2019. Vol. 14. Issue 2, Pp. 138–141

URL: http://mfs.uimech.org/mfs2019.2.019,en

DOI: 10.21662/mfs2019.2.019

URL: http://mfs.uimech.org/mfs2019.2.019,en

DOI: 10.21662/mfs2019.2.019

Longitudinal oscillation of a rod with a variable cross section

Utyashev I.M.

Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in
percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal
vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal
vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional
area as an exponential function of a polynomial of degree

rod,

eigenfrequencies,

eigenvalues,

longitudinal vibrations,

section function,

section area

**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

The study **found** that:

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

**Conclusions:**

- 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.
- 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.

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https://elibrary.ru/item.asp?id=20519086 - Akulenko, L.D., Baidulov, V.G., Georgievskii, D.V. et al. Evolution of Natural Frequencies of Longitudinal Vibrations of a Bar as Its
Cross-Section Defect Increases // Mech. Solids. 2017. No. 52. Pp. 708–714.

DOI: 10.3103/S0025654417060103 - Pavlov V.P. [Transverse vibrations of a rod with a variable cross section and the calculation of its natural frequencies by the splines method]
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DOI: 10.21662/mfs2018.4.014