Труды Института механики им. Р.Р. Мавлютова

Khakimov A.G. Determining the velocity of a moving rod and the thickness of its coating using natural frequencies of flexural vibrations Proceedings of the Mavlyutov Institute of Mechanics. 2016. 11(1). 10–15.

2016. Vol. 11. Issue 1, Pp. 10–15
URL: http://proc.uimech.org/uim2016.1.002,en
DOI: 10.21662/uim2016.1.002

Determining the velocity of a moving rod and the thickness of its coating using natural frequencies of flexural vibrations

Khakimov A.G.

Mavlyutov Institute of Mechanics, Ufa

Abstract

Research has been performed on natural transverse vibrations of a portion of a constant length in a straight thin coated rod moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the rod moving portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase in the velocity parameter. It is also shown that a decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase of the coating thickness. Using two frequencies of flexural vibrations, we can determine the velocity parameter of the moving rod and the thickness of its coating. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing the coated products: in textile industry, wire manufacturing process, metallurgy (especially for rolling metal bars and strips), wire drawing, plastic products and paper rolls manufacturing and can be used to determine the velocity and the coating thickness of a moving rod, strip or wire using two frequencies of flexural vibrations.

Keywords

rod movement,
coating,
flexural vibrations,
natural frequencies,
primal and inverse problems

Article outline

Purpose: Determining the velocity parameter of a moving rod and the thickness of its coating using two natural frequencies of flexural vibrations.

Methods: Research is performed on the natural frequencies of flexural vibrations of a rod moving between two supports in the form of a sliding connection and affected by the tensile force. No flexural rigidity is taken into account. The Kirchhoff's equation of flexural vibrations in a rod is used. In the general case the roots of the characteristic equations are found using the Ferrari’s formulae. In order to determine the velocity and force parameters, a set of two equations is written down, the solution of which is done using the method of successive approximations in the domain of unique dependence between the velocity and force parameters and the vibration frequency of the moving rod.

As a result of the research, it has been found that:

1. A decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase in the velocity parameter.

2. A decrease in the natural frequencies of flexural vibrations of the rod occurs with an increase of the coating thickness.

Thus, using two frequencies of flexural vibrations, one can determine the velocity of a moving rod and the thickness of its coating.

Conclusions: The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms: in manufacturing coated products: in the textile industry, wire manufacturing process, metallurgy (especially for rolling metal bars and strips), wire drawing, plastic products and paper rolls manufacturing can be used the velocity and the coating thickness of a moving rod, strip or wire using two frequencies of flexural vibrations.