Stability and surface errors are investigated numerically for milling operations with a helical tool. A detailed two degree of freedom mechanical model is derived that includes both surface regeneration and the helical teeth of the tool. The governing delay-differential equation is analyzed by the semi-discretization method. The surface errors are predicted based on the (stable) forced motion of the tool. New surface error parameters were introduced to characterize the properties of the spatial machined surface. The errors were calculated numerically for a given machine tool and workpiece for different axial depths of cut and spindle speeds. It is shown that both good surface properties and large material removal rate can be achieved by appropriate axial immersion in case of helical fluted tool. This phenomenon was proved analytically by means of the Fourier transformation of the cutting force.

The governing equation of milling processes is generalized with the help of accurate chip thickness derivation resulting in a state dependent delay model. This model is valid for large amplitude machine tool vibrations and uses accurate nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool relative to the workpiece. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization around these periodic solutions by means of the semidiscretization method applied for the corresponding time-periodic delay system. Predictor-corrector method is developed to continue the periodic solutions as the bifurcation parameter, that is, the axial immersion is varied. It is observed that the influence of the state dependent delay on linear stability can be significant close to resonance where large amplitude forced vibrations occur. The existence of an unusual fold bifurcation is shown where a kind of hysteresis phenomenon appears between two different stable periodic motions.

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

Summary: In this article, the dynamics and stability of a linear system with stochastic delay and additive noise are investigated. It is assumed that the delay value is sampled periodically from a stationary distribution. A semi‐discretization technique is used to time‐discretize the system and derive the mean and second‐moment dynamics. These dynamics are used to obtain the stationary moments and the corresponding necessary and sufficient stability conditions. The application of the proposed method is illustrated through the analysis of the Hayes equation with stochastic delay and additive noise. The method is also applied to the control design of a connected automated vehicle. These examples illuminate the effects of stochastic delays on the robustness of dynamical systems.