Period-1 Motions in a Periodically Forced, Nonlinear, Machine-Tool System

2019 ◽  
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Numerical Simulations ◽  
Analytical Method ◽  
Degree Of Freedom ◽  
Eigenvalue Analysis ◽  
Tool Vibrations ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this paper, period-1 motions in a two-degree-of-freedom, nonlinear, machine-tool system are investigated by a semi-analytical method. The stability and bifurcations of the period-1 motions are discussed from the eigenvalue analysis. A condition is presented for the tool-and-workpiece separation in period-1 motions. Machine-tool vibrations varying with displacement disturbance from a workpiece are discussed. Numerical simulations of period-1 motions are completed from analytical predictions.

Regenerative Cutting Dynamics for a Periodically Forced Machine-Tool System

2019 ◽  
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Numerical Simulations ◽  
Analytical Method ◽  
Orthogonal Cutting ◽  
Periodic Oscillation ◽  
Eigenvalue Analysis ◽  
Stable Period ◽  
Machine Tool Chatter ◽  
Stability And Bifurcations ◽  
Cutting System

Abstract In this paper, a nonlinear, regenerative, orthogonal cutting system with a weak periodic oscillation of workpiece is considered. Period-1 motions in such a system are studied through a semi-analytical method, and the corresponding stability and bifurcations of the period-1 motions are analyzed via the eigenvalue analysis. The vibration of machine-tool varying with excitation is studied, and excitation effects on machine-tool chatters are discussed. Numerical simulations of unstable and stable period-1 motions are completed from analytical predictions. The machine-tool chatter can emerge from the saddle-node or Neimark bifurcation of period-1 motions.

D-partition Stability Analysis of a Two-Degree-of-Freedom System with Delayed Restoring Force

1967 ◽  
Vol 9 (3) ◽  
pp. 190-197 ◽  
Author(s):  
B. Porter
Keyword(s):  
Stability Analysis ◽  
Machine Tool ◽  
Algebraic Equation ◽  
Characteristic Equation ◽  
Normal Modes ◽  
Degree Of Freedom ◽  
Central Feature ◽  
Restoring Force ◽  
The Stability ◽  
Two Degree Of Freedom

The method of D-partition is used to analyse the stability of a two-degree-of-freedom system subjected to a delayed restoring force of the kind which causes chatter in certain types of machine tool. The central feature of the analysis is the reduction of a stabliity problem involving a transcendental characteristic equation to a much simpler problem concerning the roots of a related algebraic equation. The results of the exact analysis are compared with approximate results obtained by assuming that the normal modes of the two-degree-of-freedom system can be decoupled.

Three-Dimensional Numerical Simulations of Circular Cylinders Undergoing Two Degree-of-Freedom Vortex-Induced Vibrations

2006 ◽  
Author(s):  
Juan P. Pontaza ◽  
Hamn-Ching Chen
Keyword(s):  
Numerical Simulations ◽  
Three Dimensional ◽  
Operating Conditions ◽  
Degree Of Freedom ◽  
Circular Cylinders ◽  
Time Step ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
High Reynolds ◽  
Two Degree Of Freedom

In an effort to gain a better understanding of the VIV phenomena, we present three-dimensional numerical simulations of VIV of circular cylinders. We consider operating conditions that correspond to high Reynolds number flow, low structural damping, and allow for two-degree of freedom motion. The numerical implementation makes use of overset (Chimera) grids, in a multiple block environment where the workload associated with the blocks is distributed among multiple processors working in parallel. The three-dimensional grids around the cylinder are allowed to undergo arbitrary motions with respect to fixed background grids, eliminating the need for tedious grid regeneration at every time step.

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

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.

scholarly journals Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zihan Wang ◽  
Jieqiong Xu ◽  
Shuai Wu ◽  
Quan Yuan
Keyword(s):  
Control Strategy ◽  
Stability Criterion ◽  
Control Method ◽  
Degree Of Freedom ◽  
Local Analysis ◽  
Grazing Bifurcation ◽  
Vibroimpact System ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

Three-Dimensional Numerical Simulations of Circular Cylinders Undergoing Two Degree-of-Freedom Vortex-Induced Vibrations

2006 ◽  
Vol 129 (3) ◽  
pp. 158-164 ◽  
Author(s):  
Juan P. Pontaza ◽  
Hamn-Ching Chen
Keyword(s):  
Numerical Simulations ◽  
Three Dimensional ◽  
Operating Conditions ◽  
Degree Of Freedom ◽  
Circular Cylinders ◽  
Numerical Implementation ◽  
Vortex Induced Vibrations ◽  
Multiple Processors ◽  
Structural Mass ◽  
Two Degree Of Freedom

In an effort to gain a better understanding of vortex-induced vibrations (VIV), we present three-dimensional numerical simulations of VIV of circular cylinders. We consider operating conditions that correspond to a Reynolds number of 105, low structural mass and damping (m*=1.0, ζ*=0.005), a reduced velocity of U*=6.0, and allow for two degree-of-freedom (X and Y) motion. The numerical implementation makes use of overset (Chimera) grids, in a multiple block environment where the workload associated with the blocks is distributed among multiple processors working in parallel. The three-dimensional grid around the cylinder is allowed to undergo arbitrary motions with respect to fixed background grids, eliminating the need for grid regeneration as the structure moves on the fluid mesh.

The Onset of Chaos in a Two-Degree-of-Freedom Impacting System

1989 ◽  
Vol 56 (1) ◽  
pp. 168-174 ◽  
Author(s):  
Jinsiang Shaw ◽  
Steven W. Shaw
Keyword(s):  
Piecewise Linear ◽  
Global Bifurcation ◽  
Degree Of Freedom ◽  
Chaotic Motions ◽  
Impact Damper ◽  
Onset Of Chaos ◽  
Spring System ◽  
Stability And Bifurcations ◽  
Mass Spring ◽  
Two Degree Of Freedom

The dynamic response of a two-degree-of-freedom impacting system is considered. The system consists of an inverted pendulum with motion limiting stops attached to a sinusoidally excited mass-spring system. Two types of periodic response for this system are analyzed in detail; existence, stability, and bifurcations of these motions can be explicitly computed using a piecewise linear model. The appearance and loss of stability of very long period subharmonics is shown to coincide with a global bifurcation in which chaotic motions, in the form of Smale horseshoes, arise. Application of this device as an impact damper is also briefly discussed.

Vibrations of Dynamical Systems With Quadratic Nonlinearities

1965 ◽  
Vol 32 (3) ◽  
pp. 576-582 ◽  
Author(s):  
P. R. Sethna
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Method ◽  
Degree Of Freedom ◽  
System Parameters ◽  
Quadratic Nonlinearities ◽  
The Stability ◽  
Physical Phenomena ◽  
Two Degree Of Freedom

General two-degree-of-freedom dynamical systems with weak quadratic nonlinearities are studied. With the aid of an asymptotic method of analysis a classification of these systems is made and the more interesting subclasses are studied in detail. The study includes an examination of the stability of the solutions. Depending on the values of the system parameters, several different physical phenomena are shown to occur. Among these is the phenomenon of amplitude-modulated motions with modulation periods that are much larger than the periods of the excitation forces.

Parametric Stability of a Two-Degree-of-Freedom Machine System: Part I—Equations of Motion and Stability

1987 ◽  
Vol 109 (2) ◽  
pp. 210-215 ◽  
Author(s):  
R. I. Zadoks ◽  
A. Midha
Keyword(s):  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Degree Of Freedom ◽  
Computationally Efficient ◽  
Machine System ◽  
Stable Operating ◽  
System Part ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Two Degree Of Freedom

An important question facing a designer is whether a certain machine system will have a stable operating condition. To date, the investigations which deal with this question have been scarce. This study treats an elastic two-degree-of-freedom system with position-dependent inertia and external forcing. In Part I, the nonlinear equations of motion are derived and linearized about the system’s steady-state rigid-body response. The stability of the linearized equations is examined using Floquet theory, and a computationally efficient method for approximating the monodromy matrix is presented. A specific example is proposed and the results are presented in Part II of this paper.

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