Nonlinear Dynamics of a Kite in Flight
Nonlinear Dynamics of a Kite in Flight Nonlinear Dynamics of a Kite in Flight
- Page 15 and 16: y
- Page 17 and 18: y
- Page 19 and 20: θ ∗ θ s
- Page 21 and 22: y θ
- Page 23 and 24: θ (θ(0), θ(0)) (θ
- Page 25 and 26: θ (θ(0), θ(0)) (θ
- Page 27 and 28: y
- Page 29 and 30: y
- Page 31 and 32: y Im
- Page 33 and 34: y Im
- Page 35 and 36: θ n 1 3
- Page 37 and 38: = π θ θ = 0 θ = −π Saddle Po
- Page 39 and 40: = π θ θ = 0 θ = −π Saddle Po
- Page 41 and 42: y
- Page 43 and 44: y vcos θ θ λ - vsin θ θ ^ θ
- Page 45 and 46: y vcos θ θ λ - vsin θ θ ^ θ
- Page 47 and 48: y vcos θ θ λ - vsin θ θ ^ θ
- Page 49 and 50: B(α) 1 2 1 2
- Page 51 and 52: θ = 1.0 θ = 0 θ = -1.0 Saddle Po
y<br />
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θ<br />
T<br />
T x<br />
x<br />
λ<br />
T y<br />
F L<br />
F G<br />
F D
y<br />
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θ<br />
T<br />
T x<br />
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T y<br />
F L<br />
F G<br />
F D
y<br />
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θ<br />
T<br />
T x<br />
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λ<br />
T y<br />
F L<br />
F G<br />
F D
β<br />
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θ s
θ ∗<br />
θ s<br />
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β ∗<br />
β
θ ∗<br />
θ s<br />
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β ∗<br />
β
y<br />
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θ<br />
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T<br />
x<br />
F L cosθ<br />
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θ<br />
F L<br />
θ<br />
F G<br />
F G cosθ<br />
F D s<strong>in</strong>θ<br />
θ<br />
F D<br />
θ
θ<br />
(θ(0), θ(0))<br />
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<br />
(θ(t), θ(t))<br />
θ
θ<br />
(θ(0), θ(0))<br />
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<br />
(θ(t), θ(t))<br />
θ
θ<br />
(θ(0), θ(0))<br />
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(θ(t), θ(t))<br />
θ
θ<br />
(θ(0), θ(0))<br />
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(θ(t), θ(t))<br />
θ
θ<br />
(θ(0), θ(0))<br />
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(θ(t), θ(t))<br />
θ
y<br />
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This eigenvector<br />
corresponds to the<br />
grow<strong>in</strong>g eigensolution.<br />
This eigenvector<br />
corresponds to the<br />
decay<strong>in</strong>g eigensolution.<br />
x
y<br />
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x
y<br />
Im<br />
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Re<br />
x
y<br />
Im<br />
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Re<br />
x
y<br />
Im<br />
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Re<br />
x
θ s<br />
∗ θ Saddle Po<strong>in</strong>t<br />
Unknown<br />
Stability<br />
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β ∗<br />
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Δ > 0<br />
Δ < 0<br />
β
θ n<br />
1<br />
3<br />
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2<br />
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θ n+1<br />
4
θ n<br />
1<br />
3<br />
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2<br />
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θ n+1<br />
4
= π<br />
θ<br />
θ = 0<br />
θ = −π<br />
Saddle Po<strong>in</strong>t Center<br />
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π<br />
θ =<br />
2
= π<br />
θ<br />
θ = 0<br />
θ = −π<br />
Saddle Po<strong>in</strong>t Center<br />
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π<br />
θ =<br />
2
= π<br />
θ<br />
θ = 0<br />
θ = −π<br />
Saddle Po<strong>in</strong>t Center<br />
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π<br />
θ =<br />
2
y<br />
θ<br />
λ<br />
r<br />
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θ<br />
vs<strong>in</strong> θ<br />
v<br />
θ<br />
vcos θ<br />
x
y<br />
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λ<br />
θ ^<br />
θ e<br />
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θ ^<br />
λ e<br />
θ<br />
vs<strong>in</strong> θ<br />
v<br />
θ<br />
θ<br />
vcos θ<br />
x
y<br />
θ<br />
λ<br />
r<br />
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θ<br />
vs<strong>in</strong> θ<br />
v<br />
θ<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
y<br />
vcos θ<br />
θ<br />
λ - vs<strong>in</strong> θ<br />
θ ^<br />
θ<br />
λ e<br />
θ e<br />
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λ<br />
vs<strong>in</strong> θ<br />
v<br />
vcos θ<br />
x
B(α)<br />
1<br />
2<br />
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1<br />
2<br />
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α
θ s<br />
θ∗ θb θ s<br />
θ b<br />
θ ∗<br />
Φ < 0<br />
Φ > 0<br />
Δ>0, τ 0<br />
β ∗ β<br />
Δ>0, τ>0 Unstable Node or Unstable Spiral<br />
Δ0 Saddle Po<strong>in</strong>t
θ = 1.0<br />
θ = 0<br />
θ = -1.0<br />
Saddle Po<strong>in</strong>t<br />
Stable Node<br />
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θ =<br />
π<br />
2
0.81<br />
θ Radians<br />
0.805<br />
0.81<br />
θ Radians<br />
0.805<br />
Time<br />
θ=<br />
0.001<br />
Time<br />
θ = 0.805 θ = 0.815<br />
θ=<br />
-0.001<br />
Radians<br />
0.81<br />
<br />
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Stable<br />
Node<br />
Time<br />
0.815<br />
θ Radians<br />
0.81<br />
0.815<br />
θ Radians<br />
0.81<br />
Time<br />
Time
0.81<br />
θ Radians<br />
0.805<br />
0.81<br />
θ Radians<br />
0.805<br />
Time<br />
θ=<br />
0.001<br />
Time<br />
θ = 0.805 θ = 0.815<br />
θ=<br />
-0.001<br />
Radians<br />
0.81<br />
<br />
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Stable<br />
Node<br />
Time<br />
0.815<br />
θ Radians<br />
0.81<br />
0.815<br />
θ Radians<br />
0.81<br />
Time<br />
Time
θ Radians<br />
0.447<br />
θ = 0.05<br />
0<br />
θ = 0.44<br />
θ = -0.05<br />
Time<br />
θ Radians<br />
0.81<br />
0.44733<br />
Saddle<br />
Po<strong>in</strong>t<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
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<br />
θ Radians<br />
0.81<br />
0.448<br />
θ = 0.46<br />
Time
θ<br />
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θ
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
θ Radians<br />
θ = 0.001<br />
θ = 0.649<br />
θ = -0.001<br />
0.649<br />
0<br />
Time<br />
θ Radians<br />
0.650<br />
0<br />
Unstable<br />
Node<br />
Time<br />
<br />
<br />
<br />
<br />
<br />
<br />
θ Radians<br />
0.651<br />
0<br />
θ = 0.651<br />
Time
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