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This paper is meant to model free vibration of a coupled rotor-bearing-housing system. In particular, the rotor is cyclic symmetric and spins at constant speed while the housing is stationary and flexible. The rotor and housing are assembled via multiple, linear, elastic bearings. A set of equations of motion is derived using component mode synthesis, in which the rotor and the housing each are treated as a component. The equations of motion take the form of ordinary differential equations with periodic coefficients. Analyses of the equations of motion indicate that instabilities could appear at certain spin speed in the form of combination resonances of the sum type. To demonstrate the validity of the formulation, two numerical examples are studied. For the first example, the spinning rotor is an axisymmetric disk, and the housing is a square plate with a central shaft. The rotor and the housing are connected via two linear elastic bearings. For the second example, the rotor is cyclic symmetric in the form of a disk with four evenly spaced radial slots. The housing and bearings remain the same. In both examples, instability appears as a combination resonance of the sum type between a rotor mode and an elastic housing mode. The cyclic symmetric rotor, however, has more instability zones. Finally, effects of damping are studied. Damping of the housing widens the instability zones, whereas the damping of the rotor does the opposite.

Parametric Resonances of a Spinning Cyclic Symmetric Rotor Assembled to a Flexible Stationary Housing Via Multiple Bearings