Extension of dynamic stiffness method to complicated damped structures

The increase in computing power has dramatically enabled the computations of intricate structural design models. For such purposes, dynamic stiffness method, a technique that relates the amplitudes of applied forces and responses of a harmonically vibrating continuum, has become a popular structural dynamic analysis tool. This technique works by separating the variable of the displacement function in frequency domain, the dynamic stiffness matrix and frequency equation of the structure are obtained, and the structural dynamic analysis can then be achieved by solving the transcendental frequency equation. In case of undamped systems, the frequency equation can be accurately solved by the Wittrick-Williams algorithm, however, the frequency equation of damped structures is a complex transcendental equation and many root-search techniques performing well in real field including the Wittrick-Williams algorithm are no longer applicable. Therefore, the application of dynamic stiffness in damped structures is a major challenge and has not been well resolved.

Recently, a team of Tongji University researchers led by Professor Dan Danhui from the School of Civil Engineering conducted a study where they improved the aforementioned dynamic stiffness method. Specifically, they focused on how to make the dynamic stiffness method and Wittrick-Williams algorithm still applicable to complicated damped systems. In addition, they extended dynamic stiffness method (EDSM) capable of resolving a much wider class of problem of classically damped structures. Their work is currently published in the research journal, Computers and Structures.

In brief, the research method commenced with a thorough review of the calculation process of original dynamic stiffness method. Next, the principles of extended dynamic stiffness method, for all the three cases; undamped system, viscous damping and hysteretic damping, were revisited. The researchers then engaged in calculations to determine the damping ration and that of the undamped frequency wavelength. Lastly, numerical case analysis for the dynamic stiffness matrix and damping ratios were done and verified.

The authors observed that their proposed method sufficiently covered both viscous and hysteresis damping. Moreover, they noted that the Wittrick-William algorithm could thereby be extended to complex and damped systems. Lastly, they realized that in the proposed technique the variables of structural displacement function were separated in the Laplace domain, and a unified form dynamic stiffness matrix and frequency equation of damped and undamped systems was obtained.

In summary, Prof Dan Danhui and his colleagues demonstrated extended dynamic stiffness method technique that eliminates difficulties encountered when applying dynamic stiffness method in dynamic analysis of damping system. In general, they observed that the relationship between the damped frequency and the undamped frequency could be established thereby converting the calculation problems of the damped frequency into the process of solving the damping ratio and undamped frequency. Altogether, the novel technique was seen to be advantageous in that it could be easily implemented and the advantages of the original method also retained.