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The unsafe zone in machining is a region of the parameter space where steady-state cutting operations may switch to regenerative chatter for certain perturbations, and vice versa. In the case of milling processes, this phenomenon is related to the existence of an unstable quasi-periodic oscillation, the in-sets of which limit the basin of attraction of the stable periodic motion that corresponds to the chatter-free cutting process. The mathematical model is a system of time-periodic nonlinear delay differential equations. It is studied by means of a nonlinear extension of the semidiscretization method, which enables the estimation of the parameter ranges where the unsafe (also called bistable) zones appear. The theoretical results are checked with thorough experimental work: first, step-by-step parameter variations are adapted to identify hysteresis loops, then harmonic burst excitations are used to estimate the extents of the unsafe zones. The hysteresis loops are accurately distinguished from the dynamic bifurcation phenomenon that is related to the dynamic effect of slowly varying parameters. The experimental results confirm the existence of the bistable parameter regions. This article is part of the theme issue 'Nonlinear dynamics of delay systems'. [ABSTRACT FROM AUTHOR]

The basic behavior of free vibratory nonlinear system is investigated in this work. The main approach was to reveal the order of fitting necessary to properly approximate the actual behavior of a nonlinear mechanical system. Tests were performed in an experimental setup subjected to geometric hardening nonlinearities. The investigation showed that, it is essential to approximate the nonlinearity correctly. Low order approximation can result in large errors in the predicted amplitudes. The nonlinear static stiffness characteristics was measured and fit with polynomial describing function. The free vibratory response was deviated from the one calculated by cubic fitting. The presented higher order approximation of the amplitude-frequency parametric relation is revealed so that, in this particular example, seventh degree approximation is sufficiently closer to the experienced behavior. The analytical solution including the first and second order internal resonances were checked and compared with continuation results of the backbone curve. [ABSTRACT FROM AUTHOR]

The paper deals with the automatic decomposition of experimental frequency response functions (FRF׳s) of mechanical structures. The decomposition of FRF׳s is based on the Green function representation of free vibratory systems. After the determination of the impulse dynamic subspace, the system matrix is formulated and the poles are calculated directly. By means of the corresponding eigenvectors, the contribution of each element of the impulse dynamic subspace is determined and the sufficient decomposition of the corresponding FRF is carried out. With the presented dominant modal decomposition (DMD) method, the mode shapes, the modal participation vectors and the modal scaling factors are identified using the decomposed FRF׳s. Analytical example is presented along with experimental case studies taken from machine tool industry. [ABSTRACT FROM AUTHOR]

This study gives the derivation of the Hopf bifurcation calculation for neutral delay differential equations using the centre manifold reduction theorem and normal form calculation. The whole concept was inspired by a case study where a robotic arm subjected to nonlinear stiffness with delayed acceleration feedback controller is modelled. Two different configurations are distinguished depending on the location of the acceleration sensor, a collocated and a non-collocated one. After a brief investigation of the linear stability, the bifurcation occurring at the loss of stability is calculated with the presented analytic equations for neutral delay differential equations. Then, a nonlinear term is introduced in the control law that can modify the occurring subcritical behaviour and improve the robustness of the system. The analytic results are carefully analysed and validated via a numerical continuation software, which also provided useful information about the global behaviour of the bifurcations in addition to the locally valid analytic results. • Two DoF non-linear robotic arm with additional acceleration feedback controller. • Linear stability analysis of the delayed system with the D-subdivision method. • Centre manifold reduction calculation for neutral delay differential equations. • Results for linear, non-linear controllers with different stiffness characteristics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

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 time-periodic 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. [ABSTRACT FROM AUTHOR]

Abstract: Chatter frequencies of milling operations and their strengths are analysed using the semi-discretization method. It is known that milling processes, being parametrically excited systems, are theoretically associated with infinitely many chatter frequencies that are given as a base frequency plus an integer multiplier of the tooth passing frequency (or the characteristic frequency of the cutter) according to the Floquet theory . The dominant vibration frequencies that are usually associated with chatter frequencies are, however, hidden among the infinitely many harmonics. In this paper, it is shown that the amplitudes corresponding to the individual frequency harmonics can be determined in a simple way by analysing the eigenvectors of the Floquet transition matrix obtained by the semi-discretization method without increasing the computational cost. The method can be used to determine the dominant frequency components that help in the identification of the interactions between different modes and the spindle speed. [Copyright &y& Elsevier]

Abstract: The large-amplitude motions of a one degree-of-freedom model of orthogonal cutting are analysed. The model takes the form of a delay differential equation which is non-smooth at the instant at which the tool loses contact with the workpiece, and which is coupled to an algebraic equation that stores the profile of the cut surface whilst the tool is not in contact. This system is approximated by a smooth delay differential equation without algebraic effects which is analysed with numerical continuation software. The grazing bifurcation that defines the onset of chattering motion is thus analysed as are secondary (period-doubling, etc.) bifurcations of chattering orbits, and convergence of the bifurcation diagrams is established in the vanishing limit of the smoothing parameters. The bifurcation diagrams of the smoothed system are then compared with initial value simulations of the full non-smooth delay differential algebraic equation. These simulations mostly validate the smoothing technique and show in detail how chaotic chattering dynamics emerge from the non-smooth bifurcations of periodic orbits. [ABSTRACT FROM AUTHOR]

Abstract: The machining behaviour of special serrated milling tools are investigated. These cutters are most commonly used for roughing operations of superalloys such as titanium and nickel based alloys which prevent high cutting speeds due to their high cutting forces and low thermal conductivity. During the experimental study, these drawbacks were avoided with the usage of aluminium alloy that allows more convenient machining circumstances and high tooth passing frequencies compared to the frequencies of the essential vibration modes. By means of a general cutting force model, simulations point out the fact that the serrated cutters require lower drive torque than their non-serrated counterparts, while our corresponding measurements validate our model. A regenerative dynamic model is constructed up directly in the modal space using the modal representation of the tool/toolholder/spindle structure and linear stability analyses are performed by the so-called semi-discretization method. The significantly larger parameter domains of stable cutting and their predicted feed dependency for these serrated mills are confirmed by chatter tests. As a result of these investigations, the practical advantages of the serrated cutters are confirmed: while they remove the same specific amount of materials using lower drive torque, their productivity can also be increased using higher stable depth of cuts compared to their non-serrated counterparts even in case of difficult-to-cut materials like titanium. The constructed mechanical model also provides an adequate tuning of the cutting parameters and the serration waves in order to optimize the process for easy-to-cut materials like aluminium. [Copyright &y& Elsevier]

An extensive study on the implementation of the convolution based impulse dynamic subspace (IDS) concept for machining applications is presented in this paper. The suggested method is adequate to avoid modeling based on modal parameters. Instead of parameter driven dynamical modeling, the impulse response functions (IRFs) are used for the IDS decomposition and for constructing the corresponding delayed and, on occasion, time periodic equations of motion of regenerative cutting processes. This approach makes possible to use measured frequency response functions (FRFs) in time domain based methods. This is achieved by establishing the connection between the IRF and the fundamental matrix of the corresponding linearized generally time periodic delayed system. It is shown that the concept provides the well-known analytic solution of the orthogonal turning model, demonstrating the applicability of the proposed methodology, while a specially tailored high-order semidiscretization scheme is tested on milling models. • Machining process dynamics is described using measured signals directly. • Process models are directly constructed by using impulse response functions. • Dynamics of machining processes is transformed by using impulse dynamic subspace. • Numerical method on the base of acquired frequency response functions. • Surface property, stability behavior and measure of stability are determined. • Effectivity, convergence tests and experimental validations are performed. [ABSTRACT FROM AUTHOR]

Advanced piezoelectric dynamometers with a wide frequency bandwidth are required for cutting force measurement in high-speed milling and micromilling applications. In many applications, the signal bandwidth is limited by the dynamic response of the mechanical system, thus compensation techniques are necessary. The most effective compensation techniques for a full 3D force correction require an accurate and complex identification phase. Extended Kalman filtering is a better alternative for input force estimation in the presence of unknown dynamic disturbances. The maximum bandwidth that can be currently achievable by Kalman filtering is approximately 2 kHz, due to crosstalk disturbances and complex dynamometer's dynamics. In this work, a novel upgraded Kalman filter based on a more general model of dynamometer dynamics is conceived, by also taking into account the influence of the force application point. By so doing, it was possible to extend the frequency bandwidth of the device up to more than 5 kHz along the main directions and up to more than 3 kHz along the transverse directions, outperforming state-of-the-art methods based on Kalman filtering. [ABSTRACT FROM AUTHOR]

Cutting capacity can be seriously limited in heavy duty face milling processes due to self-excited structural vibrations. Special geometry tools and, specifically, variable pitch milling tools have been extensively used in aeronautic applications with the purpose of removing these detrimental chatter vibrations, where high frequency chatter related to slender tools or thin walls limits productivity. However, the application of this technique in heavy duty face milling operations has not been thoroughly explored. In this paper, a method for the definition of the optimum angles between inserts is presented, based on the optimum pitch angle and the stabilizability diagrams. These diagrams are obtained through the brute force (BF) iterative method, which basically consists of an iterative maximization of the stability by using the semidiscretization method. From the observed results, hints for the selection of the optimum pitch pattern and the optimum values of the angles between inserts are presented. A practical application is implemented and the cutting performance when using an optimized variable pitch tool is assessed. It is concluded that with an optimum selection of the pitch, the material removal rate can be improved up to three times. Finally, the existence of two more different stability lobe families related to the saddle-node and flip type stability losses is demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, an extensive analysis of a stochastically excited one-degree-of-freedom mechanical system with cubic nonlinearity is presented. This is motivated by the need for realistic bifurcation analyses of stochastic dynamical systems, given that many physical applications contain significant time-varying uncertainty that can lead to drastically different solutions from the deterministic case. The proposed methodology is based on pseudo arc-length continuation combined with the moment-map method, which allows the investigation of dynamical systems with stochastic behaviour. It is shown that the introduction of noise in the excitation leads to the destabilisation of stable periodic orbits over time for the underlying system. For better interpretation and lower computational costs, instead of the combined stochastic continuation, the deterministic bifurcation diagram is modified with the detection and approximation of the mean first passage time of the stochastic system by introducing three different methods. Two semi-analytical approaches with Markov models, together with a completely numerical detection method for the mean first passage time, are compared to each other with similar results. The parameter sensitivity analysis and the comparison of the different methods support the proposed methodology for structural dynamic cases. • Stable oscillation of the stochastically forced mechanical system is highly time-dependent. • Improved calculation costs of the stochastic bifurcation analysis method. • Transient vibrations vary the boundary of stability loss, which leads to the jump phenomenon. • Identification of jump phenomenon in time domain simulations. • A novel, semi-analytical approximation of the first passage time (jump phenomenon). [ABSTRACT FROM AUTHOR]

This paper investigates the effect of displacement constraints on the attenuation performance of tuned mass dampers (TMDs) used in boring and turning applications. A simplified piecewise-smooth mechanical model is investigated through time domain simulations and hybrid periodic orbit continuation, first under harmonic excitation, then under regenerative cutting load. A quasi-frequency response function is derived for impacting TMDs through composition of different families of period-1 orbits, then an acceptability map for turning is formulated based on the appearance of cutting-edge contact-loss and fly-over events. The bi-stable domain boundaries are determined through two parameter continuation of contact-loss grazing events. It is shown that in both cases arising rigid body collisions can significantly hinder TMD damping performance and lead to resonance problems or machine tool chatter. [ABSTRACT FROM AUTHOR]

• The stochastic model of single-degree-of-freedom milling operations is established. • The moment dynamics, -stability and steady-state behavior of milling are derived. • The steady-state second moment is used to identify the noise-induced resonance. • The stochastic resonant vibrations are used to predict chatter. • The resonant vibrations and chatter are demonstrated through milling experiments. In this paper, the stochastic dynamical model of a single-degree-of-freedom milling operation is formulated, where a Gaussian white noise process models the high-frequency variation in the cutting force. With the help of this stochastic model, it is shown, that large-amplitude stable vibrations can occur near the critical machining parameters, due to stochastic noise-induced resonance. During the analysis, the second moment stability and stationary first and second moment behavior of the periodic stochastic delay differential equation (SDDE) describing the milling operation are investigated. The behavior of these quantities are then compared to the evolution of the so-called "chatter peak" in the Fourier-spectrum of the vibrations, that is used to experimentally determine the presence of chatter, in the stable machining parameter domain. Furthermore, it is discussed, how the statistical properties of the resonant vibrations can be used to predict the stability boundary and the formulation of chatter, while the machining parameters are kept in the safe region. The theoretical calculations are supported by experiments performed on a single-degree-of-freedom system. [ABSTRACT FROM AUTHOR]

Two different kinds of descriptions for edge geometry of harmonically varied helix tools are studied in this work. The edge geometries of the so-called lag and helix variations are used in this paper, and their equivalency is established from engineering point of view. The equivalent relation is derived analytically and the nonlinear algebraic system is described, with which the numerical equivalency properties can be determined. The equivalent description can be utilized in variable helix tool production to determine an optimized set of geometrical parameters of the edge geometry. The stability properties are shown and compared for a simple one degree-of-freedom case with the nonuniform constant helix tools. The robustness of the results related to the harmonically varied tools is critically discussed in this paper showing advantages compared to the nonuniform constant helix case. The results suggest that the more extreme the edge variation is, the more stable the process performed with the corresponding harmonically varied tool becomes. [ABSTRACT FROM AUTHOR]

The curved uncut chip thickness model is introduced to predict the cutting forces for general uncut chip geometries using the mechanistic approach. Classical geometric models assume that the cutting force is distributed along straight elementary sections of the uncut chip area, which has limited physical validity, but makes mathematical treatments easier for simple cases. The new model assumes that the flow of the material on the contact area of the tool is given by a continuous vector field, according to which the curved uncut chip thickness is measured. The cutting force is distributed along these paths, which leads to a mathematically unique and consistent solution for regular and complex cutting edge geometries. These curved paths can be generated by basic mechanical models, which mimic the more realistic motion of the chip segments along the rake face, without the need of explicit time-consuming cutting simulations. The presented computational procedure generalizes cutting force prediction based on geometric parameters, orthogonal cutting data and the orthogonal to oblique transformations only. The effectiveness of the model for various cutting edge geometries (e.g., thread turning inserts) under extreme cutting conditions is presented in case studies, laboratory and industrial experiments. [Display omitted] • The novel concept of the curved uncut chip thickness is introduced. • Cutting force is distributed along curved paths on the rake face of the tool. • The solution of the model is unique and free of inconsistency for complex geometries. • The method improves cutting force predictions at high feeds and extreme conditions. [ABSTRACT FROM AUTHOR]

In this work we present a quantitative measurement-based method to describe the stability behaviour of a periodic dynamical system. In-process response for impulse during milling is analysed to provide operational stability prediction. The proposed method combines chatter detection techniques with the theory of time-periodic delay differential equations applied for general milling operations. Signal processing based on dynamic modal decomposition approach not only presents qualitative properties (like stability/instability) but also quantifies a measure of stability through the determination of the Floquet multipliers. The corresponding monodromy operator of the milling process is approximated from the measured system's response during machining operation. By changing the technological parameters, the variation of the modulus of the Floquet multipliers can be monitored. The stability limit can precisely be interpolated with the unitary multiplier; furthermore, the stability limit can be extrapolated while the manufacturing parameters remain in the chatter-free region. The presented approach is validated by numerical results and laboratory tests. • Measurement-based method to quantify stability of milling operation. • In-process impulse response is analysed to provide operational stability prediction. • It combines chatter detection techniques with the theory of time-periodic systems. • Extrapolation of stability limit opens ways for efficient cutting parameter optimization. • Results are supported by theoretical, numerical and experimental parts. [ABSTRACT FROM AUTHOR]

In large machine tool applications, where machine's productivity is limited by structural chatter vibrations, an accurate dynamic characterization is vital for predicting reliable stability lobes. Motivated by the fact that the sweep milling force excitation method shows a great potential, this paper conducts a numerical stability analysis for optimizing the tool radial engagement employed during the characterization cutting tests. Additionally, an alternative spectra computation technique is proposed which considerably improves the signal to noise ratio over the existing technique. The receptance measurement is demonstrated using time domain simulations and applied in a large-scale three axis machining centre obtaining more accurate stability boundary predictions. The source code of the temporal milling simulation and the sweep milling signal processing is freely available to reproduce the simulation and experimental results. https://github.com/OierFranco/SMFE. [Display omitted] • Machining system dynamics are identified through an improved Sweep Milling Force Excitation. • Optimal tool radial engagement is selected based on numerical stability analysis. • Analytical approximated expression for optimal tool radial engagement calculations are provided. • High quality Frequency Response Functions of a three axis machining centre are experimentally characterized. • Process stability limit prediction is improved by using characterized dynamic responses with respect to traditional impact hammer tests. [ABSTRACT FROM AUTHOR]

A new method for calculation of chip removal machining stability diagrams is proposed. The method can be considered as an application of the Floquet theorem by repeated time integrations and represents an alternative to the previously presented time domain stability methods (semi-discretisation, time finite elements and so on), without the requirement to build the transition matrix. In this way, the computation effort is very much reduced, especially when the required number of segments is large. As a result, the computing time depends on the number of segments with an exponent 1.5, instead of 2.8 that is the exponent for the optimised semi-discretisation. This results in that the presented method is the most efficient among the previous ones. As a further advantage, the memory requirements for the method are much lower, allowing computing very-high-order stability lobes. As a drawback, for the computation of high-order lobes, the method is not as efficient as could be expected, due to the slow convergence of the eigensystem resolution when many eigenvalues of similar magnitude exist. Even in that case, the method is five times faster than the optimised semi-discretisation, but a more efficient eigenvalue resolution method is sought for. [ABSTRACT FROM AUTHOR]

Abstract: The material removal capability of machines is partially conditioned by self-excited vibrations, also known as chatter. In order to predict chatter free machining conditions, dynamic transfer function at the tool tip is required. In many applications, such as high-speed machining (HSM), the problematic modes are related to the flexibility of the tool, and experimental calculation of the Frequency Response Function (FRF) should be obtained considering every combination of tool, toolholder and machine. Therefore, it is a time consuming process which disturbs the production. The bibliography proposes the Receptance Coupling Substructure Analysis (RCSA) to reduce the amount of experimental tests. In this paper, a new approach based on the calculation of the fixed boundary dynamic behavior of the tool is proposed. Hence, the number of theoretical modes that have to be considered is low, instead of the high number of modes required for the models presented up today. This way, the Timoshenko beam theory can be used to obtain a fast prediction. The accuracy of this new method has been verified experimentally for different tools, toolholders and machines. [Copyright &y& Elsevier]