Browsing by Author "Dey, Pinakee"

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    Asymptotic Method for Time Dependent Nonlinear Differential Systems with Slowly Varying Coefficients
    (University of Rajshahi, 2013) Roshid, Harun-Or-; Ali, M. Zulfikar; Dey, Pinakee
    Almost all perturbation methods are developed to find periodic solutions of nonlinear system where transients are not considered. First Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient process. In this dissertation, we have modified and extended the KBM method to investigate some fifth order and second order nonlinear systems in both cases with constant and slowly varying coefficients. At first, a fifth order damped nonlinear autonomous differential system is considered and a perturbation solution is developed. Then a procedure is developed for the same system with damped taking three of eigenvalues are real. After then we considered fifth order systems for over damped with small nonlinearity to obtain the transient response. We also developed a formula for fifth order critically damped nonlinear systems to control micro vibration, in micro and nano-technological industries that bring the system to equilibrium as quickly as possible without oscillating. After then we presented an analytical technique based on the extended Krylov-Bogoliubov-Mitropolskii method (by Popov) to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. At last, Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The implementations of the methods are illustrated by several examples.
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    Damped Forced Vibration of Some Quasi-Linear Differential Systems
    (University of Rajshahi, 2008) Dey, Pinakee; Sattar, M A.; Ali, M. Zulfikar; Alam, M. Shamsul
    There are many approaches for approximating solutions of nonlinear vibrating problems. The most common methods for constructing approximate analytical solutions to the nonlinear vibrating problems are the perturbation methods. These methods are developed to find only periodic vibrations of the nonlinear differential systems. In order to investigate the transients of nonlinear vibrations, Krylov and BogoLiubov introduced a perturbation method to discuss the transients in the second order autonomous systems with small nonlinearities. The method is well known as an "asymptotic averaging method" in the theory of nonlinear vibrations. Then the method was amplified and justified by BogoLiubov and Mitropolskii. These methods were applied to autonomous systems. Later, Arya and Bojadziev, Bojadziev and Hung, and Shamsul extended the Krylov-Bogo Liubov-Mitropolskii (KBM) method to sometime dependent nonlinear differential systems. In this dissertation, we extend the work of KBM and investigate some other time dependent non-linear differential systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear vibrations in presence of a slowly decaying external force. We then find an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the KBM method. Later, we find the perturbation solutions of damped forced vibrations using the modified KBM method, in which the coefficients change slowly varying with time. Further, this technique is used to obtain the second approximate solution of second order forced vibrations. Finally, this technique is used to obtain the higher approximate solution of an n-th order damped forced vibrating problem in the resonance case, and the stability of the stationary regime of vibrations has also been investigated. The methods are illustrated by several examples.