Krylov-Bogoliubov-Mitropolskii Research Articles

A modified Krylov–Bogoliubov–Mitropolskii method for solving damped nonlinear oscillators with large oscillation

A modification of the extended Krylov–Bogoliubov–Mitropolskii​ method (by Popov et al.) has been presented to investigate damping effects of strongly nonlinear oscillators when the amplitude of oscillation is large. The modification is similar to the Lindstedt-Poincare method, which was early presented by Cheung et al. for handling un-damped large oscillations. The determination of solution is quite different from earlier all techniques and it is valid whether linear restring force is present or not. Moreover, the solution approaches toward modified Lindstedt-Poincare’s solution when damped force vanishes.

ASYMPTOTIC SOLUTION OF SECOND ORDER OVER-DAMPED NON-LINEAR SYSTEMS FOR SMALL RATIO OF THE EIGENVALUES

A technique is developed for obtaining the solution of second order over-damped non-linear systems based on the Krylov-Bogoliubov-Mitropolskii (KBM) method for small ratio of the eigenvalues. The solution is also useful for large ratio of the eigenvalues. The method is illustrated by an example.

ON FOURTH ORDER MORE CRITICALLY DAMPED NON-LINEAR SYSTEMS UNDER SOME CONDITIONS

Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended for solving fourth order more critically damped non-linear systems. For different damping forces, the solutions obtained by the present method show good coincidence with numerical solutions. The method is illustrated by an example.

HIGHER ORDER IMPROVED SOLUTION OF SECOND ORDER OVER-DAMPED NON-LINEAR SYSTEMS

A second order nonlinear differential system modeling non-oscillatory processes by over damping is considered. Then second order approximate solution is found by means of an extension of the Krylov-Bogoliubov-Mitropolskii (KBM) method. The method is illustrated by an example. The solutions for different initial conditions show a good agreement with those obtained by numerical solution.

Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Some novel exact solutions and approximations to the damped Duffing–Mathieu-type oscillator with cubic nonlinearity are obtained. This work is divided into two parts: in the first part, some exact solutions to both damped and undamped Mathieu oscillators are obtained. These solutions are expressed in terms of the Mathieu functions of the first kind. In the second part, the equation of motion to the damped Duffing–Mathieu equation (dDME) is solved using some effective and highly accurate approaches. In the first approach, the nonintegrable dDME with cubic nonlinearity is reduced to the integrable dDME with linear term having undermined optimal parameter (maybe called reduced method). Using a suitable technique, we can determine the value of the optimal parameter and then an analytical approximation is obtained in terms of the Mathieu functions. In the second approach, the ansatz method is employed for deriving an analytical approximation in terms of trigonometric functions. In the third approach, the homotopy perturbation technique with the extended Krylov–Bogoliubov–Mitropolskii (HKBM) method is applied to find an analytical approximation to the dDME. Furthermore, the dDME is solved numerically using the Runge–Kutta (RK) numerical method. The comparison between the analytical and numerical approximations is carried out. All obtained approximations can help a large number of researchers interested in studying the nonlinear oscillations and waves in plasma physics and many other fields because many evolution equations related to the nonlinear waves and oscillations in a plasma can be reduced to the family of Mathieu-type equation, Duffing-type equation, etc.

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.

Free vibration of nonlinear autonomous systems modeled by 4-th order ordinary differential equations including internal resonance

Abstract In this article, the dynamic behavior of nonlinear autonomous system modeled by 4-th order ordinary differential equations is considered. Based on the pioneer work of Krylov-Bogoliubov-Mitropolskii (KBM), a modified KBM method is applied to achieve analytical solutions. Two different cases such as non-resonant and resonant conditions are studied in the presence of quadratic nonlinearities. The utility of the proposed method is justified by numerical result which is generated by Runge-Kutta fourth-order procedure. The analytical solution in each case shows an excellence agreement with numerical results.

Asymptotic Solutions of Fifth Order More Critically Damped Nonlinear Systems in the Case of Four Repeated Roots

In this article, we have modified the Krylov-Bogoliubov-Mitropolskii (KBM) method, which is one of the most widely used methods to delve into the transient behavior of oscillating systems, to find out the solutions of fifth order more critically damped nonlinear systems. In this paper, we have considered the asymptotic solutions of fifth order more critically damped nonlinear systems when the four eigenvalues are equal and another one is distinct. This article suggests that the perturbation solutions obtained by the modified KBM method for both the cases (when repeated eigenvalues are greater than the distinct eigenvalue, and when the distinct eigenvalue is greater than repeated eigenvalues) satisfactorily correspond to the numerical solutions obtained by Mathematica 9.0.

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

Asymptotic Method of Krylov-Bogoliubov-Mitropolskii for Fifth Order Critically Damped Nonlinear Systems

In oscillatory problems, the method of Krylov–Bogoliubov–Mitropolskii (KBM) is one of the most used techniques to obtain analytical approximate solution of nonlinear systems with a small non-linearity. This article modifies the KBM method to examine the solutions of fifth order critically damped nonlinear systems with four pairwise equal eigenvalues and one distinct eigenvalue, in which the latter eigenvalue is much larger than the former four pairwise eigenvalues. This paper suggests that the results obtained in this study correspond accurately to the numerical solutions obtained by the fourth order Runge-Kutta method. This paper, therefore, concludes that the modified KBM method provides highly accurate results, which can be applied for different kinds of nonlinear differential systems.

Asymptotic Method for Certain over-Damped Nonlinear Vibrating Systems

Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended and applied to certain over-damped nonlinear system in which the linear equation has two almost equal roots. The method is illustrated by an example.

Approximate Solution of Fourth Order Near Critically Damped Nonlinear Systems with Special Conditions

A perturbation technique has been developed based on the Krylov-Bogoliubov-Mitropolskii (KBM) method to investigate the solution of fourth order near critically damped nonlinear systems in the case of ?1??2, ?4 ? ?3 + 2?1 but ?4<2?3 among the eigenvalues ?1, ?2, ?3, ?4. The solutions obtained by this technique were compared with those obtained by numerical method. The method has been explained by an example. DOI: http://dx.doi.org/10.3329/jbas.v36i2.12962 Journal of Bangladesh Academy of Sciences, Vol. 36, No. 2, 187-197, 2012

Approximate solutionsof Damped Nonlinear Vibrating System with Varying Coefficients under Some Conditions

Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The solution obtained for different initial conditions for a second order nonlinear system show a good coincidence with those obtained by numerical method. The method is illustrated by an example.

A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, <TEX>${\ldots}$</TEX>, order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

A Unified KBM Method for Obtaining the Second Approximate Solution of a Third Order Weakly Non-linear Differential System with Strong Damping and Slowly Varying Coefficients

To obtain the second order approximate solution of a third order weakly nonlinear ordinary differential system with strong damping and slowly varying coefficients modeling a damped oscillatory process is considered based on the extension of a unified Krylov-Bogoliubov- Mitropolskii (KBM) method. The asymptotic solution for different initial conditions shows a good coincidence with those obtained by the numerical procedure for obtaining the transient’s response. The method is illustrated by an example.DOI: http://dx.doi.org/10.3329/jbas.v35i1.7973Journal of Bangladesh Academy of Sciences, Vol.35, No.1, 77-89, 2011

Perturbation solutions of fifth order oscillatory nonlinear systems

Oscillatory systems play an important role in the nature. Many engineering problems and physical systems of fifth degrees of freedom are oscillatory and their governing equations are fifth order nonlinear differential equations. To investigate the solution of fifth order weakly nonlinear oscillatory systems, in this article the Krylov–Bogoliubov–Mitropolskii (KBM) method has been extended and desired solution is found. An example is solved to illustrate the method. The results obtain by the extended KBM method show good agreement with those obtained by numerical method.

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.

An extension of the general Struble's method for solving an nth order nonlinear differential equation when the corresponding unperturbed equation has some repeated eigenvalues

Abstract This paper attempts to show the more suitability of the extended general Struble's technique than the unified Krylov–Bogoliubov–Mitropolskii (KBM) method in solving the problems that occur during the critical conditions. Recently a critically damped condition of an n th, n =2,3, … order weakly nonlinear autonomous ordinary differential equation has been investigated by the unified KBM method, in which the corresponding unperturbed equation has some real (negative) repeated eigenvalues. But there are more important critical conditions, which are still untouched. One of them occurs when a pair of complex eigenvalues is equal to another. It is complicated to formulate as well as to utilize the KBM method to investigate this condition. However, the extended general Struble's technique is applicable to both autonomous and non-autonomous systems. Solutions obtained for different critical conditions as well as for different initial conditions show a good agreement with the numerical solutions. The method is illustrated by an example of a fourth-order nonlinear differential equation whose unperturbed equation has repeated complex eigenvalues. A steady-state solution is determined for the non-autonomous equation. Moreover, a critical condition of a fourth-order nonlinear equation is investigated when two real eigenvalues of the unperturbed equation are non-positive and equal.

Asymptotic solutions of third order over-damped non-linear systems with special conditions

An asymptotic method based on the Krylov-Bogoliubov-Mitropolskii (KBM) method has been developed in this article for obtaining the solutions of a third order over-damped non-linear system. For different sets of initial conditions the solutions obtained by this method are better than the solutions found in [13]. The method is illustrated by an example.

KBM unified method for solving an nth order non-linear differential equation under some special conditions including the case of internal resonance

The Krylov–Bogoliubov–Mitropolskii (KBM) unified method is used for obtaining the approximate solution of an nth order ( n ⩾ 4 ) ordinary differential equation with small non-linearities when a pair of eigen-values of the unperturbed equation is multiple (approximately or perfectly) of the other pair or pairs. The general solution can be used arbitrarily for over-damped, damped and undamped cases. In a damped or undamped case, one of the natural frequencies of the unperturbed equation may be a multiple of the other. Thus, the solution also covers the case of internal resonance which is an interesting and important part of non-linear oscillation. The determination of the solution is very simple and easier than the existing procedures developed by several authors (both in methods of averaging and multiple time scales) especially to tackle the case of internal resonance. The method is illustrated by an example of a fourth-order differential equation. The solution shows a good agreement with numerical solution in all of the three cases, e.g. over-damped, damped and undamped.