**Abstract. **Goal of the work is to apply periodic Ateb-functions to investigate dynamic processes of strongly nonlinear systems with a finite number of degrees of freedom. Significance. Practically all problems in mechanics and engineering related to system oscillations, when strictly formulated, are nonlinear, as they are mathematically described by nonlinear differential equations. This often poses significant challenges in studying their behaviour, as finding a solution in exact form can be difficult. One way to overcome this complexity is utilizing special functions, such as Ateb-functions. Therefore, the application of periodic Ateb-functions for investigating the dynamic processes of highly nonlinear systems with a finite number of degrees of freedom is a relevant task, as it allows for increased accuracy and efficiency of estimations. Method. The methodology is based on finding partial solutions of the "normal" oscillations form, which do not correspond to linearized systems. The normal modes of oscillation of highly nonlinear conservative systems, whose potential energy is a homogeneous function of degree ν+1, are described using periodic Ateb-functions. In cases where linearization of the original system is not possible, such an approach to studying the oscillatory processes of highly nonlinear systems with multiple degrees of freedom is the most feasible. Results. The presented methodology for investigating normal modes of oscillations can be generalized to highly nonlinear systems with small perturbations of autonomous and non-autonomous types. Scientific novelty. Mathematical relations have been established to determine the normal modes of oscillations in highly nonlinear mechanical systems. Practical significance. The application of periodic Ateb-functions for investigating dynamic processes in highly nonlinear systems will enhance the accuracy and efficiency of estimations.

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