(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.3' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 166540, 4151] NotebookOptionsPosition[ 159320, 4043] NotebookOutlinePosition[ 159843, 4063] CellTagsIndexPosition[ 159800, 4060] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ In-plane free vibration analysis of an inclined taut cable with a point mass\ \>", "Title", CellChangeTimes->{{3.8451816061502457`*^9, 3.845181668618754*^9}, { 3.8451841913333673`*^9, 3.8451842209582434`*^9}, {3.8510737396978474`*^9, 3.8510737749946957`*^9}, 3.8510738098788295`*^9, {3.8521970128807325`*^9, 3.8521971224430265`*^9}, {3.8521971616929693`*^9, 3.8521972045991507`*^9}, {3.852197237364731*^9, 3.852197253786562*^9}, { 3.8521972874445553`*^9, 3.8521972913664293`*^9}, {3.852200993358186*^9, 3.852201005186269*^9}, {3.852285474429924*^9, 3.8522854748829308`*^9}, { 3.888312175648477*^9, 3.888312195465354*^9}, {3.8915763243331537`*^9, 3.891576326981943*^9}, {3.911884252851575*^9, 3.911884258362969*^9}, { 3.9464608187491817`*^9, 3.946460832387149*^9}}, TextAlignment->Center,ExpressionUUID->"0f1abdc2-3256-4949-b403-5ac5bf2b1e3e"], Cell["Acta Mechanica", "Subtitle", CellChangeTimes->{{3.94646106657581*^9, 3.946461069455846*^9}}, TextAlignment->Center,ExpressionUUID->"408b9862-2226-4f81-99df-54363aa54aa5"], Cell[TextData[{ "Moritz ", Cell[BoxData[ FormBox[ SuperscriptBox["Patreider", RowBox[{"1", ",", "2"}]], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "f8e6a886-283d-4520-9aee-40e0418d9b19"], ", Markus ", Cell[BoxData[ FormBox[ SuperscriptBox["Wenin", "2"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "42882462-50fc-4551-8b20-81cc68e016ba"], ", Christoph ", Cell[BoxData[ FormBox[ SuperscriptBox["Adam", RowBox[{"1", ",", "*"}]], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "d5aebd6e-49d7-43e9-92ae-bb6d3df5a5cb"], " and Thomas ", Cell[BoxData[ FormBox[ SuperscriptBox["Furtm\[UDoubleDot]ller", "1"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "c7f790ad-02fd-4052-9f93-ed31cb700e9d"] }], "Author", CellChangeTimes->{{3.946460876392438*^9, 3.946460918614237*^9}, { 3.946460980896133*^9, 3.9464610055225267`*^9}},ExpressionUUID->"d8014d63-d591-4442-b8b1-\ a20cf8386c08"], Cell[TextData[{ Cell[BoxData[ FormBox[ SuperscriptBox["", "1"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "44f25c05-5435-466b-8bab-8173e533d6ac"], "Unit of Applied Mechanics, University of Innsbruck, Technikerstr. 13, \ Innsbruck, 6020, Austria\n", Cell[BoxData[ FormBox[ SuperscriptBox["", "2"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "6c052a40-1dcb-4d2a-9979-b90a3f14ca5b"], "CPE Computational Physics and Engineering, Weingartnerstr. 28, Lana, 39011, \ Italy" }], "Affiliation", CellChangeTimes->{{3.9464609266579514`*^9, 3.946460974392508*^9}, { 3.946461009693512*^9, 3.9464610238047543`*^9}},ExpressionUUID->"52a078aa-e5a0-4eac-a987-\ f0aee3d37e7e"], Cell["*Corresponding author. E-mail: christoph.adam@uibk.ac.at", "Text", CellChangeTimes->{{3.9464610852625093`*^9, 3.946461106052409*^9}},ExpressionUUID->"5a58cf02-ddc5-4275-a6c0-\ f56bf0ae549c"], Cell["\<\ An analytical theory for predicting the free in-plane vibrations of an \ inclined taut cable carrying a \\txtr{point} mass is presented in this paper. \ A linearized eigenvalue formulation of the problem is derived. It is based on \ the nonlinear equations of motion and the quasi-static stretching assumption, \ which assumes that the dynamic tension is piecewise constant to the left and \ right of the point mass. By neglecting the equation of motion for the \ longitudinal vibrations in favor of the constitutive cable equation, the \ equation governing the lateral vibrations can be solved using its boundary \ conditions. Two coupled linear equations, whose non-trivial solution gives \ the natural frequencies of the system, can be constructed from the jump and \ continuity conditions in the longitudinal direction. The comparison of the \ results with those of a known Galerkin procedure shows that the analytical \ theory is sufficiently accurate in the applicable parameter range. These \ results reproduce the curve-veering and the elastic mode transition phenomena \ known from previous studies. As such, the theory presented here extends the \ range of applicability of previous analytical cable theories to include \ inclined cables with point masses as well.\ \>", "Abstract", CellChangeTimes->{{3.9464612028098803`*^9, 3.946461266039629*^9}},ExpressionUUID->"ff58b1bf-22db-40ec-96c7-\ 3877b00f7c59"], Cell[TextData[{ "The addition of the letter ", Cell[BoxData[ FormBox["b", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "cebd0bd7-d8ec-40f5-9da7-5089fdecbb2b"], " to any variable corresponds to the dimensionless version of said variable \ that is represented by a bar in the paper. 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