On the characteristic modes of a rigid body under forces

It is recently shown in [1, 2] that the exact analytical solution of a rigid body controlled by a linear feedback law may directly be written down in terms of eigenvalues and eigenvectors. However, this control has been computed in previous work [3] for the regulation of satellite angular momentum when the flywheels are at rest. Therefore, the method will here be extended to deal with the structure of the solution when the control is not linear in the state. Moreover, we show that, for the free motion, when the initial condition coincides with an eigenvector, the solution then remains in rotation about a parallel axis to this eigenvector. The eigenvectors thus define the characteristic directions. A preliminary result is given in [4]. The characteristic directions are also here determined for the nonlinear controlled system. In addition, the eigenvalues of the system under study are real or complex. It is claimed in [12] that, in general, it is not possible to evaluate closed form characteristic solutions for complex eigenvectors. This paper and the above author's references are devoted to show that it is possible to describe characteristic solutions with complex eigenvalues and complex eigenvectors. Consider a rigid body in an inertial reference frame. Let uj\, u2l u>3 as usual denote the angular velocity components and let I\, I2, I3 be the moments of inertia of the body about the principal axes which are the body axes.