In this paper we establish a practical formula that could be used to augment existing linear stability charts for turning to include the occurrence of contact loss between tool and workpiece in turning. We show that the contact loss discontinuity in the global model is responsible for the creation of the experimentally observed coexistence of subcritical instability and hysteresis in the cutting process. Comparison of experimental data with extensive numerical simulations nicely support the theoretical findings.

Suppression of regenerative instability in a single-degree-of-freedom (SDOF) machine tool model was studied by means of targeted energy transfers (TETs). The regenerative cutting force generates time-delay effects in the tool equation of motion, which retained the nonlinear terms up to the third order in this work. Then, an ungrounded nonlinear energy sink (NES) was coupled to the SDOF tool, by which biased energy transfers from the tool to the NES and efficient dissipation can be realized whenever regenerative effects invoke instability in the tool. Shifts of the stability boundary (i.e., Hopf bifurcation point) with respect to chip thickness were examined for various NES parameters. There seems to exist an optimal value of damping for a fixed mass ratio to shift the stability boundary for stably cutting more material off by increasing chip thickness; on the other hand, the larger the mass ratio becomes, the further the occurrence of Hopf bifurcation is delayed. The limit cycle oscillation (LCO) due to the regenerative instability appears as being subcritical, which can be (locally) eliminated or attenuated at a fixed rotational speed of a workpiece by the nonlinear modal interactions with an NES (i.e., by means of TETs). Three suppression mechanisms have been identified; that is, recurrent burstouts and suppressions, partial and complete suppressions of regenerative instabilities in a machine tool model. Each suppression mechanism was characterized numerically by time histories of displacements, and wavelet transforms and instantaneous energies. Furthermore, analytical study was performed by employing the complexification-averaging technique to yield a time-delayed slow-flow model. Finally, regenerative instability suppression in a more practical machine tool model was examined by considering contact-loss conditions.

It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.

In this paper we investigate synchronization of oscillators. We use mechanical metronomes that are coupled through a mechanical medium. In passive coupling of two oscillators, the coupling medium is a one degree of freedom passive mechanical basis. The analysis of the system is shown and supported by simulations of the proposed model and experimental results. We show how the oscillators synchronize and discuss the affecting parameters in synchronization. In another case, the oscillators are forced by an external input while that input is also affected by the oscillators. This feedback loop introduces dynamics to the whole system. For this case, we place the mechanical metronomes on a one degree of freedom moving base. The movements of the base are a function of a feedback from the phases of the metronomes. We study the space of possibilities for the movements of the base and consider impacts of the base movement on the synchronization of metronomes. We also show how such a system evolves in time when we introduce an adjusting parameter that changes over time and updates based on feedbacks from the system.