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Unformatted text preview: configurations need different torques to achieve accelerations:
consider 1 below
1 Ze Xe D1 Z1 X1
Z2
D0 • Z0 X2 D2 X0 The moment of inertia is related to the mass distribution of the links and its
motions in particular coordinate frames Inertia tensor
•
• Using inertia to describe mass distribution with respect to a
coordinate
Inertia tensor in {A} can be expressed as distance × mass
, •
• , =∭ ,
, are called mass moments of inertia
are called mass products of inertia + )ρdv, =∭
=∭ •
• + )ρdv, =∭ + ρdv i, j = x, y, or z; i≠j ρ is the density of the material
If we chose the frame in such a way that the products of inertias are
all zero, the axes are called principal axes, and mass moments are
called the principal moments of inertia )ρdv Inertia tensor examples (1)
̂ Which one has greater
• ̂ , , or ? Calculate the inertia tensor for the following object and
the attached frame. Assuming density to be even ̂ Inertia tensor examples (2)
= + )ρdxdydz = + )ρwdydz = + )ρwdz =( hw + lw)ρ
= l +4 h ) • What if we remove the frame to a
new place as shown below?
• How can we make the mass products
of inertia all zero?
̂ =
= ρdxdydz...
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 Fall '13

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