Local and Global Stability and Dynamics of a Class of Nonlinear Time-Delayed One-Degree-of-Freedom Systems
Nayfeh, Nader Ali
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We investigate the dynamics and stability of nonlinear time-delayed one-degree-of-freedom systems possessing quadratic and cubic nonlinearities and subjected to external and parametric disturbances. Due to the time-delay terms, the trivial solution of the unforced system undergoes Hopf bifurcations. We use the method of multiple scales to determine the normal forms of the Hopf bifurcations and hence determine whether they are locally supercritical or subcritical. Then, we use a combination of a path following scheme, the normal forms, and the method of harmonic balance to calculate and trace small- and large-amplitude limit cycles and use Floquet theory to ascertain their stability and hence generate global bifurcation diagrams. We validate these diagrams using numerical simulations. We apply the results to two important physical problems: machine-tool chatter in lathes and control of the sway of container cranes using time-delayed position feedback. We find that the Hopf bifurcations in machine tools are globally subcritical even when they are locally supercritical. We find multiple large-amplitude solutions coexisting with the linearly stable trivial solution. Consequently, there are three operating regions for machine tools: an unconditionally stable region, an unconditionally unstable region, and a conditionally stable region. In the latter region, the multiple responses lead to hysteresis. Then, we investigate the use of bifurcation control to transform the subcritical bifurcations into supercritical ones. We find that cubic-velocity feedback with appropriate gains can shrink or even eliminate the conditionally stable region. Then, we find that time-delayed acceleration feedback with an appropriate gain can completely eliminate the linear instability region. In contrast, we find that the Hopf bifurcations in controlled cranes are locally and globally supercritical. Finally, we investigate the effectiveness of time-delayed position feedback in rejecting external and parametric disturbances in ship-mounted cranes.
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