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

2010 ◽  
Author(s):  
Brandon C. Gegg ◽  
Steve S. Suh
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
Machine Tool ◽  
General Model ◽  
Solution Structure ◽  
Work Piece ◽  
Periodic Motions ◽  
Discontinuous Systems ◽  
Intermittent Cutting

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.

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

2009 ◽  
Author(s):  
Brandon C. Gegg ◽  
Steve S. Suh
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Machine Tool ◽  
Systems Theory ◽  
Degree Of Freedom ◽  
Continuous Systems ◽  
Discontinuous Systems ◽  
Mechanical Analogy ◽  
Steady State Motion ◽  
Two Degree Of Freedom

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.

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

2008 ◽  
Author(s):  
Brandon C. Gegg ◽  
Steven C. S. Suh ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Closed Form ◽  
Systems Theory ◽  
Phase Trajectory ◽  
Degree Of Freedom ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Discontinuous Systems ◽  
Closed Form Solutions ◽  
Two Degree Of Freedom

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.

Modeling and Theory of Intermittent Motions in a Machine Tool With a Friction Boundary

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Brandon C. Gegg ◽  
Steve C. S. Suh ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Systems Theory ◽  
Frictional Force ◽  
Periodic Motion ◽  
Mechanical Model ◽  
The State ◽  
Discontinuous Systems ◽  
Intermittent Cutting ◽  
Boundary Mapping ◽  
Simplified Mechanical Model

In this paper the simplified mechanical model for a machine-tool system is presented. The state and domains are defined with respect to the (contact and frictional force) boundaries in this system. The switching sets for this machine-tool will be defined for all the boundaries considered herein. The forces and force product components at the switching points are determined according to discontinuous systems theory. The forces and force product govern the passability of the machine-tool through the respective boundary. Mapping definitions and notations are developed through the switching sets for the boundaries. A mapping structure and notation for one type of intermittent cutting periodic motion is defined as an example.

On Periodic Solutions of a Time-Delayed, Discontinuous System with a Hyperbola Switching Control Law

2021 ◽  
Vol 31 (02) ◽  
pp. 2150032
Author(s):  
Liping Li ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Complex Dynamics ◽  
Periodic Motions ◽  
Discontinuous Systems ◽  
Discontinuous System ◽  
Sliding Motion ◽  
Delay Duration ◽  
Delayed System ◽  
Specific Mapping

In this paper, the existence of periodic motions of a discontinuous delayed system with a hyperbolic switching boundary is investigated. From the delay-related [Formula: see text]-function, the crossing, sliding and grazing conditions of a flow to the switching boundary are first developed. For this time-delayed discontinuous dynamical system, there are 17 classes of generic mappings in phase plane and 66 types of local mappings in a delay duration. The generic mappings are determined by subsystems in three domains and two switching boundaries. Periodic motions in such a delay discontinuous system are constructed and predicted analytically from specific mapping structures. Three examples are given for the illustration of periodic motions with or without sliding motion on the switching boundary. This paper shows how to develop switchability conditions of motions at the switching boundary in the time-delayed discontinuous systems and how to construct the specific periodic solutions for the time-delayed discontinuous systems. This study can help us understand complex dynamics in time-delayed discontinuous dynamical systems, and one can use such analysis to control the time-delayed discontinuous dynamical systems.

scholarly journals On the periodic motions of dynamical systems

1927 ◽  
Vol 50 (0) ◽  
pp. 359-379 ◽  
Author(s):  
George D. Birkhoff
Keyword(s):  
Dynamical Systems ◽  
Periodic Motions

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Design of a Force Feedback Chatter Control System

1972 ◽  
Vol 94 (1) ◽  
pp. 5-10 ◽  
Author(s):  
C. Nachtigal
Keyword(s):  
Machine Tool ◽  
Frequency Domain ◽  
Transient Response ◽  
Force Feedback ◽  
Design Procedure ◽  
Experimental Tests ◽  
Work Piece ◽  
Regenerative Chatter ◽  
Chatter Control ◽  
The Stability

The analysis of machine tool chatter from frequency domain considerations is generally accepted as a valid representation of the regenerative chatter phenomenon. However, active control of regenerative chatter is still in its embryonic stage. It was established in reference [2] that a measurement of the cutting force could be effectively used in conjunction with a controller and a tool position servo system to increase the stability of an engine lathe and to improve its transient response. This paper presents the design basis for such a system, including both analytical and experimental considerations. The design procedure stems from a real part stability criterion based on the work by Merritt [1]. Because of the unknown variability in the dynamics of a machine tool system, the controller parameters were chosen to accomodate some mismatch between structure and tool servo dynamics. Experimental tests to determine the stability zone of the controlled machine tool system qualitatively confirmed the analytical design results. The experimental results were consistent in that the transient response tests confirmed the frequency domain stability tests. It was also demonstrated experimentally that the equivalent static stiffness of a flexible work-piece system could be substantially increased.

Dynamic Instability in Quartering Seas: The Behavior of a Ship During Broaching

1996 ◽  
Vol 40 (01) ◽  
pp. 46-59 ◽  
Author(s):  
K. J. Spyrou
Keyword(s):  
Steady State ◽  
Transient Analysis ◽  
Dynamic Instability ◽  
Dynamical Analysis ◽  
Wave Excitation ◽  
Periodic Motions ◽  
Regular Waves ◽  
Limit Sets ◽  
Diffraction Wave ◽  
Surf Riding

The dynamic stability of ships encountering large regular waves from astern is analyzed, with focus on delineating the specific conditions leading to the uncontrolled turn identified as broaching. The problem's formulation takes into account motions of the actively steered or controls-fixed vessel in surge-sway-yaw-roll with consideration of Froude-Krylov and diffraction wave excitation. Dynamical analysis of surf-riding is carried out for the general case of quartering waves, exploring the route periodic motions—surf riding, loss of stationary stability, turn, capsize. Steady-state and transient analysis is carried out in the system's multidimensional state-space in order to identify all existing limit sets and locate attracting domains. Broaching from periodic motions is also a part of the investigation.

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