# Euler Rotation

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```					Euler Rotation
Angular Momentum
   The angular momentum J is
J i  I ij  j           defined in terms of the inertia
tensor and angular velocity.
• All rotations included

p      The angular momentum
need not be collinear with
J                            the angular velocity.
r
• Not along principal axis
• Not at center of mass
Torque
   Torque N causes a change
 I ij j 
dJ i d
in angular momentum.            Ni 
• Rotational second law                dt dt

   Use the body frame for a
constant inertia tensor.
Ni 
dJ i
dt
 

 J i         
• Motion in accelerated frame
Ni 
d
I ij j    ijk  j I kll
dt
d j
N i  I ij            ijk  j I kll
dt
Euler Equations
   Select the body coordinates
to match the principal axes.       I1   0    0          J1  I11
              
• Three moments of inertia    I 0     I2   0         J 2  I 22
• Simplified angular             0          I3 
      0              J 3  I 33
momentum terms

   Redo the torque equations.      N1  I11  ( I 3  I 2 )23

N 2  I 22  ( I1  I 3 )31

   These are Euler’s equations
N 3  I 33  ( I 2  I1 )12

of motion.
Dumbbell
  sin        1 0 0
   The principal axes are along
                2 1     2            and perpendicular to the rod.
  0        I   0 2 0 ml
  cos       0 0 0
                                  Measure change in angular
          x3                momentum.

x1         
J
l          N1  I11  ( I 3  I 2 )23  0

N 2  I 22  ( I1  I 3 )31  1 ml 2 2 sin  cos
                      2

N 3  I 33  ( I 2  I1 )12  0

Euler Angles
    A rotation matrix can be
described with three free
e3
parameters.                        e3 
•   Select three separate
e2
1)   Rotation of f about e3 axis.
2)   Rotation of  about e1 axis.    e1 f
3)   Rotation of y about e3 axis.           y    e1

    These are the Euler angles.
Euler Matrices
   Any vector z can be rotated
though the Euler angles.                   cosf      sin f     0
                      
Sf    sin f   cosf      0
   The equivalent matrix                      0                   1
             0        
operation is the product of
three separate operations.                1     0            0 
                       
S   0 cos          sin  
 0  sin        cos 
                       
zi  S ij z j
 cosy      siny      0
Sij  S (y )ik S ( ) kl S (f ) lj                              
Sy    siny    cosy      0
 0                   1
             0        
Full Rotation
   Any rotation may be expressed with the three angles.
Sij  S (y )ik S ( ) kl S (f ) lj

S
 cosy cosf  siny cos sin f     cosy sin f  siny cos cosf                  siny sin  
                                                                                         
  siny cosf  cosy cos sin f    siny sin f  cosy cos cosf                cosy sin  
          sin  sin f                     sin  cosf                            cos 
                                                                                         

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