Reconstruction of Dynamic Displacement and Velocity
based on the Variational Statement of an Inverse Problem
from Measured Acceleration with Special Application to the
Extraction of Flutter Derivatives
Yun Hwa Hong
A new class of displacement reconstruction scheme is presented using only acceleration measured from a structure. For a given set of acceleration data, the reconstruction problem is formulated as a boundary value problem in which the acceleration is defined by the second-order ordinary differentiation of displacement. The displacement is reconstructed by minimizing the least squared errors between measured and approximated acceleration within a finite time interval. The displacement reconstruction problem becomes ill-posed because the boundary conditions at both ends of each time window are not known a priori. Furthermore, random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement. A Tikhonov regularization scheme is adopted to alleviate the ill-posedness. The governing equation for the reconstruction is derived by taking the variation to the regularized minimization problem, which yield beam on the elastic foundation problem. The conventional FIR (CFIR) filter directly approximates the transfer function of the governing equation, while the FDMbased FIR (FDM-FIR) and FEM-based FIR (FFIR) filter are formulated by the discretization of the minimization problem with the finite difference method and the finite element method, respectively. The FFIR filter is capable of reconstructing displacement and velocity simultaneously. The fundamental characteristics of the proposed filters are investigated in the frequency domain using the transfer and accuracy functions. It is shown that the proposed FIR filters suppress low frequency noise components in measured accelerations effectively, and reconstruct physically meaningful displacement accurately. The validity of the proposed filters is demonstrated through several examples.
In the final example, a force-acceleration-based identification of the flutter derivatives of bridge decks in a wind tunnel is presented. An equation error estimator (EEE), which is the least square residual errors of the equation of motion, is employed to formulate the force-based identification scheme. Unlike most of previously proposed methods, the acceleration of an oscillating section model is measured in wind tunnel tests. The velocity and the displacement required in the EEE are reconstructed from the measured acceleration using the FFIR filter. As the EEE is expressed as a quadratic form with respect to flutter derivatives, neither an iterative solution scheme nor a complex eigenvalue analysis is required for optimization. The EEE method is capable of identifying the representative values of the flutter derivatives by one optimization process using multiple measurements for a wind velocity in wind tunnel tests and can be generally employed for the extraction of the flutter derivatives regardless of the testing procedures.
Flutter derivatives; wind tunnel test; free-oscillation test; equation error estimator; reconstructed displacement; reconstructed velocity; FEM finite impulse response filter; Displacement reconstruction; Acceleration; Boundary value problem; Low frequency dominant structure; Central finite difference; Time window; Tikhonov regularization;