On the bistable zone of milling processes

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasi-periodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasi-periodic solutions and the checking of their grazing to a time-periodic switching surface in the infinite-dimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains

Revising mtDNA haplotypes of the ancient Hungarian conquerors with next generation sequencing

As part of the effort to create a high resolution representative sequence database of the medieval Hungarian conquerors we have resequenced the entire mtDNA genome of 24 published ancient samples with Next Generation Sequencing, whose haplotypes had been previously determined with traditional PCR based methods. We show that PCR based methods are prone to erroneous haplotype or haplogroup determination due to ambiguous sequence reads, and many of the resequenced samples had been classified inaccurately. The SNaPshot method applied with published ancient DNA authenticity criteria is the most straightforward and cheapest PCR based approach for testing a large number of coding region SNP-s, which greatly facilitates correct haplogroup determination

The Hopf-van der Pol system: Failure of a homotopy method

The purpose of this article to provide an explicit example where continuation based on the homotopy method fails. The example is a one-parameter homotopy for periodic orbits between two well-known nonlinear systems, the normal form of the Hopf bifurcation and the van der Pol system. Our analysis shows that various types of obstructions can make approximation over the whole range of the homotopy parameter impossible. The Hopf-van der Pol system demonstrates that homotopy methods may fail even for seemingly innocent systems

Persistence analysis of velocity and temperature fluctuations in convective surface layer turbulence

Persistence is defined as the probability that the local value of a fluctuating field remains at a particular state for a certain amount of time, before being switched to another state. The concept of persistence has been found to have many diverse practical applications, ranging from non-equilibrium statistical mechanics to financial dynamics to distribution of time scales in turbulent flows and many more. In this study, we carry out a detailed analysis of the statistical characteristics of the persistence probability density functions (PDFs) of velocity and temperature fluctuations in the surface layer of a convective boundary layer using a field-experimental dataset. Our results demonstrate that for the time scales smaller than the integral scales, the persistence PDFs of turbulent velocity and temperature fluctuations display a clear power-law behavior, associated with a self-similar eddy cascading mechanism. Moreover, we also show that the effects of non-Gaussian temperature fluctuations act only at those scales that are larger than the integral scales, where the persistence PDFs deviate from the power-law and drop exponentially. Furthermore, the mean time scales of the negative temperature fluctuation events persisting longer than the integral scales are found to be approximately equal to twice the integral scale in highly convective conditions. However, with stability, this mean time scale gradually decreases to almost being equal to the integral scale in the near-neutral conditions. Contrarily, for the long positive temperature fluctuation events, the mean time scales remain roughly equal to the integral scales, irrespective of stability

Homotopy continuation for characteristic roots of delay differential equations using the Lambert W function

In this work, we develop a homotopy continuation method to find the characteristic roots of delay differential equations with multiple delays. We introduce a homotopy parameter μ into the characteristic equation in such a way that for μ = 0 this equation contains only one exponential term (corresponding to the largest delay) and for μ = 1 the original characteristic equation is recovered. For μ = 0, all the characteristic roots can be expressed in terms of the Lambert W function. Pseudo-arclength continuation is then used to trace the roots as a function of μ. We demonstrate the method on several test cases. Cases where it may fail are also discussed

Nonlinear stability of a delayed feedback controlled container crane

A simplified model of a container crane subject to a delayed feedback is investigated. The conditions for a Hopf bifurcation to stable/unstable limit-cycle solutions are determined. It is shown that a subcritical Hopf bifurcation to unstable oscillations cannot be ruled out and the undesired coexistence of stable large amplitude oscillations and a stable equilibrium endangers the robustness of time-delay control strategies. The bifurcation is analyzed both analytically and numerically using a continuation method.info:eu-repo/semantics/publishe

Quantifying Rate-Dependence of a Nonlinear Hysteretic Device

Hysteretic systems are extensively used in vibration control and in a wide range of mechanical problems. The first issue is to know whether the system is rate-dependent or not. In order to address the problem for a specific device employed for vibration isolation, laboratory measurements using a uniaxial testing machine are performed. The device is represented by a wire rope spring. Different types of tests are conducted in displacement control acquiring the device restoring force: cyclic ramps with increasing amplitudes and displacement time histories are obtained applying simple and random homeomorphisms to the ramp histories. The rate-dependence is evaluated comparing the restoring forces measured for the different tests and estimating a quantitative parameter which is chosen to measure the rate-dependence degree