Complex frequency identification using real modal shapes for a structure with proportional damping

Modal identification is based on modal superposition theory. For structures with proportional damping, the structural responses are recognized as real modal shapes multiplied by real modal coordinates. However, it is not true because the actual modal shapes should be complex even though they can be normalized to real ones. If the modal shapes are recognized as real ones, the modal superposition theory cannot reflect the actual modal parameters. This paper proposes an innovative method to identify complex frequencies for a structure with proportional damping using the identified real modal shapes, where the complex modal shapes are considered. The proportional damping is ideal but always assumed in practical engineering. The characteristic of the proposed method is that the complex property of frequencies is considered along with modal identification process. First, the reason why the complex modal shapes are hidden is revealed. When the real modal shapes are obtained, the Hilbert transform of analytical responses is used to estimate complex coordinates. Then, formulas to identify the complex frequencies are derived. The k-means clustering method is employed. Finally, the proposed method is used in a numerical example and a practical bridge example. The results show that the proposed method can estimate the complex frequencies effectively, and can be used in practical engineering.