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This article discusses the stability of periodic responses for the dynamical motion of AC machines from the perspective of Lyapunov function approach. The dynamical motion of AC machines is prototypically modeled as an equivalent linear RLC series circuit with time-variant inductance represented by a linear differential equation with periodic coefficients. Based on the deduced stability conditions, some special identities among the equivalent circuit parameters to ensure the stability of responses and their periodic structures are concluded. Through these conditions, the periodic structure of responses is obtained by using the method of strained parameters. Through a comparison with the experimental results from the specialized practical literatures, a strong agreement with the obtained analytical results is achieved. In addition, from a practical point of views, some future points within the discussion are raised to improve the mathematical modeling of AC machines to obtain a better model and simulation.

In this article, the rotational motion of a non-symmetric rigid body about a fixed point under the action of multi-Newtonian centers located on one fixed axis is considered. The aim of the work is to derive a technique to obtain such new (particular) integrable cases for the rotational motion of a rigid body in newtonian field of forces whose centers located on the vertical fixed axis. This technique reduces the governing equations and their known first integrals describing the motion to the unknown fourth integral and new system of differential equations. The well-known fourth integral cases are tested to verify the validity of the derived system and some new (particular) integrable cases are obtained with their geometrical interpretations of motion. The particular scope from this work is to extend Arkangel’skii’s theorem and the other related theorems to obtain such new integrable cases of single valuedness of solutions for this problem and its related special cases. Keywords: Motion of a rigid body about a fixed point, Non-linear ODE equations and systems, Integrable cases of motion, 2000 Mathematics subject classification: 70E17, 34A34, 70E40

This paper presents the nonlinear dynamic analysis of energy-based models arisen from the applied systems characterized by the energy transport in the presence of fractional order derivative and time delay. The studied model is the fractional version of Bianca-Ferrara-Dalgaard-Strulik (BFDS) model of economy which is viewed as a transport network for energy in which the law of motion of capital occurs. By considering the time delay as bifurcation parameter, a proof to investigate the existence of Hopf bifurcation and the phase lock solutions using the Poincare-Linstedt and the harmonic balance methods is given. At definite values of time delay, period-doubling bifurcations followed up by the consequences of chaotic states are detected. Simulation results assure that the BFDS model can generate new (hyper) chaotic attractors beyond half order derivatives through the effect of the time delay on that system. In accordance with the literatures related to the problem of chaos, the concluded results confirm the proposed theorem by El-Borhamy in which the time delay possesses the ability to change the dynamic state of nonlinear systems from regular to chaotic within the fractional order derivative domain.

In this work, numerical schemes for the generalized stationary regularized viscoplastic fluids equations with highly nonlinear viscosities are analyzed. The discretisation using certain high order finite elements (Q2P1disc) is achieved to transform the nonlinear continuous form to the corresponding nonlinear discretized one. The solution process is analyzed in the sense of successive continuous Newton method (CNM) with minimal residual method as a linear solver. The convergence of the method is proved using the sense of the minimization technique due to the monotone decreasing feature of their viscosities. It is shown that, the convergence process of the successive nonlinear iterative method is mostly approaching a semi-quadratic behavior for slow flow cases. The numerical tests are achieved to have good fitting with the theoretical results using a self implemented Matlab code for the interested problem. Keywords: Generalized viscoplastic fluid, Finite element method, Posteriori error estimate, Newton iterative method, Minimum residual method

In this article, a semi-analytical technique is proposed to predict stable sustained periodic responses of AC electrical machines. Based on such desired outputs, the proper selections of machine variables are captured, such as the perturbation parameter arisen from the relative movement between the stationary and rotating parts. Compared to the experimental results, the derived analytical results are relatively well-fitted with the studied practical cases.