| The most common and most widely studied methods of all analytical approximation methods for nonlinear differential equations are the perturbation methods. These methods involve the expansion of a solution to a differential equation in a series in a small parameter. They include the L-P method, the KBM method and the multiple scales method and so on. However, these methods apply to weakly nonlinear oscillations only. The method of harmonic balance is another procedure for determining analytical approximations to the solutions of differential equations by using a truncated Fourier series representation. An important advantage of the method is its applicability to nonlinear oscillatory problems for which the nonlinear terms are not "small", i.e., no perturbation parameter needs to exist. However, applying the method of harmonic balance to construct higher-order approximate analytical solutions is also very difficult, since they require analytical solution of sets of algebraic equation with very complex nonlinearity. In this thesis, a new analytical approximate method is presented to solve large amplitude nonlinear oscillations of single degree of freedom non-natural conservative systems. The most interesting features of the new method are its simplicity and its excellent accuracy in a wide range of values of oscillations amplitudes. These analytical approximate periods and corresponding periodic solutions are valid for small as well as large amplitudes of oscillation, including the case of amplitude of oscillation tending to infinity. Consider the non-linear oscillator E (u ) ddt2 u2+ F(u ) ddt2u2+G(u ) =0 (1) u (0 ) = A, ddut (0 ) =0 (2) where E (u ), F(u ), G(u )satisfy E (? u) =E(u ), F (? u) =?F(u ),G (? u) =?G(u ),and they are polynomials of u . Introduce a new independent variable, τ= ωt, then Eqs.(1) and (2) become ? [E (u )u& & +F(u )u&2 ] +G(u ) =0 (3) u (0 ) = A, u& (0 ) =0 (4) where ? =ω2 and over-dot denotes differentiation with respect to τ. The new independent variable is chosen in a way such that the solution of (3) is a periodic function of τof period 2π. The corresponding period of nonlinear oscillation is given by T = 2π?. Here, both periodic solution u (τ) and frequency ωdepend on A . Base on Eq.(3), the periodic solution u (τ) has a Fourier series expansion: (τ) cos[( 2 1)τ]0= ∑2 1+∞u n = hn + n (5) Which contains only odd multiples of τ. Following the single term HB approximation, first we setwhich satisfies the initial conditions in (4). Substituting (6) into (3), and equating the coefficient of cos τto zero yield which can be solved for ? 0 as a function of A . Therefore, the first approximation to the period and periodic solution of non-linear oscillator are Here, u 0 in the main part of periodic solution u (τ),and ? u0 is the correction part, ? 0is the main part of ? , ?? 0 is the correction part . Substituting(10) and (11) into(3), making linearization with respect to the correction terms ? u0 and ?? 0 leads to...
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