Dynamical Systems with Applications Using MAPLE
About: This article is published in Shock and Vibration.The article was published on 2001-01-01 and is currently open access. It has received 36 citations till now. The article focuses on the topics: Maple & Dynamical systems theory.
Citations
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Dissertation•
01 Sep 2009
TL;DR: In this article, the authors considered the limitations of resonant linear devices and investigated two non-linear generators to see if they can outperform the linear devices in certain situations, and found that the performance of a nonlinear device in terms of the power harvested is at most 4π greater than that of a tuned linear system and is strongly dependent upon the type of the nonlinearity used.
Abstract: Vibration-based energy harvesting devices have received much attention over the past few years due to
the need to power wireless devices in remote or hostile environments. To date, resonant linear generators
have been the most common type of generators used in harvesting energy for such devices. Simple tuning
and modelling methods make it a more favourable solution theoretically if not practically. This thesis
considers the limitations of resonant linear devices and investigates two non-linear generators to see if
they can outperform the linear devices in certain situations. So far, in most of the literature, the energy
harvester is assumed to be very small dynamically compared to the source so the source is not affected
by the presence of the device. This thesis considers how the dynamics of the source is affected by the
device if its impedance is significant compared to the source. A tuning condition for maximum power
transfer from the source to the device is derived. This tuning condition converges to the one presented
in most of the literature when the impedance of the device is assumed to be very small compared to that
of the source i.e. tuned so that the natural frequency of the device equals the excitation frequency. For
the case when the impedance of the device has a negligible effect on the source, the performance of the
device is only limited to a narrow frequency band and drops off rapidly if mistuned. To accommodate the
mistuning limitations, new types of generators are proposed mainly by using a non-linear mechanism.
These mechanisms are made up of a non-linear spring connected together with a mass and a linear
viscous damper i.e. the energy harvesting component. The analysis of the fundamental performance
limit of any non-linear device compared to that of a tuned linear device is carried out using the principal
of conservation of energy. The analysis reveals that the performance of a non-linear device in terms of the
power harvested is at most 4/π greater than that of a tuned linear system and is strongly dependent upon
the type of the non-linearity used. Two types of non-linear mechanisms are studied in this thesis. The
first one is a non-linear bi-stable mechanism termed a snap-through mechanism which rapidly moves the
mass between two stable states. The aim is to steepen the displacement response curve as a function of
time which results in the increase of velocity for a given excitation, thus increasing the amount of power
harvested. This study reveals that the performance of the mechanism is better than a linear system
when the natural frequency of the system is much higher than the excitation frequency. The study also
shows that the power harvested by this mechanism rolls off at a slower rate compared to that of the
linear system. Another non-linear mechanism described in this thesis uses a hardening-type spring. The
aim of this mechanism is to provide a wider bandwidth over which the power can be harvested. This
thesis presents numerical solutions and approximate analytical solutions for the bandwidth and effective
viscous damping of a non-linear device employing a hardening-type stiffness. Unlike the linear system ,
in which the bandwidth is only dependent on the damping ratio, it is found that the bandwidth of the
nonlinear device depends on both the strength of the nonlinearity and the damping ratio. Experimental
results are presented to validate the theoretical results. This thesis also investigates the benefits of the
non-linear device for a low frequency and high amplitude application using the measured vibration inputs
from human motion such as walking and running. The effect of harmonics on the power harvested is also
studied. Numerical simulations are carried out using measured input vibrations from human motion to
study the best placement of the natural frequency of the device across the range of harmonics.
17 citations
Cites background from "Dynamical Systems with Applications..."
...Chapter 3 Non-linear snap-through mechanism 80 the point in the Poincaré map from the origin [105]...
[...]
28 Sep 2010
TL;DR: To evaluate this encryption method’s performance, a statistical analysis was applied, and histograms of the original and encrypted images were developed, and the behavior of the algorithm to the most common cryptographic attacks were analyzed.
Abstract: In this work, we propose a color image encryption of variable sizes using logistic chaotic map. We considered a process of permutation, followed by a diffusion algorithm. To evaluate this encryption method’s performance, we applied a statistical analysis, and developed histograms of the original and encrypted images. For the correlation analysis, we evaluate the behavior of two adjacent pixels on horizontal, vertical and diagonal directions. We also analyzed the behavior of the algorithm to the most common cryptographic attacks. Numerical simulations and graphic results are presented for image encryption and decryption using MatLab software.
14 citations
Cites background or methods from "Dynamical Systems with Applications..."
...system used to model a single species is that known as the logistic map [16] given by model equation (1): xn+1= fμ (xn) = μxn (1- xn), (1) 0 ≤ xn ≤ 1 (2) initial value x (0) = 0....
[...]
...Theorem 2 [16] ,“If at least one of the average Lyapunov exponent is positive, then the system is chaotic; if the average Lyapunov exponent is negative, then the orbit is periodic; and when the average Lyapunov exponent is zero, a bifurcation occurs”....
[...]
01 Jan 2007
TL;DR: The researcher characterized the pragmatic reasoning of students' mathematics learning using certain technology, and NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out) were seen to be important contributors to the students' pragmatic reasoning.
Abstract: The researcher characterized the pragmatic reasoning of students' mathematics learning using certain technology. A “Natural Mathematics computer Interface” designation, NMI, was introduced and predicated on its virtual use of things like compass-rule, or pencil-paper traditional mathematical inscriptions. The NMI provided capacities for manipulative geometric constructions and transformations, or symbolic interfacing to a Computer Algebra System. Two separate case studies facilitated empirically-based characterization and reflection concerning students' explorations, experimentations, and deductions in this NMI use setting. Over the course of a semester, one student studied Geometry proof (an elementary education major), and one Markov Chains (a lower division mathematics major). Four distinctive types of perceived mathematical embodiments were observed to be used by the students. These abstract embodiments, and related reasoning acts were described in the context of C. S. Peirce's Pragmatic Reasoning theory. NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out), were seen to be important contributors to the students' pragmatic reasoning. The abstract types of mathematical embodiments revealed were named: (i) Interface-procedural, (ii) Natural Mathematics Computational, (iii) Applications , and (iv) Generic. These mean, respectively, (i) interpreted merely as memorized interface procedures, (ii) resultant from interactive computation, interpreted as mathematical in a (sometimes) surface sense, as the student may not understand the underlying mathematics directing the computation, (iii) resultant from a student's interest in a real-world application used to analogously consider a mathematical model and its interpretation, and (iv) clearly abstracted and generalized, internal or mentalesque mathematical explanations or systematizations.
14 citations
TL;DR: In this paper, the dynamics of an ion confined in a nonlinear Paul trap was investigated, and the equation of motion for the ion was shown to be consistent with the equation describing a damped, forced Duffing oscillator.
Abstract: This paper investigates the dynamics of an ion confined in a nonlinear Paul trap The equation of motion for the ion is shown to be consistent with the equation describing a damped, forced Duffing oscillator All perturbing factors are taken into consideration in the approach Moreover, the ion is considered to undergo interaction with an external electromagnetic field The method is based on numerical integration of the equation of motion, as the system under investigation is highly nonlinear Phase portraits and Poincare sections show that chaos is present in the associated dynamics The system of interest exhibits fractal properties and strange attractors The bifurcation diagrams emphasize qualitative changes of the dynamics and the onset of chaos
14 citations
Cites background or methods from "Dynamical Systems with Applications..."
...In order to illustrate the dynamics of the trapped ion we have represented the trajectories in the two-dimensional phase space (phase portraits) [19] and extended phase space as seen in Fig....
[...]
...We have performed a numerical integration of the equation of motion, using the fourth order Runge-Kutta method [19]....
[...]
01 Jan 2008
TL;DR: In this paper, the authors describe a world in which the real growth of the world economy is zero and remains zero and describe a story of a world that is so different from what we take for granted that today's economic systems, political systems and social systems will no longer work.
Abstract: What if we were to have 100 years of no growth? What if conditions were such that there is no future scenario under which growth will ever occur again? We might characterize this as impossible, as a vision that violates the outcome that we as innovative people must realize. In the document that follows I will show you our world as it must be sometime in the future. I will describe a world in which the real growth of the world economy is zero and remains zero. I will tell a story of a world that is so different from what we take for granted that today's economic systems, political systems, and social systems will no longer work. Importantly (and unique to this research), this story will be told from within the boundaries of modern economic growth theory. That is, rather than follow an ecological and/or geographical path to explore limits to growth, this research is an "inside job" that suggests that when modern growth theories are decoupled from assumptions that have no basis in how the real world is developing but are, for the most part, mathematical conveniences applied for the sake of "stability," then the long-run economic outcome is no longer capitalism. In the shadows beneath the foundations of capitalism lurk assumptions that are so ubiquitous as to be almost invisible. This research works back to the source of the myth of endless growth and suggests that the source is simply something we have made up. Furthermore, with increasing rigor, it exposes the fallacies that allow our world-view to take endless growth as a given and natural state upon which we can make choices; upon which, in the aggregate, are taking humankind on a very bad trip. Unaware, we are blinded from knowledge because we do not question the assumption of more forever. The regime of endless growth is a sort of fission-like chain reaction in which, depending upon one's perspective, the by-products are desirable or toxic. This research shows that some of the by-products are social and ecological anti-matter.
11 citations
References
More filters
TL;DR: In this article, the authors provide a detailed review of the conceptual foundation and physical mechanisms governing the complex optical response of deterministic aperiodic nano structures in relation to the engineering of novel devices and functionalities, mainly focusing on passive and active planar structures with enhanced light-matter coupling for photonics and plasmonics technologies.
Abstract: This review focuses on the optical properties and device applications of deterministic aperiodic media generated by mathematical rules with spectral features that interpolate in a tunable fashion between periodic crystals and disordered random media. These structures are called Deterministic Aperiodic Nano Structures (DANS) and can be implemented in different materials (linear and nonlinear) and physical systems as diverse as dielectric multilayers, optical gratings, photonic waveguides and nanoparticle arrays. Among their distinctive optical properties are the formation of multi-fractal bandgaps and characteristic optical resonances, called critical modes, with unusual localization, scaling and transport properties. The goal of the paper is to provide a detailed review of the conceptual foundation and the physical mechanisms governing the complex optical response of DANS in relation to the engineering of novel devices and functionalities. The discussion will mostly focus on passive and active planar structures with enhanced light-matter coupling for photonics and plasmonics technologies.
233 citations
Dissertation•
01 Sep 2009
TL;DR: In this article, the authors considered the limitations of resonant linear devices and investigated two non-linear generators to see if they can outperform the linear devices in certain situations, and found that the performance of a nonlinear device in terms of the power harvested is at most 4π greater than that of a tuned linear system and is strongly dependent upon the type of the nonlinearity used.
Abstract: Vibration-based energy harvesting devices have received much attention over the past few years due to
the need to power wireless devices in remote or hostile environments. To date, resonant linear generators
have been the most common type of generators used in harvesting energy for such devices. Simple tuning
and modelling methods make it a more favourable solution theoretically if not practically. This thesis
considers the limitations of resonant linear devices and investigates two non-linear generators to see if
they can outperform the linear devices in certain situations. So far, in most of the literature, the energy
harvester is assumed to be very small dynamically compared to the source so the source is not affected
by the presence of the device. This thesis considers how the dynamics of the source is affected by the
device if its impedance is significant compared to the source. A tuning condition for maximum power
transfer from the source to the device is derived. This tuning condition converges to the one presented
in most of the literature when the impedance of the device is assumed to be very small compared to that
of the source i.e. tuned so that the natural frequency of the device equals the excitation frequency. For
the case when the impedance of the device has a negligible effect on the source, the performance of the
device is only limited to a narrow frequency band and drops off rapidly if mistuned. To accommodate the
mistuning limitations, new types of generators are proposed mainly by using a non-linear mechanism.
These mechanisms are made up of a non-linear spring connected together with a mass and a linear
viscous damper i.e. the energy harvesting component. The analysis of the fundamental performance
limit of any non-linear device compared to that of a tuned linear device is carried out using the principal
of conservation of energy. The analysis reveals that the performance of a non-linear device in terms of the
power harvested is at most 4/π greater than that of a tuned linear system and is strongly dependent upon
the type of the non-linearity used. Two types of non-linear mechanisms are studied in this thesis. The
first one is a non-linear bi-stable mechanism termed a snap-through mechanism which rapidly moves the
mass between two stable states. The aim is to steepen the displacement response curve as a function of
time which results in the increase of velocity for a given excitation, thus increasing the amount of power
harvested. This study reveals that the performance of the mechanism is better than a linear system
when the natural frequency of the system is much higher than the excitation frequency. The study also
shows that the power harvested by this mechanism rolls off at a slower rate compared to that of the
linear system. Another non-linear mechanism described in this thesis uses a hardening-type spring. The
aim of this mechanism is to provide a wider bandwidth over which the power can be harvested. This
thesis presents numerical solutions and approximate analytical solutions for the bandwidth and effective
viscous damping of a non-linear device employing a hardening-type stiffness. Unlike the linear system ,
in which the bandwidth is only dependent on the damping ratio, it is found that the bandwidth of the
nonlinear device depends on both the strength of the nonlinearity and the damping ratio. Experimental
results are presented to validate the theoretical results. This thesis also investigates the benefits of the
non-linear device for a low frequency and high amplitude application using the measured vibration inputs
from human motion such as walking and running. The effect of harmonics on the power harvested is also
studied. Numerical simulations are carried out using measured input vibrations from human motion to
study the best placement of the natural frequency of the device across the range of harmonics.
17 citations
28 Sep 2010
TL;DR: To evaluate this encryption method’s performance, a statistical analysis was applied, and histograms of the original and encrypted images were developed, and the behavior of the algorithm to the most common cryptographic attacks were analyzed.
Abstract: In this work, we propose a color image encryption of variable sizes using logistic chaotic map. We considered a process of permutation, followed by a diffusion algorithm. To evaluate this encryption method’s performance, we applied a statistical analysis, and developed histograms of the original and encrypted images. For the correlation analysis, we evaluate the behavior of two adjacent pixels on horizontal, vertical and diagonal directions. We also analyzed the behavior of the algorithm to the most common cryptographic attacks. Numerical simulations and graphic results are presented for image encryption and decryption using MatLab software.
14 citations
01 Jan 2007
TL;DR: The researcher characterized the pragmatic reasoning of students' mathematics learning using certain technology, and NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out) were seen to be important contributors to the students' pragmatic reasoning.
Abstract: The researcher characterized the pragmatic reasoning of students' mathematics learning using certain technology. A “Natural Mathematics computer Interface” designation, NMI, was introduced and predicated on its virtual use of things like compass-rule, or pencil-paper traditional mathematical inscriptions. The NMI provided capacities for manipulative geometric constructions and transformations, or symbolic interfacing to a Computer Algebra System. Two separate case studies facilitated empirically-based characterization and reflection concerning students' explorations, experimentations, and deductions in this NMI use setting. Over the course of a semester, one student studied Geometry proof (an elementary education major), and one Markov Chains (a lower division mathematics major). Four distinctive types of perceived mathematical embodiments were observed to be used by the students. These abstract embodiments, and related reasoning acts were described in the context of C. S. Peirce's Pragmatic Reasoning theory. NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out), were seen to be important contributors to the students' pragmatic reasoning. The abstract types of mathematical embodiments revealed were named: (i) Interface-procedural, (ii) Natural Mathematics Computational, (iii) Applications , and (iv) Generic. These mean, respectively, (i) interpreted merely as memorized interface procedures, (ii) resultant from interactive computation, interpreted as mathematical in a (sometimes) surface sense, as the student may not understand the underlying mathematics directing the computation, (iii) resultant from a student's interest in a real-world application used to analogously consider a mathematical model and its interpretation, and (iv) clearly abstracted and generalized, internal or mentalesque mathematical explanations or systematizations.
14 citations
TL;DR: In this paper, the dynamics of an ion confined in a nonlinear Paul trap was investigated, and the equation of motion for the ion was shown to be consistent with the equation describing a damped, forced Duffing oscillator.
Abstract: This paper investigates the dynamics of an ion confined in a nonlinear Paul trap The equation of motion for the ion is shown to be consistent with the equation describing a damped, forced Duffing oscillator All perturbing factors are taken into consideration in the approach Moreover, the ion is considered to undergo interaction with an external electromagnetic field The method is based on numerical integration of the equation of motion, as the system under investigation is highly nonlinear Phase portraits and Poincare sections show that chaos is present in the associated dynamics The system of interest exhibits fractal properties and strange attractors The bifurcation diagrams emphasize qualitative changes of the dynamics and the onset of chaos
14 citations