Natural small vibrations of a flat viscoelastic spiral spring

NATURAL SMALL VIBRATIONS OF A FLAT VISCOELASTIC SPIRAL SPRING
I.I. Safarov¹, M.Kh. Teshaev^2,3*, Sh.I.Djuraev⁴, F.F.Khomidov^3
1Department of Mathematics, Tashkent Institute of Chemical Technology, Tashkent, Uzbekistan
²Bukhara branch, Institute of Mathematics named after. V.I.Romanovsky Academy of Sciences of Uzbekistan, Bukhara, Uzbekistan
³Department of Mathematics, Bukhara Engineering and Technology Institute, Bukhara, Uzbekistan
⁴Department of Differential Equations, Bukhara State University, Bukhara, Uzbekistan
^*Corresponding author: [e-mail]
Abstract. Curved pipe systems are widely used in mechanical engineering, nuclear industry, offshore oil production, and aerospace engineering. The purpose of the work is to study small vibrations of a viscoelastic helical spring. Small vibrations of a thin curved rod, the elastic line of which is a flat curve and one of the main directions of the cross section of which lies in the plane of the curve, breaks down into two types: vibrations with displacements in the plane of the curve and with displacements perpendicular to the plane of the curve. The viscoelastic properties of materials are taken into account using complex elastic modulus. Asymptotic expansions are constructed for the eigenfunctions and eigenfrequencies corresponding to both types of oscillations of a repeatedly twisted flat spiral spring with fixed ends. A technique for obtaining resolving equations has been developed.
Keywords: small vibrations, spiral springs, viscoelastic properties, displacements, eigenfunction, frequency
1. Introduction
Curvilinear pipe systems are widely used in mechanical engineering and in many fields of technology, as well as for transporting liquids, as one of the typical and simplest systems for the interaction of fluid structures, constantly encountered in various areas of the national economy. For example, in the nuclear industry, offshore oil production, aerospace engineering, and so on [1-4]. However, due to the oscillatory properties of the liquid medium and the pipeline material, the pipeline system may suffer. Therefore, the study of fluid flow oscillations or forced oscillations of a pipeline under the dynamic action of internal flow is an urgent task [5-7]. In addition, the system of curved pipelines for transporting liquids has been studied very little to date, but is being intensively developed. can display rich dynamic characteristics. The dynamics of pipes transporting fluid has been mainly studied in the last decade [8,9]. Existing literature on this line of research mainly concerns the interaction of straight or curved pipes transporting fluid. For straight pipes transmitting liquid, a linear theoretical model was first developed [10,11]. As an example, the dynamic characteristics of a cantilever pipe were studied, and attention was drawn to the flutter phenomena that occurs in the pipeline when exposed to air flow [12,13]. For a long time, no theoretical model has been developed to study the dynamic behavior of curved pipelines. The developed model deals with a situation where the pipe deformation was considered to be small.