In this paper a complex-valued formulation of the modal superposition equation is provided and shown to be equivalent to the original, real-valued Blind Modal IDentification (BMID) problem. The complex-valued variant involves the analytic form of the physical and modal responses. The formulation is shown to be more concise and straight-forward than the original. It is noted that complex-valued mode shapes can be obtained using a complex version of the two-step Joint Approximate Diagonalization (JAD) algorithm. Using this approach the modal response pairing step of the original BMID method is eliminated.
Since the development of the original BMID method, several new, one-step JAD algorithms were devised. Many of the algorithms can be extended to identify complex mixing matrices. A complex version of the one-step JAD method known as the Weighted Exhaustive Diagonalization with Gauss itErations (WEDGE) algorithm is utilized to solve for the complex mode shapes and modal responses. By using this simplified formulation, the whitening step is eliminated as well as the modal response pairing step that is necessary in the original BMID algorithm.
Performance of the new Complex BMID (CBMID) algorithm is evaluated by application to synthesized data from a 3 DOF system with complex modes, application to measured laboratory data on a structural frame and application to measured output-only data from the Heritage Court Tower building. It is seen that the CBMID method results in essentially the same estimates of modal responses, complex mode shapes, natural frequencies and modal damping compared to results from BMID. Furthermore, it is shown that modal parameters from BMID and CBMID are very consistent with those obtained from state of the art methods, such as the Eigensystem Realization Algorithm (ERA) and the covariance-driven Stochastic Subspace Identification (SSI) method.
McNeill, S., “An Analytic Formulation for Blind Modal Identification,” Journal of Vibration and Control 18(14), 2111-2121, 2012, DOI: 10.1177/1077546311429146.