Volume 1: Active Materials, Mechanics and Behavior; Modeling, Simulation and Control, 2009
This paper describes a new approach to examine the stability of delay differential equations that... more This paper describes a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems-systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.
We extend the temporal spectral element method further to study the periodic orbits of general au... more We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is ...
The last two decades have witnessed several advances in microfabrication technologies and electro... more The last two decades have witnessed several advances in microfabrication technologies and electronics, leading to the development of small, low-power devices for wireless sensing, data transmission, actuation, and medical implants. Unfortunately, the actual implementation of such devices in their respective environment has been hindered by the lack of scalable energy sources that are necessary to power and maintain them. Batteries, which remain the most commonly used power sources, have not kept pace with the demands of these devices, especially in terms of energy density. In light of this challenge, the concept of vibratory energy harvesting has flourished in recent years as a possible alternative to provide a continuous power supply. While linear vibratory energy harvesters have received the majority of the literature's attention, a significant body of the current research activity is focused on the concept of purposeful inclusion of nonlinearities for broadband transduction. ...
This paper investigates the linear response of an archetypal energy harvester that uses electroma... more This paper investigates the linear response of an archetypal energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. In contrast with most prior works, the influence of the circuit inductance is not assumed negligible. Instead, we highlight parameter regimes where the inductance can alter resonance and derive an expression for the resonant frequency. The governing equations consider the case of a vibratory generator directly powering a resistive load. These equations are non-dimensionalized and analytical solutions are obtained for the system's response to single harmonic, periodic, and stochastic environmental excitations. The presented analytical solutions are then used to study the power delivered to an electrical load.
This paper investigates the design and analysis of a novel energy harvesting device that uses mag... more This paper investigates the design and analysis of a novel energy harvesting device that uses magnetic levitation to produce an oscillator with a tunable resonance. The governing equations for the mechanical and electrical domains are derived to show the designed system reduces to the form of a Duffing oscillator under both static and dynamic loads. Thus, nonlinear analyses are required to investigate the energy harvesting potential of this prototypical nonlinear system. Theoretical investigations are followed by a series of experimental tests that validate the response predictions. The motivating hypothesis for the current work was that nonlinear phenomenon could be exploited to improve the effectiveness of energy harvesting devices.
In this paper we present a systematic experimental study of two one-degree-of-freedom nonlinear d... more In this paper we present a systematic experimental study of two one-degree-of-freedom nonlinear devices using the newly introduced control-based continuation method of Sieber and Krauskopf. By considering hardening, softening and bistable spring characteristics, we demonstrate the versatility and power of the control-based continuation method for investigating nonlinear experiments. We show that, using this method, it is possible to track the stable orbits of the devices through a saddle-node bifurcation (fold) ...
Journal of Manufacturing Science and Engineering, 2005
Optimizing the milling process requires a priori knowledge of many process variables. However, th... more Optimizing the milling process requires a priori knowledge of many process variables. However, the ability to include both milling stability and accuracy information is limited because current methods do not provide simultaneous milling stability and accuracy predictions. The method described within this paper, called Temporal Finite Element Analysis (TFEA), provides an approach for simultaneous prediction of milling stability and surface location error. This paper details the application of this approach to a multiple mode system in two orthogonal directions. The TFEA method forms an approximate analytical solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. The formulated dynamic map is then used to determine stability, steady-state surface location error, and to reconstruct the time series for a stable cutting process. Solution convergence is ...
International Journal of Machine Tools and Manufacture, 2006
This paper provides two new analytical solutions for the computation of surface location errors, ... more This paper provides two new analytical solutions for the computation of surface location errors, or part geometry errors that arise due to forced vibrations of the cutting tool, during stable milling. Prior studies of surface location error have applied numerical simulation and semi-analytical methods (e.g., temporal finite element analysis) to this problem. This work provides new results that show analytical
A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable pe... more A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC). The spectral approach does not require solving time-dependent eigenproblems that existing methods require. Instead, the spectral approach determines the stability of the delay differential equations of the system by numerical approximation. The method is capable of analyzing systems whose UPOs arise from bifurcations other than period-doubling. Results are presented for stabilizing UPOs in Duffing systems. The FEs calculated by the spectral approach are compared to published results for two examples. In both cases, the spectral method results agree well with those determined by previous methods. In addition, the spectral method was used to analyze a high-dimensional, asymmetrical system with a UPO in chaos arising from tori doubling following a Hopf bifurcation.
This paper provides the first comparison of the semi-discretization, spectral element, and Legend... more This paper provides the first comparison of the semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example, the delayed damped Mathieu equation, and a helicopter rotor control problem. For each study, we present one or more stability diagrams in the parameter space and one or more convergence plots. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computational time. The semi-discretization method on the other hand has both the slowest convergence rate and requires the most computational time.
This paper investigates the stability of a pivoting cylindrical container that is slowly filled w... more This paper investigates the stability of a pivoting cylindrical container that is slowly filled with fluid. The action of filling the container with fluid causes the system's potential energy to evolve and modify the stability of equilibria. We analyze the stability behavior of this system and find distinct regions where edge and spill conditions require alternative expressions for the system's potential energy. The stability of the upright and tilt angle equilibria are studied using the Lagrange-Dirichlet theorem. We provide exact expressions for the potential energy of the system and bifurcation diagrams that compactly represent the stability behavior of the upright equilibria and additionally predict the presence of non-trivial or tilted equilibria. Theoretical investigations are then compared with a series of experimental tests that validate the container upright and tilted equilibria stability.
International Journal of Machine Tools & Manufacture, 2006
This paper provides two new analytical solutions for the computation of surface location errors, ... more This paper provides two new analytical solutions for the computation of surface location errors, or part geometry errors that arise due to forced vibrations of the cutting tool, during stable milling. Prior studies of surface location error have applied numerical simulation and semi-analytical methods (e.g., temporal finite element analysis) to this problem. This work provides new results that show analytical
The electromagnetic induction of voltage across a coil due to the motion of a magnet is among the... more The electromagnetic induction of voltage across a coil due to the motion of a magnet is among the fundamental problems of physics, and it has a broad range of practical applications. While Maxwell's equations exactly describe this phenomenon, the physical complexity inherent in most realistic situations often prevents the generation of closed-form expressions for the electromagnetic coupling. This paper uses basic principles to develop an approximate analytical expression for the induced voltage in terms of a set of physical parameters, and experimental results demonstrate a high level of validity in the model over the parameter values tested. For oscillatory magnet motion about a point on a coil's axis, it is shown that the induced voltage is an infinite sum of harmonics at integer multiples of the oscillation frequency; the relative amplitudes of these harmonics vary as the magnet's equilibrium position migrates along the coil's axis, causing the odd and even harmonics to vanish, reappear and reach peak values at predictable locations. Several simplifications to the model are considered, and their validity is investigated analytically over a range of parameters.
The stability of interrupted cutting in a single degree of freedom milling process was studied ex... more The stability of interrupted cutting in a single degree of freedom milling process was studied experimentally. An instrumented flexure was used to provide a flexible workpiece with a natural frequency comparable to the tooth pass frequency, mimicking high speed milling dynamics. The displacement of the system was sampled continuously and periodically once per cutter revolution. These data samples were used to asses the stability of the system. Results confirm the theoretical predictions obtained in Part 1.
This paper investigates the underlying physics of a gravity car race. This work seeks to provide ... more This paper investigates the underlying physics of a gravity car race. This work seeks to provide a sound theoretical basis to elucidate the design considerations that maximize performance while simultaneously dispelling false assertions that may arise from incomplete analyses. The governing equations are derived and solved analytically to predict race times; trend analyses are then performed along with a sensitivity analysis to ascertain the most important factors that influence performance. The inferences from a conservative energy balance are then compared with the predictions from the full set of differential equations, which include the dissipative terms associated with air resistance and friction.
Time finite element analysis (TFEA) is used to determine the accuracy, stability, and limit cycle... more Time finite element analysis (TFEA) is used to determine the accuracy, stability, and limit cycle behavior of the milling process. Predictions are compared to traditional Euler simulation and experiments. The TFEA method forms an approximate solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. Stability is then determined from the characteristic multipliers of the map. Map fixed points correspond to stable periodic solutions which are used to evaluate surface location error. Bifurcations and limit cycle behavior are predicted from a non-linear TFEA formulation. Experimental cutting tests are used to confirm theoretical predictions. r
Vibrational energy harvesters are often linear mass-spring-damper-type devices, which have their ... more Vibrational energy harvesters are often linear mass-spring-damper-type devices, which have their resonant frequency tuned to the dominant vibration frequency of their host environment. As such, they can be highly sensitive to uncertainties, which may arise from the imprecise characterization of the host environment or, alternatively, from manufacturing defects and tolerances. It has previously been claimed that the use of nonlinear energy harvesters may be one way to alleviate the problems of these uncertainties. This article presents a systematic uncertainty propagation study of a prototypical electromagnetic energy harvester. More specifically, the response of a linear harvester in the presence of parametric uncertainty is compared to the response of harvesters containing some common forms of nonlinearity, that is, hardening, softening, or bistability. Analytical solutions are used in combination with presumed levels of parametric uncertainty to quantify the resulting uncertainty in the power output. Consequently, these studies can determine the regions in the parameter space where a nonlinear strategy may outperform a more traditional linear approach.
This paper describes an approach for recovering a signal, along with the derivatives of the signa... more This paper describes an approach for recovering a signal, along with the derivatives of the signal, from a noisy time series. To mimic an experimental setting, noise was superimposed onto a deterministic time series. Data smoothing was then used to successfully recover the derivative coordinates; however, the appropriate level of data smoothing must be determined. To investigate the level of smoothing, an information theoretic is applied to show a loss of information occurs for increased levels of noise; conversely, we have shown data smoothing can recover information by removing noise. An approximate criterion is then developed to balance the notion of information recovery through data smoothing with the observation that nearly negligible information changes occur for a sufficiently smoothed time series.
In this paper we describe a new approach to examine the stability of delay differential equations... more In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.
Volume 1: Active Materials, Mechanics and Behavior; Modeling, Simulation and Control, 2009
This paper describes a new approach to examine the stability of delay differential equations that... more This paper describes a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems-systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.
We extend the temporal spectral element method further to study the periodic orbits of general au... more We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is ...
The last two decades have witnessed several advances in microfabrication technologies and electro... more The last two decades have witnessed several advances in microfabrication technologies and electronics, leading to the development of small, low-power devices for wireless sensing, data transmission, actuation, and medical implants. Unfortunately, the actual implementation of such devices in their respective environment has been hindered by the lack of scalable energy sources that are necessary to power and maintain them. Batteries, which remain the most commonly used power sources, have not kept pace with the demands of these devices, especially in terms of energy density. In light of this challenge, the concept of vibratory energy harvesting has flourished in recent years as a possible alternative to provide a continuous power supply. While linear vibratory energy harvesters have received the majority of the literature's attention, a significant body of the current research activity is focused on the concept of purposeful inclusion of nonlinearities for broadband transduction. ...
This paper investigates the linear response of an archetypal energy harvester that uses electroma... more This paper investigates the linear response of an archetypal energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. In contrast with most prior works, the influence of the circuit inductance is not assumed negligible. Instead, we highlight parameter regimes where the inductance can alter resonance and derive an expression for the resonant frequency. The governing equations consider the case of a vibratory generator directly powering a resistive load. These equations are non-dimensionalized and analytical solutions are obtained for the system's response to single harmonic, periodic, and stochastic environmental excitations. The presented analytical solutions are then used to study the power delivered to an electrical load.
This paper investigates the design and analysis of a novel energy harvesting device that uses mag... more This paper investigates the design and analysis of a novel energy harvesting device that uses magnetic levitation to produce an oscillator with a tunable resonance. The governing equations for the mechanical and electrical domains are derived to show the designed system reduces to the form of a Duffing oscillator under both static and dynamic loads. Thus, nonlinear analyses are required to investigate the energy harvesting potential of this prototypical nonlinear system. Theoretical investigations are followed by a series of experimental tests that validate the response predictions. The motivating hypothesis for the current work was that nonlinear phenomenon could be exploited to improve the effectiveness of energy harvesting devices.
In this paper we present a systematic experimental study of two one-degree-of-freedom nonlinear d... more In this paper we present a systematic experimental study of two one-degree-of-freedom nonlinear devices using the newly introduced control-based continuation method of Sieber and Krauskopf. By considering hardening, softening and bistable spring characteristics, we demonstrate the versatility and power of the control-based continuation method for investigating nonlinear experiments. We show that, using this method, it is possible to track the stable orbits of the devices through a saddle-node bifurcation (fold) ...
Journal of Manufacturing Science and Engineering, 2005
Optimizing the milling process requires a priori knowledge of many process variables. However, th... more Optimizing the milling process requires a priori knowledge of many process variables. However, the ability to include both milling stability and accuracy information is limited because current methods do not provide simultaneous milling stability and accuracy predictions. The method described within this paper, called Temporal Finite Element Analysis (TFEA), provides an approach for simultaneous prediction of milling stability and surface location error. This paper details the application of this approach to a multiple mode system in two orthogonal directions. The TFEA method forms an approximate analytical solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. The formulated dynamic map is then used to determine stability, steady-state surface location error, and to reconstruct the time series for a stable cutting process. Solution convergence is ...
International Journal of Machine Tools and Manufacture, 2006
This paper provides two new analytical solutions for the computation of surface location errors, ... more This paper provides two new analytical solutions for the computation of surface location errors, or part geometry errors that arise due to forced vibrations of the cutting tool, during stable milling. Prior studies of surface location error have applied numerical simulation and semi-analytical methods (e.g., temporal finite element analysis) to this problem. This work provides new results that show analytical
A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable pe... more A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC). The spectral approach does not require solving time-dependent eigenproblems that existing methods require. Instead, the spectral approach determines the stability of the delay differential equations of the system by numerical approximation. The method is capable of analyzing systems whose UPOs arise from bifurcations other than period-doubling. Results are presented for stabilizing UPOs in Duffing systems. The FEs calculated by the spectral approach are compared to published results for two examples. In both cases, the spectral method results agree well with those determined by previous methods. In addition, the spectral method was used to analyze a high-dimensional, asymmetrical system with a UPO in chaos arising from tori doubling following a Hopf bifurcation.
This paper provides the first comparison of the semi-discretization, spectral element, and Legend... more This paper provides the first comparison of the semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example, the delayed damped Mathieu equation, and a helicopter rotor control problem. For each study, we present one or more stability diagrams in the parameter space and one or more convergence plots. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computational time. The semi-discretization method on the other hand has both the slowest convergence rate and requires the most computational time.
This paper investigates the stability of a pivoting cylindrical container that is slowly filled w... more This paper investigates the stability of a pivoting cylindrical container that is slowly filled with fluid. The action of filling the container with fluid causes the system's potential energy to evolve and modify the stability of equilibria. We analyze the stability behavior of this system and find distinct regions where edge and spill conditions require alternative expressions for the system's potential energy. The stability of the upright and tilt angle equilibria are studied using the Lagrange-Dirichlet theorem. We provide exact expressions for the potential energy of the system and bifurcation diagrams that compactly represent the stability behavior of the upright equilibria and additionally predict the presence of non-trivial or tilted equilibria. Theoretical investigations are then compared with a series of experimental tests that validate the container upright and tilted equilibria stability.
International Journal of Machine Tools & Manufacture, 2006
This paper provides two new analytical solutions for the computation of surface location errors, ... more This paper provides two new analytical solutions for the computation of surface location errors, or part geometry errors that arise due to forced vibrations of the cutting tool, during stable milling. Prior studies of surface location error have applied numerical simulation and semi-analytical methods (e.g., temporal finite element analysis) to this problem. This work provides new results that show analytical
The electromagnetic induction of voltage across a coil due to the motion of a magnet is among the... more The electromagnetic induction of voltage across a coil due to the motion of a magnet is among the fundamental problems of physics, and it has a broad range of practical applications. While Maxwell's equations exactly describe this phenomenon, the physical complexity inherent in most realistic situations often prevents the generation of closed-form expressions for the electromagnetic coupling. This paper uses basic principles to develop an approximate analytical expression for the induced voltage in terms of a set of physical parameters, and experimental results demonstrate a high level of validity in the model over the parameter values tested. For oscillatory magnet motion about a point on a coil's axis, it is shown that the induced voltage is an infinite sum of harmonics at integer multiples of the oscillation frequency; the relative amplitudes of these harmonics vary as the magnet's equilibrium position migrates along the coil's axis, causing the odd and even harmonics to vanish, reappear and reach peak values at predictable locations. Several simplifications to the model are considered, and their validity is investigated analytically over a range of parameters.
The stability of interrupted cutting in a single degree of freedom milling process was studied ex... more The stability of interrupted cutting in a single degree of freedom milling process was studied experimentally. An instrumented flexure was used to provide a flexible workpiece with a natural frequency comparable to the tooth pass frequency, mimicking high speed milling dynamics. The displacement of the system was sampled continuously and periodically once per cutter revolution. These data samples were used to asses the stability of the system. Results confirm the theoretical predictions obtained in Part 1.
This paper investigates the underlying physics of a gravity car race. This work seeks to provide ... more This paper investigates the underlying physics of a gravity car race. This work seeks to provide a sound theoretical basis to elucidate the design considerations that maximize performance while simultaneously dispelling false assertions that may arise from incomplete analyses. The governing equations are derived and solved analytically to predict race times; trend analyses are then performed along with a sensitivity analysis to ascertain the most important factors that influence performance. The inferences from a conservative energy balance are then compared with the predictions from the full set of differential equations, which include the dissipative terms associated with air resistance and friction.
Time finite element analysis (TFEA) is used to determine the accuracy, stability, and limit cycle... more Time finite element analysis (TFEA) is used to determine the accuracy, stability, and limit cycle behavior of the milling process. Predictions are compared to traditional Euler simulation and experiments. The TFEA method forms an approximate solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. Stability is then determined from the characteristic multipliers of the map. Map fixed points correspond to stable periodic solutions which are used to evaluate surface location error. Bifurcations and limit cycle behavior are predicted from a non-linear TFEA formulation. Experimental cutting tests are used to confirm theoretical predictions. r
Vibrational energy harvesters are often linear mass-spring-damper-type devices, which have their ... more Vibrational energy harvesters are often linear mass-spring-damper-type devices, which have their resonant frequency tuned to the dominant vibration frequency of their host environment. As such, they can be highly sensitive to uncertainties, which may arise from the imprecise characterization of the host environment or, alternatively, from manufacturing defects and tolerances. It has previously been claimed that the use of nonlinear energy harvesters may be one way to alleviate the problems of these uncertainties. This article presents a systematic uncertainty propagation study of a prototypical electromagnetic energy harvester. More specifically, the response of a linear harvester in the presence of parametric uncertainty is compared to the response of harvesters containing some common forms of nonlinearity, that is, hardening, softening, or bistability. Analytical solutions are used in combination with presumed levels of parametric uncertainty to quantify the resulting uncertainty in the power output. Consequently, these studies can determine the regions in the parameter space where a nonlinear strategy may outperform a more traditional linear approach.
This paper describes an approach for recovering a signal, along with the derivatives of the signa... more This paper describes an approach for recovering a signal, along with the derivatives of the signal, from a noisy time series. To mimic an experimental setting, noise was superimposed onto a deterministic time series. Data smoothing was then used to successfully recover the derivative coordinates; however, the appropriate level of data smoothing must be determined. To investigate the level of smoothing, an information theoretic is applied to show a loss of information occurs for increased levels of noise; conversely, we have shown data smoothing can recover information by removing noise. An approximate criterion is then developed to balance the notion of information recovery through data smoothing with the observation that nearly negligible information changes occur for a sufficiently smoothed time series.
In this paper we describe a new approach to examine the stability of delay differential equations... more In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.
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