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Amplitudes are displayed for a wide range of masses \ and stiffnesses .", FontWeight->"Bold"] }], "Text"], Cell[TextData[StyleBox["Haiduke Sarafian\nPennsylvania State University\n\ Commonwealth College\nYork, PA 17403", FontSize->14]], "Text", TextAlignment->Left, TextJustification->0], Cell["\<\ A recent study [Clark and Hill 1999] measured the effect of hardening of one \ of the ideal springs of a double-mass double-spring system on vibrating \ amplitudes of masses. Their analysis is confined to a special case of \ identical unit mass. Although this restriction simplifies the formulation of \ the problem it takes away the generality of the analysis. By relaxing this \ condition we reformulate the problem and show the difference between the \ masses plays a crucial role in controlling the vibrations. For a wide range \ of masses and stiffnesses we display the amplitudes. We provide visual means \ to relate the derived theoretical equations to actual situations. We show \ without graphical presentations most of the amplitudes are intuitively \ impossible to predict.\ \>", "Text"], Cell[TextData[{ "Figure 1 shows a double-mass double-spring system. The two ideal springs \ with un-stretched length of ", Cell[BoxData[ \(TraditionalForm\`L\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`L\_2\)]], " assumed to have stiffness of ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], " are connected in series to two masses ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], " and are hung vertically. The static vertical equilibrium position of the \ masses measured from the support are ", Cell[BoxData[ \(TraditionalForm\`y\_10\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\_20\)]], ". These are measured in a coordinate system with the origin at the \ support with positive direction consistent with the direction of gravity - \ from north to south. The system is put in vertical vibrations by releasing \ the pulled down masses from equilibria. The vertical position of the masses \ at any given time from the support are assumed ", Cell[BoxData[ \(TraditionalForm\`y\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\_2\)]], ". 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A double-mass double-spring system with spring \ stiffness of ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox[\(k\_\(\(\ \)\(1\)\)\), FontWeight->"Bold"], TraditionalForm]]], StyleBox[" and ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox[\(k\_\(\(\ \)\(2\)\)\), FontWeight->"Bold"], TraditionalForm]]], StyleBox[". The ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)], FontWeight->"Bold"], " ", StyleBox["and ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)], FontWeight->"Bold"], " ", StyleBox["are the displacements from static equilibria at any given time.", FontWeight->"Bold"] }], "Text"], Cell[BoxData[ RowBox[{ StyleBox["{", SpanMaxSize->Infinity], GridBox[{ {\(F\_1 = \(-\ k\_1\) \((y\_1 - L\_1)\) + \(k\_2\) \((y\_2 - y\_1 - L\_2)\) + \(m\_1\) g\)}, {\(F\_2 = \(-\ k\_2\) \((y\_2 - y\_1 - L\_2)\) + \(m\_2\) g\)} }]}]], "Text", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "The unbalanced forces acting on the masses are the cause of the \ accelerations - we equate the rhs of the these equations to ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], Cell[BoxData[ \(TraditionalForm\`\(d\^2\/dt\^2\) y\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], Cell[BoxData[ \(TraditionalForm\`\(d\^2\/dt\^2\) y\_2\)]], " respectively. Since the objective of our analysis is to study the \ vibrating amplitude of the masses, we introduce two new coordinates: ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(y\_1\), "TraditionalForm"], "-", \(y\_10\)}], TraditionalForm]]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(y\_2\), "TraditionalForm"], "-", \(y\_20\)}], TraditionalForm]]], ". These are displacements of the masses from their respective static \ equilibria. The last equations in terms of ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], " yield:" }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{"{", GridBox[{ {\(\(m\_1\) \(d\^2\/dt\^2\) \[Xi]\_1 = \(-\((k\_1 + k\_2)\)\) \[Xi]\_1 + \(k\_2\) \[Xi]\_2\)}, {\(\(m\_2\) \(d\^2\/dt\^2\) \[Xi]\_2 = \(k\_2\) \[Xi]\_1 - \(k\_2\ \) \[Xi]\_2\)} }]}], SpanMaxSize->Infinity]], "Text"], Cell[TextData[{ "According to linear transformation ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\({y\_1\), "TraditionalForm"], ",", \(y\_2\)}], TraditionalForm]]], "} \[DoubleLongRightArrow] ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\({\[Xi]\_1\), "TraditionalForm"], ",", \(\[Xi]\_2\)}], TraditionalForm]]], "} the equations of motion are considerably simplified - constant \ quantities ", Cell[BoxData[ \(TraditionalForm\`L\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`L\_2\)]], " and gravitational acceleration g are all dropped. " }], "Text"], Cell[TextData[{ "The last set of equations in terms of the natural angular frequency of \ individual spring-mass system ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_i\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\@\(K\_i\/m\_i\)\)]], " and the ratio of the masses \[Mu] = ", Cell[BoxData[ \(TraditionalForm\`m\_2\/m\_1\)]], " can be written as " }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{"{", GridBox[{ {\(\(d\^2\/dt\^2\) \[Xi]\_1 = \(-\((\[Omega]\_1\%2 + \[Mu]\ \ \[Omega]\_2\%2)\)\)\ \[Xi]\_1 + \[Mu]\ \(\[Omega]\_2\%2\) \[Xi]\_2\)}, {\(\(d\^2\/dt\^2\) \[Xi]\_2 = \(\[Omega]\_2\%2\) \[Xi]\_1 - \(\ \[Omega]\_2\%2\) \[Xi]\_2\)} }]}], SpanMaxSize->Infinity]], "Text"], Cell[TextData[{ "These are two coupled linear second-order Ordinary Differential Equations \ (ODEs) and one needs to adopt a method to solve them - the solutions of these \ equations are the vibration amplitudes. Traditionary, one replaces these \ equations with four coupled linear first-order ODEs and forms an eigenvalue \ equation. The vibration amplitudes are linked to the eigenvalues - this is \ discussed in method I. By applying ", StyleBox["Mathematica,", FontSlant->"Italic"], " I have shown that the amplitudes deduced by this method converge to those \ derived analytically by Clark and Hill. Rather than relying solely on \ traditional approach, I present alternative solutions. It is shown that the \ coupled equations of motion can be decoupled, resulting in two independent \ fourth-order ODEs - this is discussed in method II. By applying ", StyleBox["Mathematica,", FontSlant->"Italic"], " the consistency of these two methods has been confirmed. These two \ independent mathematical approaches provide two different technical views. \ They also establish a basis for a reliable and quick solution based on the \ direct application of ", StyleBox["Mathematica ", FontSlant->"Italic"], "using", " the un-manipulated equations of motion", StyleBox[" ", FontSlant->"Italic"], "- as discussed in method III. The details of these methods are presented \ below. " }], "Text"], Cell[TextData[StyleBox["Method I - the reduction procedure", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Following the reduction prescription, we introduce two new variables; ", Cell[BoxData[ \(TraditionalForm\`\[Eta]\_1\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(d\/dt\), "TraditionalForm"], \(\[Xi]\_1\)}], TraditionalForm]]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Eta]\_2\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(d\/dt\), "TraditionalForm"], \(\[Xi]\_2\)}], TraditionalForm]]], " - these are the speeds of the masses about their respective static \ equilibria. The above set of equations of motion in terms of ", Cell[BoxData[ \(TraditionalForm\`{\[Xi]\_1, \[Xi]\_2, \)]], Cell[BoxData[ \(TraditionalForm\`\[Eta]\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Eta]\_2\)]], "} yields a set of four coupled linear first-order ordinary differential \ equations with constant coefficients. " }], "Text"], Cell[BoxData[ StyleBox[ RowBox[{"{", GridBox[{ {\(\(d\/dt\) \[Xi]\_1 = \[Eta]\_1\)}, {\(\(d\/dt\) \[Xi]\_2 = \[Eta]\_2\)}, {\(\(d\/dt\) \[Eta]\_1 = \(-\((\[Omega]\_1\%2 + \[Mu]\ \ \[Omega]\_2\%2)\)\)\ \[Xi]\_1 + \[Mu]\ \(\[Omega]\_2\%2\) \[Xi]\_2\)}, {\(\(d\/dt\) \[Eta]\_2 = \(\[Omega]\_2\%2\) \[Xi]\_1 - \(\[Omega]\ \_2\%2\) \[Xi]\_2\)} }]}], SpanMaxSize->Infinity]], "Text"], Cell["Rewriting these equations in a matrix form we have ", "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(d\/dt\), RowBox[{"(", GridBox[{ {\(\[Xi]\_1\)}, {\(\[Xi]\_2\)}, {\(\[Eta]\_1\)}, {\(\[Eta]\_2\)} }], ")"}]}], "=", RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "1", "0"}, {"0", "0", "0", "1"}, {\(-\((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2)\)\), \(\[Mu]\ \ \[Omega]\_2\%2\), "0", "0"}, {\(\[Omega]\_2\%2\), \(-\ \[Omega]\_2\%2\), "0", "0"} }], ")"}], RowBox[{"(", GridBox[{ {\(\[Xi]\_1\)}, {\(\[Xi]\_2\)}, {\(\[Eta]\_1\)}, {\(\[Eta]\_2\)} }], ")"}]}]}]], "Text"], Cell[TextData[{ "The last equation can be written in compact form ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(d\/dt\), "TraditionalForm"], \(Q\& \[Rule] \)}], TraditionalForm]]], " = \[CapitalOmega] ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], " where ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], " = ", Cell[BoxData[ RowBox[{"(", GridBox[{ {\(\[Xi]\_1\)}, {\(\[Xi]\_2\)}, {\(\[Eta]\_1\)}, {\(\[Eta]\_2\)} }], ")"}]]], " and \[CapitalOmega] = ", Cell[BoxData[ RowBox[{"(", GridBox[{ {"0", "0", "1", "0"}, {"0", "0", "0", "1"}, {\(-\((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2)\)\), \(\[Mu]\ \ \[Omega]\_2\%2\), "0", "0"}, {\(\[Omega]\_2\%2\), \(-\ \[Omega]\_2\%2\), "0", "0"} }], ")"}]]], ". The formal solution of this linear differential equation is ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], "(t) = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Sum]\+\(i = 1\)\%4 c\_i\ \(\(K\_i\)\& \[Rule] \)\), "TraditionalForm"], \(\[ExponentialE]\^\(\(\[Lambda]\_i\) t\)\)}], TraditionalForm]]], " i.e. the general solution is a linear combination of the particular \ solutions. The ", Cell[BoxData[ \(TraditionalForm\`\(K\& \[Rule] \)\_i\)]], " are 4x1 eigenvectors, ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_i\)]], " are the corresponding eigenvalues of \[CapitalOmega] and ", Cell[BoxData[ \(TraditionalForm\`c\_i\)]], " are constants - all three quantities are time-independent. Values of ", Cell[BoxData[ \(TraditionalForm\`c\_i\)]], " can be determined by applying a set of desired initial conditions." }], "Text"], Cell[TextData[{ "With the help of ", StyleBox["Mathematica", FontSlant->"Italic"], " we search for the eigenvalues and eigenvectors of \[CapitalOmega]. " }], "Text"], Cell[BoxData[ \(Off[General::Spell, General::Spell1]\)], "Input"], Cell[BoxData[ RowBox[{"\[CapitalOmega]", ":=", RowBox[{"(", GridBox[{ {"0", "0", "1", "0"}, {"0", "0", "0", "1"}, {\(-\((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2)\)\), \(\[Mu]\ \ \[Omega]\_2\%2\), "0", "0"}, {\(\[Omega]\_2\%2\), \(-\[Omega]\_2\%2\), "0", "0"} }], ")"}]}]], "Input"], Cell["The eigenvalues of \[CapitalOmega] are:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eigval = Eigenvalues[\[CapitalOmega]] // Simplify\)], "Input"], Cell[BoxData[ \({\(-\(\@\(\(-\[Omega]\_1\%2\) - \((1 + \[Mu])\)\ \[Omega]\_2\%2 - \ \@\(\(-4\)\ \[Omega]\_1\%2\ \[Omega]\_2\%2 + \((\[Omega]\_1\%2 + \((1 + \ \[Mu])\)\ \[Omega]\_2\%2)\)\^2\)\)\/\@2\)\), \@\(\(-\[Omega]\_1\%2\) - \((1 + \ \[Mu])\)\ \[Omega]\_2\%2 - \@\(\(-4\)\ \[Omega]\_1\%2\ \[Omega]\_2\%2 + \((\ \[Omega]\_1\%2 + \((1 + \[Mu])\)\ \[Omega]\_2\%2)\)\^2\)\)\/\@2, \(-\(\@\(\(-\ \[Omega]\_1\%2\) - \((1 + \[Mu])\)\ \[Omega]\_2\%2 + \@\(\(-4\)\ \ \[Omega]\_1\%2\ \[Omega]\_2\%2 + \((\[Omega]\_1\%2 + \((1 + \[Mu])\)\ \ \[Omega]\_2\%2)\)\^2\)\)\/\@2\)\), \@\(\(-\[Omega]\_1\%2\) - \((1 + \[Mu])\)\ \ \[Omega]\_2\%2 + \@\(\(-4\)\ \[Omega]\_1\%2\ \[Omega]\_2\%2 + \ \((\[Omega]\_1\%2 + \((1 + \[Mu])\)\ \[Omega]\_2\%2)\)\^2\)\)\/\@2}\)], \ "Output"] }, Open ]], Cell[TextData[{ "Inspection of these eigenvalues reveals irrespective the value of \[Mu] \ all the eigenvalues are purely complex quantities. For a special case of ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(m\_1\), "TraditionalForm"], "=", \(m\_2\)}], TraditionalForm]]], "= 1 these are reduced to the values derived by Clark and Hill." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eigvalspe = eigval /. {\[Omega]\_1 -> \@k\_1, \[Omega]\_2 -> \@k\_2, \[Mu] \[Rule] 1} // Simplify\)], "Input"], Cell[BoxData[ \({\(-\(\@\(\(-k\_1\) - 2\ k\_2 - \@\(k\_1\%2 + 4\ k\_2\%2\)\)\/\@2\)\), \ \@\(\(-k\_1\) - 2\ k\_2 - \@\(k\_1\%2 + 4\ k\_2\%2\)\)\/\@2, \(-\(\@\(\(-k\_1\ \) - 2\ k\_2 + \@\(k\_1\%2 + 4\ k\_2\%2\)\)\/\@2\)\), \@\(\(-k\_1\) - 2\ k\_2 \ + \@\(k\_1\%2 + 4\ k\_2\%2\)\)\/\@2}\)], "Output"] }, Open ]], Cell["\<\ Similarly we identify the eigenvectors of \[CapitalOmega] - because of the \ lengthiness of the expressions we suppress the output.\ \>", "Text"], Cell[BoxData[ \(\(eigvec = Eigenvectors[\[CapitalOmega]] // Simplify;\)\)], "Input"], Cell[TextData[{ "For a special case of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], "= ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], "= 1 these are reduced to Clark and Hill expressions as well. We show only \ one of these special eigenvectors" }], "Text"], Cell[BoxData[ \(\(eigvecspe = eigvec /. {\[Omega]\_1 -> \@k\_1, \[Omega]\_2 -> \@k\_2, \[Mu] \ \[Rule] 1};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(eigvecspe\[LeftDoubleBracket]1\[RightDoubleBracket] // Simplify\)], "Input"], Cell[BoxData[ \({\(k\_1 + \@\(k\_1\%2 + 4\ k\_2\%2\)\)\/\(\@2\ k\_2\ \@\(\(-k\_1\) - 2\ \ k\_2 - \@\(k\_1\%2 + 4\ k\_2\%2\)\)\), \(-\(\@2\/\@\(\(-k\_1\) - 2\ k\_2 - \@\ \(k\_1\%2 + 4\ k\_2\%2\)\)\)\), \(-\(\(k\_1\%2 + k\_1\ \((k\_2 + \@\(k\_1\%2 + 4\ k\_2\%2\))\) + k\_2\ \((2\ k\_2 + \@\(k\_1\%2 + 4\ k\_2\%2\))\)\)\/\(k\_2\ \ \((k\_1 + 2\ k\_2 + \@\(k\_1\%2 + 4\ k\_2\%2\))\)\)\)\), 1}\)], "Output"] }, Open ]], Cell[TextData[{ "Having identified the eigenvalues and eigenvectors of \[CapitalOmega], \ and according to the format of the formal solution of the equation in hand, \ i.e. ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], "(t) = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Sum]\+\(i = 1\)\%4 c\_i\ \(\(K\_i\)\& \[Rule] \)\), "TraditionalForm"], \(\[ExponentialE]\^\(\(\[Lambda]\_i\) t\)\)}], TraditionalForm]]], " we construct the components of ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], "." }], "Text"], Cell[BoxData[ \(c := {c1, c2, c3, c4}\)], "Input"], Cell[BoxData[{ \(\(\[Xi]\_1 = \[Sum]\+\(i = 1\)\%4 c\[LeftDoubleBracket] i\[RightDoubleBracket]\ eigvec\[LeftDoubleBracket]i, 1\[RightDoubleBracket]\ \[ExponentialE]\^\(eigval\ \[LeftDoubleBracket]i\[RightDoubleBracket]\ t\);\)\), "\n", \(\(\[Xi]\_2 = \[Sum]\+\(i = 1\)\%4 c\[LeftDoubleBracket] i\[RightDoubleBracket]\ eigvec\[LeftDoubleBracket]i, 2\[RightDoubleBracket]\ \[ExponentialE]\^\(eigval\ \[LeftDoubleBracket]i\[RightDoubleBracket]\ t\);\)\), "\[IndentingNewLine]", \(\(\[Eta]\_1 = \[Sum]\+\(i = 1\)\%4 c\[LeftDoubleBracket] i\[RightDoubleBracket]\ eigvec\[LeftDoubleBracket]i, 3\[RightDoubleBracket]\ \[ExponentialE]\^\(eigval\ \[LeftDoubleBracket]i\[RightDoubleBracket]\ t\);\)\), "\n", \(\(\[Eta]\_2 = \[Sum]\+\(i = 1\)\%4 c\[LeftDoubleBracket] i\[RightDoubleBracket]\ eigvec\[LeftDoubleBracket]i, 4\[RightDoubleBracket]\ \[ExponentialE]\^\(eigval\ \[LeftDoubleBracket]i\[RightDoubleBracket]\ t\);\)\)}], "Input"], Cell[TextData[{ "According to method I this procedure concludes the formal mathematical \ solution of the problem.\nTo comprehend the physical meaning of these \ solutions and to visualize the vibrating amplitudes of the masses we now \ apply reasonable practical initial conditions. We set the system in motion \ by freely releasing the pulled down masses from their static equilibria by ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_10\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_20\)]], " - this implies zero initial speeds. With these four initial conditions \ we identify the constant coefficients ", Cell[BoxData[ \(TraditionalForm\`c\_i\)]], ". The procedure is shown below:" }], "Text"], Cell[BoxData[{ \(\(eq1 = \[Xi]\_1 /. t \[Rule] 0;\)\), "\[IndentingNewLine]", \(\(eq2 = \[Xi]\_2 /. t \[Rule] 0;\)\), "\[IndentingNewLine]", \(\(eq3 = \[Eta]\_1 /. t \[Rule] 0;\)\), "\[IndentingNewLine]", \(\(eq4 = \[Eta]\_2 /. t \[Rule] 0;\)\)}], "Input"], Cell[BoxData[ \(\(sol1 = Solve[{eq1 == \[Xi]\_10, eq2 == \[Xi]\_20, eq3 \[Equal] 0, eq4 \[Equal] 0}, {c1, c2, c3, c4}];\)\)], "Input"], Cell[TextData[{ "The solution of the above equation is the coefficients ", Cell[BoxData[ \(TraditionalForm\`c\_i\)]], " for i = 1...4. By substituting these coefficients for the components of \ ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], ", we have " }], "Text"], Cell[BoxData[{ \(\(\[Xi]\_11 = \[Xi]\_1 /. sol1;\)\), "\n", \(\(\[Xi]\_12 = \[Xi]\_2 /. sol1;\)\), "\n", \(\(\[Eta]\_11 = \[Eta]\_1 /. sol1;\)\), "\n", \(\(\[Eta]\_12 = \[Eta]\_2 /. sol1;\)\)}], "Input"], Cell[TextData[{ "To study the amplitudes of the vibrations we consider a wide range of mass \ and stiffness parameters. The commonly used parameters are stored in ", StyleBox["valcom", FontWeight->"Bold"], " and parameters associated with the rest are stored in various ", StyleBox["vaL", FontWeight->"Bold"], "* listings; only vaL1 is shown below - we assume CGS units." }], "Text"], Cell[BoxData[{ \(\(valcom = {\[Xi]\_10 \[Rule] 5. , \[Xi]\_20 \[Rule] 4. , \[Omega]\_1 -> \@\(k\_1\/m\_1\), \[Omega]\_2 -> \ \@\(k\_2\/m\_2\), \[Mu] -> m\_2\/m\_1};\)\), "\[IndentingNewLine]", \(\(vaL1 = {k\_1 \[Rule] 50. , k\_2 \[Rule] 50. , m\_1 \[Rule] 1. , m\_2 \[Rule] 1. };\)\)}], "Input"], Cell[TextData[{ "In general, the pure complex eigenvalues of \[CapitalOmega] ensure the \ vibrating nature of the masses. Furthermore, since the components of the \ eigenvectors ", Cell[BoxData[ FormBox[ SubscriptBox[ FormBox[\(K\& \[Rule] \), "TraditionalForm"], "i"], TraditionalForm]]], " and coefficients ", Cell[BoxData[ \(TraditionalForm\`c\_i\)]], " are complex, to assure the realness of the amplitudes, we inspect \ individually their characters. " }], "Text"], Cell[TextData[{ "First we evaluate the amplitude of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " for the parameters specified by vaL1. By applying ExpToTrig command we \ convert the exponentials to a combination of sine and cosine functions - the \ zero output ensures its realness. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ComplexExpand[Im[ExpToTrig[\(\[Xi]\_11 /. valcom\) /. vaL1]]] // Chop\)], "Input"], Cell[BoxData[ \({0}\)], "Output"] }, Open ]], Cell[TextData[{ "Following a similar procedure we identify ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_i\)]], "'s pure real expression. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Xi]11vaL1 = ComplexExpand[Re[ExpToTrig[\(\[Xi]\_11 /. valcom\) /. vaL1]]] // Chop\)], "Input"], Cell[BoxData[ \({3.1708203932499357`\ Cos[4.37016024448821`\ t] + 1.8291796067500625`\ Cos[11.441228056353685`\ t]}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(plo1 = Plot[\[Xi]11vaL1, {t, 0, 5}, FrameLabel \[Rule] {"\", \*"\"\<\!\(\[Xi]\_1\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ \*"\"\<\!\(k\_1\)=\!\(k\_2\),\!\(m\_1\)=\!\(m\_2\)=1\>\"", DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[TextData[{ "By repeating the above three-step procedure we verified the realness of \ the amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " for other sets of parameters of interest. 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2.0254, 0.00931199, 0.00931199}, {{86.125, 165.688}, {339.188, 290}} -> \ {-16.9199, 72.0452, 0.123393, 0.394997}, {{173.625, 253.188}, {339.188, 290}} -> \ {-30.4159, 72.7752, 0.123393, 0.399298}, {{261.125, 340.688}, {339.188, 290}} -> \ {-43.912, 72.6227, 0.123393, 0.398399}, {{348.625, 428.188}, {339.188, 290}} -> \ {-57.4081, 72.7708, 0.123393, 0.399272}, {{436.125, 515.688}, {339.188, 290}} -> \ {-7.55059, 42.3706, 0.0135331, 0.218939}, {{86.125, 165.688}, {285.063, \ 235.938}} -> {-16.9755, 45.6446, 0.123746, 0.397873}, {{173.625, 253.188}, \ {285.063, 235.938}} -> {-30.5102, 45.845, 0.123746, 0.399835}, {{261.125, \ 340.688}, {285.063, 235.938}} -> {-44.0449, 45.7272, 0.123746, 0.398682}, \ {{348.625, 428.188}, {285.063, 235.938}} -> {-57.5796, 45.8726, 0.123746, \ 0.400105}, {{436.125, 515.688}, {285.063, 235.938}} -> {-7.56347, 27.5601, \ 0.0135547, 0.2188}, {{86.125, 165.688}, {231, 181.813}} -> {-17.4294, 18.756, \ 0.123393, 0.398706}, {{173.625, 253.188}, {231, 181.813}} -> {-30.9255, \ 18.7247, 0.123393, 0.397799}, {{261.125, 340.688}, {231, 181.813}} -> \ {-44.4215, 18.766, 0.123393, 0.398995}, {{348.625, 428.188}, {231, 181.813}} -> \ {-57.7325, 18.7427, 0.123393, 0.398319}, {{436.125, 515.688}, {231, 181.813}} -> \ {-7.67507, 12.9576, 0.0137423, 0.222307}}], Cell[TextData[{ StyleBox["Figure 2. Vibrating amplitudes of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_1\)], FontWeight->"Bold"], " ", StyleBox["vs. time for various values of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_1\)], FontWeight->"Bold"], ",", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_2\)], FontWeight->"Bold"], " ", StyleBox["and ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`k\_1\)], FontWeight->"Bold"], ",", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`k\_2\)], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "Figure 2 consists of a 3x5 graphic matrix. Each matrix element depicts a \ vibrating amplitude of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " for the chosen set of mass and stiffness. For a chosen relationship \ between the stiffness, the columns of the graphic matrix display ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], " versus time, describing the effects of the masses. Conversely, for a \ chosen relationship between the masses, the rows of the graphic matrix depict \ ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], " versus time, describing the effects of the stiffness of the springs. In \ creating these amplitudes, we used the following stiffness: the first column \ corresponds to ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(k\_1\), "TraditionalForm"], "=", \(k\_2\)}], TraditionalForm]]], "=50, the second column is ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], "=50 and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], "=200, the third column corresponds to ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], "=200 and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], "=50, the fourth column is ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], "=50 and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], "=5000, and the last column corresponds to ", Cell[BoxData[ \(TraditionalForm\`k\_1\)]], "=5000 and ", Cell[BoxData[ \(TraditionalForm\`k\_2\)]], "=50.\nThe fourth element of the first row describes a situation where the \ bottom spring, relative to the top one, is completely hardened. Since the \ masses are equal to unity and are small, the vibrations of the top spring \ are not influenced by the bottom massless stiffened spring. This is the only \ case studied by Clark and Hill. However, even in this case, as depicted by \ the second and third graphic elements of the fourth column, changing the \ masses changes the situation.\nThe time scale used displaying the elements of \ the fifth column is chosen appropriately differently from the rest to show \ the effects of the fast oscillations of the situation.\nBy looking over the \ graphs presented in figure 2, one realizes how important the role of unequal \ masses are." }], "Text"], Cell[TextData[{ "We repeat the above analysis to identify and then display collectively the \ vibrations of ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ComplexExpand[Im[ExpToTrig[\(\[Xi]\_12 /. valcom\) /. vaL1]]] // Chop\)], "Input"], Cell[BoxData[ \({0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Xi]12vaL1 = ComplexExpand[Re[ExpToTrig[\(\[Xi]\_12 /. valcom\) /. vaL1]]] // Chop\)], "Input"], Cell[BoxData[ \({5.1304951684997055`\ Cos[4.37016024448821`\ t] - 1.130495168499705`\ Cos[11.441228056353685`\ t]}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(plo10 = Plot[\[Xi]12vaL1, {t, 0, 5}, FrameLabel \[Rule] {"\", \*"\"\<\!\(\[Xi]\_2\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ \*"\"\<\!\(k\_1\)=\!\(k\_2\),\!\(m\_1\)=\!\(m\_2\)=1\>\"", DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["The results are shown in figure 3.", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .36624 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.190476 0.00872006 0.190476 [ [ 0 0 0 0 ] [ 1 .36624 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .36624 L 0 .36624 L closepath clip newpath % Start of sub-graphic p 0.0238095 0.00872006 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{{173.625, 254.625}, {216.563, \ 166.5}} -> {-29.7995, 10.2942, 0.118649, 0.418155}, {{262.75, 343.75}, \ {216.563, 166.5}} -> {-43.0178, 10.9534, 0.118649, 0.444524}, {{351.875, \ 432.875}, {216.563, 166.5}} -> {-56.0581, 9.42935, 0.118649, 0.383997}, \ {{441, 522}, {216.563, 166.5}} -> {-69.2764, 7.74323, 0.118649, 0.31471}}], Cell[TextData[{ StyleBox["Figure 3. Vibrating amplitudes of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_2\)], FontWeight->"Bold"], " ", StyleBox["vs. time for various values of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_1\)], FontWeight->"Bold"], ",", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_2\)], FontWeight->"Bold"], " ", StyleBox["and ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`k\_1\)], FontWeight->"Bold"], ",", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`k\_2\)], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "Figure 3 is formed similar to figure 2 and consists of a 3x5 graphic \ matrix. Each matrix element depicts a vibrating amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], " for the chosen mass and stiffness, Clark and Hill considered only the \ amplitude displayed by the fourth element of the first row." }], "Text"], Cell["\<\ This graph explicitly shows the effects of the differences between the masses \ on vibrating amplitudes of the second mass. With a few exceptions, most of \ the amplitudes displayed in figure 2 and 3 are hard if not impossible to \ predict and visualize intuitively.\ \>", "Text"], Cell[TextData[{ "Since all the components of ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], " are identified and because ", Cell[BoxData[ \(TraditionalForm\`\(Q\& \[Rule] \)\)]], "'s upper two components namely ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\({\[Xi]\_1\), "TraditionalForm"], ",", \(\[Xi]\_2\)}], TraditionalForm]]], "} are related to its lower two ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\({\[Eta]\_1\), "TraditionalForm"], ",", \(\[Eta]\_2\)}], TraditionalForm]]], "} components by time-derivative i.e. ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\({\[Eta]\_1\), "TraditionalForm"], ",", \(\[Eta]\_2\)}], TraditionalForm]]], "} = ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ FormBox[\(d\/dt\), "TraditionalForm"], \({\[Xi]\_1\)}], "TraditionalForm"], ",", \(\[Xi]\_2\)}], TraditionalForm]]], "}, we independently verify the accuracy and consistency of the solutions \ presented above. We call for instance the speed of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " for a set of parameters defined by vaL1. " }], "Text"], Cell[BoxData[{ \(\(\[Eta]11vaL1 = ComplexExpand[ Re[ExpToTrig[\(\[Eta]\_11 /. valcom\) /. vaL1]]];\)\), "\n", \(\(plo19 = Plot[\[Eta]11vaL1, {t, 0, 5}, FrameLabel \[Rule] {"\", \*"\"\<\!\(\[Eta]\_1\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ \*"\"\<\!\(k\_1\)=\!\(k\_2\),\!\(m\_1\)=\!\(m\_2\)=1\>\"", DisplayFunction \[Rule] Identity];\)\)}], "Input"], Cell[TextData[{ "We then evaluate the speed of the same mass directly by differentiating ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], " with respect to time. " }], "Text"], Cell[BoxData[ \(\(plo20 = Plot[Evaluate[\(D[\[Xi]\_11, t] /. valcom\) /. vaL1], {t, 0, 5}, PlotStyle \[Rule] {Dashing[{0.02}]}, FrameLabel \[Rule] {"\", \ \*"\"\<\!\(d\/dt\)\!\(\[Xi]\_1\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ \*"\"\<\!\(k\_1\)=\!\(k\_2\),\!\(m\_1\)=\!\(m\_2\)=1\>\"", DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell["\<\ These two results are then compared graphically as shown in figure 4 - \ results are identical.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(s1920 = Show[GraphicsArray[{plo19, plo20}], DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .2943 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 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o`03001oo`0000=oo`03001oo`0000]oo`04001oogoo0002Ool00`00Ool0000Ool200001Goo000007oo000047oo00<007ooOol0KGoo002LOol00`00Ool00003000? Ool00`00Ool00003000=Ool200001Goo001oogoo00003Woo0P0000Eoo`00Oomoo`00015oo`03001o ogoo0?moo`moo`03001oo`0000<000moo`03001oo`0000<000eoo`800005Ool007ooOol0000>Ool2 00001Goo001oogoo00004Goo00<007ooOol0KGoo002LOol00`00Oomoo`0BOol00`00Oomoo`0kOol3 003oOolAOol00`00Oomoo`0BOol00`00Oomoo`0kOol3001_Ool009]oo`8001=oo`8003moo`03001o ogoo0?moo`ioo`8001=oo`8003moo`03001oogoo06eoo`00ogooogoobGoo003oOoooOoo9Ool00?mo oomoolUoo`00ogooogoobGoo003oOoooOoo9Ool00?mooomoolUoo`00ogooogoobGoo0000\ \>"], ImageRangeCache->{{{85.0625, 539.563}, {458.75, 325.75}} -> {-0.451282, \ 1.49901, 0.00371751, 0.00371751}, {{97.0625, 302.063}, {455.563, 328.875}} -> \ {-4.74141, 197.15, 0.027174, 0.616488}, {{322.5, 527.5}, {455.563, 328.875}} -> \ {-12.5009, 197.15, 0.027174, 0.616488}}], Cell[TextData[{ StyleBox["Figure 5. Speeds of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(\(m\_2\)\(\ \)\)\)], FontWeight->"Bold"], StyleBox[" are compared independently. The solid curve (the left graph) is \ ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\[Eta]\_2\)], FontWeight->"Bold"], " ", StyleBox["and the dashed curve (the right graph) is ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(d\/dt\), "TraditionalForm"], \(\[Xi]\_2\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[". ", FontWeight->"Bold"] }], "Text"], Cell[TextData[StyleBox["Method II - the decoupling procedure", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Here we show an alternative method to solve the coupled equations of \ motion. Starting with the original set of second-order differential \ equations given in terms of ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], ", \[Omega]'s and \[Mu] , we differentiate both sides of the first equation \ twice with respect to time. This gives ", Cell[BoxData[ \(TraditionalForm\`\(d\^4\/dt\^4\) \[Xi]\_1\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(-\((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2\)\), "TraditionalForm"], ")"}], " "}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\(d\^2\/dt\^2\) \[Xi]\_1\)]], " +\[Mu] ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_2\%2\)]], " ", Cell[BoxData[ \(TraditionalForm\`\(d\^2\/dt\^2\) \[Xi]\_2\)]], ". Then we substitute ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(d\^2\/dt\^2\), "TraditionalForm"], \(\[Xi]\_2\)}], TraditionalForm]]], " from the second equation. This yields ", Cell[BoxData[ \(TraditionalForm\`\(d\^4\/dt\^4\) \[Xi]\_1\)]], "= ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[\(-\((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2\)\), "TraditionalForm"], ")"}], " "}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\(d\^2\/dt\^2\) \[Xi]\_1\)]], " +\[Mu] ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_2\%2\)]], "( ", Cell[BoxData[ \(TraditionalForm\`\(\[Omega]\_2\%2\) \[Xi]\_1 - \(\[Omega]\_2\%2\) \ \[Xi]\_2\)]], ") and by substituting ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Xi]\_2\)\(\ \)\)\)]], "from the first equation in the last one we get ", Cell[BoxData[ \(TraditionalForm\`\(d\^4\/dt\^4\) \[Xi]\_1\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\([\[Omega]\_1\%2\)\)]], " + (1+\[Mu]) ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_2\%2\)]], "] ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(d\^2\/dt\^2\) \[Xi]\_1\)\)\)]], "+ ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ FormBox[\(\[Omega]\_1\%2\), "TraditionalForm"], \(\[Omega]\_2\%2\)}], "TraditionalForm"], \(\[Xi]\_1\)}], TraditionalForm]]], "= 0 . Following identical steps, one can also show that ", Cell[BoxData[ \(TraditionalForm\`\[Xi]\_2\)]], " is subject to an identical equation - in other words ", Cell[BoxData[ \(TraditionalForm\`\(d\^4\/dt\^4\) \[Xi]\_i\)]], "+ ", Cell[BoxData[ \(TraditionalForm\`\([\[Omega]\_1\%2\)\)]], " + (1+\[Mu]) ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_2\%2\)]], "] ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\(d\^2\/dt\^2\) \[Xi]\_i\)\)\)]], "+ ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Omega]\_1\%2\), "TraditionalForm"], \(\[Omega]\_2\%2\), \(\[Xi]\_i\)}], TraditionalForm]]], "= 0; i =1,2. Therefore, we show that instead of reducing the problem to a \ set of four coupled linear first-order differential equations, we can present \ the problem in terms of two decoupled fourth-order homogenous linear ordinary \ differential equations with constant coefficients. " }], "Text"], Cell["\<\ To solve these equations we also need to provide consistent initial \ conditions. Each of these fourth-order equations require four independent \ initial conditions. In addition to the first two initial conditions used in \ method I, we consistently consider two more. The table below provides the \ initial values in need.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t\[Xi]11 = TableForm[\(\(Table[ Flatten[{n, D[{\[Xi]\_11, \[Xi]\_12}, {t, n}]}], {n, 0, 3}] /. valcom\) /. vaL1\) /. t \[Rule] 0 // Chop, TableHeadings \[Rule] {Automatic, {"\", \ \*"\"\<\!\(d\^n\/dt\^n\)\!\(\[Xi]\_1\)\>\"", \ \*"\"\<\!\(d\^n\/dt\^n\)\!\(\[Xi]\_2\)\>\""}}]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\"\"\>", "\<\"n\"\>", \ "\<\"\\!\\(d\\^n\\/dt\\^n\\)\\!\\(\[Xi]\\_1\\)\"\>", \ "\<\"\\!\\(d\\^n\\/dt\\^n\\)\\!\\(\[Xi]\\_2\\)\"\>"}, {"1", "0", "4.999999999999997`", "4.000000000000001`"}, {"2", "1", "0", "0"}, {"3", "2", \(-299.9999999999999`\), "49.99999999999993`"}, {"4", "3", "0", "0"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #, TableHeadings -> {Automatic, {"n", "\!\(d\^n\/dt\^n\)\!\(\[Xi]\_1\)", "\!\(d\^n\/dt\^n\)\!\(\[Xi]\_2\)"}}]&)]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Table 1. The third and fourth columns are the values of ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[ FractionBox[ StyleBox[\(d\^n\), FontWeight->"Plain"], \(dt\^n\)], "TraditionalForm"], \(\[Zeta]\_i\)}], TraditionalForm]], FontWeight->"Bold"], StyleBox[" for i = 1,2 at t = 0. The n's are the derivative order and are \ given in the second column.", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "By applying ", StyleBox["Mathematica'", FontSlant->"Italic"], "s DSolve command we solve these 4th-order ODEs. To identify the \ amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " we apply the initial values in accordance with the third column of the \ table 1. " }], "Text"], Cell[BoxData[ \(\(dsol\[Xi]11 = DSolve[{D[ y[t], {t, 4}] + \((\[Omega]\_1\%2 + \((1 + \[Mu])\)\ \[Omega]\_2\%2\ \ )\)\ D[y[t], {t, 2}] + \(\[Omega]\_\(\(1\)\(\ \ \)\)\%2\) \ \(\[Omega]\_2\%2\) y[t] \[Equal] 0, y[0] \[Equal] 5, \(y'\)[0] \[Equal] 0, \(y''\)[0] \[Equal] \(-300\), \(y'''\)[0] \[Equal] 0}, y[t], t];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(dsol\[Xi]11vaL1 = \(ComplexExpand[ Re[ExpToTrig[\(\(y[t] /. dsol\[Xi]11\) /. valcom\) /. vaL1]]] // Simplify\) // Chop\)], "Input"], Cell[BoxData[ \({3.1708203932499375`\ Cos[4.370160244488211`\ t] + 1.829179606750064`\ Cos[11.441228056353687`\ t]}\)], "Output"] }, Open ]], Cell["\<\ This solution is identical to one provided by method I, \[Zeta]11vaL1.\ \>", "Text"], Cell[TextData[{ "By applying the initial conditions given by the fourth column of the table \ 1 we identify the amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], " this yields" }], "Text"], Cell[BoxData[ \(\(dsol\[Xi]12 = DSolve[{D[ y[t], {t, 4}] + \((\[Omega]\_1\%2 + \((1 + \[Mu])\)\ \ \[Omega]\_2\%2)\)\ D[y[t], {t, 2}] + \(\[Omega]\_1\%2\) \(\[Omega]\_2\%2\) y[t] \[Equal] 0, y[0] \[Equal] 4, \(y'\)[0] \[Equal] 0, \(y''\)[0] \[Equal] 50, \(y'''\)[0] \[Equal] 0}, y[t], t];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(dsol\[Xi]12vaL1 = \(ComplexExpand[ Re[ExpToTrig[\(\(y[t] /. dsol\[Xi]12\) /. valcom\) /. vaL1]]] // Simplify\) // Chop\)], "Input"], Cell[BoxData[ \({5.130495168499706`\ Cos[4.370160244488211`\ t] - 1.1304951684997053`\ Cos[11.441228056353687`\ t]}\)], "Output"] }, Open ]], Cell["\<\ This solution is also identical to one provided by the method I, \ \[Zeta]12vaL1. \ \>", "Text"], Cell[TextData[{ StyleBox["Method III - ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" procedure", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "As a third alternative method, unique to ", StyleBox["Mathematica", FontSlant->"Italic"], ", without any manipulation we directly solve the original coupled \ equations of motion given in terms of ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Xi]\_\(\(1\)\(\ \)\)\), "TraditionalForm"], ",", \(\[Xi]\_2\)}], TraditionalForm]]], ",\[Omega]'s and \[Mu]. In preparing the equations for ", StyleBox["Mathematica", FontSlant->"Italic"], " we replace ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ FormBox[\(\[Xi]\_1\), "TraditionalForm"], ",", \(\[Xi]\_2\)}]}], TraditionalForm]]], "} by {x,z}. For the sake of consistency and comparison, we identify the \ amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], " for the same set of parameters used in the previous two methods. We \ apply the DSolve command and implement the initial conditions used in method \ I. This yields " }], "Text"], Cell[BoxData[ \(\(sol\[Xi]1112 = DSolve[{D[ x[t], {t, 2}] + \((\[Omega]\_1\%2 + \[Mu]\ \[Omega]\_2\%2)\)\ x[ t] - \[Mu]\ \[Omega]\_2\%2\ z[t] \[Equal] 0, D[z[t], {t, 2}] - \[Omega]\_2\%2\ x[t] + \[Omega]\_2\%2\ z[ t] \[Equal] 0, x[0] \[Equal] 5, z[0] \[Equal] 4, \(x'\)[0] \[Equal] 0, \(z'\)[0] \[Equal] 0}, {x[ t], z[t]}, t];\)\)], "Input"], Cell[TextData[{ "The amplitudes of ", Cell[BoxData[ \(TraditionalForm\`m\_1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`m\_2\)]], " are given below:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\[Xi]11 = \(TrigFactor[ ComplexExpand[ Re[ExpToTrig[\(\(x[t] /. sol\[Xi]1112\) /. valcom\) /. vaL1]]]] // Simplify\) // Chop\)], "Input"], Cell[BoxData[ \({3.170820393249937`\ Cos[4.370160244488211`\ t] + 1.829179606750063`\ Cos[11.441228056353687`\ t]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(z\[Xi]12 = \(TrigFactor[ ComplexExpand[ Re[ExpToTrig[\(\(z[t] /. sol\[Xi]1112\) /. valcom\) /. vaL1]]]] // Simplify\) // Chop\)], "Input"], Cell[BoxData[ \({5.130495168499706`\ Cos[4.370160244488211`\ t] - 1.1304951684997055`\ Cos[11.441228056353687`\ t]}\)], "Output"] }, Open ]], Cell["\<\ These results are identical to the ones presented by the other two methods.\ \>", "Text"], Cell["\<\ In conclusions, to show the consistency of the three independent solution \ methods described in the text, the amplitudes of individual masses \ graphically are compared separately in figures 6 and 7.\ \>", "Text"], Cell[BoxData[{ \(\(plod4\[Xi]11 = Plot[dsol\[Xi]11vaL1, {t, 0, 5}, FrameLabel \[Rule] {"\", \*"\"\<\!\(\[Xi]\_1\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ \*"\"\<\!\(k\_1\)=\!\(k\_2\),\!\(m\_1\)=\!\(m\_2\)=1\>\"", DisplayFunction \[Rule] Identity];\)\), "\n", \(\(plod4\[Xi]12 = Plot[dsol\[Xi]12vaL1, {t, 0, 5}, FrameLabel \[Rule] {"\", \*"\"\<\!\(\[Xi]\_2\)\>\""}, Frame \[Rule] {True, True, True, True}, PlotLabel \[Rule] \ 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{-11.4632, -11.1074, 0.0444057, 0.17747}, {{510.125, 733}, {141.125, 3.4375}} -> \ {-20.1712, -11.1074, 0.0444057, 0.17747}}] }, Open ]], Cell[TextData[{ StyleBox["Figure 7. Comparison of amplitudes of ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`m\_2\)], FontWeight->"Bold"], StyleBox[" derived by three methods described in the text. From left to \ right these correspond to methods I, II and III.", FontWeight->"Bold"], " " }], "Text"], Cell[TextData[{ "In general, all three methods discussed in the text can easily be extended \ to describe vibrations of systems with more than two springs and masses. For \ instance, according to method I, the number of coupled differential linear \ equations of motion for a three-mass three-spring will increase from four to \ six. The corresponding \[CapitalOmega] matrix describing the situation \ becomes\n\n", Cell[BoxData[ RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "1", "0"}, {"0", "0", "0", "0", "0", "1"}, {\(-\((\[Omega]\_1\%2 + \(\[Mu]\_21\) \[Omega]\_2\%2)\)\), \(\(\ \[Mu]\_21\) \[Omega]\_2\%2\), "0", "0", "0", "0"}, {\(\[Omega]\_2\%2\), \(-\((\[Omega]\_2\%2 + \(\[Mu]\_32\) \ \[Omega]\_3\%2)\)\), \(\(\[Mu]\_32\) \[Omega]\_3\%2\), "0", "0", "0"}, {"0", \(\[Omega]\_3\%2\), \(-\[Omega]\_3\%2\), "0", "0", "0"} }], ")"}], ",", " "}]], "Text"] }], "Text"], Cell[TextData[{ "the natural angular frequency of individual masses are ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_i\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\@\(k\_i\/m\_i\)\)]], ", and ratio of the masses is given by ", Cell[BoxData[ \(TraditionalForm\`\[Mu]\_ij\)]], " = ", Cell[BoxData[ \(TraditionalForm\`m\_i\/m\_j\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Mathematica,", FontSlant->"Italic"], " will identify the eigenvalues and eigenvectors of the corresponding 6x6 \ matrix. Consequently, following detailed descriptions in the text, \ vibrating amplitudes of the masses can easily be graphed and analyzed. " }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["REFERENCE", FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ "Clark, K.E., Hill, S. The Effects of a Stiffening Spring. ", StyleBox["College Mathematics Journal", FontSlant->"Italic"], " 30:5 (1999) 379-382." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["ABOUT THE AUTHOR", "Subsection"], Cell["\<\ Haiduke Sarafian Pennsylvania State University Commonwealth College York, PA 17403\ \>", "Text", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell["ELECTRONIC SUBSCRIPTIONS", "Subsection"], Cell[TextData[{ "Included in the distribution for each electronic subscription is the file \ ", StyleBox["Spring5.nb", "Input", FontWeight->"Plain"], " containing ", StyleBox["Mathematica", FontSlant->"Italic"], " code for the material described in this article." }], "Text"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{"RulerBar", "EditBar"}, WindowSize->{1016, 636}, WindowMargins->{{-8, Automatic}, {Automatic, 2}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.25 ] (*********************************************************************** Cached data follows. 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