Initial-condition consideration by transferring and loading reconstruction for the dynamic analysis of linear structures in the frequency domain

Abstract Traditional frequency-domain dynamic analysis methods can only compute the steady-state responses of the system, typically by employing the fast Fourier transform (FFT). In this paper, a new frequency-domain method that can consider the initial conditions is proposed. External loadings are decomposed into Prony series, and the initial conditions are transferred from the system into the reconstructed loadings, which are combinations of the initial loadings and the corresponding initial conditions in the Laplace domain. Similar to the traditional frequency-domain method, the inverse Fourier transform (IFT) is used to compute the responses of the system in the time domain. One theoretical development is that the initial conditions can be combined into the corresponding reconstructed loadings, which avoids the required periodic assumption for the FFT. The other improvement is that the decomposition of the initial external loadings requires only a short duration of measurements, which indicates a good potential capability of the computing efficiency for complex systems. The numerical results from a single degree-of-freedom (DOF) indicate that the proposed method can perform similar to the frequency-domain method if only the initial conditions are set to zero values because the IFT is used. When non-zero initial conditions are considered, the proposed method has a similar accuracy to the time-domain method, but one should note that the proposed method has an error on the response at time zero. One can draw identical conclusions from the numerical results of a four-DOF system, which demonstrates the ability to address multiple-DOF systems. The third example is a three-dimensional (3D) frame structure subjected to more general external loadings by randomly generating different components; the initial conditions are simulated using two series of random numbers: one is for the initial displacements and the other is for the initial velocities. The results show that the proposed method obtains consistent responses with the time-domain method, but it has better computation cost because the equations of motion are solved in the frequency domain. In addition, the reason and corresponding measurement on most relative errors at the early time are also provided.