Perturbation Methods in Nonlinear Dynamics—Applications to Machining Dynamics

Abstract

The role of perturbation methods and bifurcation theory in predicting the stability and complicated dynamics of machining is discussed using a nonlinear single-degree-of-freedom model that accounts for the regenerative effect, linear structural damping, quadratic and cubic nonlinear stiffness of the machine tool, and linear, quadratic, and cubic regenerative terms. Using the width of cut w as a bifurcation parameter, we find, using linear theory, that disturbances decay with time and hence chatter does not occur if w < wc and disturbances grow exponentially with time and hence chatter occurs if w > wc. In other words, as w increases past wc, a Hopf bifurcation occurs leading to the birth of a limit cycle. Using the method of multiple scales, we obtained the normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that the bifurcation is supercritical; that is, local disturbances decay for w < wc and result in small limit cycles (periodic motions) for w > wc. Using a six-term harmonic-balance solution, we generated a bifurcation diagram describing the variation of the amplitude of the fundamental harmonic with the width of cut. Using a combination of Floquet theory and Hill’s determinant, we ascertained the stability of the periodic solutions. There are two cyclic-fold bifurcations, resulting in large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability. As the width of cut w increases, the periodic solutions undergo a secondary Hopf bifurcation, leading to a two-period quasiperiodic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the torus doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portraits, Poincare´ sections, and power spectra. The results indicate the importance of including the nonlinear stiffness terms.

Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

“…The multiple scales perturbation analysis was carried out by using a specialized symbolically based computational solver, Khanin et al [33] and Khanin and Cartmell [34], and has led to an analytical criterion, in the form of formulas (31) and (32), for the definition of stable, or unstable, dry friction chatter vibrations in orthogonal metal cutting. Compared with numerical methods, including neural network procedures, this is a fast and elegant approach to such a criterion, although it is clear that a good dynamical model and physically realistic data are important pre-requisites.…”

“…This numerical study revealed that a different cutting thicknesses can result in qualitatively different behaviour of the system. A non-linear single-degree-of-freedom model originally proposed by Hanna and Tobias [31] has been discussed by Nayfeh et al [32]. This model included linear structural damping, quadratic and cubic non-linear stiffness of the machine tool, and linear, quadratic and cubic regenerative terms, as required to define the regenerative cutting process in dynamical terms.…”

Influence of frictional mechanism on chatter vibrations in the cutting process–analytical approach

“…An analytical solution of nonlinear problems is not exact but approximate and difficult to obtain. Nonetheless, it is frequently used due to its universality [19]. Sometimes, the impact of ploughing mechanism on chatter stability is presented as well [20].…”

“…The nonlinear spring is sometimes used in the literature (e.g. [19,24]) to model the nonlinear properties of the tool and tool holder, although a linear spring …”

Suppression of Hopf bifurcation in metal cutting by extrusion machining

“…Shi and Tobias [16] extended the work of Hanna and Tobias to analyze the process whereby the cutting tool leaves the surface of materials. Based on the model of Hanna and Tobias, Nayfeh et al [17] used the method of multiple scales to research the behavior of the Hopf bifurcations in metal cutting. Pratt and Nayfeh [18] proposed a twodegrees-of-freedom system with time delay to examine the subcritical instabilities in boring bars.…”