Application of Ateb and generalized trigonometric functions for nonlinear oscillators
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R A P I D C O M M U N I C AT I O N
L. Cveticanin
· S. Vujkov · D. Cveticanin
Application of Ateb and generalized trigonometric functions for nonlinear oscillators
Received: 7 May 2020 / Accepted: 20 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 1 Ateb function and of the special GTF function with parameters a = 21 and b = α+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. Keywords Ateb function · Generalized trigonometric function · Purely nonlinear oscillator 1 Introduction Recently, in the era of computers, vibration of the purely nonlinear oscillator (differential equation of motion is without linear deflection term) is obtained numerically. The computed approximate numeric solution has high accuracy, but is valid only for certain parameter values and initial conditions and is, in general, not sufficient for motion analysis. To overcome this lack, the closed-form analytic solution for the purely nonlinear oscillator has to be introduced. It is known that the solution has to be periodical, dependent on parameters of the oscillator and initial conditions, and with constant amplitude. Mathematically, the solution of the purely nonlinear oscillator is the inverse version of the incomplete Beta function which is defined as the generalized trigonometric function (GTF) or Ateb function dependently on the initial conditions. Thus, the solution in the form of GTF or Ateb function, which satisfies the equation of motion, gives in the explicit form the dependence of vibration on the variation of oscillator parameters and initial conditions. L. Cveticanin (B) University of Novi Sad, Novi Sad, Serbia E-mail: [email protected] S. Vujkov University of Novi Sad, Faculty of Medicine, Novi Sad, Serbia L. Cveticanin Obuda University, Budapest, Hungary D. Cveticanin Remming, Novi Sad, Serbia
L. Cveticanin et al.
Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are related to the p-Laplacian which is known as a typical nonlinear differential operator. First, the one parameter GTFs were developed [1], but a few years later GTFs with 2 parameters were introduced [2]
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