In this paper, we deal with the control of a transport partial differential equation/ nonlinear o... more In this paper, we deal with the control of a transport partial differential equation/ nonlinear ordinary differential equation (PDE/nonlinear ODE) cascade system in which the transport coefficient depends on the ODE state. We develop a PDE-based predictor-feedback boundary control law, which compensates the transport dynamics of the actuator and guarantees global asymptotic stability of the closed-loop system. The stability proof is based on an infinite-dimensional backstepping transformation and a Lyapunov-like argument. The relation of the PDE-ODE cascade with a state-dependent propagation speed to an ODE system with a state-dependent input delay, which is defined implicitly via an integral of past values of the ODE state, is also highlighted and the corresponding equivalent predictor-feedback design is presented together with an alternative proof of global asymptotic stability of the closedloop system based on the construction of a Lyapunov functional. The practical relevance of our control framework is illustrated in an example that is concerned with the control of a metal rolling process. Index Terms-Boundary control, metal rolling, nonlinear control, PDE-ODE cascade systems, predictor-feedback, state-dependent delay. I. INTRODUCTION T HE problem of stabilization of coupled transport partial differential equation/ordinary differential equation (PDE/ODE) systems in which the transport coefficient or the boundary of the PDE domain varies with time is currently attracting considerable attention. This is attributed to the fact that such systems occur in a large number of challenging
Vibrations are limiting the productivity and the process quality of cutting machine tools. For th... more Vibrations are limiting the productivity and the process quality of cutting machine tools. For the monitoring of these vibrations, often external sensors, such as acceleration sensors, are used. These external systems require additional cost and maintenance effort. This paper presents a virtual sensor, which is capable of detecting vibrations at the tool center point, based on internal machine data. External sensors are only necessary once for model identification. This reduces the overall cost of the system significantly. The virtual sensor uses the high-quality data of the linear position encoder near the ball screw nut and calculates the vibrations at the tool tip by using transmissibility functions. This paper explains the theory behind the used transmissibility functions and describes how they are measured, by comparing different experimental approaches to identify the modal parameters of cutting machine tools. After the identification of the sensor, a dynamical test cycle is u...
Chatter detection from sensor signals has been an active field of research. While some success ha... more Chatter detection from sensor signals has been an active field of research. While some success has been reported using several featurization tools and machine learning algorithms, existing methods have several drawbacks such as manual preprocessing and requiring a large data set. In this paper, we present an alternative approach for chatter detection based on K-Nearest Neighbor (kNN) algorithm for classification and the Dynamic Time Warping (DTW) as a time series similarity measure. The used time series are the acceleration signals acquired from the tool holder in a series of turning experiments. Our results, show that this approach achieves detection accuracies that in most cases outperform existing methods. We compare our results to the traditional methods based on Wavelet Packet Transform (WPT) and the Ensemble Empirical Mode Decomposition (EEMD), as well as to the more recent Topological Data Analysis (TDA) based approach. We show that in three out of four cutting configurations our DTW-based approach attains the highest average classification rate reaching in one case as high as 99% accuracy. Our approach does not require feature extraction, is capable of reusing a classifier across different cutting configurations, and it uses reasonably sized training sets. Although the resulting high accuracy in our approach is associated with high computational cost, this is specific to the DTW implementation that we used. Specifically, we highlight available, very fast DTW implementations that can even be implemented on small consumer electronics. Therefore, further code optimization and the significantly reduced computational effort during the implementation phase make our approach a viable option for in-process chatter detection.
The International Journal of Advanced Manufacturing Technology, 2022
Machining processes are most accurately described using complex dynamical systems that include no... more Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining to the analysis of real cutting signals remains challenging. Some studies have focused on studying the time series of cutting processes using machine learning algorithms with the goal of identifying and predicting undesirable vibrations during machining referred to as chatter. These tools typically decompose the signal using Wavelet Packet Transforms (WPT) or Ensemble Empirical Mode Decomposition (EEMD). However, these methods require a significant overhead in identifying the feature vectors before a classifier can be trained. In this study, we present an alternative approach based on featurizing the time series of the cutting process using its topological features. We first embed the time series as a point cloud using Takens embedding. We then utilize Support Vector Machine, Logistic Regression, Random Forest and Gradient Boosting classifier combined with feature vectors derived from persistence diagrams, a tool from persistent homology, to encode chatter's distinguishing characteristics. We present the results for several choices of the topological feature vectors, and we compare our results to the WPT and EEMD methods using experimental turning data. Our results show that in two out of four cutting configurations the TDA-based features yield accuracies as high as 97%. We also show that combining Bézier curve approximation method and parallel computing can reduce runtime for persistence diagram computation of a single time series to less than a second thus making our approach suitable for online chatter detection.
Recently, it was shown that certain systems with large time-varying delay exhibit different types... more Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional Turbulent Chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional Laminar Chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [J.
CIRP Journal of Manufacturing Science and Technology, 2020
The increasing availability of sensor data at machine tools makes automatic chatter detection alg... more The increasing availability of sensor data at machine tools makes automatic chatter detection algorithms a trending topic in metal cutting. Two prominent and advanced methods for feature extraction via signal decomposition are Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). We apply these two methods to time series acquired from an acceleration sensor at the tool holder of a lathe. Different turning experiments with varying dynamic behavior of the machine tool structure were performed. We compare the performance of these two methods with Support Vector Machine (SVM), Logistic Regression, Random Forest Classification and Gradient Boosting combined with Recursive Feature Elimination (RFE). We also show that the common WPT-based approach of choosing wavelet packets with the highest energy ratios as representative features for chatter does not always result in packets that enclose the chatter frequency, thus reducing the classification accuracy. Further, we test the transfer learning capability of each of these methods by training the classifier on one of the cutting configurations and then testing it on the other cases. It is found that when training and testing on data from the same cutting configuration both methods yield high accuracies reaching in one of the cases as high as 94% and 95%, respectively, for WPT and EEMD. However, our experimental results show that EEMD can outperform WPT in transfer learning applications with accuracy of up to 95%.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020
The expansion of inverter-connected generation facilities (i.e. wind and photovoltaics) and the r... more The expansion of inverter-connected generation facilities (i.e. wind and photovoltaics) and the removal of conventional power plants is necessary to mitigate the impacts of climate change. Whereas conventional generation with large rotating generator masses provides stabilizing inertia, inverter-connected generation does not. Since the underlying power
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019
Time delays play an important role in many fields such as engineering, physics or biology. Delays... more Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue ‘Nonlinear dynamics of delay systems’, which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue ‘Nonlinear dynamics of delay...
Synchronization in networks of identical oscillators with heterogeneous coupling delays is studie... more Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delaycoupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.
We show that the output of systems with time-varying delay can exhibit a new kind of chaotic beha... more We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems, the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.
International Journal of Dynamics and Control, 2017
In this paper we investigate the effect of the toolworkpiece mode interaction on the stability of... more In this paper we investigate the effect of the toolworkpiece mode interaction on the stability of a flexible-tool, flexible-workpiece turning process. The workpiece dynamics are modeled by classical beam theory while for the tool we consider a description in terms of eigenmodes with an arbitrary orientation. The focus here is on elucidating the coupling effects due to the flexibility of both structures. Specifically, we show how the tool location along the workpiece and the dynamics of both the workpiece and the tool affect stability. Another contribution of this paper is the utilization of a novel analytical force model for cutting with round inserts. Using this force model, we further show that when round inserts are present, commonly used frequency domain methods can no longer be utilized to capture the system stability. Consequently, we use the spectral element approach, a highly-efficient time-domain approach, for studying the system stability. Keywords Machining • Stability • Time-domain • Turning 1 Introduction Regenerative chatter is one of the important challenges in metal removal processes [1]. Its detrimental effects on surface quality, tool wear, and machine/workpiece integrity has led B Firas A. Khasawneh
In the present paper we investigate the influence of the retarded access by a time-varying delay ... more In the present paper we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In the present paper we extend our previous work [Otto, Müller, and Radons, Phys. Rev. Lett. 118, 044104 (2017)], by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.
We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the... more We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.
CIRP Journal of Manufacturing Science and Technology, 2013
In the present paper the chatter instability of variable speed machining is studied. Though, ther... more In the present paper the chatter instability of variable speed machining is studied. Though, there exist numerical methods for the computation of the stability lobes for variable speed machining, especially in turning processes the potential of an active spindle speed variation for chatter suppression is mostly unexploited. In the case of a slowly time-varying spindle speed, which is practicable on a real machine tool, the stability behavior with a time-varying spindle speed is connected to the stability behavior with constant spindle speeds. This so-called frozen time approximation helps to understand the stabilizing mechanism of turning with spindle speed variation. Strategies for tuning the parameters of the speed variation for an optimal stabilization are developed. The results presented here are useful for a practical implementation of variable speed machining to increase the productivity without any negative effect due to the variation of the spindle speed.
The International Journal of Advanced Manufacturing Technology, 2015
A universal frequency domain approach for studying the dynamics of metal cutting with a flexible ... more A universal frequency domain approach for studying the dynamics of metal cutting with a flexible workpiece and a compliant tool is derived. The method is used for the identification of the position-dependent stability lobes in turning. It enables a fast, accurate, and systematic stability analysis of turning processes in the parameter space, which is not restricted to simple dynamic models with only a specific number of modes in one spatial direction. In particular, a combination of experimental data for the tool tip dynamics with analytical or numerical data for the workpiece dynamics is possible, which is demonstrated by a concrete example. The effect of the mode interaction between tool and workpiece modes via the cutting process is illustrated. Counterintuitively, the possibility of a process destabilization for an increased workpiece stiffness was observed, which can be explained by the mode interaction. The presented methods and results can be efficiently used for optimizing machine tool development and process planning.
International Journal of Machine Tools and Manufacture, 2014
Chatter vibrations in cutting processes are studied in the present paper. A unified approach for ... more Chatter vibrations in cutting processes are studied in the present paper. A unified approach for the calculation of the stability lobes for turning, boring, drilling and milling processes in the frequency domain is presented. The method can be used for a fast and reliable identification of the stability lobes and can take into account nonlinear shearing forces, as well as process damping forces. The applicability of Tlusty's law, which is a simple scalar relationship between the real part of the oriented transfer function of the structure and the limiting chip width, is extended to milling and any other multi-dimensional chatter problem without neglecting the coupled dynamics. The given analysis is suitable for getting a deep understanding of the chatter stability dependent on the parameters of the cutting process and the structure. Basic examples based on experimental data of real machine tools include the dependence of the stability behavior on the rotational direction in turning, the effect of axial-torsional structural coupling in drilling, and the dynamics of slot milling.
In this paper, we deal with the control of a transport partial differential equation/ nonlinear o... more In this paper, we deal with the control of a transport partial differential equation/ nonlinear ordinary differential equation (PDE/nonlinear ODE) cascade system in which the transport coefficient depends on the ODE state. We develop a PDE-based predictor-feedback boundary control law, which compensates the transport dynamics of the actuator and guarantees global asymptotic stability of the closed-loop system. The stability proof is based on an infinite-dimensional backstepping transformation and a Lyapunov-like argument. The relation of the PDE-ODE cascade with a state-dependent propagation speed to an ODE system with a state-dependent input delay, which is defined implicitly via an integral of past values of the ODE state, is also highlighted and the corresponding equivalent predictor-feedback design is presented together with an alternative proof of global asymptotic stability of the closedloop system based on the construction of a Lyapunov functional. The practical relevance of our control framework is illustrated in an example that is concerned with the control of a metal rolling process. Index Terms-Boundary control, metal rolling, nonlinear control, PDE-ODE cascade systems, predictor-feedback, state-dependent delay. I. INTRODUCTION T HE problem of stabilization of coupled transport partial differential equation/ordinary differential equation (PDE/ODE) systems in which the transport coefficient or the boundary of the PDE domain varies with time is currently attracting considerable attention. This is attributed to the fact that such systems occur in a large number of challenging
Vibrations are limiting the productivity and the process quality of cutting machine tools. For th... more Vibrations are limiting the productivity and the process quality of cutting machine tools. For the monitoring of these vibrations, often external sensors, such as acceleration sensors, are used. These external systems require additional cost and maintenance effort. This paper presents a virtual sensor, which is capable of detecting vibrations at the tool center point, based on internal machine data. External sensors are only necessary once for model identification. This reduces the overall cost of the system significantly. The virtual sensor uses the high-quality data of the linear position encoder near the ball screw nut and calculates the vibrations at the tool tip by using transmissibility functions. This paper explains the theory behind the used transmissibility functions and describes how they are measured, by comparing different experimental approaches to identify the modal parameters of cutting machine tools. After the identification of the sensor, a dynamical test cycle is u...
Chatter detection from sensor signals has been an active field of research. While some success ha... more Chatter detection from sensor signals has been an active field of research. While some success has been reported using several featurization tools and machine learning algorithms, existing methods have several drawbacks such as manual preprocessing and requiring a large data set. In this paper, we present an alternative approach for chatter detection based on K-Nearest Neighbor (kNN) algorithm for classification and the Dynamic Time Warping (DTW) as a time series similarity measure. The used time series are the acceleration signals acquired from the tool holder in a series of turning experiments. Our results, show that this approach achieves detection accuracies that in most cases outperform existing methods. We compare our results to the traditional methods based on Wavelet Packet Transform (WPT) and the Ensemble Empirical Mode Decomposition (EEMD), as well as to the more recent Topological Data Analysis (TDA) based approach. We show that in three out of four cutting configurations our DTW-based approach attains the highest average classification rate reaching in one case as high as 99% accuracy. Our approach does not require feature extraction, is capable of reusing a classifier across different cutting configurations, and it uses reasonably sized training sets. Although the resulting high accuracy in our approach is associated with high computational cost, this is specific to the DTW implementation that we used. Specifically, we highlight available, very fast DTW implementations that can even be implemented on small consumer electronics. Therefore, further code optimization and the significantly reduced computational effort during the implementation phase make our approach a viable option for in-process chatter detection.
The International Journal of Advanced Manufacturing Technology, 2022
Machining processes are most accurately described using complex dynamical systems that include no... more Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining to the analysis of real cutting signals remains challenging. Some studies have focused on studying the time series of cutting processes using machine learning algorithms with the goal of identifying and predicting undesirable vibrations during machining referred to as chatter. These tools typically decompose the signal using Wavelet Packet Transforms (WPT) or Ensemble Empirical Mode Decomposition (EEMD). However, these methods require a significant overhead in identifying the feature vectors before a classifier can be trained. In this study, we present an alternative approach based on featurizing the time series of the cutting process using its topological features. We first embed the time series as a point cloud using Takens embedding. We then utilize Support Vector Machine, Logistic Regression, Random Forest and Gradient Boosting classifier combined with feature vectors derived from persistence diagrams, a tool from persistent homology, to encode chatter's distinguishing characteristics. We present the results for several choices of the topological feature vectors, and we compare our results to the WPT and EEMD methods using experimental turning data. Our results show that in two out of four cutting configurations the TDA-based features yield accuracies as high as 97%. We also show that combining Bézier curve approximation method and parallel computing can reduce runtime for persistence diagram computation of a single time series to less than a second thus making our approach suitable for online chatter detection.
Recently, it was shown that certain systems with large time-varying delay exhibit different types... more Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional Turbulent Chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional Laminar Chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [J.
CIRP Journal of Manufacturing Science and Technology, 2020
The increasing availability of sensor data at machine tools makes automatic chatter detection alg... more The increasing availability of sensor data at machine tools makes automatic chatter detection algorithms a trending topic in metal cutting. Two prominent and advanced methods for feature extraction via signal decomposition are Wavelet Packet Transform (WPT) and Ensemble Empirical Mode Decomposition (EEMD). We apply these two methods to time series acquired from an acceleration sensor at the tool holder of a lathe. Different turning experiments with varying dynamic behavior of the machine tool structure were performed. We compare the performance of these two methods with Support Vector Machine (SVM), Logistic Regression, Random Forest Classification and Gradient Boosting combined with Recursive Feature Elimination (RFE). We also show that the common WPT-based approach of choosing wavelet packets with the highest energy ratios as representative features for chatter does not always result in packets that enclose the chatter frequency, thus reducing the classification accuracy. Further, we test the transfer learning capability of each of these methods by training the classifier on one of the cutting configurations and then testing it on the other cases. It is found that when training and testing on data from the same cutting configuration both methods yield high accuracies reaching in one of the cases as high as 94% and 95%, respectively, for WPT and EEMD. However, our experimental results show that EEMD can outperform WPT in transfer learning applications with accuracy of up to 95%.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020
The expansion of inverter-connected generation facilities (i.e. wind and photovoltaics) and the r... more The expansion of inverter-connected generation facilities (i.e. wind and photovoltaics) and the removal of conventional power plants is necessary to mitigate the impacts of climate change. Whereas conventional generation with large rotating generator masses provides stabilizing inertia, inverter-connected generation does not. Since the underlying power
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019
Time delays play an important role in many fields such as engineering, physics or biology. Delays... more Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue ‘Nonlinear dynamics of delay systems’, which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue ‘Nonlinear dynamics of delay...
Synchronization in networks of identical oscillators with heterogeneous coupling delays is studie... more Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delaycoupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.
We show that the output of systems with time-varying delay can exhibit a new kind of chaotic beha... more We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems, the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.
International Journal of Dynamics and Control, 2017
In this paper we investigate the effect of the toolworkpiece mode interaction on the stability of... more In this paper we investigate the effect of the toolworkpiece mode interaction on the stability of a flexible-tool, flexible-workpiece turning process. The workpiece dynamics are modeled by classical beam theory while for the tool we consider a description in terms of eigenmodes with an arbitrary orientation. The focus here is on elucidating the coupling effects due to the flexibility of both structures. Specifically, we show how the tool location along the workpiece and the dynamics of both the workpiece and the tool affect stability. Another contribution of this paper is the utilization of a novel analytical force model for cutting with round inserts. Using this force model, we further show that when round inserts are present, commonly used frequency domain methods can no longer be utilized to capture the system stability. Consequently, we use the spectral element approach, a highly-efficient time-domain approach, for studying the system stability. Keywords Machining • Stability • Time-domain • Turning 1 Introduction Regenerative chatter is one of the important challenges in metal removal processes [1]. Its detrimental effects on surface quality, tool wear, and machine/workpiece integrity has led B Firas A. Khasawneh
In the present paper we investigate the influence of the retarded access by a time-varying delay ... more In the present paper we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In the present paper we extend our previous work [Otto, Müller, and Radons, Phys. Rev. Lett. 118, 044104 (2017)], by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.
We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the... more We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.
CIRP Journal of Manufacturing Science and Technology, 2013
In the present paper the chatter instability of variable speed machining is studied. Though, ther... more In the present paper the chatter instability of variable speed machining is studied. Though, there exist numerical methods for the computation of the stability lobes for variable speed machining, especially in turning processes the potential of an active spindle speed variation for chatter suppression is mostly unexploited. In the case of a slowly time-varying spindle speed, which is practicable on a real machine tool, the stability behavior with a time-varying spindle speed is connected to the stability behavior with constant spindle speeds. This so-called frozen time approximation helps to understand the stabilizing mechanism of turning with spindle speed variation. Strategies for tuning the parameters of the speed variation for an optimal stabilization are developed. The results presented here are useful for a practical implementation of variable speed machining to increase the productivity without any negative effect due to the variation of the spindle speed.
The International Journal of Advanced Manufacturing Technology, 2015
A universal frequency domain approach for studying the dynamics of metal cutting with a flexible ... more A universal frequency domain approach for studying the dynamics of metal cutting with a flexible workpiece and a compliant tool is derived. The method is used for the identification of the position-dependent stability lobes in turning. It enables a fast, accurate, and systematic stability analysis of turning processes in the parameter space, which is not restricted to simple dynamic models with only a specific number of modes in one spatial direction. In particular, a combination of experimental data for the tool tip dynamics with analytical or numerical data for the workpiece dynamics is possible, which is demonstrated by a concrete example. The effect of the mode interaction between tool and workpiece modes via the cutting process is illustrated. Counterintuitively, the possibility of a process destabilization for an increased workpiece stiffness was observed, which can be explained by the mode interaction. The presented methods and results can be efficiently used for optimizing machine tool development and process planning.
International Journal of Machine Tools and Manufacture, 2014
Chatter vibrations in cutting processes are studied in the present paper. A unified approach for ... more Chatter vibrations in cutting processes are studied in the present paper. A unified approach for the calculation of the stability lobes for turning, boring, drilling and milling processes in the frequency domain is presented. The method can be used for a fast and reliable identification of the stability lobes and can take into account nonlinear shearing forces, as well as process damping forces. The applicability of Tlusty's law, which is a simple scalar relationship between the real part of the oriented transfer function of the structure and the limiting chip width, is extended to milling and any other multi-dimensional chatter problem without neglecting the coupled dynamics. The given analysis is suitable for getting a deep understanding of the chatter stability dependent on the parameters of the cutting process and the structure. Basic examples based on experimental data of real machine tools include the dependence of the stability behavior on the rotational direction in turning, the effect of axial-torsional structural coupling in drilling, and the dynamics of slot milling.
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Papers by Andreas Otto