Application of Ateb and generalized trigonometric functions for nonlinear oscillators

Abstract

In this paper, the new application of the generalized trigonometric function (GTF) and of the Ateb function in strong nonlinear dynamic systems is considered. It is found that the GTF and the Ateb function represent the closed-form solution of the purely nonlinear one-degree of freedom oscillator with specific initial conditions. Definition of the GTF and Ateb functions is introduced. In spite of the fact that both functions use the incomplete Beta function and its inverse form, the difference exists according to the definition of both of these functions. The correlation between these two types of functions is exposed. Main properties of the Ateb function and of the special GTF function with parameters $$a = \frac{1}{2}$$ and $$b = \frac{1}{\alpha +1}$$ , which are the solution of the pure nonlinear oscillator, are compared and the value of the functions are calculated. Special attention is directed toward the sine GTF and the cosine Ateb function. Advantages and disadvantages of the both type of solutions are discussed.