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Angular momentum about center of mass of
a rigid body
Recall that: Torque
is the rate of change of angular momentum.
To define inertia, we need to be able to write angular momentum
(a vector) in terms of something
(inertia, which is a matrix) that multiplies angular velocity.
Note
that the inertia matrix, when written about the center of
mass, is always symmetric.
The object shown
here is flat and thin. It
is made of aluminum and weighs 19 grams, and is uniform.
Our task is to find its center of mass and its inertia
about its center of mass.
•
Divide up the piece into equal size squares of 1x1 cm.
•
Each small square is assumed to be a particle of mass 1
with its location specified by the center of square.
Size of the
upper arm: 32.5x10x7.5
cm
density:
1.05 g/cm3
“particle”
size: 2.5x2.5x2.5
cm
When the
object is rotating at its center of mass:
When
rotation is occurring about another point on the object:
Inertia
remains symmetric
even if we express it about a point other than center of mass.
Mass appears
linearly in all expressions of inertia.
When the body
is rotating, the vector sxc will
have to be written in terms of angular position of the body
with respect to the point of rotation.
Therefore, Is will
be a function of position of the body.
Example:
Effect of interaction torques
Command a
flexion torque on the elbow
Torque on
the elbow causes motion of the elbow and shoulder
Example:
Effect of interaction torques
Command
an extension torque on the elbow
Command a
flexion torque on shoulder
1.
Immobilize the second link.
Inertia seen
by the first joint when the second joint is immobilized.
Reaches maximum value when the arm is open.
2.
Immobilize the first link.
Reaction
forces acting on the first link when the second link
accelerates.
3.
Centripetal forces acting on link 2 due to rotation
of link 1
Objective:
Is there a simple relation between torques when the
speed of movements scales up?
Example:
Imagine that one makes a movement from point 1 to 2
in 1 second. How
much more joint torque would be needed to make the same
movement in 0.5 seconds?
To move
twice as fast, you’d need four times as much torque to
compensate for inertial dynamics
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