrave]}, {i, 1, travi}], TableSpacing -> {4, 2, 2}]\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ {"0"}, {"0"}, {\(\(\[Theta]b[1]\)["-"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}], GridBox[{ {\(-\(ub\_d[1]\)["+"]\)}, {"0"}, {\(\(\[Theta]b[1]\)["+"]\)} }, RowSpacings->2, ColumnSpacings->1, RowAlignments->Baseline, ColumnAlignments->{Left}]} }, RowSpacings->4, ColumnSpacings->2, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {{{0, 0, \[Theta]b[ 1][ "-"]}, { Times[ -1, Subscript[ ub, d][ 1][ "+"]], 0, 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\[ScriptCapitalD][1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(\(-\(1\/12\)\)\ \[ScriptB]\ \[Zeta]\^4\ Cos[\ \[ScriptA]] + \[Zeta]\^2\ sMo[1] - 1\/3\ \[Zeta]\^3\ sQo[1]\)\/\(2\ \ \[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[{\(\(-\(1\/2\)\)\ \[ScriptB]\ \((\[ScriptCapitalL] - 2\ \[Zeta])\)\ Sin[\[ScriptA]]\), "==", FractionBox[ RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_1[1]\), "\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}], \(\[ScriptCapitalL]\^2\ \[Kappa]\)]}], ",", RowBox[{\(1\/2\ \[ScriptB]\ \((\[ScriptCapitalL] - \[Zeta])\)\ \ \[Zeta]\ Cos[\[ScriptA]]\), "==", RowBox[{"\[ScriptCapitalY]\[ScriptCapitalJ]", " ", RowBox[{ SuperscriptBox[\(u\_2[1]\), "\[Prime]\[Prime]", MultilineFunction->None], "[", "\[Zeta]", "]"}]}]}]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(spsolD = \(DSolve[eqnsp, Flatten[Table[{u\_1[i], u\_2[i]}, {i, 1, travi}]], \[Zeta], DSolveConstants \[Rule] \[ScriptCapitalD]]\)\[LeftDoubleBracket]1\ \[RightDoubleBracket] // Simplify\)], "Input"], Cell[BoxData[ \({u\_1[1] \[Rule] Function[{\[Zeta]}, \(\(-\[ScriptB]\)\ \[ScriptCapitalL]\^3\ \[Zeta]\ \ \[Kappa]\ Sin[\[ScriptA]] + \[ScriptB]\ \[ScriptCapitalL]\^2\ \[Zeta]\^2\ \ \[Kappa]\ Sin[\[ScriptA]]\)\/\(2\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \ \[ScriptCapitalD][1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(-\(\(\(-\(1\/6\)\)\ \[ScriptB]\ \ \[ScriptCapitalL]\ \[Zeta]\^3\ Cos[\[ScriptA]] + 1\/12\ \[ScriptB]\ \[Zeta]\^4\ Cos[\[ScriptA]]\)\/\(2\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\) + \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][3]]}\)], "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]]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolDO // Simplify\)], "Input"], Cell[BoxData[ \({\(\[ScriptB]\ \[ScriptCapitalL]\^2\ \[Zeta]\^2\ \[Kappa]\ Sin[\ \[ScriptA]]\)\/\(2\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + \ \(\[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[1]\)\/\[ScriptCapitalY]\ \[ScriptCapitalJ] + \[ScriptCapitalD][ 1], \(-\(\(\[Zeta]\^2\ \((\[ScriptB]\ \[Zeta]\^2\ Cos[\[ScriptA]] - 12\ sMo[1] + 4\ \[Zeta]\ sQo[ 1])\)\)\/\(24\ \[ScriptCapitalY]\[ScriptCapitalJ]\)\)\ \) + \[ScriptCapitalD][ 2] + \[Zeta]\ \[ScriptCapitalD][ 3], \(\(-\[ScriptB]\)\ \[Zeta]\^3\ Cos[\[ScriptA]] + 6\ \[Zeta]\ \ sMo[1] - 3\ \[Zeta]\^2\ sQo[1] + 6\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \ \[ScriptCapitalD][3]\)\/\(6\ \[ScriptCapitalY]\[ScriptCapitalJ]\)}\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(splist /. vinBer\) /. spsolD // Simplify\)], "Input"], Cell[BoxData[ \({\(\(-\[ScriptB]\)\ \[ScriptCapitalL]\^2\ \((\[ScriptCapitalL] - \ \[Zeta])\)\ \[Zeta]\ \[Kappa]\ Sin[\[ScriptA]] + 2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\ \[ScriptCapitalD][1]\)\/\(2\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\), \(\[ScriptB]\ \((2\ \[ScriptCapitalL] - \[Zeta])\)\ \ \[Zeta]\^3\ Cos[\[ScriptA]] + 24\ \[ScriptCapitalY]\[ScriptCapitalJ]\ \((\ \[ScriptCapitalD][2] + \[Zeta]\ \[ScriptCapitalD][3])\)\)\/\(24\ \ \[ScriptCapitalY]\[ScriptCapitalJ]\), \(\[ScriptB]\ \((3\ \[ScriptCapitalL] - \ 2\ \[Zeta])\)\ \[Zeta]\^2\ Cos[\[ScriptA]] + 12\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\ \[ScriptCapitalD][3]\)\/\(12\ \[ScriptCapitalY]\ \[ScriptCapitalJ]\)}\)], "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]}\)], "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] Function[{\[Zeta]}, \(-\(\(\(-\(1\/2\)\)\ \[ScriptB]\ \ \[ScriptCapitalL]\^2\ \[Zeta]\^2\ \[Kappa]\ Sin[\[ScriptA]] - \ \[ScriptCapitalL]\^2\ \[Zeta]\ \[Kappa]\ sNo[ 1]\)\/\[ScriptCapitalY]\[ScriptCapitalJ]\)\) + uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(\(-\(1\/12\)\)\ \[ScriptB]\ \[Zeta]\^4\ Cos[\ \[ScriptA]] + \[Zeta]\^2\ sMo[1] - 1\/3\ \[Zeta]\^3\ sQo[1]\)\/\(2\ \ \[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]}, \(\(-\[ScriptB]\)\ \[ScriptCapitalL]\^3\ \[Zeta]\ \ \[Kappa]\ Sin[\[ScriptA]] + \[ScriptB]\ \[ScriptCapitalL]\^2\ \[Zeta]\^2\ \ \[Kappa]\ Sin[\[ScriptA]]\)\/\(2\ \[ScriptCapitalY]\[ScriptCapitalJ]\) + uo\_1[1]], u\_2[1] \[Rule] Function[{\[Zeta]}, \(-\(\(\(-\(1\/6\)\)\ \[ScriptB]\ \ \[ScriptCapitalL]\ \[Zeta]\^3\ Cos[\[ScriptA]] + 1\/12\ \[ScriptB]\ \[Zeta]\^4\ Cos[\[ScriptA]]\)\/\(2\ \ \[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[#]]])\)] &)\)\n\t\t}, vincoli] // Simplify\)], "Input"], Cell[BoxData[ \({\(u\_1[1]\)[0] == 0, \(u\_2[1]\)[0] == 0, Sin[\[ScriptA]]\ \(u\_1[1]\)[\[ScriptCapitalL]] + 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