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. ...
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
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
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. ...
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
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
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