Chatter mitigation using the nonlinear tuned vibration absorber

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability properties of the machining operation, while its nonlinear properties are chosen in order to control the bifurcation behavior of the system and guarantee robustness of stable operation. In this study, the NLTVA is applied to turning machining.     

Dynamics of milling processes with variable time delays

A milling-process model with a variable time delay associated with each cutting tooth is presented in this article. The source of this variable time delay is the feed rate. The effect of the feed motion on the entry cutting angle, the exit cutting angle, and the amplitude of feed mark is also discussed. Loss-of-contact effects are also considered. The system dynamics is described by a set of delay differential equations with periodic coefficients and variable time delays. A semi-discretization scheme is presented for analyzing the stability of periodic orbits of this system. The analysis provides evidence of period-doubling bifurcations and secondary Hopf bifurcations. Good agreement is found between the numerical results obtained from this work and the results of related experimental studies.     

A Mechanics Based Model for Study of Dynamics of Milling Operations

A unified mechanics based model with multiple degrees of freedom is developed and numerically simulated to study workpiece-tool interactions during milling of ductile workpieces with helical tools. A refined orthogonal cutting model is used at each section of the tool, and the milling forces are determined by using a spatial integration scheme along the axis of the tool. Both regenerative and loss of contact effects are considered in determining the cutting forces, which makes the model well suited for a wide range of milling operations. The model also allows for partial engagement of a tool with a workpiece, which is an important feature needed for milling operations with helical tools. Time domain simulations are carried out by using the developed model to predict the stability boundaries in the space of the tool spindle speed and the axial depth of cut. Poincaré sections are used to determine loss of stability from period-one motions to other motions such as two-period quasiperiodic motions, as a control parameter is varied.     

Delay equations with fluctuating delay related to the regenerative chatter I: reduced bifurcation equation

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. The suppression of regenerative chatter by spindle speed variation is attracting increasing attention. In this paper, we study nonlinear delay differential equations with periodic delays which models the machine tool chatter with continuously modulated spindle speed. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov–Schmidt Reduction method.     

Analysis of Machining Stability for a Parallel Machine Tool

Abstract

Machine tool chatter is a self-excited vibration generated by chip thickness variation. It severely degrades the quality of the machined surface. The incidence of chatter is greatly affected by the dynamic characteristics of machine tool structure. This article extends chatter stability analysis to a machine tool equipped with a parallel mechanism. The vibration model of a parallel machine tool is derived, in which the legs of the parallel mechanism are considered as spring-damper systems. Then, the regenerative cutting dynamics is combined with the vibration model and stability analysis is performed. The chatter stability charts for various machining parameters are examined, with the example of the cubic parallel mechanism that is specially designed for machine tool use.     

Nonlinear dynamics of a machining system with two interdependent delays

The dynamics of turning by a tool head with two rows, each containing several cutters, is considered. A mathematical model of a process with two interdependent delays with the possibility of cutting discontinuity is analyzed. The domains of dynamic instability are derived, and the influence of technological parameters on system response is presented. The numeric analysis show that there exists specific conditions for given regimes in which one row of cutters produces an intermittent chip form while the other row produces continuous chips. It is demonstrated that the contribution of parametric excitation by shape roughness of an imperfect (unmachined) cylindrical workpiece surface is not substantial due to the special filtering properties of cutters that are uniformly distributed circumferentially along the tool head.     

Hopf bifurcation for delay differential equation with application to machine tool chatter

In the machining process, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece, hence the relevant measures must be taken to predict and avoid this phenomenon of instability. In this paper, we propose a weakly nonlinear model with square and cubic terms in both structural stiffness and regenerative terms, to represent self-excited vibrations in machining. It is proved that Hopf bifurcation exists when bifurcation parameter equals a critical value, a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are given by using the normal form method and center manifold theorem. Numerical simulations show excellent agreement with the theoretical results.     

Cutting process of composite materials: An experimental study

This paper focuses on experimental research of milling process of the epoxide-polymer matrix composite reinforced carbon fibers (EPMC—carbon composite). An influence of two control parameters, namely feed and rotational speed, on cutting forces is investigated. The experiment is conducted on a CNC machine with feed rate ranging from 200 to 720 mm/min and rotational speed from 2000 to 8000 rpm. The experimental time series are analysed by means of the delay coordinates method in order to find stable cutting regions and to recognize the kind of behaviour. Using this information, a new model for the cutting forces is proposed that can be used to build a new regenerative vibration model for EPMC milling.     

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.     

Stability analysis of a thin-walled cylinder in turning operation using the semi-discretization method

In this work, the stability of a flexible thin cylindrical workpiece in turning is analyzed. A process model is derived based on a finite element representation of the workpiece flexibility and a nonlinear cutting force law. Repeated cutting of the same surface due to overlapping cuts is modeled with the help of a time delay. The stability of the so obtained system of periodic delay differential equations is then determined using an approximation as a time-discrete system and Floquet theory. The time-discrete system is obtained using the semi-discretization method. The method is implemented to analyze the stability of two different workpiece models of different thicknesses for different tool positions with respect to the jaw end. It is shown that the stability chart depends on the tool position as well as on the thickness.