Analysis of periodic systems with time delay via Chebyshev spectral collocation with application to milling

In this dissertation, two semi-analytical techniques based on Chebyshev spectral collocation are used for the analysis of time-periodic delay differential equations (DDEs).;First, the Chebyshev collocation method is used for the stability analysis of linear time-periodic DDEs by converting them into maps thus introducing a finite approximation to the infinite-dimensional monodromy operator. This method is used for the linear stability analysis of zero- and nonzero-helix milling models. In particular, for the case of down-milling in the zero-helix milling model, an interval of optimal stable immersion levels is found where the stability of the system is maximized. In addition, the effect on stability of multiple cutting tooth engagement in the cut is studied. In the nonzero-helix milling model, the effect of multiple flute engagement in both axial and circumferential directions is considered, and the simultaneous effect of the helix angle and radial immersion on the stability is investigated.;Second, a new technique called the Chebyshev spectral continuous time approximation for time-periodic DDE analysis is introduced. This technique allows for conversion of a periodic DDE system into a larger order system of ordinary differential equations whose stability properties are equivalent to those of the initial DDE system. The advantages of this technique include the removal of the equality requirement between the parametric period and the delay, which is a limitation for the Chebyshev collocation method, as well as the ability to easily include nonlinearities and external excitation. Also, this technique is applicable to periodic DDEs with multiple delays. The proposed method is utilized for applying the Liapunov-Floquet transformation to time-periodic DDEs and for analysis of DDEs with discontinuous distributed delay. It is shown that even though this technique is not an efficient computational tool for the stability analysis in the form it is initially presented, additional tools can be introduced to considerably decrease the necessary time of computation.