A solution procedure based on the Ateb function for a two-degree-of-freedom oscillator

In this paper vibration of a two mass system with two degrees of freedom is considered. Two equal harmonic oscillators are coupled with a strong nonlinear viscoelastic connection. Mathematical model of the system is two coupled second-order strong nonlinear differential equations. Introducing new variables the system transforms into two uncoupled equations: one of them is linear and the other with a strong nonlinearity. In the paper a method for solving the strong nonlinear equation is developed. Based on the exact solution of a pure nonlinear differential equation, we assumed a perturbed version of the solution with time variable parameters. Due to the fact that the solution is periodical, the averaging procedure is introduced. As a special case vibrations of harmonic oscillators with fraction order nonlinear connection are considered. Depending on the order and coefficient of nonlinearities bounded and unbounded motion of masses is determined. Besides, the conditions for steady-state periodical solution are discussed. The procedure given in the paper is applied for investigation of the vibration of a vocal cord, which is modeled with two harmonic oscillators with strong nonlinear fraction order viscoelastic connection. Using the experimental data for the vocal cord the parameters for the steady-state solution which describes the flexural vibration of the vocal cord is analyzed. The influence of the order of nonlinearity on the amplitude and frequency of vibration of the vocal cord is obtained. The analytical results are close to those obtained experimentally.