Class 43: Eulers Angles and Equations of Class 43: Euler s Angles and - PowerPoint PPT Presentation

Class 43: Euler’s Angles and Equations of Class 43: Euler s Angles and Equations of Motion

General motion of a spinning top precession   nutation spin   p   

Euler’s Angles Every infinitesimal motion of a spinning top composes of variation of three y p g p p angles (generalized coordinates): 1. Rotation about e 3 by d Φ - precession 2 Rotation about the new e 1 (e 1 ’)by d θ - nutation 2. Rotation about the new e 1 (e 1 )by d θ nutation 3. Rotation about the new e 3 (e 3 ’’)by d ψ - spin

ω in terms of Euler’s angles ( e 2 ) ( e 3 )     ( e 1 )                        ˆ ˆ ˆ ˆ z sin sin e sin cos e cos e 1 2 3           ˆ ˆ x ˆ ' cos e sin e 1 2        ˆ z z ˆ ' ' e e 3            z ˆ x ˆ ' z ˆ '                                      ˆ ˆ ˆ ( ( sin i sin i cos ) ) e ( ( sin i cos sin i ) ) e ( ( cos ) ) e 1 2 3            sin sin cos 1              sin i cos sin i 2         cos 3

Symmetric top in Earth’s gravitational field I      For symmetric top, 1 2 1 1 1          2 2 2 T 1 1 2 2 3 3 2 2 2 2 2 2 1 1 1                            2 2  2 ( sin sin cos ) ( sin cos sin ) ( cos ) 3 2 2 2 1 1                       2 2 2 2 2 2  2 2 ( ( sin i ) ) ( ( cos ) ) 3 2 2   V MgL g cos L  1 1                   Mg 2 2 2  2 L T - V ( sin ) ( cos ) MgL cos 3 2 2