(*ASYMMETRICALTETHERMODEL*)
​
Out[]=
{50000,50000,50000,1000,1000,5000,970,0.00006283,0.5,0.5,0.5,3.98778×
14
10
,5000000}
Out[]=
0.00447207
Out[]=
86400
(Dialog) In[]:=
(*INPUTPARTSREQUIREDTOCREATEKINEICANDPOTENITALENERGYEQUATIONS*)​​​​​​
R
p1
[t]=√((R[t])^2+
L
1
^2+2*R[t]*
L
1
*(Sin[θ[t]]*Sin[ψ[t]]+Cos[θ[t]]*Cos[ψ[t]]));​​
R
p2
[t]=√((R[t])^2+
L
2
^2+2*R[t]*
L
2
*(Sin[θ[t]]*Sin[ψ[t]]+Cos[θ[t]]*Cos[ψ[t]]));​​
R
m
[t]=√(R[t])^2+(
L
p2
-
L
2
)^2+2*R[t]*(
L
p2
-
L
2
)*(Sin[θ[t]]*Sin[ψ[t]]+Cos[θ[t]]*Cos[ψ[t]]);​​
R
t1
[t]=√((R[t])^2+(
L
1
/2)^2+R[t]*
L
1
*(Sin[θ[t]]*Sin[ψ[t]]+Cos[θ[t]]*Cos[ψ[t]]));​​
R
t1
[t]=√((R[t])^2+(
L
2
/2)^2+R[t]*
L
2
*(Sin[θ[t]]*Sin[ψ[t]]+Cos[θ[t]]*Cos[ψ[t]]));​​​​
i
p1
=(
M
p1
*
r
p1
^2)2;​​
i
p2
=(
M
p2
*
r
p2
^2)2;​​
i
m
=(
M
m
*
r
m
^2)/2;​​
i
t1
=ρ*A*
L
1
*(3*rt^2+
L
1
^2)/12;​​
i
t2
=ρ*A*
L
2
*(3*rt^2+
L
2
^2)/12;​​​​​​(*KINETICENERGYEQUATION*)​​T=(
M
p1
*(
R
p1
'[t])^2)2+(
M
p2
*(
R
p2
'[t])^2)2+((
M
m
*(
R
m
'[t])^2)/2)+((ρ*A*
L
1
*(
R
t1
'[t])^2)/2)+((ρ*A*
L
2
*(
R
t2
'[t])^2)/2)+
i
p1
+
i
p2
+
i
m
+
i
t1
+
i
t2
*(ψ'[t]+θ'[t])^22;​​TraditionalForm[T]​​​​(*POTENTIALENERGYEQUATION*)​​n=3;​​U=-(μ*
M
p1
)
((R[t])^2)+(
L
1
^2)+(2*R[t]*
L
1
*Cos[ψ[t]])
-(μ*
M
p2
)
((R[t])^2)+(
L
2
^2)+(2*R[t]*
L
2
*Cos[ψ[t]])
-(μ*
M
m
)√(((R[t])^2)+((
L
p2
-
L
2
)^2)+(2*R[t]*(
L
p2
-
L
2
)*Cos[ψ[t]]))-
n
∑
j=1
(μ*ρ*A*
L
1
)n*
2
(R[t])
+
2
((2*j)-1)*
L
1
(2*n)
+
2*R[t]*
L
1
*((2*j)-1)
2*n
*Cos[ψ[t]]-
n
∑
j=1
(μ*ρ*A*
L
2
)n*
2
(R[t])
+
2
((2*j)-1)*
L
2
(2*n)
-
2*R[t]*
L
2
*((2*j)-1)
2*n
*Cos[ψ[t]];​​TraditionalForm[U]
(Dialog) Out[]//TraditionalForm=
1250
2
′
R
m
(t)
+250
2
′
R
p1
(t)
+250
2
′
R
p2
(t)
+761.814
2
′
R
t1
(t)
+761.814
2
′
R
t2
(t)
+7.93556×
10
10
2
(
′
θ
(t)+
′
ψ
(t))
(Dialog) Out[]//TraditionalForm=
-
2.0253×
17
10
-
125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9
-
2.0253×
17
10
-25000R(t)cos(ψ(t))+
2
R(t)
+156250000
-
2.0253×
17
10
-
25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9
-
2.0253×
17
10
25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9
-
2.0253×
17
10
25000R(t)cos(ψ(t))+
2
R(t)
+156250000
-
2.0253×
17
10
125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9
-
3.98778×
17
10
50000R(t)cos(ψ(t))+
2
R(t)
+625000000
-
9.96946×
17
10
2
R(t)
(Dialog) In[]:=
(*CONSTRUCTINGLAGRANGEEQUATIONS*)​​Clear[eqn1,eqn2,eqn3]​​
R
1
=D[T,ψ'[t]];​​
R
2
=D[
R
1
,t];​​
R
3
=D[T,ψ[t]];​​
R
4
=D[U,ψ[t]];​​eqn1=
R
2
-
R
3
+
R
4
-τ;​​TraditionalForm[eqn1]​​​​(*ConstructingtheEoMinR[t]*)​​
R
5
=D[T,R'[t]];​​
R
6
=D[
R
5
,t];​​
R
7
=D[T,R[t]];​​
R
8
=D[U,R[t]];​​eqn2=
R
6
-
R
7
+
R
8
;​​TraditionalForm[%]​​​​(*ConstructingtheEoMinθ[t]*)​​
R
9
=D[T,θ'[t]];​​
R
10
=D[
R
9
,t];​​
R
11
=D[T,θ[t]];​​
R
12
=D[U,θ[t]];​​eqn3=
R
10
-
R
11
+
R
12
;​​TraditionalForm[%]
(Dialog) Out[]//TraditionalForm=
4.21937×
21
10
R(t)sin(ψ(t))
3/2
-
125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9

+
2.53162×
21
10
R(t)sin(ψ(t))
3/2
(-25000R(t)cos(ψ(t))+
2
R(t)
+156250000)
+
8.43875×
20
10
R(t)sin(ψ(t))
3/2
-
25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9

-
8.43875×
20
10
R(t)sin(ψ(t))
3/2

25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9

-
2.53162×
21
10
R(t)sin(ψ(t))
3/2
(25000R(t)cos(ψ(t))+
2
R(t)
+156250000)
-
4.21937×
21
10
R(t)sin(ψ(t))
3/2

125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9

-
9.96946×
21
10
R(t)sin(ψ(t))
3/2
(50000R(t)cos(ψ(t))+
2
R(t)
+625000000)
+1.58711×
11
10
(
′′
θ
(t)+
′′
ψ
(t))-5000000
(Dialog) Out[]//TraditionalForm=
1.01265×
17
10
2R(t)-
125000
3
cos(ψ(t))
3/2
-
125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9

+
1.01265×
17
10
(2R(t)-25000cos(ψ(t)))
3/2
(-25000R(t)cos(ψ(t))+
2
R(t)
+156250000)
+
1.01265×
17
10
2R(t)-
25000
3
cos(ψ(t))
3/2
-
25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9

+
1.01265×
17
10
2R(t)+
25000
3
cos(ψ(t))
3/2

25000
3
R(t)cos(ψ(t))+
2
R(t)
+
156250000
9

+
1.01265×
17
10
(2R(t)+25000cos(ψ(t)))
3/2
(25000R(t)cos(ψ(t))+
2
R(t)
+156250000)
+
1.01265×
17
10
2R(t)+
125000
3
cos(ψ(t))
3/2

125000
3
R(t)cos(ψ(t))+
2
R(t)
+
3906250000
9

+
1.99389×
17
10
(2R(t)+50000cos(ψ(t)))
3/2
(50000R(t)cos(ψ(t))+
2
R(t)
+625000000)
+
9.96946×
17
10
R(t)
3/2
(
2
R(t)
)
(Dialog) Out[]//TraditionalForm=
1.58711×
11
10
(
′′
θ
(t)+
′′
ψ
(t))
​
​
(Dialog) In[]:=
(*CONSTANTSDEFININGPROBLEM*)​​{
L
1
,
L
2
,
L
p2
,
M
p1
,
M
p2
,
M
m
,ρ,A,
r
p1
,
r
p2
,
r
m
,μ,τ}=25000,25000,25000,500,500,2500,970,62.83*
-6
10
,0.5,0.5,0.5,3.9877848*
14
10
,5000000;​​​​rt=
A
π
;​​
t
end
=86400;​​​​vars={ψ,R,θ};​​domain=Reals;​​equations={eqn1==0,eqn2==0,eqn3==0};​​If[AllTrue[equations,Head[#]===Equal&],Print["eqns are valid"],Print["eqns not valid"]];​​​​s=NDSolve[{eqn1==0,eqn2==0,eqn3==0,ψ[0]==0,ψ'[0]0,R[0]35000000,R'[0]0,θ[0]97.1877,θ'[0]0.000093742},{ψ,R,θ},{t,0,
t
end
},MaxStepsInfinity,AccuracyGoalAutomatic,PrecisionGoalAutomatic]​​​​(*Extractthesolutionsol={ψ[t],R[t],θ[t]}/.First@s;​​​​(*Definetheresidualfunction*)​​residualFunction[t_?NumericQ]:={eqn1,eqn2,eqn3}/.Thread[{ψ[t],R[t],θ[t]}->sol]//Norm;​​​​Evaluatetheresidualfunctionatdifferentpoints​​residualValues=Table[residualFunction[t],{t,0,
t
end
,0.1}];​​​​Analyzethebehavioroftheresidualvalues​​ListLinePlot[residualValues,DataRange->{0,
t
end
}]*)​​​​(*
R
apo
:=NMaximize[{Evaluate[R[t]/.s[[1]]],(
t
end
-20000)<t<
t
end
},t];​​
R
per
:=NMinimize[{Evaluate[R[t]/.s[[1]]],(
t
end
-20000)<t<
t
end
},t];​​e=(
R
apo
-
R
per
)/(
R
apo
+
R
per
);​​First[e]​​(*2*First[e]isthewaytouseeinlatercalculations*)​​Plot[Evaluate[ψ[t]/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"ψ[t] (rad)"}]​​Plot[Evaluate[ψ'[t]/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"ψ'[t] (rad/s)"}]​​PlotEvaluate[(0.001*L*ψ'[t])/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabeltime,"
V
pay
​[t] (km/s)"​​Plot[Evaluate[(0.000000001*
M
p
*L*((ψ'[t])^2)/A)/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"
σ
cent
​[t] (GPa)"}]​​Plot[Evaluate[(0.001*R[t])/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"R[t] (km)"}]​​Plot[Evaluate[(0.001*R'[t])/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"R'[t] (km/s)"}]​​Plot[Evaluate[θ[t]/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"θ[t] (rad)"}]​​Plot[Evaluate[θ'[t]/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"θ'[t] (rad/s)"}]​​Plot[Evaluate[(
M
p
*L*((ψ'[t])^2)+(ρ*A*(L^2)/2)*((ψ'[t])^2))/.s],{t,0,
t
end
},FrameTrue,FrameTicksAutomatic,GridLinesAutomatic,FrameLabel{time,"
F
cent
​[t] (N)"}]*)