Journal of Computational and Nonlinear Dynamics, 2006
In this paper we present stability analysis of a nonlinear model for chatter vibration in a drill... more In this paper we present stability analysis of a nonlinear model for chatter vibration in a drilling operation. The results build our previous work [1,, where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.
Journal of Computational and Nonlinear Dynamics, 2006
In this paper we present stability analysis of a nonlinear model for chatter vibration in a drill... more In this paper we present stability analysis of a nonlinear model for chatter vibration in a drilling operation. The results build our previous work [1,, where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.
... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal C... more ... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal Cutting Emily Stone Department of Mathematical Sciences, The University of Montana-Missoula Missoula, MT 59812-0864, USA ... Mechanical engineers Page 2. 60 Emily Stone et. al. ...
We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, ... more We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, Dyn. Syst. 17 (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the ...
A rather general 'explanation' is outlined of intermittent turbulence production in flo... more A rather general 'explanation' is outlined of intermittent turbulence production in flow dominated, at the larger scales, by coherent structures. Symmetries in a 'perfect' underlying flow lead to bundles of recurrent solutions which circulate near a heteroclinic attractor in the ...
Video data from experiments on the dynamics of two-dimensional flames are analyzed. The Karhunen-... more Video data from experiments on the dynamics of two-dimensional flames are analyzed. The Karhunen-Lo6ve (KL) analysis is used to identify the dominant spatial structures and their temporal evolution for several dynamical regimes of the flames. A data analysis procedure to extract and process the boundaries of flame cells is described. It is shown how certain spatial structures are associated with certain temporal events. The existence of small scale, high frequency, turbulent background motion in almost all regimes is revealed.
The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensio... more The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.
The application of archetypal analysis to high-dimensional data arising from video-taped images i... more The application of archetypal analysis to high-dimensional data arising from video-taped images is presented. Included in the analysis are intermittent regimes which have not been analyzed previously by other statistical methods such as principal component analysis (PCA). A hybrid PCA/archetypes technique has been developed to overcome the difficulties of applying archetypes to data sets with points living in a space of dimension higher than about 500. The advantages of the method lie in the creation of patterns typical of the set as a whole, and an expression of the dynamics in terms of these patterns. Archetypes are particularly useful in identifying intermittent regimes, where low energy events that might be missed by a severe principal component truncation are none-the-less crucial to understanding the dynamics. They are part of a suite of data analysis techniques that can be used on dynamic data sets (such as FFT, PCA and other spectral decompositions). This hybrid method extends the application of archetypes to spatio-temporal dynamics in two-dimensional patterns.
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particul... more The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random ...
A comparison is made between the principal component or Karhunen-Loève decomposition of two sets ... more A comparison is made between the principal component or Karhunen-Loève decomposition of two sets of spatio-temporal data (one numerical, the other experimental) and a new procedure called archetypal analysis (Cutler and Breiman, 1994). Archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values. Archetypes may be more appropriate than KL when the data do not have elliptical distributions, and are often well-suited to studying regimes in which the system spends a long time near a “steady” state, punctuated with quick excursions to outliers in the data set, which may represent intermittency. Other advantages and disadvantages of each method are discussed.
The presence e[ attracting heteroclinic cycles in a set of truncated ordinary differential equati... more The presence e[ attracting heteroclinic cycles in a set of truncated ordinary differential equations for the modal amplil ades in a proper orthogonal decomposition of the Navier-Stokes equations leads to "burst"-like, intermittent phenomenon iu the reconstructed velocity field. The time between bursts, or the cycle time, is effectively randomized by the addition of small random perturbations, in this case taking the form of irregular pressure fluctuations from the outer flow. This suggests that interaction of the inner and outer flow is important in the generation and triggering of these burst events. A scaring law is developed for the time between burst events as well as an expression for the probability distribution of these time intervals.
We show that solutions of ordinary differential equations possessing attracting heteroclinic cycl... more We show that solutions of ordinary differential equations possessing attracting heteroclinic cycles and subject to small random or periodic perturbations typically exhibit sharp events whose separations are well characterizedby a probability distribution with an exponential tail. We review experimental evidence forsuch distributions in turbulent flows and indicate how our simple theory might be used to identify instability mechanisms. * Research partially supported by AFOSR 0226A (Wall i.ay-nections to infinity" of the Zakharov equations which ers) and the UK Science and Engineering Research Council. model Langmuir plasma turbulence.
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
Using finite element software developed for metal cutting by Third Wave Systems we investigate th... more Using finite element software developed for metal cutting by Third Wave Systems we investigate the forces involved in chatter, a self-sustained oscillation of the cutting tool. The phenomena is decomposed into a vibrating tool cutting a flat surface work piece, and motionless tool cutting a work piece with a wavy surface. While cutting the wavy surface, the shearplane was seen to oscillate in advance of the oscillation of the depth of cut, as were the cutting, thrust, and shear plane forces. The vibrating tool was used to investigate process damping through the interaction of the relief face of the tool and the workpiece. Crushing forces are isolated and compared to the contact length between the tool and workpiece. We found that the wavelength dependence of the forces depended on the relative size of the wavelength to the length of the relief face of the tool. The results indicate that the damping force from crushing will be proportional to the cutting speed for short tools, and inversely proportional for long tools.
The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a ... more The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a particular stoma is open, CO 2 is absorbed through its aperture, but H 2 O is also lost due to evaporation. Thus a plant will seek a stomatal aperture that balances its need for CO 2 ...
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
16451 that a population of globally coupled, identical, integrate-and-fire oscillators will almos... more 16451 that a population of globally coupled, identical, integrate-and-fire oscillators will almost always become entrained. We find that the inclusion of a refractory period in the cycle of each oscillator can result in open sets of initial configurations evolving to asynchronous states. 0 1997 Published by Elsevier Science B.V.
In this paper the frequencyentrainment of the Rössler system to a periodic pulse is discussed. It... more In this paper the frequencyentrainment of the Rössler system to a periodic pulse is discussed. It was found that a periodic orbit will entrain to weak periodic pulses with frequency near the natural frequency of oscillation, but rather more surprisingly, the phase coherent strange attractor can also be entrained, on average, to small pulses near the peak frequency in the broad band power spectra of the attractor. 0375-9601/92/s 05.00
We show that solutions of ordinary differential equations possessing attracting heteroclinic cycl... more We show that solutions of ordinary differential equations possessing attracting heteroclinic cycles and subject to small random or periodic perturbations typically exhibit sharp events whose separations are well characterizedby a probability distribution with an exponential tail. We review experimental evidence forsuch distributions in turbulent flows and indicate how our simple theory might be used to identify instability mechanisms. * Research partially supported by AFOSR 0226A (Wall i.ay-nections to infinity" of the Zakharov equations which ers) and the UK Science and Engineering Research Council. model Langmuir plasma turbulence.
Journal of Computational and Nonlinear Dynamics, 2006
In this paper we present stability analysis of a nonlinear model for chatter vibration in a drill... more In this paper we present stability analysis of a nonlinear model for chatter vibration in a drilling operation. The results build our previous work [1,, where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.
Journal of Computational and Nonlinear Dynamics, 2006
In this paper we present stability analysis of a nonlinear model for chatter vibration in a drill... more In this paper we present stability analysis of a nonlinear model for chatter vibration in a drilling operation. The results build our previous work [1,, where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.
... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal C... more ... vol. 2, no. 1-2, 2006, pp. 59-79 ISSN 15740404 Investigations of Nonlinear Forces in Metal Cutting Emily Stone Department of Mathematical Sciences, The University of Montana-Missoula Missoula, MT 59812-0864, USA ... Mechanical engineers Page 2. 60 Emily Stone et. al. ...
We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, ... more We consider here a model from Stone and Askari [Nonlinear models of chatter in drilling process, Dyn. Syst. 17 (2002), pp. 65-85] for regenerative chatter in a drilling process. The model is a nonlinear delay differential equation where the delay arises from the fact that the ...
A rather general 'explanation' is outlined of intermittent turbulence production in flo... more A rather general 'explanation' is outlined of intermittent turbulence production in flow dominated, at the larger scales, by coherent structures. Symmetries in a 'perfect' underlying flow lead to bundles of recurrent solutions which circulate near a heteroclinic attractor in the ...
Video data from experiments on the dynamics of two-dimensional flames are analyzed. The Karhunen-... more Video data from experiments on the dynamics of two-dimensional flames are analyzed. The Karhunen-Lo6ve (KL) analysis is used to identify the dominant spatial structures and their temporal evolution for several dynamical regimes of the flames. A data analysis procedure to extract and process the boundaries of flame cells is described. It is shown how certain spatial structures are associated with certain temporal events. The existence of small scale, high frequency, turbulent background motion in almost all regimes is revealed.
The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensio... more The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.
The application of archetypal analysis to high-dimensional data arising from video-taped images i... more The application of archetypal analysis to high-dimensional data arising from video-taped images is presented. Included in the analysis are intermittent regimes which have not been analyzed previously by other statistical methods such as principal component analysis (PCA). A hybrid PCA/archetypes technique has been developed to overcome the difficulties of applying archetypes to data sets with points living in a space of dimension higher than about 500. The advantages of the method lie in the creation of patterns typical of the set as a whole, and an expression of the dynamics in terms of these patterns. Archetypes are particularly useful in identifying intermittent regimes, where low energy events that might be missed by a severe principal component truncation are none-the-less crucial to understanding the dynamics. They are part of a suite of data analysis techniques that can be used on dynamic data sets (such as FFT, PCA and other spectral decompositions). This hybrid method extends the application of archetypes to spatio-temporal dynamics in two-dimensional patterns.
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particul... more The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching are found, depending on the details of the underlying deterministic dynamics: random ...
A comparison is made between the principal component or Karhunen-Loève decomposition of two sets ... more A comparison is made between the principal component or Karhunen-Loève decomposition of two sets of spatio-temporal data (one numerical, the other experimental) and a new procedure called archetypal analysis (Cutler and Breiman, 1994). Archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values. Archetypes may be more appropriate than KL when the data do not have elliptical distributions, and are often well-suited to studying regimes in which the system spends a long time near a “steady” state, punctuated with quick excursions to outliers in the data set, which may represent intermittency. Other advantages and disadvantages of each method are discussed.
The presence e[ attracting heteroclinic cycles in a set of truncated ordinary differential equati... more The presence e[ attracting heteroclinic cycles in a set of truncated ordinary differential equations for the modal amplil ades in a proper orthogonal decomposition of the Navier-Stokes equations leads to "burst"-like, intermittent phenomenon iu the reconstructed velocity field. The time between bursts, or the cycle time, is effectively randomized by the addition of small random perturbations, in this case taking the form of irregular pressure fluctuations from the outer flow. This suggests that interaction of the inner and outer flow is important in the generation and triggering of these burst events. A scaring law is developed for the time between burst events as well as an expression for the probability distribution of these time intervals.
We show that solutions of ordinary differential equations possessing attracting heteroclinic cycl... more We show that solutions of ordinary differential equations possessing attracting heteroclinic cycles and subject to small random or periodic perturbations typically exhibit sharp events whose separations are well characterizedby a probability distribution with an exponential tail. We review experimental evidence forsuch distributions in turbulent flows and indicate how our simple theory might be used to identify instability mechanisms. * Research partially supported by AFOSR 0226A (Wall i.ay-nections to infinity" of the Zakharov equations which ers) and the UK Science and Engineering Research Council. model Langmuir plasma turbulence.
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
Using finite element software developed for metal cutting by Third Wave Systems we investigate th... more Using finite element software developed for metal cutting by Third Wave Systems we investigate the forces involved in chatter, a self-sustained oscillation of the cutting tool. The phenomena is decomposed into a vibrating tool cutting a flat surface work piece, and motionless tool cutting a work piece with a wavy surface. While cutting the wavy surface, the shearplane was seen to oscillate in advance of the oscillation of the depth of cut, as were the cutting, thrust, and shear plane forces. The vibrating tool was used to investigate process damping through the interaction of the relief face of the tool and the workpiece. Crushing forces are isolated and compared to the contact length between the tool and workpiece. We found that the wavelength dependence of the forces depended on the relative size of the wavelength to the length of the relief face of the tool. The results indicate that the damping force from crushing will be proportional to the cutting speed for short tools, and inversely proportional for long tools.
The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a ... more The process of photosynthesis is facilitated by pores on the leaf surface called stomata. When a particular stoma is open, CO 2 is absorbed through its aperture, but H 2 O is also lost due to evaporation. Thus a plant will seek a stomatal aperture that balances its need for CO 2 ...
1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections,... more 1. Introduction. It is well known that homoclinic and heteroclinic orbits, or saddle connections, are important in determining the global behavior of dynamical systems. Under appropriate conditions, transverse homoclinic orbits lead to chaotic motions via the presence of ...
16451 that a population of globally coupled, identical, integrate-and-fire oscillators will almos... more 16451 that a population of globally coupled, identical, integrate-and-fire oscillators will almost always become entrained. We find that the inclusion of a refractory period in the cycle of each oscillator can result in open sets of initial configurations evolving to asynchronous states. 0 1997 Published by Elsevier Science B.V.
In this paper the frequencyentrainment of the Rössler system to a periodic pulse is discussed. It... more In this paper the frequencyentrainment of the Rössler system to a periodic pulse is discussed. It was found that a periodic orbit will entrain to weak periodic pulses with frequency near the natural frequency of oscillation, but rather more surprisingly, the phase coherent strange attractor can also be entrained, on average, to small pulses near the peak frequency in the broad band power spectra of the attractor. 0375-9601/92/s 05.00
We show that solutions of ordinary differential equations possessing attracting heteroclinic cycl... more We show that solutions of ordinary differential equations possessing attracting heteroclinic cycles and subject to small random or periodic perturbations typically exhibit sharp events whose separations are well characterizedby a probability distribution with an exponential tail. We review experimental evidence forsuch distributions in turbulent flows and indicate how our simple theory might be used to identify instability mechanisms. * Research partially supported by AFOSR 0226A (Wall i.ay-nections to infinity" of the Zakharov equations which ers) and the UK Science and Engineering Research Council. model Langmuir plasma turbulence.
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