Closed-form exact solutions for periodic motions of a nonlinear oscillator, which contains a quadratic mixed-parity restoring force, are derived analytically. This oscillator is characterised by two ...real parameters, which are the coefficients of the linear and the quadratic terms, and has single-well potential. All possible combinations of positive and negative values of these coefficients providing periodic motions are considered, and two families of exact solutions are obtained in terms of Jacobi elliptic functions. The periods are given in terms of the complete elliptic integral of the first kind, the behaviour of these periods as a function of the initial amplitude is analysed, and the exact solutions for certain values of these parameters are plotted.
•We study a family of conservative oscillators with integer or non-integer order nonlinearity.•We obtain the Fourier series expansion of the exact solution, though the analytical expression of this ...solution is unknown.•We compute the Fourier coefficients as an integral expression in which a regularized incomplete Beta function appears.•We apply this technique to build up accurate analytical approximate solutions for this type of nonlinear oscillators.•We show that a possible application of this technique is to test the effectiveness of analytical approximate methods.
A family of conservative, truly nonlinear, oscillators with integer or non-integer order nonlinearity is considered. These oscillators have only one odd power-form elastic-term and exact expressions for their period and solution were found in terms of Gamma functions and a cosine-Ateb function, respectively. Only for a few values of the order of nonlinearity, is it possible to obtain the periodic solution in terms of more common functions. However, for this family of conservative truly nonlinear oscillators we show in this paper that it is possible to obtain the Fourier series expansion of the exact solution, even though this exact solution is unknown. The coefficients of the Fourier series expansion of the exact solution are obtained as an integral expression in which a regularized incomplete Beta function appears. These coefficients are a function of the order of nonlinearity only and are computed numerically. One application of this technique is to compare the amplitudes for the different harmonics of the solution obtained using approximate methods with the exact ones computed numerically as shown in this paper. As an example, the approximate amplitudes obtained via a modified Ritz method are compared with the exact ones computed numerically.
Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the ...behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at x=0 are also considered.
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