Transcript Physics 321 Hour 31 Euler’s Angles

Physics 321
Hour 31
Euler’s Angles
Space and Body Coordinates
𝑒3
𝑧
𝑒2
𝑦
𝑥
𝑒1
• Body coordinates are on principal axes
Euler’s Equations – No Torques
𝐼11 𝜔1 = 𝐼22 − 𝐼33 𝜔2 𝜔3
𝐼22 𝜔2 = 𝐼33 − 𝐼11 𝜔3 𝜔1
𝐼33 𝜔3 = 𝐼11 − 𝐼22 𝜔1 𝜔2
Euler’s Equations – No Torques, I11=I22
𝐼11 𝜔1 = 𝐼11 − 𝐼33 𝜔2 𝜔3
𝐼22 𝜔2 = 𝐼33 − 𝐼11 𝜔3 𝜔1
𝐼33 𝜔3 = 0
𝜔3 is a constant
𝐼11 − 𝐼33
𝜔1 =
𝜔2 𝜔3 ≡ Ω𝑏 𝜔2
𝐼11
𝐼11 − 𝐼33
𝜔2 = −
𝜔3 𝜔1 ≡ −Ω𝑏 𝜔1
𝐼11
2
𝜔2 = −Ω𝑏 𝜔1 = −Ω𝑏 𝜔2
Euler’s Equations – No Torques, I11=I22
𝜔0 cos Ω𝑏 𝑡
𝜔 = −𝜔0 sin Ω𝑏 𝑡
𝜔3
𝐼11 𝜔0 cos Ω𝑏 𝑡
𝐿 = −𝐼11 𝜔0 sin Ω𝑏 𝑡
𝐼33 𝜔3
In the space axes (𝐿 = 𝐿𝑧):
𝐿
Ω𝑠 =
𝐼11
𝜔0 sin α cos Ω𝑠 𝑡
𝜔 = 𝜔0 sin 𝛼 sin Ω𝑠 𝑡
𝜔0 cos 𝛼
sin 𝜃 cos Ω𝑠 𝑡
𝑒3 = sin 𝜃 sin Ω𝑠 𝑡
cos 𝜃
Prolate object: Ωb<0, Ωs>0
Example
football.nb
Constants of the Motion, No Torque
𝐼11 𝜔0 cos Ω𝑏 𝑡
𝐿 = −𝐼11 𝜔0 sin Ω𝑏 𝑡
𝐼33 𝜔3
Ω𝑏 =
𝐼11 −𝐼33
𝜔3
𝐼11
𝜔0 cos Ω𝑏 𝑡
𝜔 = −𝜔0 sin Ω𝑏 𝑡 in body
𝜔3
is constant
ω and L are constants
𝜔3 and 𝐿3 are constants
𝐿 in space is constant, usually 𝐿 = 𝐿𝑧 (no torques)
cos 𝜃 = 𝐿3 /𝐿 so θ is constant
cos 𝛼 = 𝜔3 /𝜔 so α is constant
Constants of the Motion, No Torque
𝐼11 𝜔0 cos Ω𝑏 𝑡
𝐿 = −𝐼11 𝜔0 sin Ω𝑏 𝑡
𝐼33 𝜔3
Ω𝑏 =
𝐼11 −𝐼33
𝜔3
𝐼11
𝜔0 cos Ω𝑏 𝑡
𝜔 = −𝜔0 sin Ω𝑏 𝑡 in body
𝜔3
is constant
ω and L are constants
𝜔3 and 𝐿3 are constants
𝐿 in space is constant, usually 𝐿 = 𝐿𝑧 (no torques)
cos 𝜃 = 𝐿3 /𝐿 so θ is constant
cos 𝛼 = 𝜔3 /𝜔 so α is constant
Unit Vectors
𝑒′3 = 𝑒3 = sin 𝜃 cos 𝜑 𝑥 + sin 𝜃 sin 𝜑 𝑦 + cos 𝜃 𝑧
𝑒′2 = cos 𝜑 𝑦 −sin 𝜑 𝑥
𝑒′1 = cos 𝜃 cos 𝜑 𝑥 + cos 𝜃 sin 𝜑 𝑦 − sin 𝜃 𝑧
Angular Velocities
𝜓 is spin about the body 3-axis
𝜃 is tipping of the body 3-axis
𝜑 is precession about the space z-axis
𝜔 = 𝜑𝑧 + 𝜃𝑒′2 + ψ𝑒3
𝑧 = 𝑒3 cos 𝜃 − 𝑒′1 sin 𝜃
𝜔 = −𝜑 sin 𝜃 𝑒′1 + 𝜃𝑒′2 + (ψ + 𝜑 cos 𝜃)𝑒′3
Example
HW31 Answers.nb
Angular Momentum
𝐿 = −𝐼11 𝜑 sin 𝜃 𝑒′1 + 𝐼22 𝜃𝑒′2
+𝐼33 (ψ + 𝜑 cos 𝜃)𝑒′3
𝐿3 = 𝐼33 ψ + 𝜑 cos 𝜃
𝐿𝑧 = 𝐿 ∙ 𝑧 = 𝐼11 𝜑sin2 𝜃 + 𝐿3 cos 𝜃
𝐿𝑧 − 𝐿3 cos 𝜃
→𝜑=
𝐼11 sin2 𝜃
For torque-free systems, Lz and L3 are constants,
so 𝜑 is also constant.
Kinetic Energy
1
1
1
𝑇 = 𝐼11 𝜔1 2 + 𝐼22 𝜔2 2 + 𝐼33 𝜔3 2
2
2
2
1
1
= 𝐼11 𝜑 2 sin2 𝜃 + 𝜃 2 + 𝐼33 ψ + 𝜑 cos 𝜃
2
2
2