Nonstationary vibration analysis in the smoothed Wigner domain

Dynamical systems represent fundamental models for vibration analysis. When the input of such dynamical systems is nonstationary also the output is nonstationary, and its frequency content changes with time. Time-frequency analysis provides an effective representation of this timevarying spectrum. Even more effective is the direct time-frequency representation of dynamical systems. We first show how to transform a dynamical system in the domain of the smoothed Wigner distribution. The result is a time-frequency dynamical system whose input and output are the smoothed Wigner distributions of the input and output signals in time, respectively. Then, we illustrate how to compute the smoothed Wigner output when the input to the dynamical system in time belongs to a class of common nonstationary inputs, including a delta function and a short duration sinusoid. Finally, we show how to obtain the smoothed Wigner output when the input is a linear combination of these nonstationary signals. We provide a series of examples that show how the time-frequency representation of dynamical systems can unveil the spectral structure of nonstationary vibrations.