When a point of observation is occupied, there is also information to Specify the observer himself, and this information cannot be shared by other observers. "Axis of percussion" defines the application line for a force that rotationally accelerates a free body about "Specific Point".

1. The principal screw of inertia
TWO

2. Mapping of an exterior point P to a
line L through a conic C

3. Inertia Mappings and Decomposition
• In projective geometry a conic section C maps
P into lines L=CP.
• In dynamics an inertia matrix I maps
acceleration twists a into force wrenches f=Ia
as if the body is at rest and v=0.
• The process is identical, except the inertia
matrix I is usually positive definite, and always
positive semi-definite.

4. A positive definite matrix
• One intuitive definition is as follows. Multiply any
vector with a positive definite matrix. The angle
between the original vector and the resultant
vector will always be less than π/2. The positive
definite matrix tries to keep the vector within a
certain half space containing the vector. This is
analogous to what a positive number does to a
real variable. Multiply it and it only stretches or
contracts the number but never reflects it about
the origin.

5. Inertia mappings
• Inertia mappings define a unique force f=Ia for every
acceleration a. The acceleration a also defines a point
in homogeneous coordinates called the pole of a. The
force f defines a line in homogeneous coordinates
called the polar of f. The point uniquely maps to the
line using the inertia matrix. In dynamics, the polar f is
called the axis of percussion for the pole of a.
• A simple planar example demonstrates how to locate
the axis of percussion. A coordinate frame is placed at
the acceleration pole a and with the local x-axis along
the line that connects a and the center of gravity.

6. Inertia maps acceleration into forces

7. The inertia matrix
• If a body has mass m, radius of gyration ,
and the center of gravity is at distance d then
the inertia matrix is
• Accelerations are points and forces are lines
on the plane. The line is the axis of percussion
of the point.
2 2
0 0
0
0 ( )
m
I m dm
dm m d
 
 
 
  


8. • and the force required is
The force f needed to accelerate the body about a is
f=Ia if the body is at rest. If the angular acceleration
is  then the acceleration twist is
0
0a

 
 
 
  
2 2
0
( )
f md
m d

 
 
 
 
  

9. Axis of percussion
Obviously f has no x-axis component and its axis of
application is located by normalizing the force
wrench. Dividing the wrench by its magnitude md
yields the shortest distance l of the application axis
as
2
l d
d

 

10. • The force lies on the so-called “axis of percussion.”
This axis defines the application line for a force that
rotationally accelerates a free body about a specified
point. In the world of sports, the effect of the axis of
percussion is called the “sweet spot;” where all of
the energy of the athlete is transferred most
effectively to the ball. In this case, the rotation
caused by f occurs about a. The minimum distance r
of the axis of percussion to the center of gravity is
2
r l d
d

 


12. Sport Engineering
• A ball hitting a baseball bat at the center of percussion
creates negligible translation at another point inside
the region where the bat is gripped.
• The linear impulsive force imparted to the grip is
minimized so that the sensation of “sting (touch)” is
reduced.
• Given the center of mass and an expected center of
impact (sweet spot), a baseball bat (or a tennis racket,
golf club, sledge hammer, etc.) is designed so that the
area of grip is centered around the pole.

13. Where  is the radius of gyration of the rigid body
and d the distance from a to the center of gravity.
The farther away a is, the closer f is and vice versa.
There are two limiting cases with d=0 or d=. In
the first case, an acceleration twist about the center
of gravity needs a force wrench at infinity which
represents a pure couple, and in the second case,
an acceleration twist at infinity which represents a
pure translation needs a force wrench through the
center of gravity.

14. Mappings in power relations

15. Mapping in power relations
• It demonstrates mapping in power relations.
• A powerless force f passes through the rigid
body velocity v such that
• If f represents a force applied on a single rigid
body, it relates this fore to an acceleration a
such that
0T
v f 
( )T
f Ia v Iv  

16. The power relationship
• The power relationship expands to
• where
• is the momentum wrench.
( )
( )
0
T T T T
T
T
v f v Ia v v Iv
Iv a
p a
  

 
p Iv

17. • In fact, in general the power is defined as
• since both calculations are equivalent.
• A reaction force f that produces no power has
an application axis that passes through the
instant center of motion v.
T
T
P v f
p a



18. Powerless acceleration
• Equivalently the resulting acceleration a lies on
the application axis of the momentum h.
• Powerless forces define reaction forces, and
powerless acceleration define reactive
accelerations.
• This equivalency is used to decompose forces and
acceleration according to reactive and active
components.
• Mapping in power relationships are very
important because the intertwines forces,
velocities, accelerations, and momenta.

19. Meaningful Information
• In dynamics it is always meaningful to classify
quantities in terms of power.
• Reaction forces are wrenches that provide
zero power.
• The subspace are defined according to their
power relation.
• With mappings these power relations help
define subspace for any wrench or twist.