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

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Aug 02, 2023 813 likes •885 views

Class 43: Eulers Angles and Equations of Class 43: Euler s Angles and Equations of Motion General motion of a spinning top precession nutation spin p Eulers Angles Every infinitesimal motion of a spinning

SLIDE 1

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

SLIDE 2

General motion of a spinning top

precession

 

nutation spin



p

  

SLIDE 3

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 e3 by dΦ - precession

2 Rotation about the new e1 (e1’)by dθ - nutation

2. Rotation about the new e1 (e1 )by dθ

nutation

3. Rotation about the new e3 (e3’’)by dψ - spin

SLIDE 4

ω in terms of Euler’s angles

(e2) (e1) (e3)

          

e ˆ ' z ˆ e ˆ sin e ˆ cos ' x ˆ e ˆ cos e ˆ cos sin e ˆ sin sin z ˆ

2 1 3 2 1

                              ˆ ) ( ˆ ) i i ( ˆ ) i i ( ' z ˆ ' x ˆ z ˆ e ' z

                                i i cos sin sin e ) cos ( e ) sin cos sin ( e ) cos sin sin (

1 3 2 1

                                   cos sin cos sin

3 2

               

SLIDE 5

Symmetric top in Earth’s gravitational field I

         2 1 2 1 2 1 T top, symmetric For

2 3 3 2 2 2 2 1 1 2 1

                            ) ( 1 ) i ( 1 ) cos ( 2 1 ) sin cos sin ( 2 1 ) cos sin sin ( 2 1 2 2 2

2 2 2 2 2 3 2 2

                         cos MgL V ) cos ( 2 ) sin ( 2

2 3 2 2 2

              cos MgL ) cos ( 2 1 ) sin ( 2 1 V

L g