ABSTRACT
This chapter establishes the connection between causality and the Hilbert transform relation. It is important to point out that the theorems hold only if the technicalities of Titchmarsh’s theorem are satisfied. The chapter shows how the Hilbert transform relations are applied to functions which do not satisfy the necessary conditions. It demonstrates a nonlinear system which was nonetheless causal and therefore satisfied Hilbert transform relations. The chapter shows the extent to which the Hilbert transforms are affected, and there is noticeable distortion at low frequencies, and around resonance for higher damping. It can be concluded that hysteretic damping should only be used with caution in simulations where the object is to investigate Hilbert transform distortion as a result of nonlinearity. A Hilbert transform of the low excitation data was calculated using the fast Fourier transform-based technique on an frequency response function using a range of 0–50 Hz in order to minimize truncation errors in the calculation.
