L_19_dynamics of General Rigid Bodies.ppt

Classical mechanics

1. Undergraduate Classical Mechanics Spring 2017
Physics 319
Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 19

2. Undergraduate Classical Mechanics Spring 2017
•Setup: let the unprimed coordinate system be those in an inertial
frame and the primed coordinates be a coordinate set tied to the
rigid body.
•Terminology/Jargon
–Unprimed Coordinate System: Space-fixed Frame
–Primed Coordinate System: Body-fixed Frame
•For simplicity, the origin of the primed coordinate system is almost
always the center of mass of the rigid body
General Rigid Bodies
1
ˆ ˆe x
2
ˆ ˆe y
3
ˆˆe z
3
ˆˆe z 
1
ˆ ˆe x 
2
ˆ ˆe y r t


o
r t

r t

3. Undergraduate Classical Mechanics Spring 2017
Motion of Center of Mass
•Use Space-fixed coordinate system. Newton’s Second Law
applies
•Integrate equations of motion in the space-fixed frame
•In many cases the motion of the Center of Mass is simply
solved, e.g. Coin flip or motion of a baton in gravity. Center
of mass follows usual motion in a gravitational field.
cm ext
cm cm
v F dt
r v dt





 
o cm ext
d d
mr mr F
dt dt
 
 
 

4. Undergraduate Classical Mechanics Spring 2017
Body-Fixed Coordinates
•Discuss rotational part of the problem using the primed, body-fixed
frame
•Need general method for transforming vectors and vector
coordinates between coordinate systems

3
1
0
ˆ
cm
cm
i i
i
r t
drdr dr
r e
dt dt dt

 

   

 

1
ˆ ˆe x
2
ˆ ˆe y
3
ˆˆe z
3
ˆˆe z 
1
ˆ ˆe x 
2
ˆ ˆe y r t


cm
r t

r t


5. Undergraduate Classical Mechanics Spring 2017
•Because scalar product symmetric
•Going unprime to prime back to unprime leaves you where
you started
Transformation of Unit Vectors
1 1 1 1 2 1 3 1 11 12 13 1
2 2 1 2 2 2 3 2 21 22 23 2
3 3 1 3 2 3 3 3 31 32 33 3
1
2
3
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ
ˆ
ˆ
e e e e e e e e A A A e
e e e e e e e e A A A e
e e e e e e e e A A A e
e
e
e
            
       
       
       
       
     
       
 
 
 
 
 
1 1 1 2 1 3 1 11 12 13 1
2 1 2 2 2 3 2 21 22 23 2
3 1 3 2 3 3 3 31 32 33 3
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
e e e e e e e B B B e
e e e e e e e B B B e
e e e e e e e B B B e
           
     
        
     
     
      
     
ˆ ˆ ˆ ˆ
t
i j j i ij ji ij
e e e e A A B      
3 3 3
1 1 1
3
1
ˆ ˆ ˆ
i ij j ij jk k
j j k
il ij jl
j
e B e B A e
B A
  

 
 
 


6. Undergraduate Classical Mechanics Spring 2017
Orthogonal Matrices Yield Rotations
•In matrix form
•Such matrices are called orthogonal. Any transformation
between right-handed sets of orthonormal unit vectors will lead
to an orthogonal matrix with determinate 1. They preserve the
scalar product when (matrix) multiplied by coordinate vectors.
•Euler’s theorem: in three dimensions any orthogonal matrix
with unit determinate is of the form
a rotation through ψ in 3 dimensions about a unit vector

11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
1 0 0
0 1 0
0 0 1
det 1
t
B B B A A A
BA B B B A A A I AB
B B B A A A
AA I A
    
    
   
    
    
    
  
ˆ,O R u
ˆu

7. Undergraduate Classical Mechanics Spring 2017
Rotation Matrices
•Expression for the rotation matrix (for your future
reference!)
•Euler’s theorem follows by showing any unit determinate
orthogonal matrix has an eigenvector with eigenvalue 1
  
0
ˆ ˆˆ ˆˆ, cos 0 sin
0
ˆˆ
z y
z x
y x
x x x y x z
y x y y y z
z x z y z z
u u
R u uu I uu u u
u u
u u u u u u
uu u u u u u u
u u u u u u
  
  
 
    
 
 
 
 
 

 
 
 

0
ˆ, cos 0 sin
0
x x x x x
y y y y y
z z z z z
u u u u u
R u u u u u u
u u u u u
  
            
            
    
            
            
            

8. Undergraduate Classical Mechanics Spring 2017
•In 1 dimension when an object rotates there is a linear
relationship between angular momentum and angular velocity
In 3 dimensions both l and ω are vectors. M is a new type of
object called a (Cartesian) tensor.
•Thought experiment: rotate a point mass
•Need to provide a torque!
Inertia Tensor
l M



0!
r mv r m r
d
dt
l
l
    

   


0
r

l


9. Undergraduate Classical Mechanics Spring 2017
•How can we make ? Add a balancing mass!
•No torque needed to have this motion. Angular momentum
conserved and directed along angular velocity vector. E.g. a baton
will rotate with constant angular momentum around its CM
•Definition of Principal Axis: Rotation direction of a rigid body
such that angular velocity and angular momentum align
Dynamic Spin Balancing
/ 0dl dt

  
   

1 1 2 2
2 2
1 1 1 2 2 2
2
2
1 1
ˆ
2
0!
tot
l r m r r m r
m r r r m r r r
m r r
dl
dt
     
     
 
   
  

     
      




0
1
r

l


10. Undergraduate Classical Mechanics Spring 2017
•Because angular momentum is additive, simply sum over all masses in the body. Integrate for continuous
mass distributions
•As above, angular momentum will rotate along with the mass of the rigid body, unless the angular
momentum is aligned with the angular velocity, i.e., the rotation is around a principal axis

General Inertia Tensor
  
2
2 2 2
2 2 2
2 2 2

0 0
0 0
0 0
a
x x
y y
z z
l r m v r m r m r r r
l
l M
l
x y z x x x y x z
M m x y z y x y y y z
x y z
        
  
        
        

  
        
 
   
   
 
   
   

   
  
 
   
 
   
 
  

        
z x z y z z

     
   
   
   
  
  

11. Undergraduate Classical Mechanics Spring 2017
Terminology
•Diagonal elements of the inertia tensor are called the moments
of inertia. Same concept/expression as from 1 D case in
elementary physics
•The non-diagonal matrix elements are called the products of
inertia.
•In body-fixed frame the inertia tensor is constant. It is usually
(not always!) evaluated with the origin at the center of mass
, ,

ij i j
M m x x i j
  

 
 
 
 
2 2
11
2 2
22
2 2
33
x
y
z
M M m y z
M M m x z
M M m y z
  

  

  

  
  
  




12. Undergraduate Classical Mechanics Spring 2017
Finding Principal Axes
•For alignment of the angular momentum and angular velocity
•This is only possible if there are solutions to the problem
As mentioned previously, if there are solutions to this problem,
λ is called an eigenvalue for the matrix M. The vector direction
associated with the solution of the problem with eigenvalue λ
generally depends on the eigenvalue. This eigenvector
direction will provide a principal axis direction.
•Linear algebra teaches us when this is possible
l M 
  
,M 
 
 
11 12 13
21 22 23
31 22 33
det det 0
M M M
M I M M M
M M M

 

 
 
   
 
 

 

13. Undergraduate Classical Mechanics Spring 2017
Some Math
•Note that M is real and symmertic. This means the matrix
is Hermitian. Hermitian matrices have a very important
property. The eigenvalues must be real (not true for
rotation matrices!) and the eigenvectors corresponding to
distinct eigenvalues must be perpendicular.
•Subtract and consider cases and
 
   
 
2
*
j i j i i j i j j i
i j j i
x
t t
y x y z
z
t t t t
i i
t
j
v
v v v v v v vv v
v
v Mv v v v M v v Mv v v
v Mv v v
         
   
 

 
 
    
 
 
 
     
 
  
         
   
i j i j

14. Undergraduate Classical Mechanics Spring 2017
Body Frame Coordinate Choice
•Align the coordinate axes for body frame along the principal
axes directions. This is the same as performing a similarity
transformation with the matrix of eigenvectors; M’=E
-1
ME
becomes diagonal. The diagonal elements are called the
principal moments of inertia
•The inertial tensor in this frame is
•For axially symmetrical masses I
2
is equal to I
3
and the
principal axes can be any pair of orthogonal vectors in the
plane normal to the eigenvector for I
1
(the symmetry axis)
1
2
3
0 0
0 0
0 0
I
M I
I
 
 

 
 
 

15. Undergraduate Classical Mechanics Spring 2017
Baton Point Mass
•Point masses of mass m at x ± L/2
•No products of inertia because y and z coordinates are all
zero
•What does M
11= 0 mean?
11
2
22
2
33
0
2 /4
2 /4
M
M mL
M mL



2
2
0 0 0
0 /2 0
0 0 /2
M mL
mL
 
 

 
 
 

16. Undergraduate Classical Mechanics Spring 2017
Baton Spherical Masses
•Moment of inertia for sphere about its center is 2mR
2
/5.
Use parallel axis theorem (a HW problem generalizing the
1 D result!) to get
for each sphere.
2
2 2
2 2
4 /5 0 0
0 /2 4 /5 0
0 0 /2 4 /5
mR
M mL mR
mL mR
 
 
 
 
  
 
2
/4MOI MOI mL 

17. Undergraduate Classical Mechanics Spring 2017
Thin Coin
•Let the z-axis be perpendicular to the coin through the
center, with the origin on the coin. All z-values are 0.
•The M
12 is zero by symmetry
 
 
 
4 2
2 2
11 2 2 2
4 2
2 2
22 2 2 2
4 2
2 2 2 2
33 1 2 2
cos 0
4 4
0 cos
4 4
2
cos sin
4 2
m m R R
M r rdrd m I
R R
m m R R
M r rdrd m I
R R
m m R R
M r r rdrd m I
R R

 
 

 
 

  
 
    
    
    



2
1 0 0
0 1 0
4
0 0 2
R
M m
 
 

 
 
 

18. Undergraduate Classical Mechanics Spring 2017
Thick Coin
•Need the additional integral
To be added to the I
2 integral
2 2
2 2
2
0 0
4 12
0 0
4 12
0 0
2
R L
m m
R L
M m m
R
m
 
 
 
 
 
 
 
 
 
 
 
3 3
2 2
2
/2 /2
3 3 12
L Lm m m
M z rdrd dz L
R L L


  
      
  


19. Undergraduate Classical Mechanics Spring 2017
Cube
•Origin at corner. Use integrals like
•Origin at center of mass. Use integrals like
 
2
2 2 3 2
0 0 0 0 0 0
2
/2 /3
8 3 3
3 8 3
12
3 3 8
a a a a a a
xdx ydy dz a a x dx dy dz a a
a
M m
 
  
 
  
 
 
 
 
     
 
/2 /2 /2 /2 /2 /2
2 3 2
/2 /2 /2 /2 /2 /2
2
0 /12
1 0 0
0 1 0
6
0 0 1
a a a a a a
a a a a a a
xdx ydy dz x dx dy dz a a
a
M m
     
 
 
 

 
 
 
     