A Linearized Theory on Ground-Based Vibration Response of Rotating Asymmetric Flexible Structures

This paper is to develop a unified algorithm to predict vibration of spinning asymmetric rotors with arbitrary geometry and complexity. Specifically, the algorithm is to predict vibration response of spinning rotors from a ground-based observer. As a first approximation, the effects 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. The equation of motion governing the modal response is derived through use of the Lagrange equation. To construct the equation of motion, explicitly, the results from the finite element 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. Results of the algorithm indicate that Campbell diagrams of spinning asymmetric rotors will not only have traditional forward and backward primary resonances as in axisymmetric rotors, but also have secondary resonances caused by higher harmonics resulting from the mode shapes. Finally, the algorithm is validated through a calibrated experiment using rotating disks with evenly spaced radial slots. Qualitatively, all measured vibration spectra show significant forward and backward primary resonances as well as secondary resonances as predicted in the theoretical analysis. Quantitatively, measured primary and secondary resonance frequencies agree extremely well with those predicted from the algorithm with mostly $<3.5%$ difference.