An adaptive algorithm for mid-frequency response of a proportional damping system

The computational cost of mid-frequency response could be considerably high for finite element systems because each response evaluation needs the numerical solving of a system of complex equations. The mode superposition method requires to calculate various modes within a range that is several times of the given excitation frequency interval, yet the accuracy of result is still unknown. In this paper, a new and innovative method for calculating the mid-frequency response of a proportional damping system is proposed. First, the middle modes and frequencies to be calculated are adaptively determined. Then, the unknown lower-order mode and higher-order mode contribution is approximated by the partial sum of a convergent power series. By using an iterative algorithm and considering only the left or right end of the excitation frequency interval, the number of terms in the sum is adaptively determined. A remarkable conclusion is established here that the resulting expression of frequency response is perfectly valid for the entire excitation frequency interval. The frequency response analysis can thus be performed by using this expression, and by changing the excitation frequency only. Finally, two numerical examples are presented to illustrate the accuracy and effectiveness of the proposed new method. The result shows that the proposed method can significantly reduce CPU (central processing unit) computational time with respect to the direct method for mid-frequency response analysis.