Vibration analysis of rotating asymmetric structures

This paper is to develop a unified algorithm to analyze vibration of spinning asymmetric rotors with arbitrary geometry and complexity. As a first approximation, the affects of housings and bearings are not included in this analysis. The unified algorithm consists of three steps. The first step is to conduct a finite element analysis on the corresponding stationary rotor to extract natural frequencies and mode shapes. The second step is to represent the vibration of the spinning rotor in terms of the mode shapes and their modal response in a coordinate system that is rotating with the spinning rotor Through use of the Lagrange equation, one can derive the modal equation of motion. To construct the equation of motion explicitly, the results from the finite elentent analysis will be used to calculate the gyroscopic matrix, centrifugal stiffening (or softening) matrix, and generalized modal excitation vector The third step is to solve the equation of motion to obtain the modal response, which, in turn, will lead to physical response of the rotor for a rotor-based observer or for a ground-based observer through a coordinate transformation. Finally, application of the algorithm to rotationally periodic rotors indicates that Campbell diagrams of the rotors will not only have traditional forward and backward branches as in axisymmetric rotors, but also have secondary resonances caused by higher harmonics resulting from the mode shapes. Calibrated experiments were conducted on an air bearing spindles carrying slotted circular disks to verify the theoretical predictions in the ground-based coordinates.