Non-iterative mode shape expansion for three-dimensional structures based on coordinate decomposition

The direct mode shape expansion method is an iterative technique, one can conclude that the convergence performance maybe challenged when applied to three-dimensional structures. In addition, mode shape values at different DOFs (degrees-of-freedom) sometimes are not in a same order of magnitude, which will produce much error for the estimation of small values of unmeasured mode components. Therefore this paper proposed a non-iterative mode shape expansion method based on coordinate decomposition technique. The advantage of coordinate decomposition is that the unmeasured components of mode shape values could be estimated with different weighting coefficients, even in a physical meaningful interval. Numerical studies in this paper are conducted for a 30-DOF cantilever beam with multiple damaged elements, as the measured modes are synthesized from finite element models. The numerical results show that the approach can estimate unmeasured mode shape values at translational and rotational DOFs in x, y and z directions with different weighting coefficients, respectively; and better mode shape expansion results can be obtained when proper constraints are employed. A numerical three dimensional structure is also investigated, and results indicate that the estimation of unmeasured components can be improved by imposing reasonable constraints based on the coordinate decomposition technique, even only translational DOFs of two diagonal nodes of the first floor are measured.