Brandon C. Gegg

Periodic Motions of the Machine Tools in Cutting Process

In this paper, a two-degree of freedom dynamical system with discontinuity is developed to describe the vibration in the cutting process. The analytical solutions for the switchability of motion on the discontinuous boundary are presented. the switching sets based on the discontinuous boundary is introduced and the basic mappings are introduced to investigate periodic motion in such a mechanical model. The mapping structures for the stick and non-stick motions are discussed. Numerical predictions of motions of the machine tool in the cutting process are presented through the two-degree of freedom system with discontinuity. The phase trajectory and velocity and force responses are presented and the switchability of motion on the discontinuous boundary is illustrated through force distribution and force product on the boundary.© 2007 ASME

Analytical Prediction of Interrupted Cutting Periodic Motions in a Machine Tool

The methodology for prediction of interrupted cutting periodic motions in a machining system is developed. The interrupted cutting mappings in the vicinity of the system constraints are defined. The criteria for the interrupted cutting periodic motions are developed through the state variables and mapping forms. The periodic interrupted cutting motions in a two-degree-of-freedom model are predicted numerically and analytically via closed form solutions. The chip and tool-piece seizure in the machine-tool system is also discussed. The bifurcations are caused by interactions of continuous dynamical systems in the neighborhood of the boundary.

A Parameter Study of a Machine Tool with Multiple Boundaries

The parameter study of a machine-tool with intermittent cutting is completed for eccentricity frequency and amplitude. The effects with respect to chip length are also incorporated, such that comparisons of the parameter maps can be accomplished. Specific areas within the parameter maps are studied, via switching components, to explain the complicated motions within. In such a case, the switching characteristics are shown in relation to the eccentricity frequency. The complexity of the periodic solution structure, with regard to the vector fields and mapping quantities, is discussed. Furthermore, the traditional definition of a stability boundary is extended beyond that in literature. The most useful data is the overlay of the number of mappings and minimum switching force product record. This aspect illustrates the extent and location of complexity in the machine-tool model studied herein.

Characteristics and Effects on a Machine-Tool Due to Grazing Motions of a Friction Boundary

The machine-tool interactions are studied with respect to the chip/tool friction boundary on the tool-piece rake face. The grazing motions with this chip/tool friction boundary are the focus of the study and how the effects are identified quantitatively. This study considers the transient motion of the machine-tool and will provide a framework for prediction of such grazing motions. The proposed study will include prediction through a range of excitation and system parameters. The significance of this lay within the application and the general approach to the definition of grazing via discontinuous systems theory.Copyright

Sensitivity of Steady State Intermittent Cutting Motion to Work-Piece Characteristics

The tool and work-piece interactions will be modeled via discontinuous systems to study the effects of work-piece characteristics the on sensitivity of steady state motions. The general model will be presented through the domains of continuous dynamical systems for this machine-tool. The periodic motions of intermittent cutting will be developed and implemented to describe the solution structure. The switching components at the chip/tool friction boundary will be discussed in regard to work-piece characteristics.Copyright

Effects of Chip Seizure on Steady State Motion of a Machine-Tool

The steady state motion of a machine-tool is numerically predicted with interaction of the chip/tool friction boundary. The chip/tool friction boundary is modeled via a discontinuous systems theory in effort to validate the passage of motion through such a boundary. The mechanical analogy of the machine-tool is shown and the continuous systems of such a model are governed by a linear two degree of freedom set of differential equations. The domains describing the span of the continuous systems are defined such that the discontinuous systems theory can be applied to this machine-tool analogy. Specifically, the numerical prediction of eccentricity amplitude and frequency attribute the chip seizure motion to the onset or route to unstable interrupted cutting.Copyright

Analytical Prediction of Periodic Motions in a Machine Tool With Loss of Effective Chip Contact

This study applies a discontinuous systems theory by Luo (2005) to an approximate machine-tool model. The machine-tool is modeled by a two-degree of freedom forced switching oscillator. The switching of the model emulates the various types of dynamics in a machine-tool system. The main focus of this study is the loss of effective chip contact and boundaries of this motion. The periodic motions will be studied through the mappings developed for this machine-tool. The periodic motions will be numerically and analytically predicted via closed form solutions. The phase trajectory, velocity, and force responses are presented.Copyright

Chip Stick and Slip Motions of a Machine Tool in the Cutting Process

The two-degree of freedom oscillator is presented to model the dynamics of a machine tool system in the cutting process. The chip dynamics are presented through a discontinuous system with a velocity boundary (frictional force). The closed form solutions are presented for the normalized linear set of ordinary differential equations. The basic mappings are introduced to investigate the stick-slip motion in such a mechanical model. The mapping structures for the periodic motions will be developed. The numerical prediction of the phase trajectory, over a range of the excitation frequency is presented through the discontinuity. The predictions are verified by numerical illustrations of the phase trajectory, velocity and forces time histories. The main contributions are the forces and force product distribution at the switching points for this machine-tool.Copyright

The Stick Motion of a Harmonically Excited, Friction-Induced Oscillator on a Sinusoidally Traveling Surface

In this paper, the methodology is presented through investigation of a periodically, forced linear oscillator with dry friction, resting on a traveling surface varying with time. The switching conditions for stick motions in non-smooth dynamical systems are obtained. From defined generic mappings, the corresponding criteria for the stick motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the stick motions is illustrated. Finally, numerical simulations of stick motions are carried out to verify the analytical prediction. The achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces possessing a CO - discontinuity.Copyright