This is to certify that the thesis work entitled "Approximate Analytic Solutions of the Inverse Cubic Truly N-linear Oscillator by Iterative Method" was carried out by Md. A new technique based on the Mickens iterative method was presented to approximate analytical solutions of the Inverse cubic truly nonlinear oscillator. Mominur Rahman, 2016, "Mickens iteration-like method for approximate solutions of the Inverse Cubic Truly Nonlinear Oscillator", British Journal of Mathematics & Computer Science, Vol.
At first Van der Pol [1] paid attention to new oscillations (self-excitations) and showed that their existence is inherent in the nonlinearity of the differential systems that characterize the procedure. This nonlinearity appears, therefore, as the very essence of these phenomena and by linearizing the differential equation in the sense of the small oscillation method, the possibility of investigating such problems is simply eliminated. The KBM method starts with the solution of the linear equation (sometimes called the generative solution of the linear equation), assuming that in the nonlinear case, the amplitude and phase variables in the solution of the linear differential equations are time-dependent functions instead of constants. .
An important advantage of the method is that it can be used in nonlinear oscillation problems, in which the nonlinear terms are not "small", i.e. there need not be a disturbance parameter. The disadvantage of the method is that for a given nonlinear differential equation it is difficult to predict in advance whether the calculation of the first-order harmonic balance will provide a sufficiently accurate approximation to the periodic solution. Chapter IV described the iteration method for obtaining approximate analytical solutions of an inverse cubic truly nonlinear oscillator.
If the trajectories on both sides of the closed trajectory diverge as t , then the closed trajectory is unstable. For each numbertt0 and each pair ( ,x y0 0) of the real number, there is a unique solution of the equation (2.6), we get. If both f and g are not constant functions, then equation (2.7) defines a curve in the phase plane, which is called the path or orbit or trajectory of the system.
If we were to integrate equation (2.15) numerically, we would see that the period of the periodic motions depended on the closed curve in the phase plane we were in. Inspection of equation (2.24) shows that each of the classical methods has as its basis the explicit assumption that when 0 the resulting “core” equation is the differential equation of the linear harmonic oscillator, namely: It follows that some aspects of the behavior of a nonlinear system usually appears chaotic, unpredictable, or counterintuitive.
By replacing equation (3.2) with equation (3.1) and equating the coefficients for the different powers of , the following equations are obtained. An important consequence of this property is that the corresponding Fourier expansions of the periodic solutions only contain odd harmonics (Mickens [70]), i.e. In general, the amplitudes and the angular frequency will be expressed in terms of the parameters 1 and 2.
Inspection of equation (3.35) shows that the second approximation for the angular frequency is a modification of the first approximation result.
Introduction
The method
The method can proceed in any order of approximation; but due to increasing algebraic complexity, the solution is limited to a lower order, usually the second [49].
An Example
Solution Procedure
Therefore, the angular frequency, “” appearing on the right-hand side of equation (4.4) in the function tk( ), is k. Secular terms can be eliminated if the coefficient of cos is set to zero, i.e., expanding the second term on the right-hand side of equation (4.17) in a truncated Fourier series, we get. 4.20) This is the second approximate frequency of the oscillator (4.7).
Now expanding the term on the right-hand side of equation (4.25) into a truncated Fourier series, we get
Results and discussions
Table
W., 2006, “An analytical approximation technique for a class of strongly nonlinear oscillators”, International Journal of Nonlinear Mechanics, Vol. P., 2006, “Higher accuracy analytical approaches to the Duffing harmonic oscillator”, Journal of Sound and Vibration, Vol. W., 2006, “Harmonic balance approach to constrain the cycle for nonlinear shock equation”, Journal of Sound and Vibration, Vol.
Hu, H., 2006, "Solutions of a quadratic nonlinear oscillator by the method of harmonic balance", Journal of Sound and Vibration, Vol. 2006, "Solutions of Duffing harmonic oscillator by the method of harmonic balance", Journal of Sound and Vibration, Vol. E., 2007, "Harmonic balance and iteration calculations of periodic solutions for y y1 0", Journal of Sound and Vibration, Vol.
B., 2007, "A new analytical technique for finding periodic solutions of nonlinear systems", International Journal of Nonlinear Mechanics, Vol. Hosen M.A., 2013, "Exact approximate analytical solutions for an anti-symmetric quadratic nonlinear oscillator", African Journal of Mathematics and Computer Science Research, Vol. A., 2009, "Fourth order asymptotic solution near critically damped nonlinear systems", Journal of information and Mathematical Science, Vol.
H., 2011, “Perturbation resolution of fourth-order critically damped oscillatory nonlinear systems”, International Journal of Basic and Applied Science, Vol. E., 1987, "Iteration Procedure for Determining Approximate Solutions of the Nonlinear Oscillator Equation", Journal of Sound and Vibration, Vol. E., 2005, "A general procedure for computing approximations to periodic solutions of truly nonlinear oscillators", Journal of Sound and Vibration, Vol.
K., 2006, “Higher order exact analytical approximate solutions for non-conservative nonlinear oscillators and application to oscillators with van der Pol damping”, International Journal of Mechanical Science, Vol. Hu, H., 2006, “Solutions of the Duffing Harmonic Oscillator by an Iterative Procedure”, Journal of Sound and Vibration, Vol. H., 2006, “A classical iteration procedure valid for some strongly nonlinear oscillators”, Journal of Sound and Vibration, Vol.
