Page 95 - The Mechatronics Handbook

0066_Frame_C07 Page 5 Wednesday, January 9, 2002 3:39 PM

FIGURE 7.4 Magnetically levitated rigid body (HSST MagLev prototype vehicle, 1998, Nagoya, Japan).

of the center of mass of a body with mass m is given by
mv ˙ c = F (7.4)

If r is a vector to some point in the rigid body, we define a local position vector ρ by r P = r c + ρ. If a
force F i acts at a point r i in a rigid body, then we define the moment of the force M about the fixed
origin by
M i = r i × F i (7.5)

The total force moment is then given by the sum over all the applied forces as the body

=
M = ∑ r i × F i = r c × F + M c where M c ∑ r i × F i (7.6)
We also define the angular momentum of the rigid body by the product of a symmetric matrix of second
moments of mass called the inertia matrix I c . The angular momentum vector about the center of mass
is defined by

H c = I c w (7.7)
⋅
Since I c is a symmetric matrix, it can be diagonalized with principal inertias (or eigenvalues) {I ic } about
principal directions (eigenvectors) {e 1 , e 2 , e 3 }. In these coordinates, which are attached to the body, the
angular momentum about the center of mass becomes

H c = I 1c w 1 e 1 + I 2c w 2 e 2 + I 3c w 3 e 3 (7.8)

where the angular velocity vector is written in terms of principal eigenvectors {e 1 , e 2 , e 3 } attached to the
rigid body.
Euler’s extension of Newton’s law for a rigid body is then given by

˙
H c = M c (7.9)

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