Complex modal analysis and interpretation for rotating machinery

The use of complex variables to represent rotating vectors which describe natural modes and forced responses in rotor dynamic systems is investigated. It is shown that general planar motions and forces can be represented as the linear superposition of forward and backward rotating vectors. It is also shown that this allows frequency response functions to implicitly include directionality. Homogeneous solutions of the complex formulation lead to clear physical definitions of forward and backward modes. Substitution of general forms of response and excitation, a process analogous to that in the classical vibration problem, is shown to result in directional frequency response functions (dFRFs) which provide modal directivity and separation in the frequency domain. The paper establishes the fact that dFRFs relate the forward and backward components of the complex displacement and excitation functions. All concepts in the procedure are directly traceable to real measurable signals, which indicates that complex vibration analysis is a generalized case of classical vibration analysis of non-rotating structures.