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An Approximate-Analytical Approach to Nonlinear FDEs under Generalized Differentiability
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O.S. Fard, T.A. Bidgoli, A.H. Borzabadi
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Volume 2, Issue 1, 2010
pp.
56 - 74
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Received
19 February 2010,
Accepted
07 June 2010
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Abstract.
The current research attempts to offer an approximate-analytical scheme to solve nonlinear fuzzy differential equations under generalized differentiability. In comparison with existing numerical methods, one may find the better capability and efficiency of the given scheme. The proposed method is illustrated by some numerical examples.
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Keywords.
Nonlinear fuzzy differential equation; Generalized differentiability; Local variational iteration method.
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Relative Probability Synchronization for Dynamical Systems Created by Homeomorphisms
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Mohammad Reza Molaei, Omur Umut
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Volume 2, Issue 1, 2010
pp.
49 - 55
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Received
14 January 2010,
Accepted
08 April 2010
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Abstract.
In this paper, using the relative probability measure, the notion of synchronized dynamical systems from the viewpoint of an observer for discrete topological dynamical systems created by the homeomorphisms of a metric space, is considered. A kind of equivalence relation up to the suitable observers, is deduced. It is proved that: topological conjugacy implies to the relative probability synchronization. The notion of relative probability synchronization for the ideal observers on the probability spaces, is studied. The relative probability synchronization of the time one maps of the solutions of Chua's diode and its driven system by the different observers, are studied.
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Keywords.
Synchronization; Chuas's diode; Relative Probability Measure; Observer; Relative Conjugacy; L$\ddot{u}$ and Chen System.
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Modeling and Simulation of Transformers
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Marius-Constantin Popescu
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Volume 2, Issue 1, 2010
pp.
27 - 48
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Received
17 December 2009,
Accepted
20 March 2010
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Abstract.
The current and power (active and reactive parts) at the terminals of the step-down transformer are positive if the transit is in line busbar to charge. The study of the dynamic behavior of the system developed in the later, will use numerical simulations and the calculation of eigen values. Theoretical results were compared with data from transformer manufacturers and the good agreement between both validates theoretical results is accomplished.
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Keywords.
Step-up and step-down transformers; Power model and current; Modeling and simulation.
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Time-like Salkowski Curves in Minkowski Space E_1^3
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Ahmad Tawfik Ali
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Volume 2, Issue 1, 2010
pp.
17 - 26
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Received
07 November 2009,
Accepted
19 January 2010
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Abstract.
One century ago, Salkowski introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\mathbf{E}^3$ \cite{salkow}. In this paper, we adapt the definition of such curves to timelike curves in Minkowski 3-space $\mathbf{E}_1^3$. Thereafter, we introduce an explicit parametrization of a timelike Salkowski curve and a timelike anti-Salkowski curve in Minkowski space $\mathbf{E}_1^3$. Also, we characterize them as space curves with constant curvature or constant torsion and whose normal vector makes a constant angle with a fixed line.
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Keywords.
Salkowski curves; Time-like Curves; Constant curvature; Minkowski 3-space.
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Vector Constants of Motion for Dynamical Systems with Nonlocal Symmetries and Applications
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Halvard White
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Volume 2, Issue 1, 2010
pp.
1 - 16
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Received
10 August 2009,
Accepted
26 November 2009
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Abstract.
We consider the class of dynamical systems $\ddot{\boldsymbol{x}}=r^{-3}I_1 \boldsymbol{x}+r^{-2}I_2 \dot{\boldsymbol{x}}$, $\boldsymbol{x} \in R^{3}$, $r=\left| \boldsymbol{x} \right|$, where $I_1, I_2$ are each functions of $\boldsymbol{x}$ and $\dot{\boldsymbol{x}}$ and invariants of the nonlocal symmetry transformation $\bar{\boldsymbol{x}}=h^{-1}\boldsymbol{x}$, ${d\bar{t} \mathord{\left/{\vphantom{d\bar{t}dt}}\right.\kern-\nulldelimiterspace} dt} = h^{-2}$, $h=1+\boldsymbol{q.x}$ and $\boldsymbol{q}$ is any constant vector. It is shown in a recent paper by the author, that this equation is the most general form of the (second-order) dynamical system which admits the above transformation as a symmetry transformation. We show that for $I_2 \ne 0$ such systems possess (a) a constant of motion $w(\alpha,L)$ which satisfies the equation $\partial_{\alpha}w+I_2 \partial_{L}w=0$, where $(r,\alpha)$ are polar coordinates in the plane of motion, $L=\left|\boldsymbol{L} \right|$, $\boldsymbol{L} = \boldsymbol{x} \wedge \dot{\boldsymbol{x}}$ and any other such function is some function of $w$ (b) a Laplace-Runge-Lenz vector constant of motion $\boldsymbol{J}$ given by $\boldsymbol{J}=\hat{\boldsymbol{L}} \wedge \boldsymbol{K}$ where $\boldsymbol{K}=L^{-1}\dot{\boldsymbol{x}}+\boldsymbol{W}$, $\hat{\boldsymbol{L}}=L^{-1} \boldsymbol{L}$ and $\boldsymbol{W}$ is a function of $w$ and $\alpha$ with $\boldsymbol{L.W}=0$. We compute the Ermanno-Bernoulli constants and use them to obtain nonlocal symmetry transformation of such systems. We then use the analysis of K. Andriopoulos and P.G.L. Leach (2002) on minimal generating sets of symmetry vectors which specify the harmonic oscillator to obtain complete symmetry groups for these systems.
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Keywords.
Vector constants; Laplace-Runge-Lenz; Kepler; Nonlocal; Symmetries; Reduction; Complete.
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Adaptive Control of Robot Manipulator Having Friction and Uncertainties
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Jyoti Ohri, Lillie Dewan
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Volume 1, Issue 2, 2009
pp.
46 - 61
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Received
19 February 2009,
Accepted
14 July 2009
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Abstract.
A novel approach for adaptive control of robot manipulators having friction and other uncertainties using exponential estimation laws has been proposed. Proposed estimation law is based on time varying parameters and depends upon the system dynamics and tracking error. Friction is an important aspect for control design of mechanical systems, including robotics, because it can lead to tracking errors, limit cycles, and undesired stick-slip motion. The developed error derived adaptive compensator ensures global position tracking when applied to an n degree of freedom manipulator perturbed by friction forces, random external disturbances, measurement noise, and all the system parameters (robot and friction model) unknown.
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Keywords.
Friction; Adaptive controller; Dynamic; Exponential law; Tracking control; Non-linear system; Uncertainties.
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On the Static and Dynamic Response of Resonant Microbeams
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Samir A. Emam, Mahmoud E. Khater, Emil H. Gad
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Volume 1, Issue 2, 2009
pp.
31 - 45
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Received
22 April 2009,
Accepted
07 July 2009
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Abstract.
An investigation into the response of microbeams to DC and AC electric actuation is presented. The beam is modeled according to the Euler-Bernoulli beam theory and small strains and moderate rotation approximations are assumed. The governing equation is a nonlinear integral-partial-differential equation in space and time. The model accounts for mid-plane stretching, applied axial load, DC electrostatic forces, and AC harmonic forces. A reduced-order model based on the Galerkin discretization technique is introduced to simulate the behavior of microswitches and resonant sensors. The static behavior of the microbeam under electrostatic forces is studied and compared to the results available in the literature. The dynamic behavior of resonant microbeams under AC harmonic forces is investigated. An analytical solution for the vibration modes and natural frequencies of the microbeam around its statically deflected position is obtained. A shooting method is used to numerically integrate the nonlinear discretized equations and obtain periodic orbits of the response. The stability of these periodic orbits is investigated using Floquet theory. The sensitivity of the device to small-amplitude excitations is also investigated.
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Keywords.
MEMS; Microswitch; Resonant microbeam; Nonlinear dynamics.
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Response of Microbeam-Based Devices Accounting for the Electrostatic Force and Geometric Nonlinearities
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Samir A. Emam, Eihab M. Abdel-Rahman
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Volume 1, Issue 2, 2009
pp.
18 - 30
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Received
21 April 2009,
Accepted
27 June 2009
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Abstract.
A model of electrically actuated microbeam-based MEMS devices incorporating the nonlinearities associated with moderately large displacements and electrostatic forces is presented. The parallel-plate restriction used to calculate the electrostatic force between the two electrodes is relaxed. Boundary-element method is used to solve for the integral equation describing the capacitance established between the moving and the stationary electrodes and hence the electrostatic force is numerically cmputed. This gives us the ability to change the fixed electrode configuration and to have gaps between its segments. Therefore, the model can handle any capacitor configuration disposing of the complete electrode-overlapping, parallel-plate theory, restriction. The model accounts for the geometric nonlinearity arising from the midplane stretching of the microbeam. The boundary-value problem describing the static deflection of the microbeam under the electrostatic loading is solved numerically. The eigenvalue problem describing the vibration of the microbeam around its statically deflected position is solved numerically for the natural frequencies and mode shapes. Results generated by our model for the parallel-plate case are in agreement with published results. Our results show that the underlying assumption of the closed-form formula of the parallel-plate case underestimates the electrostatic force and leads to an overestimation of the pull-in voltage. The model provides an analytical tool to predict the static and dynamic response of any electrically actuated MEMS device based on clamped-clamped microbeams.
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Keywords.
MEMS; BEM; Microbeam; Nonlinear vibration.
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On Linearizations of Dynamical Systems Admitting a Laplace-Runge-Lenz Vector
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Halvard White
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Volume 1, Issue 2, 2009
pp.
1 - 17
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Received
18 February 2009,
Accepted
23 June 2009
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Abstract.
It is shown in Leach PGL et al; J. Math. Phys., 44 (2003) 4090-4106, that the classical Kepler problem is reducible to a linear system consisting of two isotropic harmonic oscillators and a conservation law, the variables in which are related to the Ermanno-Bernouilli constants and the components of the angular momentum vector. We show that a reduction to such a linear system is also possible for all dynamical systems which (i) admit a Laplace-Runge-Lenz vector (ii) describe motion in a plane, and for which the equation of motion for the angular momentum admits a first integral. We show that a number of choices exist for the variables in the reduced system. We show also that this reduction is possible for all such dynamical systems which, instead of planar motion, describe conical motion associated with a Poincar{\'e} vector constant of motion.
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Keywords.
Vector constants; Laplace-Runge-Lenz; Poincare; Symmetries; Groups; Realizations; Oscillators.
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Use of Piecewise Linear Models for Optimal Control of Nonlinear Systems
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Jose Luis Figueroa, Andres Garcia, Osvaldo Agamennoni
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Volume 1, Issue 1, 2009
pp.
75 - 83
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Received
23 April 2009,
Accepted
22 June 2009
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Abstract.
The optimal feedback control of nonlinear process is attacked in this paper. The solution of this problem is numerically computed using a Continuous Piecewise Linear (CPWL) approximation of the Ordinary Differential Equations (ODEs) system which describes the dynamics of the plant to control. In order to obtain this solution, the optimal regulation problem of an affine system is obtained. A numerical simulation example of a nonlinear chemical reactor is presented to shown the quality of the obtained response.
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Keywords.
Optimal Control; Nonlinear Systems; Piecewise Linear Functions.
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