Article outline

Purpose: Determination of natural frequencies of longitudinal vibrations of an elastic rod with a variable cross-sectional area.

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in shock mechanisms (hubs). The strength of such parts directly depends on the natural frequencies of longitudinal vibrations.

Methodology: Longitudinal vibrations of an elastic rod are described by a second-order differential equation, where the cross-sectional area is a function of the coordinate and varies along the axis. The main difference between this work and the works of other authors is that the area function is an exponential function of a polynomial of degree n. This view has several advantages. For example, the cross-sectional area will always be non-negative. We will seek a general solution in the form u(x,t)=y(x)cos(wt), substituting which in the differential equation, we obtain the Sturm-Liouville problem (eigenvalue problem). The second advantage of the exponential function of the polynomial is that the derivative of its function always exists and in the formulation of the Sturm-Liouville problem, the cross-sectional area enters the equation as the derivative of the polynomial, and the exponent is reduced. Further, expanding the linearly independent functions of the fundamental system of solutions in a power series in the variables x and λ, we obtain a general solution. Substituting which into the boundary conditions, we obtain the frequency equation. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given.

The study found that:

1. For a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod.
2. With an increase in the rigidity of fixing at the right end, the natural frequencies increase for all cross-sectional profiles.

Conclusions:

1. The method of modeling the cross-sectional area presented in this work allows one to approximately determine the natural frequencies of the longitudinal vibrations of the rod. An exponential function of a polynomial of degree n approximates smooth section functions well. If the section function is piecewise defined, then the degree of the polynomial increases greatly, which leads to the accumulation of calculation errors.
2. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.