Output-only modal identification based on the variational mode decomposition (VMD) framework

This paper proposes a novel modal identification variational mode decomposition (MIVMD) for output-only modal identification (OMI). The proposed MIVMD is more direct and elegant than most existing VMD-related OMI works that repeatedly use the conventional VMD to each channel of multivariate vibration signals and need post-processing to extract mode shapes. Firstly, OMI is expressed as a constrained variational problem by using the modal superposition equation. Secondly, as in VMD, the squared L₂-norm of the gradient of the modal response is employed to evaluate its bandwidth. Then, the quadratic penalty term and Lagrangian multipliers are used to render the problem unconstrained. Finally, the alternate direction method of multipliers (ADMM) is utilized to solve this unconstrained optimization problem. The results demonstrate that the proposed MIVMD can concurrently identify natural frequencies, mode shapes, and modal responses from multivariate signals without multiple decompositions and post-processing. Because VMD-based methods are restricted to decompose narrowband modes, a short-time counterpart of MIVMD (ST-MIVMD), which can identify time-varying systems involving wideband modes or closely-spaced modes, is also presented. In the end, a series of numerical and experimental examples are performed to verify the effectiveness and advantages of the proposed method in addressing the OMI problem.