Zoltan Dombovari

Estimates of the bistable region in metal cutting

The classical model of regenerative vibration is investigated with new kinds of nonlinear cutting force characteristics. The standard nonlinear characteristics are subjected to a critical review from the nonlinear dynamics viewpoint based on the experimental results available in the literature. The proposed nonlinear model includes finite derivatives at zero chip thickness and has an essential inflexion point. In the case of the one degree-of-freedom model of orthogonal cutting, the existence of unstable self-excited vibrations is proven along the stability limits, which is strongly related to the force characteristic at its inflexion point. An analytical estimate is given for a certain area below the stability limit where stable stationary cutting and a chaotic attractor coexist. It is shown how this domain of bistability depends on the theoretical chip thickness. The comparison of these results with the experimental observations and also with the subcritical Hopf bifurcation results obtained for standard nonlinear cutting force characteristics provides relevant information on the nature of the cutting force nonlinearity.

The Effect of Helix Angle Variation on Milling Stability

Helical milling tools of nonuniform helix angles are widely used in manufacturing industry. While the milling tools with these special cutting edges are already available in the market, their cutting dynamics has not been fully explored. Also, there have been several attempts to introduce complex harmonically varied helix tools, but the manufacturing of harmonic edges is extremely difficult, and their effect on cutting dynamics is not clear either. In this study, a general mechanical model is introduced to predict the linear stability of these special cutters with optional continuous variation of the helix angle. It is shown that these milling tools cause distribution in regeneration. The corresponding timeperiodic distributed delay differential equations are investigated by semi-discretization. This work points out how the nonuniform and harmonically varied helix cutters behave in case of high and low cutting speed applications. [DOI: 10.1115/1.4007466]

INTERACTION BETWEEN MULTIPLE MODES IN MILLING PROCESSES

The productivity of many industrial cutting processes is limited by high amplitude chatter vibrations. An optimization technique based on the use of the stability lobes helps to increase the productivity of these processes, improving the life of machine elements and reducing the tool wear as well. The best-known lobes correspond to Hopf bifurcations. However, in case of interrupted cutting, additional lobes appear due to period doubling or flip bifurcation. When the system has more than one dominant vibration mode, important variations can appear in stability due to interaction between modes. The basic mathematics for the appearance of these new lobes are shown in this article. The frequency domain study shows that lobes related to flip bifurcation are a special case of the interaction between modes. The results of these interactions are verified by comparison with semi-discretization method and time domain simulations, respectively.

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.

Fixed Boundaries Receptance Coupling Substructure Analysis for Tool Point Dynamics Prediction

In many applications, chatter free machining is limited by the flexibility of the tool. Estimation of that capacity requires to obtain the dynamic transfer function at the tool tip. Experimental calculation of that Frequency Response Function (FRF) is a time consuming process, because it must be done using an impact test for any combination of tool, toolholder and machine. The bibliography proposes the Receptance Coupling Substructure Analysis (RCSA) to reduce the number of experimental test. A new approach consisting of calculating the fixed boundary dynamic behaviour of the tool is proposed in the paper. This way the number of modes that have to be considered is low, just one or two for each bending plane, and it supposes an important improvement in the application of the RCSA to the calculation of stability diagrams. The predictions of this new method have been verified experimentally.

On the effective Young’s modulus of metal matrix syntactic foams

ABSTRACT The effective Young’s modulus of aluminium matrix syntactic foams was determined by modal analysis. Two different matrix materials (Al99.5 and AlSi12) were used, and they were reinforced by Globocer grade ceramic hollow spheres. In order to validate the results, a full-scale finite element model was also created. A new algorithm was developed to place the spheres in a proper, probabilistic spatial distribution. Finite element simulations were carried out in modal analysis and compression test senses. In addition, three different analytical methods were studied to estimate the effective Young’s modulus. The measured values were compared with the finite element and analytical results. The determined effective Young’s moduli showed good agreement. This paper is part of a thematic issue on Light Alloys.

The Effect of Harmonic Helix Angle Variation on Milling Stability

The linearly varied helix tool is widely used in manufacturing industry and milling tools are available in the market with these special cutting edges. There were several attempts to introduce complex harmonically varied helix tools, but the manufacturing of sinusoid edges is extremely difficult and its effect on cutting dynamics is not clear yet. In this study a mechanical model is introduced to predict the linear stability of these special cutters. It is shown that these milling tools cause distribution in regeneration and the corresponding time periodic distributed delay differential equations are investigated by semi-discretization. This work points out how the harmonically varied helix cutters behave in case of high and low cutting speed applications.Copyright

Experimental and Theoretical Study of Distributed Delay in Machining

Abstract In this work, an experimental and theoretical studies are given for the well-known process damping effect arisen in turning processes. This effect generally pushes the stability chart (lobes) to higher depth of cuts in low spindle speed domain and causes difficulties to predict the stability of stationary cutting (equilibrium of the process). The orthogonal cutting model presented and investigated here contains distributed delays due to the cutting force distribution on the rake face. The paper presents measurement set up to create and to track the oscillating cutting force for the identification of the model parameters. An extended model also investigated that explains the experimental observations.

On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction

The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.

On the robustness of stable turning processes

Self-excited non-linear vibrations occurring in the machining processes are investigated in this paper. Our treatment applies analytical techniques to a one Degree of Freedom (DOF) but strongly non-linear mechanical model of the turning process. This tool enables us to describe and analyse the highly non-linear dynamics of the appearing periodic motions. Using normal form calculations for the Delay-Differential Equation (DDE) model, we prove that the low-amplitude vibrations are unstable all along the stability lobes due to the subcriticality of Hopf bifurcations. This means that self-excited vibrations of the machine tool may occur below the stability boundaries predicted by the linear theory. Consequently, stable stationary cutting may not be robust enough for external perturbations close to the linear stability limits determined during the parameter optimisation of turning processes. Robustness is characterised by the amplitude of unstable oscillations along the stability lobes for non-linear cutting force characteristics having essential inflection points against chip thickness.