**Reconstruction of dynamic displacement and velocity from measured accelerations using the variational statement of an inverse problem**

**Yun Hwa Hong, Ho-Kyung Kim, Hae Sung Lee**

**ABSTRACT**

This paper presents two types of ?nite impulse response (FIR) ?lters to reconstruct dynamic displacement induced by structural vibration from measured acceleration. The governing equation for the reconstruction is derived by taking the variation of a minimization problem, which de?nes an inverse problem on displacement. A regularization function for overcoming the ill-posedness of the inverse problem is included in the minimization problem. The governing equation of the inverse problem becomes the same type of differential equation as that of a beam on an elastic foundation. The conventional FIR (CFIR) ?lter directly approximates the transfer function of the governing equation, while the FEM-based FIR (FFIR) ?lter is formulated by the discretization of the minimization problem with the ?nite element method. For the ?nite element discretization, the Hermitian shape function is utilized. The proposed FFIR ?lter is capable of reconstructing displacement and velocity simultaneously. The fundamental characteristics of the proposed ?lters are investigated in the frequency domain using the transfer and accuracy functions. It is shown that the proposed FIR ?lters suppress low frequency noise components in measured accelerations effectively, and reconstruct physically meaningful displacement accurately. The validity of the proposed ?lters is demonstrated through a numerical simulation study, a ?eld experiment and an evaluation of ?utter derivatives using measurements taken from a wind tunnel test.

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