Modal correlation approaches for general second-order systems: Matching mode pairs and an application to Campbell diagrams

Modal correlation is well developed for undamped and proportionally damped vibrating systems. It is less well defined for generally damped linear systems. This paper addresses the fundamental problem of comparing two general second-order linear systems through modal information. It considers precisely the problem of how to achieve matching of modes (mode pairs).There are several possible motivations for modal correlation of which the most important is probably the model updating application. In that application, one set of modes derives from a numerical model and the other from measured data. This paper focuses mainly on a different application—constructing Campbell diagrams for rotating machines. There are two significant differences here: (a) the two sets of modes being compared at any one time are from the same numerical model but for different spin speeds and (b) there is generally a strong distinction between the left and right modes of the system. Without some modal correlation approach, the Campbell diagram is constructed simply as a set of points on the frequency-speed graph. With modal correlation, the eigenvalue problem can be solved at far fewer speeds and the points can be joined meaningfully.A dimensionless (n×n) modal-matching array is produced whose entries indicate which pairs of modes from the first system best correlate with any particular pair of modes from the second system. The presented work is motivated mainly by the application of developing Campbell diagrams for rotating machines by means which are more effective than simply plotting a large set of discrete points. Wider applications of this paper include model updating procedures where mode pairs must be matched initially to ensure convergence towards the exact system.

► This paper examines ways to best match (pairs of) modes from different generally damped systems.
► These are logical extensions of mass- and stiffness-weighted cross-orthogonality.
► The methods are particularly suited to drawing Campbell diagrams.
► The new measure gives an (n × n) mode-matching array (MMA)—the identity when systems identical.
► (MMA—I) varies with the SQUARE of distance between the two systems.