Abstract:The instantaneous frequency of modal response is not equal to the instantaneous frequency of the structural system for a time-varying and nonlinear structure. In this paper, the structural instantaneous frequencies are directly derived from the decomposed modal responses for single- and multiple- degree-of-freedom systems with both free and force vibrations. The theoretical result shows that the slow-varying component of the signal’s instantaneous frequency is approximately equal to the systems instantaneous frequency for slowly time-varying linear structures and weakly nonlinear structures. The result is validated by a numerical simulation of a Duffing system and an experimental test of a cable with time-varying tension force and frequency. For a time-varying and nonlinear structure with closely-spaced modes, this paper extended analytical mode decomposition into time signals with time-varying frequencies. The mathematical foundation for this new extension is provided and an approach with wavelet transform is developed for the selection of time-varying bisecting frequencies. Finally, a two-story time-varying structure with closely-spaced modes subjected both white noise and earthquake excitations are analyzed, the result show that proposed method can effectively decomposed modal responses and accurately estimated the instantaneous frequency of time-varying and nonlinear structures.