|
|
| (13 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| | + | <center><math>\underline{\textbf{Navigation}}</math> |
| | | | |
| − | We can define the constraints of the plane the DC is in
| + | [[Wire_Number_Function|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[The_Wires|<math>\vartriangleright </math>]] |
| | | | |
| − | <pre> | + | </center> |
| − | right = ContourPlot[
| |
| − | x2 == Cot[29.5 \[Degree]] y + .09156, {y, -1, 1}, {x2, 0, 1.8},
| |
| − | Frame -> {True, True, False, False},
| |
| − | PlotLabel -> "Right side limit of DC as a function of X and Y",
| |
| − | FrameLabel -> {"y (meters)", "x (meters)"}, ContourStyle -> Black,
| |
| − | PlotLegends -> Automatic];
| |
| | | | |
| − | left = ContourPlot[
| + | <center>[[File:Part1d1.png]]</center> |
| − | x2 == -Cot[29.5 \[Degree]] y + .09156, {y, -1, 1}, {x2, 0, 1.8},
| |
| − | Frame -> {True, True, False, False},
| |
| − | PlotLabel -> "Right side limit of DC as a function of X and Y",
| |
| − | FrameLabel -> {"y (meters)", "x (meters)"}, ContourStyle -> Black,
| |
| − | PlotLegends -> Automatic];
| |
| | | | |
| − | </pre>
| |
| | | | |
| | + | ---- |
| | | | |
| − | We can define the x coordinate of the wires as they cross the midpoint plane as shown earlier.
| |
| | | | |
| − | <pre> | + | <center><math>\underline{\textbf{Navigation}}</math> |
| − | x0forWires[number_] := .23168 + .01337*(number);
| |
| − | </pre> | |
| | | | |
| | + | [[Wire_Number_Function|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[The_Wires|<math>\vartriangleright </math>]] |
| | | | |
| − | We can define the point midway between two parallel lines as the point where one wire is recorded versus its next highest neighbor
| + | </center> |
| − | | |
| − | <pre>
| |
| − | x0forWireMiddles[
| |
| − | number_] := ((.23168 + .01337*(number)) + (.23168 + .01337*(number \
| |
| − | + 1)))/2;
| |
| − | </pre>
| |
| − | | |
| − | All of the conditions dependent on <math> \theta </math> and <math>\phi</math>
| |
| − | | |
| − | <pre>
| |
| − | | |
| − | \[CapitalDelta]a :=
| |
| − | FullSimplify[(R Sin[\[Theta] \[Degree]])/
| |
| − | 2 (Csc[65 \[Degree] - \[Theta] \[Degree]] -
| |
| − | Csc[115 \[Degree] - \[Theta] \[Degree]]), \[Theta] > 0];
| |
| − | e := Sin[25 \[Degree]]/Cos[\[Theta] \[Degree]];
| |
| − | a := FullSimplify[(R Sin[\[Theta] \[Degree]])/
| |
| − | 2 (Csc[65 \[Degree] - \[Theta] \[Degree]] +
| |
| − | Csc[115 \[Degree] - \[Theta] \[Degree]]), \[Theta] > 0];
| |
| − | rD1 := Simplify[(a e - \[CapitalDelta]a) Tan[
| |
| − | 65 \[Degree]] Cos[\[Theta] \[Degree]], \[Theta] > 0];
| |
| − | rD2 := Simplify[(a e + \[CapitalDelta]a) Tan[
| |
| − | 65 \[Degree]] Cos[\[Theta] \[Degree]], \[Theta] > 0];
| |
| − | xD1 := Simplify[rD1 Cos[\[Phi] \[Degree]]];
| |
| − | yD1 := Simplify[rD1 Sin[\[Phi] \[Degree]]];
| |
| − | zD1 := Simplify[rD1 Cot[\[Theta] \[Degree]], \[Theta] > 0];
| |
| − | xD2 := Simplify[rD2 Cos[\[Phi] \[Degree]], \[Theta] > 0];
| |
| − | yD2 := Simplify[rD2 Sin[\[Phi] \[Degree]], \[Theta] > 0];
| |
| − | zD2 := Simplify[rD2 Cot[\[Theta] \[Degree]], \[Theta] > 0];
| |
| − | xP := Simplify[(R Cos[\[Phi] \[Degree]])/(Cot[\[Theta] \[Degree]] +
| |
| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
| |
| − | yP := Simplify[(R Sin[\[Phi] \[Degree]])/(Cot[\[Theta] \[Degree]] +
| |
| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
| |
| − | zP := Simplify[(R Cot[\[Theta] \[Degree]])/(Cot[\[Theta] \[Degree]] +
| |
| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
| |
| − | x1 := Simplify[(rD2^2 - rD1^2 +
| |
| − | Cot[\[Theta] \[Degree]]^2 (rD2^2 - rD1^2) - 2 xP (xD2 - xD1) -
| |
| − | 2 yP (yD2 - yD1) - 2 zP (zD2 - zD1))/(4 a e) - a e, \[Theta] >
| |
| − | 0];
| |
| − | x := Simplify[x1 - \[CapitalDelta]a + a e, \[Theta] > 0];
| |
| − | xCenter := x + \[CapitalDelta]a;
| |
| − | n := -957.412/(Tan[\[Theta] \[Degree]] + 2.14437) + 430.626;
| |
| − | D2P := Simplify[((xD2 - xP)^2 + (yD2 - yP)^2 + (zD2 -
| |
| − | zP )^2)^.5, \[Theta] > 0] // N
| |
| − | D1P := Simplify[((xP - xD1)^2 + (yP - yD1)^2 + (zP -
| |
| − | zD1)^2)^.5, \[Theta] > 0] // N;
| |
| − | y := Simplify[(D1P^2 - x1^2)^.5, \[Theta] > 0] // N;
| |
| − | b := Simplify[a Sqrt[1 - e^2], \[Theta] > 0] // N;
| |
| − | R = 2.52934271645;
| |
| − | </pre>
| |
| − | | |
| − | We can define the x-y position on the DC plane as a function of <math>\phi</math> for and limit the angles with the wall on the right and left hand sides. By symmetry these angles are equal, but opposite in sign.
| |
| − | | |
| − | <pre>
| |
| − | Limits = Table[\[Phi] /.
| |
| − | NSolve[Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a ==
| |
| − | Cot[29.5 \[Degree]] y + .09156 , \[Phi]], {\[Theta], 5, 40}];
| |
| − | Lim = Table[0, {rows, 1, 36}];
| |
| − | For[rows = 1, rows < 37, rows++,
| |
| − | For[i = 1, i < 5, i++,
| |
| − | If[Limits[[rows, i]] > 0 && Limits[[rows, i]] < 40,
| |
| − | Lim[[rows]] = Limits[[rows, i]];
| |
| − | ];
| |
| − | ];
| |
| − | ];
| |
| − | </pre>
| |
| − | | |
| − | The angle must have 4 subtracted to equal the line element numbering
| |
| − | | |
| − | <pre>
| |
| − | In[33]:= Lim[[7 - 4]]
| |
| − | | |
| − | Out[33]= 23.4101
| |
| − | </pre>
| |
| − | | |
| − | The limit of <math>\phi</math> increases in the plane of the DC sector changes as <math>\theta</math> increases.
| |
| − | | |
| − | <pre>
| |
| − | In[34]:= Lim[1]
| |
| − | | |
| − | Out[34]= {20.076, 22.024, 23.4101, 24.448, 25.255, 25.9008, 26.4297,
| |
| − | 26.8711, 27.2453, 27.5667, 27.846, 28.0911, 28.3079, 28.5013,
| |
| − | 28.675, 28.8319, 28.9744, 29.1044, 29.2237, 29.3336, 29.4352,
| |
| − | 29.5294, 29.6171, 29.699, 29.7757, 29.8477, 29.9155, 29.9795,
| |
| − | 30.0399, 30.0973, 30.1517, 30.2034, 30.2528, 30.2999, 30.3449,
| |
| − | 30.388}[1]
| |
| − | </pre>
| |
| − | | |
| − | This limiting condition will record the angle <math>\phi</math> in degrees within an array of wire number(1-112) horizontally and angle <math>\theta (5^{\circ}-40^{\circ} )</math> vertically
| |
| − | | |
| − | <pre>
| |
| − | RightSolutions =
| |
| − | Table[{\[Phi] /.
| |
| − | Solve[Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a ==
| |
| − | Tan[6 \[Degree]] y +
| |
| − | x0forWireMiddles[number] , \[Phi]]}, {\[Theta], 5,
| |
| − | 40}, {number, 1, 112}];
| |
| − | LineRight = Table[{\[Phi], columns}, {rows, 1, 36}, {columns, 1, 112}];
| |
| − | For[rows = 1, rows < 37, rows++,
| |
| − | For[columns = 1, columns < 113, columns++,
| |
| − | For[i = 1, i < 5, i++,
| |
| − | If[RightSolutions[[rows, columns, 1, i]] > 0 &&
| |
| − | RightSolutions[[rows, columns, 1, i]] < Lim[[rows]],
| |
| − | LineRight[[rows,
| |
| − | columns]] = {RightSolutions[[rows, columns, 1, i]],
| |
| − | columns + .5};
| |
| − | ];
| |
| − | ];
| |
| − | ];
| |
| − | ];
| |
| − | </pre>
| |
| − | | |
| − | At <math>\phi </math>=0, the condition of n=-959.637/(tan <math>\theta^{\circ}</math> +2.14437)+430.189 should be met
| |
| − | | |
| − | <pre>
| |
| − | In[38]:= f[\[Theta]for\[Phi]at0_] := -959.637/(
| |
| − | Tan[\[Theta]for\[Phi]at0 \[Degree]] + 2.14437) + 430.189
| |
| − | | |
| − | In[39]:= f[40]
| |
| − | | |
| − | Out[39]= 108.538
| |
| − | </pre>
| |
| − | | |
| − | | |
| − | This implies that for <math>\theta=40^{\circ}</math>, we know that the wire number to be 108.538. This corresponds to the position being in between 108.5 and 109.5, hence it falls into the 109 "bin".
| |
| − | | |
| − | Testing the geometry
| |
| − | <pre>
| |
| − | ClearAll[\[Theta]];
| |
| − | \[Theta] = 40;
| |
| − | ellipse40 =
| |
| − | ContourPlot[(x + \[CapitalDelta]a)^2/a^2 + y^2/b^2 == 1, {y, -1,
| |
| − | 1}, {x, 1.2, 1.7}, Frame -> {True, True, False, False},
| |
| − | PlotLabel ->
| |
| − | "XY position on DC as a function of \[Phi] for \[Theta]=40\
| |
| − | \[Degree]", FrameLabel -> {"y (meters)", "x (meters)"},
| |
| − | ContourStyle -> Red, PlotLegends -> Automatic];
| |
| − | Show[Table[
| |
| − | ContourPlot[
| |
| − | xWire == Tan[6 \[Degree]] yWire + x0forWires[number], {yWire, -1,
| |
| − | 1}, {xWire, 1.2, 1.7},
| |
| − | FrameLabel -> {"y(meters)", "x(meters)"}], {number, 70, 109}],
| |
| − | Table[ContourPlot[
| |
| − | xWire ==
| |
| − | Tan[6 \[Degree]] yWire + x0forWireMiddles[number2], {yWire, -1,
| |
| − | 1}, {xWire, 1.2, 1.7},
| |
| − | ContourStyle -> {Dashing[Large]}], {number2, 70,
| |
| − | 109}], bottom, right, left, ellipse40]
| |
| − | </pre> | |
| − | | |
| − | [[File:40EllipseTest.png]]
| |
