EXPERIMENTAL CHARACTERIZATION OF QUASIPERIODICITY AND CHAOS IN A MECHANICAL SYSTEM WITH DELAY

1999 ◽  
Vol 09 (01) ◽  
pp. 49-65 ◽  
Author(s):  
M. A. JOHNSON ◽  
F. C. MOON
Keyword(s):  
Machine Tool ◽  
Mechanical System ◽  
Experimental Technique ◽  
Poincaré Sections ◽  
Chaotic Vibrations ◽  
Tool Vibrations ◽  
Delay Differential ◽  
Poincare Sections ◽  
Equation With Delay

We present an electro-mechanical system with finite delay whose construction was motivated by delay differential equations used to describe machine tool vibrations [Johnson, 1996; Moon & Johnson, 1998]. We show that the electro-mechanical system is capable of exhibiting periodic, quasiperiodic and chaotic vibrations. We provide a novel experimental technique for creating real-time Poincaré sections for systems with delay. This experimental technique was also applied to machine tool vibrations [Johnson, 1996]. Experimental Poincaré sections clearly show the existence of tori, and reveal the tori bifurcation sequence which leads to chaotic vibrations. The electro-mechanical system can be modeled by a single second-order differential equation with delay and a cubic nonlinearity. We show that the simple mathematical model fully replicates the bifurcation sequence seen in the electro-mechanical system.

NONLINEAR TECHNIQUES TO CHARACTERIZE PRECHATTER AND CHATTER VIBRATIONS IN THE MACHINING OF METALS

2001 ◽  
Vol 11 (02) ◽  
pp. 449-467 ◽  
Author(s):  
M. A. JOHNSON ◽  
F. C. MOON
Keyword(s):  
Phase Plane ◽  
Nearest Neighbor ◽  
Equation Model ◽  
Poincaré Sections ◽  
Chatter Vibrations ◽  
Experimental Time Series ◽  
Tool Vibrations ◽  
Delay Differential ◽  
Poincare Sections ◽  
False Nearest Neighbors

In this paper we report on the use of Poincaré sections, phase plane portraits and false nearest neighbor techniques to understand bifurcations in the lathe cutting of aluminum. The experiments presented here offer further evidence of deterministic tool vibrations below the classical onset of chatter. While classical chatter exhibits strong periodic content, these prechatter dynamics exhibit both quasiperiodic and chaotic vibrations. The method of false nearest neighbors predicts an embedding dimenion between 4 and 6 for prechatter cutting conditions while showing a random characteristic for no-cutting running of the machine. A one-dimensional delay-differential equation model is presented which exhibits some of the characteristics of the Poincaré sections of the experimental time series.

BIFURCATION STRUCTURE OF VIBRATIONS IN AN AGRICULTURAL TRACTOR-VIBRATING SUBSOILER SYSTEM

1999 ◽  
Vol 09 (10) ◽  
pp. 2091-2098 ◽  
Author(s):  
KENSHI SAKAI ◽  
KAZUYUKI AIHARA
Keyword(s):  
Strange Attractor ◽  
Field Experiments ◽  
Period Doubling ◽  
Bifurcation Structure ◽  
Impact Oscillator ◽  
Poincaré Sections ◽  
Chaotic Vibrations ◽  
Route To Chaos ◽  
Poincare Sections ◽  
Agricultural Tractor

The agricultural tractor-vibrating subsoiler system behaves as an impact oscillator. The bifurcation structure of vibrations of the system are investigated by field experiments by varying the forcing frequency and observing the periodic, period-doubling and chaotic vibrations. The position-triggered Poincaré sections are obtained by generating signals for a sampling timing and the structure of a strange attractor is clearly demonstrated on the Poincaré sections. The period-doubling route to chaos is identified experimentally for the tested tractor-implement system.

Use of Fractal Dimension in the Characterization of Chaotic Structural Dynamic Sytems

1991 ◽  
Vol 44 (11S) ◽  
pp. S107-S113 ◽  
Author(s):  
E. Hall ◽  
S. Kessler ◽  
S. Hanagud
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Specific Class ◽  
Chaotic Motions ◽  
Structural Dynamic ◽  
Poincaré Sections ◽  
Pde Model ◽  
Forcing Function ◽  
Poincare Sections

The purpose of this paper is to investigate the use of fractal dimensions in the characterization of chaotic systems in structural dynamics. The investigation focuses on the example of a simply-supported, Euler-Bernoulli beam which when subjected to a transverse forcing function of a particular amplitude responds chaotically. Three different nonlinear models of the system are studied: a complex partial differential equation (PDE) model, a simplified PDE model, and a Galerkin approximation to the simpler PDE model. The responses of each model are examined through zero velocity Poincare´ sections. To characterize and compare the chaotic trajectories, the box counting fractal dimension of the Poincare´ sections are computed. The results demonstrate that the fractal dimension is a spatial invariant along the length of the beam for the specific class of forcing function studied, and thus it can be used to characterize chaotic motions. In addition, the three models yield different fractal dimensions for the same forcing which indicates that fractal dimensions can also be used to quantify whether a simplification of a chaotic model accurately predicts the chaotic behavior of the full-blown model. Thus the conclusion of the paper is that fractal dimensions may play an important role in the characterization of chaotic structural dynamic systems.

NOISE-SENSITIVITY IN MACHINE TOOL VIBRATIONS

2006 ◽  
Vol 16 (08) ◽  
pp. 2407-2416 ◽  
Author(s):  
E. BUCKWAR ◽  
R. KUSKE ◽  
B. L'ESPERANCE ◽  
T. SOO
Keyword(s):  
Machine Tool ◽  
Scaling Laws ◽  
Additive Noise ◽  
Noise Sensitivity ◽  
Coherence Resonance ◽  
Regenerative Chatter ◽  
Stochastic Delay ◽  
Material Parameters ◽  
Tool Vibrations ◽  
Delay Differential

We consider the effect of random variation in the material parameters in a model for machine tool vibrations, specifically regenerative chatter. We show that fluctuations in these parameters appear as both multiplicative and additive noise in the model. We focus on the effect of additive noise in amplifying small vibrations which appear in subcritical regimes. Coherence resonance is demonstrated through computations, and is proposed as a route for transitions to larger vibrations. The dynamics also exhibit scaling laws observed in the analysis of general stochastic delay differential models.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

scholarly journals Influence of machine tool vibrations on geometric specifications of balls

2021 ◽  
Vol 55 ◽  
pp. 568-575
Author(s):  
Michal Richtarik ◽  
Martin Gavlas ◽  
Mário Drbúl ◽  
Andrej Czán ◽  
Milan Sága
Keyword(s):  
Machine Tool ◽  
Tool Vibrations
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