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k\_c\ \((u\_2 - u\_3)\)\^2 + k\_a\ u\_3\%2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(grad\[CapitalPi]\ = \ \(Table[D[\[CapitalPi], u\_i], {i, 3}] // Expand\) // Simplify\)], "Input", FontSize->18], Cell[BoxData[ \({\(-P\_1\) + \((k\_d + k\_e)\)\ \((u\_1 - u\_2)\) + k\_b\ \((u\_1 - u\_3)\), \(-P\_2\) - k\_e\ u\_1 + k\_c\ u\_2 + k\_e\ u\_2 + k\_d\ \((\(-u\_1\) + u\_2)\) - k\_c\ u\_3, \(-P\_3\) - k\_c\ u\_2 + k\_a\ u\_3 + k\_c\ u\_3 + k\_b\ \((\(-u\_1\) + u\_3)\)}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Let's check that the two sets of equations are the same:\ \>", \ "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[grad\[CapitalPi][\([i]\)] + loadbalance\_i, {i, 3}] // Simplify\)], "Input"], Cell[BoxData[ \({0, 0, 0}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Let's find the solution", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(solution\ = \ \(Solve[ grad\[CapitalPi] \[Equal] 0, {u\_1, u\_2, u\_3}] // Simplify\) // Flatten\)], "Input", FontSize->18], Cell[BoxData[ \({u\_1 \[Rule] \(k\_a\ \((k\_c\ P\_1 + \((k\_d + k\_e)\)\ \((P\_1 + \ P\_2)\))\) + \((k\_c\ \((k\_d + k\_e)\) + k\_b\ \((k\_c + k\_d + k\_e)\))\)\ \ \((P\_1 + P\_2 + P\_3)\)\)\/\(k\_a\ \((k\_c\ \((k\_d + k\_e)\) + k\_b\ \ \((k\_c + k\_d + k\_e)\))\)\), u\_2 \[Rule] \(\((k\_d + k\_e)\)\ \((k\_a\ \((P\_1 + P\_2)\) + k\_c\ \ \((P\_1 + P\_2 + P\_3)\))\) + k\_b\ \((k\_e\ P\_1 + k\_a\ P\_2 + k\_e\ P\_2 + \ k\_e\ P\_3 + k\_c\ \((P\_1 + P\_2 + P\_3)\) + k\_d\ \((P\_1 + P\_2 + P\_3)\))\ \)\)\/\(k\_a\ \((k\_c\ \((k\_d + k\_e)\) + k\_b\ \((k\_c + k\_d + \ k\_e)\))\)\), u\_3 \[Rule] \(P\_1 + P\_2 + P\_3\)\/k\_a}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ (c) Special solutions \ti.)\ \>", "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(solution\ /. \ {k\_a \[Rule] \ k, \ k\_b \[Rule] \ k, \ k\_c \[Rule] \ k, \ k\_d \[Rule] \ k, \ k\_e \[Rule] \ k, P\_1 \[Rule] \ 0, \ P\_2 \[Rule] \ 0\ , P\_3 \[Rule] P}\)], "Input", FontSize->18], Cell[BoxData[ \({u\_1 \[Rule] P\/k, u\_2 \[Rule] P\/k, u\_3 \[Rule] P\/k}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "This corresponds to the case of all the springs being equal and the only \ load being applied at point 3. The displacement of the spring ", Cell[BoxData[ \(TraditionalForm\`k\_a\)]], " is only affected by this load and the result is ", Cell[BoxData[ \(TraditionalForm\`u\_\(\(3\)\(\ \)\) = \ P\/k\)]], "the rest of the springs don't deform, they displace rigidly.\n\tii.)\n\t" }], "Subtitle"], Cell[CellGroupData[{ Cell[BoxData[ \(solution\ /. \ {k\_a \[Rule] \ k, \ k\_b \[Rule] \ k, \ k\_c \[Rule] \ k, \ k\_d \[Rule] \ k, \ k\_e \[Rule] \ k, P\_1 \[Rule] \ P, \ P\_2 \[Rule] \ 0\ , P\_3 \[Rule] 0}\)], "Input"], Cell[BoxData[ \({u\_1 \[Rule] \(8\ P\)\/\(5\ k\), u\_2 \[Rule] \(7\ P\)\/\(5\ k\), u\_3 \[Rule] P\/k}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ This cooresponds to the case of all the springs being equal and the \ only load being applied at point 1. 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