An important precursor to the dynamic analysis of rotating machinery, in either the frequency or time domain, is the extraction of its frequencies and mode shapes. This is often presented as a Campbell diagram, which plots the frequency of each mode as a function of the rotor speed (or a ‘reference’ rotor speed, in case of multiple rotors). In addition, Campbell diagrams may be used to ascertain the critical speeds of the rotor as large vibration amplitudes are encountered while operating at these speeds. On a Campbell diagram, it is typical for frequencies to vary with rotor speeds leading to switches between two neighboring modes, and this requires the engineer to monitor the mode shapes using a suitable post-processing tool. This might prove cumbersome, particularly for large industrial models or numerous rotor operating speeds. Therefore, an automatic mode tracking mechanism will be of significant aid to to construct a useful Campbell diagram in addition to a complex eigenvalue extraction procedure.

Mode tracking may be broadly defined as a procedure to map the mode shapes of a system from one state to another. It finds applicability in three major areas of aerospace engineering: rotordynamics (tracking the modes with respect to the rotor speeds), aeroelastic flutter (tracking modes with respect to the airspeed) and optimization (tracking the mode shapes with optimization iterations). Owing to its practical importance, the quest for a robust mode tracking algorithm continues to be an important area of both research and structural solver technology development.

In this paper, several mode tracking methods are implemented and assessed for practical rotor dynamic analysis in a modern finite-element based structural solver, OptiStruct. Rotor dynamics functionalities in OptiStruct include a Hessenberg based complex eigenvalue solver to compute the frequencies and mode shapes of systems that possess rotors, and this solution sequence has been enhanced with mode tracking algorithms to keep record of various mode switches, in order to finally produce a Campbell diagram where modes are sorted based on the continuity of physical characteristics and not numerical values. Using various practical test cases, it is demonstrated that, despite some methods being more mathematically and numerically rigorous than others, no single mode tracking algorithm finds universal applicability, and practical guidelines are prescribed for a judicious selection of the most appropriate method for the system of interest under prescribed operating speeds.

The complex eigenvalue solver operates on reduced system matrices projected onto a subspace determined by the undamped, non-rotating system. For rotor dynamics, the damping and stiffness matrices are augmented by the gyroscopic and circulatory terms respectively.

The classic approach for mode tracking is the Modal Assurance Criteria (MAC), which is based on the orthogonality of mode shapes. However, it is essential to note that the orthogonality property does not hold between eigenvectors when the system matrices are not symmetric. Therefore in addition to the MAC, another method is implemented which uses the orthogonality between the left- and right-eigenvectors – that still holds when the system is complex and unsymmetric. In a similar vein, numerical and geometric considerations are used to construct two more methods of tracking modes. All these methods can be easily triggered to output a fully tracked set of modes when using the complex eigenvalue solver for rotor dynamics problems in OptiStruct.

The full paper will present a complete view of the various rotor dynamic functionalities in OptiStruct with focus on the capabilities of the complex eigenvalue solution sequence. Each mode tracking approach will be discussed starting from its mathematical foundations to its numerical implementation. Their practical applicability with limitations, if any, is underlied via various test cases, whose results will be presented with suitable discussions to drive home the takeaway points. Finally, the advantage of having all these options (with mix and match facilities and diagnostic controls) in a single user-friendly framework allows us to present some general guidelines for practicing rotor dynamicists.


An Evaluation of Mode Tracking Methods for Practical Rotor Dynamic Analysis

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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