In the present paper the separation variables method is applied to finding an exact solution of the sixth-order mathematical model with nonsmooth solutions. Analyzing arising difficulties, in particular the spectral problem, we use a pointwise method of interpretation of solutions proposed by Yu.V. Pokornyi. This method showed its effectiveness in constructing of an exact parallel to the classical theory of differential equations, including oscillation theorems, both second and fourth orders.

In the present paper, the growth rate of eigenvalues for the fourth-order spectral problem with nonsmooth solutions is obtained. It arises when we apply the Fourier method to the mathematical model, describing small free vibrations of a mechanical system consisting of pivotally connected rods. We assume that at the connection points there are springs that respond to rotation, while the system is in the external environment with localized features, leading to a loss of smoothness of the solution. The analysis of the problem is based on the pointwise approach proposed by Yu. V. Pokorny, and proved to be effective in studying linear boundary problems of the second and fourth orders with continuous solutions (an exact parallel with oscilation theory of ordinary differential equations is constructed).