A force identification method using cubic B-spline scaling functions

Abstract

For force identification, the solution may differ from the desired force seriously due to the unknown noise included in the measured data, as well as the ill-posedness of inverse problem. In this paper, an efficient basis function expansion method based on wavelet multi-resolution analysis using cubic B-spline scaling functions as basis functions is proposed for identifying force history with high accuracy, which can overcome the deficiency of the ill-posed problem. The unknown force is approximated by a set of translated cubic B-spline scaling functions at a certain level and thereby the original governing equation of force identification is reformulated to find the coefficients of scaling functions, which yields a well-posed problem. The proposed method based on wavelet multi-resolution analysis has inherent numerical regularization for inverse problem by changing the level of scaling functions. The number of basis functions employed to approximate the identified force depends on the level of scaling functions. A regularization method for selecting the optimal level of cubic B-spline scaling functions by virtue of condition number of matrix is proposed. In this paper, the validity and applicability of the proposed method are illustrated by two typical examples of Volterra–Fredholm integral equations that are both typical ill-posed problems. Force identification experiments including impact and harmonic forces are conducted on a cantilever beam to compare the accuracy and efficiency of the proposed method with that of the truncated singular value decomposition (TSVD) technique.