ISSN 2658–5782

DOI 10.21662

DOI 10.21662

Electronic Scientific Journal

2019. Vol. 14. Issue 3, Pp. 214–217

URL: http://mfs.uimech.org/mfs2019.3.028

DOI: 10.21662/mfs2019.3.028

URL: http://mfs.uimech.org/mfs2019.3.028

DOI: 10.21662/mfs2019.3.028

Determination of the dimensions of the cylindrical weight at the end of the rod

Aitbaeva A.A.

Mavlyutov Institute of Mechanics UFRC RAS, Ufa

This article discusses free transverse vibrations of a homogeneous rod. The left end of the rod is clamped, and a cylindrical weight is concentrated at the right end. The eigenfrequencies of the rod vibration are known. The purpose of this work is to determine the parameters of the end cylindrical weight of the rod (mass, moment of inertia, length and radius) by the natural frequencies of the rod vibrations. We use a partial differential equation derivative of the fourth order to solve this problem. This equation and boundary conditions are reduced to a spectral problem. To find the mass and moment of inertia of the weight, the «Method of an additional unknown» was applied. In the characteristic determinant of the spectral problem, there are terms that contain products of unknown coefficients. The essence of the «Method of an additional unknown» is that some of these products are proposed to be considered new additional unknowns, through which the rest can be expressed. It is shown that the mass and moment of inertia of the weight can be found using the three natural frequencies of the rod vibrations. Formulas for finding the length and radius of a cylindrical weight are obtained, and corresponding examples of finding unknown parameters are considered.

eigenvalues,

natural frequencies,

cylindrical weight,

rod

**Problem:** To determine the parameters of the end cylindrical weight using the natural
frequencies of the rod’s vibrations. The parameters are the mass and moment of inertia, length and radius of the cylindrical end weight.

**Methods:** This article discusses the free bending vibrations of a homogeneous rod. The left end of the rod is sealed,
and a cylindrical weight of mass _{1} and moment of inertia _{1} is concentrated on the right end.
Length _{2} are the dimensions of the cylindrical weight. These parameters are considered unknown.
The eigenfrequencies of vibrations of the rod is used as known acoustic data.

To solve this problem, we use a fourth-order partial differential equation that describes the bending vibrations of the rod that
has a constant bending stiffness. The equation and boundary conditions are reduced to the problem with the spectral parameter _{1} = _{2} = _{1}/(_{3} = _{1}_{2}. Substituting three known eigenvalues into the characteristic
determinant we obtain a system of three equations from three unknowns. To solve the system, we use Kramer's method and find the
dyenhrtcoefficients _{1} and _{2}. Using the known formulas for determining the mass and moment of
inertia of the cylinder, as well as knowing the coefficients _{1} and _{2}, we find the length and radius of the weight:

_{1}^{2}_{1}^{2}_{2}_{1}^{2} -
2_{1}^{2}_{1}_{1}),
_{2} = √((2_{2}^{2} -
_{1}^{2}_{1})/_{1}),

where _{1} is the density of material of weight, _{1}is the radius of the rod (inner radius of the cylinder).

In a study was **determined**:

The mass and moment of inertia of the cylindrical weight attached to the right end of the rod are found by three natural frequencies of bending vibrations of the rod.

Formulas for finding the length and radius of the end cylindrical weight are obtained.

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https://www.ugatu.su/assets/files/documents/dissov/06/2018/AitbaevaAA/Autoref_AitbaevaAA.pdf