Techno Press

Smart Structures and Systems   Volume 17, Number 5, May 2016, pages 753-771
Free and transient responses of linear complex stiffness system by Hilbert transform and convolution integral
S.H. Bae, J.R. Cho and W.B. Jeong

Abstract     [Full Text]
    This paper addresses the free and transient responses of a SDOF linear complex stiffness system by making use of the Hilbert transform and the convolution integral. Because the second-order differential equation of motion having the complex stiffness give rise to the conjugate complex eigen values, its time-domain analysis using the standard time integration scheme suffers from the numerical instability and divergence. In order to overcome this problem, the transient response of the linear complex stiffness system is obtained by the convolution integral of a green function which corresponds to the unit-impulse free vibration response of the complex system. The damped free vibration of the complex system is theoretically derived by making use of the state-space formulation and the Hilbert transform. The convolution integral is implemented by piecewise-linearly interpolating the external force and by superimposing the transient responses of discretized piecewise impulse forces. The numerical experiments are carried out to verify the proposed time-domain analysis method, and the correlation between the real and imaginary parts in the free and transient responses is also investigated.
Key Words
    linear complex stiffness system; free and transient responses; time domain analysis; Hilbert transform; state-space formulation; convolution integral
S.H. Bae and W.B. Jeong: School of Mechanical Engineering, Pusan National University, Busan 609-735, Korea
J.R. Cho: Department of Naval Architecture and Ocean Engineering, Hongik University, Sejong 339-701, Korea

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