Analytical Studies of SWCNTs Embedded in Nonlinear Viscous Elastic Media and the Chaotic Effect of Its Various Parameters

The analytical studies and chaotic behavior of forced vibration on Single-wall Carbon Nanotubes (SWCNTs) embedded in nonlinear viscous elastic medium subjected to parametric excitation are investigated. The analytical solution of the amplitude of nonlinear vibration is studied using Krylov–Bogoliubov–Mitropolsky (KBM) method. Both resonant and nonresonant cases are deduced. The computational techniques are used to draw graphs of time series, phase plot and Poincaré surface of section to analyze the chaotic behavior of the system considered. The plots are drawn for various values of different parameters like linear damping, nonlinear damping and amplitude of external forces in the considered model of SWCNTs. This work could be helpful in differentiating various elements of Carbon Nanotubes (CNTs) into the chaotic elements and controlling elements. The chaotic elements contributes to increase in the aging of CNTs while controlling elements can be used to control the irregular behavior of CNTs.

Approximate solution of nonlinear differential system with time variation

this paper develops a reliable algorithm based on the general Struble’s technique and extended KBM method for solving nonlinear differential systems. Moreover, we find a solution based on the KBM and general Struble’s technique of nonlinear autonomous systems with time variation, which is more powerful than the existing perturbation method. Finally, results are discussed, primarily to enrich the physical prospects, and shown graphically by utilizing MATHEMATICA and MATLAB software. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 121-130, 2020

Asymptotic Solutions of Fifth Order Overdamped-Oscillatory Nonlinear Systems

Combo overdamp-oscillatory system plays an important role in natural phenomena in many engineering problems. In this paper, fifth order nonlinear damped-oscillatory differential system is studied to investigate an asymptotic analytical approximate solution in the fashion of overdamp-oscillations via an extension of the Krylov-Bogoliubov-Mitropolskii (KBM) method. The proposed method is demonstrated by its applications on a Duffing oscillators in the combined form of overdamp and oscillatory effects. The result obtained by the presented extended technique good agreement with the numerical solutions of the fourth order Runge-Kutta method.

APPROXIMATE SOLUTIONS OF DAMPED NON LINEAR SYSTEM WITH VARYING PARAMETER AND DAMPING FORCE

The study of second-order damped nonlinear differential equations is important in the development of the theory of dynamical systems and the behavior of the solutions of the over-damped process depends on the behavior of damping forces. We aim to develop and represent a new approximate solution of a nonlinear differential system with damping force and an approximate solution of the damped nonlinear vibrating system with a varying parameter which is based on Krylov–Bogoliubov and Mitropolskii (KBM) Method and Harmonic Balance (HB) Method. By applying these methods we solve and also analyze the finding result of an example. Moreover, the solutions are obtained for different initial conditions, and figures are plotted accordingly where MATHEMATICA and C++ are used as a programming language.

Perturbation Solutions to Fifth Order Over-damped Nonlinear Systems

Fifth order over-damp nonlinear differential systems can be used to describe many engineering problems and physical phenomena occur in the nature. In this article, the Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to investigate the solution of a certain fifth order over-damp nonlinear systems and desired result has been found. The implementation of the presented method is illustrated by an example. The first order analytical approximate solutions obtained by the method for different initial conditions show a good agreement with those obtains by numerical method.

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

An approximate analytical technique for solving second order strongly nonlinear generalized duffing equation with small damping

In this paper, He’s homotopy perturbation method has been extended for obtaining the analytical approximate solution of second order strongly nonlinear generalized duffing oscillators with damping based on the extended form of the Krylov-Bogoliubov-Mitropolskii (KBM) method. Accuracy and validity of the solutions obtained by the presented method are compared with the corresponding numerical solutions obtained by the well-known fourth order Rangue-Kutta method. The method has been illustrated by examples.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 1, 103-114, 2015