Robust Stability Limit of Delayed Dynamical Systems

Abstract

Keywords Robust stability, Multi-Dimensional Bisection Method (MDBM), turning, process damping IntroductionThe determination of the stability of dynamical systems with time delay is of high importance for many industrial and research applications. The investigation of the dynamics of turning processes is such a typical industrial problem. In order to reach high process efficiency the material removal rate (MRR) has to be maximized. The major limitation for increasing the MRR is the so-called chatter vibration. The origin of these self-excited vibrations is the surface regeneration effect, which can be described by linear delayed differential equations (DDEs) [1,2]. Another example is human balancing, which is often modelled by a simple inverted pendulum in the presence of reflex delay [3,4,5], which is a relevant issue to human motion control [6]. A similar problem is the remote control of periodic robotic motions, when the delay in the information transmission system is not negligible [7,8].The most important qualitative property of the corresponding dynamical system is the stability of the equilibrium or periodic motions. This is usually presented in the form of the socalled stability chart, that identifies those ranges of parameters where the linear system is stable.Many well-known computational techniques are available to determine the stability chart of DDEs [9,10,11,12,13,14].In many cases, especially for machining operations at low spindle speed, the computation of the stability boundary requires very high computational effort and unnecessarily high resolution, due to the dense and sharp line segments of the stability boundary (see Fig. 1). In these cases, the computation of the lower envelope of the dense stability lobe structure would be adequate.In case of time-domain computations [10,13,15,16,17], really high degree of discretization is required for proper results. In frequency domain computations [14,18,19,20], the determination of the very dense lobe structure in the unstable domain is unnecessary and also high resolution is required to determine the chatter frequency parameter, which is also a required computational step.Traditional computational methods consider exact models of the system, and do not take into account the uncertainty of

On the global dynamics of connected vehicle systems

“…We prefer those robust control parameters for which the system is stable for any available headway h * ∈ [5,55] m. The corresponding stability boundaries can be found by following the edges of the droplets where the tangent is horizontal (at h * = 30 m in our case) [23]. That is, if we change parameter h * , the real parts of the critical characteristic roots remain zero (Reλ = 0).…”

“…Thus, the robust parameter boundaries can be calculated by solving the real and imaginary parts of Eqs. (16) and (23) together while setting λ = iω cr .…”

“…In the 4-dimensional parameter space (h * , β 1 , α, ω cr ), Eqs. (16) and (23) give us a 1-dimensional object (curve). Still multi-dimensional bisection method can be used to find the corresponding solutions.…”

Algorithm for Robust Stability of Delayed Multi-Degree-of-Freedom Systems

“…3 was already time consuming, even for only 30 different τ values. To overcome this problem, a new idea was shown in [3], according to which the set of boundary curves can be connected to a surface by means of an additional parameter, which is defined as the regenerative phase shift parameter Φ := mod(τω c , 2π) .…”

Optimization of the Robust Stability Limit for Multi-Cutter Turning Processes

“…Inaccurate natural frequency and spindle speed can endanger safe machining in the stability pockets, since they severely influence the horizontal position of the machining parameter points relative to the lobe curves. To eliminate the risk of sliding into an unstable machining parameter domain, in industrial applications, one has to be aware of the lower envelope of the stability boundaries, which can, for instance, be computed by the Robust Stability Computation Method [13]. The main steps of this method are summarized below.…”