Railgun Force Computation With Simple Magnetic Field

Mathematica Analytic Force Attempt - Recovery2.nb

Analytic Force Computation Comparison Code

Define Constants

In[15]:= ClearAll[F, Lp, Ic, I0, I1, I2, I3, \[Omega], \[Tau], t, t1, t2, t3, m, V0, \
V2, imp, d, w, x]
\[Mu]0 = 4*\[Pi] *10^(-7);

r = 0.005;
wR = r * 2;
d = sep + r;
w = d + r;

(* Case 1 ICL Force *)
m = 0.00556; (* kg *)
V0 = 0; (* m/s *)
I0 = 20000; (* A *)
t1 = 0.5*^-3;    t2 = 0.5*^-3;   t3 = 2.5*^-3;
sep = 0.012;
L = 0.2;
wR = 0.01;
Lp = 4.8973*^-7;(*0.9*^-6;*) 
\[Tau] = 0.0007;

f = 1/(4*t1);
\[Omega] = 2*\[Pi]*f;

[S1] Simple Force and Velocity Computation - Constant Current

In[32]:= SimpleInt = Integrate[I0^2, {tt, 0, t3}];
Fsimple = Lp/2*
  I0^2;                                                                       \
     FsimpleDisp = HoldForm[Lp/2*I0^2];
ImpSimpleInt = 
 Integrate[Fsimple, {tt, 0, t3}];                    ImpsimpleIntDisp1 = 
 HoldForm[  Integrate[Fsimple, {t, 0, t3}]];
													      
ImpsimpleIntDisp2 = HoldForm[  Integrate[Lp/2*I0^2, {t, 0, t3}] ];
ImpSimple = 
 Fsimple*t3;                                                                  \
     ImpsimpleDisp = HoldForm[Lp/2*I0^2*t3];
V2simple = 
 ImpSimple / 
  m;                                                                     \
V2simpleDisp = HoldForm[Lp*I0^2*t3/(2*m)];


Results

In[38]:= Print[" ::::::::::::::::: [S1] Simple Results - Constant Current \
::::::::::::::::: "];
Print["The charge used was              \!\(\*SuperscriptBox[\(I\), \(2\)]\)t \
= ", I0^2*t3, "[C]"]
Print[" Simple Force                     \!\(\*SubscriptBox[\(F\), \(S\)]\) = \
", FsimpleDisp, " = ", Fsimple];
Print[" Simple Integrated Impulse    \!\(\*SubscriptBox[\(Imp\), \(S\\\ \
Int\)]\) = ", ImpsimpleIntDisp1, " = ", ImpsimpleIntDisp2, " = ", ImpSimple];
Print[" Simple Impulse                 \!\(\*SubscriptBox[\(Imp\), \(S\)]\) = \
", ImpsimpleDisp, " = ", ImpSimple];
Print[" Simple Velcotiy                 \!\(\*SubscriptBox[\(V2\), \(S\)]\) = \
", V2simpleDisp, " = ", V2simple];

During evaluation of In[38]:=  ::::::::::::::::: [S1] Simple Results - Constant Current ::::::::::::::::: 
During evaluation of In[38]:= The charge used was              I^2t = 1.*10^6[C]
During evaluation of In[38]:=  Simple Force                     Subscript[F, S] = (Lp I0^2)/2 = 97.946
During evaluation of In[38]:=  Simple Integrated Impulse    Subscript[Imp, S Int] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t3\)]\(Fsimple \[DifferentialD]t\)\) = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t3\)]\(\*FractionBox[\(Lp\ 
\*SuperscriptBox[\(I0\), \(2\)]\), \(2\)] \[DifferentialD]t\)\) = 0.244865
During evaluation of In[38]:=  Simple Impulse                 Subscript[Imp, S] = 1/2 Lp I0^2 t3 = 0.244865
During evaluation of In[38]:=  Simple Velcotiy                 Subscript[V2, S] = (Lp I0^2 t3)/(2 m) = 44.0405

[S3] Simple Force and Velocity Computation with Circular Wire Assumption - Constant Current
In[54]:= Bc[x_] := \[Mu]0 * I0 / (2*\[Pi]*x);
FsimpleCDisp = 
  HoldForm[ Integrate[I0*\[Mu]0 * I0 / (2*\[Pi]*x) , {x, r, (w - r)}] ];
FsimpleC = Integrate[I0*Bc[x], {x, r, (w - r)}] // N;
ImpSimpleCInt = Integrate[FilrA, {tt, 0, t3}];        	ImpSimpleCIntDisp =  HoldForm[  Integrate[Integrate[I0 * B, {x, r, w - r}], {t, 0, t3}] ];
ImpSimpleC =  FsimpleC*t3;           			ImpSimpleCdisp = HoldForm[Lp/2*I0^2*t3];
V2SimpleC =  ImpSimpleC /   m;    			V2SimpleCDisp = HoldForm[FsimpleC*t3/m];
tRail = L / (V2SimpleC /      2);   (*                 Time the armature spends in the barrel *)
Iused = I0^2 *    tRail;            (*                    Current used in the shot *)

Results
In[62]:= Print[" ::::::::::::::::: [ILR2] ILR Equivalent Analytical Model Results - \
Constant Current  ::::::::::::::::: "];
Print["The charge used was                 \!\(\*SuperscriptBox[\(I\), \(2\)]\
\)t = ", Iused, "[C]"]
Print["Simple Circular Force               \!\(\*SubscriptBox[\(F\), \
\(SC\)]\) = ", FsimpleCDisp, " = ", FsimpleC];
Print["Simple Circular Impulse           \!\(\*SubscriptBox[\(Imp\), \
\(SC\)]\) = ", ImpSimpleCIntDisp, " = ", ImpSimpleCInt];
Print["Simple Circular Impulse           \!\(\*SubscriptBox[\(Imp\), \
\(SC\)]\) = ", ImpSimpleCdisp, " = ", ImpSimpleC];
Print["Simple Circular Velcotiy           \!\(\*SubscriptBox[\(V\), \(2\\\ SC\
\)]\) = ", V2SimpleCDisp, " = ", V2SimpleC];

During evaluation of In[62]:=  ::::::::::::::::: [ILR2] ILR Equivalent Analytical Model Results - Constant Current  ::::::::::::::::: 
During evaluation of In[62]:= The charge used was                 I^2t = 3.63465*10^6[C]
During evaluation of In[62]:= Simple Circular Force               Subscript[F, SC] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(r\), \(w - r\)]\(
\*FractionBox[\(I0\ \[Mu]0\ I0\), \(2\ \[Pi]\ x\)] \[DifferentialD]x\)\) = 97.902
During evaluation of In[62]:= Simple Circular Impulse           Subscript[Imp, SC] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t3\)]\(\((
\*SubsuperscriptBox[\(\[Integral]\), \(r\), \(w - r\)]\(I0\ B\) \[DifferentialD]x)\) \[DifferentialD]t\)\) = 0.0025 FilrA
During evaluation of In[62]:= Simple Circular Impulse           Subscript[Imp, SC] = 1/2 Lp I0^2 t3 = 0.244755
During evaluation of In[62]:= Simple Circular Velcotiy           Subscript[V, 2 SC] = (FsimpleC t3)/m = 44.0207

[ILR1] Analytically Compute Force and Velocity from ILR Model (1D Biot-Savart) - Constant Current
In[68]:= B[x_] := \[Mu]0 * I0 / (4*\[Pi]*x) * (L / Sqrt[L^2 + x^2]);
FilrADisp = 
  HoldForm[ Integrate[
    2*I0*\[Mu]0 * I0 / (4*\[Pi]*x) * (L / Sqrt[L^2 + x^2]), {x, r, (w - r)}] ];
FilrA = Integrate[I0*2*B[x], {x, r, (w - r)}] // N;

ImpILRaInt = Integrate[FilrA, {tt, 0, t3}];        ImpILRaIntDisp = 
 HoldForm[  Integrate[Integrate[I0 * B, {x, r, w - r}], {t, 0, t3}] ];
ImpILRa = Filr*
  t3;                                                               \
ImpILRadisp = HoldForm[Lp/2*I0^2*t3];
V2ilrA = ImpILRa / 
  m;                                                          V2ilrADisp = 
 HoldForm[FilrA*t3/m];
tRail = L / (V2ilr / 
     2);      (*                  Time the armature spends in the barrel *)
Iused = I0^2 * 
   tRail;            (*                  Current used in the shot *)

Results
In[115]:= Print[" ::::::::::::::::: [ILR1] ILR Analytical Model Results - Constant \
Current  ::::::::::::::::: "];
Print["The charge used was             \!\(\*SuperscriptBox[\(I\), \(2\)]\)t \
= ", Iused, "[C]"]
Print[" ILR Analytic Force         2*\!\(\*SubscriptBox[\(F\), \(ILR\\\ \
A\)]\) = ", FilrADisp, " = ", FilrA];
Print[" ILR Integrated Impulse   \!\(\*SubscriptBox[\(Imp\), \(ILR\\\ A\\\ \
Int\)]\) = ", ImpILRaIntDisp, " = ", ImpILRaInt];
Print[" ILR Analytic Impulse       \!\(\*SubscriptBox[\(Imp\), \(ILR\\\ \
A\)]\) = ", ImpILRadisp, " = ", ImpILRa];
Print[" ILR Analytic Velcotiy       \!\(\*SubscriptBox[\(V\), \(2\\\ ILR\\\ A\
\)]\) = ", V2ilrADisp, " = ", V2ilrA];
Plot[B[x], {x, r, sep}, 
 AxesLabel -> {"Distance [m]", "Magnetic Field Strength [T]"}, 
 PlotLabel -> "Magnetic Field Strength"]

During evaluation of In[115]:=  ::::::::::::::::: [ILR1] ILR Analytical Model Results - Constant Current  ::::::::::::::::: 

During evaluation of In[115]:= The charge used was             I^2t = 1.6*10^8/V2ilr[C]

During evaluation of In[115]:=  ILR Analytic Force         2*Subscript[F, ILR A] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(r\), \(w - r\)]\(
\*FractionBox[\(2\ I0\ \[Mu]0\ I0\ L\), \(\((4\ \[Pi]\ x)\)\ 
\*SqrtBox[\(
\*SuperscriptBox[\(L\), \(2\)] + 
\*SuperscriptBox[\(x\), \(2\)]\)]\)] \[DifferentialD]x\)\) = 97.7704

During evaluation of In[115]:=  ILR Integrated Impulse   Subscript[Imp, ILR A Int] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t3\)]\(\((
\*SubsuperscriptBox[\(\[Integral]\), \(r\), \(w - r\)]\(I0\ B\) \[DifferentialD]x)\) \[DifferentialD]t\)\) = 0.244426

During evaluation of In[115]:=  ILR Analytic Impulse       Subscript[Imp, ILR A] = 1/2 Lp I0^2 t3 = 0.0025 Filr

During evaluation of In[115]:=  ILR Analytic Velcotiy       Subscript[V, 2 ILR A] = (FilrA t3)/m = 0.44964 Filr

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PlotLabel->FormBox["\"Magnetic Field Strength\"", TraditionalForm],
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PlotRangeClipping->True,
PlotRangePadding->{{Scaled[0.02], Scaled[0.02]}, {Scaled[0.05], Scaled[0.05]}},Ticks->{Automatic, Automatic}]\)

[ILR2] Use Analytic Equivalent to Compute Force and Velocity from ILR Model (1D Biot-Savart) - Constant Current
In[84]:= F1 = Log[(L + Sqrt[L^2 + d^2])/d];
F2 = Log[(L + Sqrt[L^2 + r^2])/r];
F3 = Log[(L + Sqrt[L^2 + (w - r)^2])/(w - r)];
F4 = Log[(L + Sqrt[L^2 + (w - d)^2])/(w - d)];
FilrE =  \[Mu]0 * I0^2/ (4*\[Pi])  * ( -F1 + F2 - F3 + F4) // N;
FilrEDisp    = 
  HoldForm[  \[Mu]0 * I0^2/ (4*\[Pi])  * (    -Log[(L + Sqrt[L^2 + d^2])/d]
      										+ Log[(L + Sqrt[L^2 + r^2])/r]
      										- Log[(L + Sqrt[L^2 + (w - r)^2])/(w - r)]
      										+ Log[(L + Sqrt[L^2 + (w - d)^2])/(w - d)])];

ImpILReInt = Integrate[FilrE, {tt, 0, t3}];        ImpILReIntDisp = 
 HoldForm[  Integrate[Integrate[I0 * B, {x, r, w - r}], {t, 0, t3}] ];
ImpILRe = Filr*
  t3;                                                              \
ImpILRedisp = HoldForm[Lp/2*I0^2*t3];
V2ilrE = ImpILR / 
  m;                                                           V2ilrEDisp = 
 HoldForm[Filr*t3/m];
tRail = L / (V2ilrE / 
     2);      (*                Time the armature spends in the barrel *)
Iused = I0^2 * tRail;             (*                Current used in the shot *)

Results

In[95]:= Print[" ::::::::::::::::: [ILR2] ILR Equivalent Analytical Model Results - \
Constant Current  ::::::::::::::::: "];
Print["The charge used was                 \!\(\*SuperscriptBox[\(I\), \(2\)]\
\)t = ", Iused, "[C]"]
Print[" ILR Equivalent Force             \!\(\*SubscriptBox[\(F\), \(ILR\\\ E\
\)]\) = ", FilrEDisp, " = ", FilrE];
Print[" ILR Integrated Impulse       \!\(\*SubscriptBox[\(Imp\), \(ILR\\\ E\\\
\ Int\)]\) = ", ImpILReIntDisp, " = ", ImpILReInt];
Print[" ILR Equivalent Impulse          \!\(\*SubscriptBox[\(Imp\), \(ILR\\\ \
E\)]\) = ", ImpILRedisp, " = ", ImpILRe];
Print[" ILR Equivalent Velcotiy          \!\(\*SubscriptBox[\(V\), \(2\\\ ILR\
\\\ E\)]\) = ", V2ilrEDisp, " = ", V2ilrE];



During evaluation of In[95]:=  ::::::::::::::::: [ILR2] ILR Equivalent Analytical Model Results - Constant Current  ::::::::::::::::: 
During evaluation of In[95]:= The charge used was                 I^2t = 889600./ImpILR[C]
During evaluation of In[95]:=  ILR Equivalent Force             Subscript[F, ILR E] = (\[Mu]0 I0^2 (-Log[(L+Sqrt[L^2+d^2])/d]+Log[(L+Sqrt[L^2+r^2])/r]-Log[(L+Sqrt[L^2+(w-r)^2])/(w-r)]+Log[(L+Sqrt[L^2+(w-d)^2])/(w-d)]))/(4 \[Pi]) = 97.7704
During evaluation of In[95]:=  ILR Integrated Impulse       Subscript[Imp, ILR E Int] = \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t3\)]\(\((
\*SubsuperscriptBox[\(\[Integral]\), \(r\), \(w - r\)]\(I0\ B\) \[DifferentialD]x)\) \[DifferentialD]t\)\) = 0.244426
During evaluation of In[95]:=  ILR Equivalent Impulse          Subscript[Imp, ILR E] = 1/2 Lp I0^2 t3 = 0.0025 Filr
During evaluation of In[95]:=  ILR Equivalent Velcotiy          Subscript[V, 2 ILR E] = (Filr t3)/m = 179.856 ImpILR