**Exact Enforcement of the Causality Condition on the Aerodynamic Impulse Response Function Using a Truncated Fourier Series**

**Jinwook Park, Kilje Jung, Yun Hwa Hong, Ho-Kyung Kim, M.ASCE; and Hae Sung Lee**

**ABSTRACT**

This paper presents the exact relation between the real and imaginary parts of aerodynamic transfer functions for deriving impulse response functions that satisfy the causality condition. A truncated Fourier series is used to express the aerodynamic transfer functions, and the causality condition is defined in terms of the coefficients of a Fourier cosine and sine series, which represent the real and imaginary parts of the aerodynamic transfer functions, respectively. The impulse response functions that satisfy the causality condition are obtained through the inverse Fourier transform of the aerodynamic transfer functions that conform to the exact relation. The coefficients of the Fourier series are determined by minimizing the error between the transfer functions formed by measured flutter derivatives and by the Fourier series. Because the impulse response functions become a series of Dirac delta functions in the proposed method, the aerodynamic forces are easily evaluated as the sum of current and past displacements with the same number of the terms in the Fourier series. The validity of the proposed method is demonstrated for two types of bluff sections: a rectangular section with a width-to-depth ratio of 5 and an H-type section. Time-domain aeroelastic analyses are performed for an elastically supported system with each section. The proposed method yields stable and accurate solutions for the examples efficiently.

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