The rotor is modeled as an elastic solid of arbitrary geometry with its centroid $G$ located on the spin axis (i.e., rotor is perfectly balanced). Moreover, the rotor is subject to free-body boundary conditions and can undergo rigid-body motion as well as elastic deformation. To formulate the motion of the rotor, let us first define a virgin state for reference. In the virgin state, the rotor-bearing-housing system experiences no elastic deformation and no rigid-body motion except the spin of the rotor. Moreover, let $G'$ denote the location of the centroid in the virgin state; see Fig. 1. (Note that $G$ and $G'$ coincide in the virgin state but will correspond to two different points in actual motion.) With $G'$ as the origin, one can define an inertia Cartesian coordinate system $X\u02dcY\u02dcZ\u02dc$ with $Z\u02dc$ being the spin axis in the virgin state. (Note that the rotor might wobble in actual motion; therefore, $Z\u02dc$ is not the spin axis in actual motion.) Also, one can define a rotating coordinate system $xyz$ with constant angular velocity $\omega 3$. It is also convenient to assume that $xyz$ coincides with the principal axes of the rotor in the virgin state. Note that $xyz$ axes do not attach to the rotor because the rotor can rock and translate, but $xyz$ axes cannot. Nonetheless, $xyz$ axes are the best coordinate system to describe the motion of the rotor. Let $i$, $j$, and $k$ be the unit vectors of the $xyz$ coordinate system and $I$, $J$, and $K$ be the unit vectors of the inertia frame $XYZ$ (or $X\u02dcY\u02dcZ\u02dc$) coordinate system. Then