Some Analytical Approximate Methods For Large Amplitude Nonlinear Oscillations

Physical and mechanical oscillatory systems are often governed by nonlineardifferential equations. In many cases, it is possible to replace a nonlineardifferential equation by a corresponding linear differential equation thatapproximates the original nonlinear equation closely enough to give useful results.Often such linearization is not feasible and for this situation the original nonlineardifferential equation itself must be dealt with. The general theory and method ofdealing with linear differential equations are highly developed branches ofmathematics, whereas very little of a general nature is known about arbitrarynonlinear differential equations. In general, the study of nonlinear differentialequations is restricted to a variety of special classes of the equations and themethod of solution usually involves one or more of a limited number oftechniques to achieve analytical approximations to the solutions. The most common and most widely studied methods of all analyticalapproximation methods for nonlinear differential equations are the perturbationmethods. These methods involve the expansion of a solution to a differentialequation in a series in a small parameter. They include the L-P method, the KBM 7Abstractmethod and the Multi-time expansion. However, these methods apply to weaklynonlinear oscillations only. The method of harmonic balance is another procedurefor determining analytical approximations to the solutions of differentialequations by using a truncated Fourier series representation. An importantadvantage of the method is its applicability to nonlinear oscillatory problems forwhich the nonlinear terms are not "small", i.e., no perturbation parameter needsto exist. In general, the success of harmonic balance method requires that thenonlinear restoring force ? f (x) is an odd function of x , where x represents thedisplacement measured from the stable equilibrium position. If this condition isnot satisfied, the method of harmonic balance, when used in lowest order, leads toinconsistencies. In addition, applying the method of harmonic balance or itsvarious generalization to construct higher-order approximate analytical solutionsis also very difficult, since they require analytical solution of sets of algebraicequation with very complex nonlinearity. In this dissertation, some analytical approximate methods are presented tosolve large amplitude nonlinear oscillations of single-degree-of-freedomconservative systems. The most interesting features of these new methods are itssimplicity and its excellent accuracy in a wide range of values of oscillationsamplitudes. These analytical approximate periods and corresponding periodicsolutions are valid for small as well as large amplitudes of oscillation, includingthe case of amplitude of oscillation tending to infinity.1. A modified method of harmonic balance for large amplitude nonlinear oscillations Consider a single-degree-of-freedom conservative system governed by 8Abstract d2x + f (x)= 0, (1a) dt2 x(0) = β , dx (0)= 0. (1b) dt By introducing an independent variableτ = ωt , and equation (1a,b) can berewritten as ω2x′′+ f (x)= 0, (2a) x(0) = β , x′(0) = 0 (2b)where a prime represents derivative with respect toτ . The new independentvariable is chosen in such a way that the solution to Eq.(2a,b), is a periodicfunction ofτ of period 2π . The corresponding period of the nonlinear oscillationis given byT = 2π ω . Here, both the periodic solution x(τ ) and frequencyωdepend on β . Let x(τ ) = x0(τ )+ ?x0(τ ), where x0(τ ) is an initial approx...