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S.B. Karavashkin, O.N. Karavashkina

Bend in elastic lumped line and its effect on vibration pattern

S.B. Karavashkin and O.N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

e-mail: selftrans@yandex.ru , selflab@mail.ru

March 02, 2002

Abstract

We prove that the bend in an elastic line does not effect on the solution pattern only, if the longitudinal and transverse stiffnesses of a line were equal. Basing on the proved theorem, we consider some models typical for the applications, particularly, models of a semi-finite elastic bended line, a homogeneous closed-loop elastic line and an elastic line having inequal longitudinal and transverse stiffness coefficients. We show that in the lines obeying the theorem conditions, with the remaining general solution, the features of vibration processes are conditioned by the regularities of the coordinate system transformation. In case of inequal stiffness coefficients in the bend region, the complex dynamical thrusts and vibration breakdowns take place, and the vibration amplitude grows. In the bend region the resonance peaks arise; their frequencies do not coincide for longitudinal and transverse components of the wave process. This last is the cause that in one and the same elastic line, with an invariable angle of external force inclination, dependently on frequency, the longitudinal, transverse or inclined waves can propagate along the line. With it, the wave inclination does not coincide with the external force inclination, as it takes place in the lines having equal stiffness coefficients. As the examples we will consider some aspects of application of these models to the geophysical problems.

Keywords: Mathematical physics; Wave physics; Theory of many-body systems; ODE systems; Dynamics; Heterogeneous dynamical systems; Elastic bended systems; Nonlinear vibration systems; Wave propagation in nonlinear media; Geophysics; Tectonics; Seismology

Classification by MSC 2000: 34A34, 34C15, 37N05, 37N15, 70E55, 70J40, 70K30, 70K40, 70K75, 74H45.

Classification by PASC 2001: 02.60.Lj; 05.45.-a; 45.05.+x; 45.10.-b; 45.20.-d; 45.20.Dd; 45.30.+s; 45.50.-j; 45.50.Jf; 46.15.-x; 46.25.Cc; 46.40.-f; 46.40.Cd; 46.40.Ff; 46.50.+a; 46.70.-p; 46.70.Lk

1. Introduction

With all importance for the practice of modelling of elastic lumped lines having one or few bends, more or less general approaches to their solution are still absent. In the literature (see, e.g., [1]-[4] etc.) we can find mainly approximate investigations of 1D elastic lines basing directly or indirectly on the matrix methods to find the characteristic frequencies. This occurs because, though “exact analytical methods are preferable in analysis, however to obtain analytical formulas for solutions of even comparably simple differential equations meets sometimes great difficulties” [5, p.10]. Furthermore, “the lumped system in which the impedances, compliances and masses can be chosen arbitrarily is quite complicated problem. It is much more complicated than the system having distributed parameters of mass, compliance, resistance. Actually, the system with lumped parameters corresponds to that distributed which can be loaded at different points by a mass and can consist of connected equal parts. So the system with lumped parameters is hardly analysable theoretically” [1, p.156]. We know analytical investigations of elastic systems having three or four hinged rods [6], when the modelling system of differential equations offers after the separation of variables to find the natural frequencies directly. In more general cases the quasi-static approximations are used by the impact coefficient [1], [2], or using St. Venant principle [7], [8]. The described approaches naturally have all defects inherent in quasi-static approaches. The influence coefficient introduced without knowing all phase and amplitude distinctions of dynamical process at the line heterogeneities disables us to take into account correctly the phase dependencies, when summing the point deviation conditioned by the effect of all rest elastic system elements. And St. Venant principle disables us to investigate the action of far elements of the system.

The problem of bended systems has a great application. As in this paper we will present its application to geophysics, we give here the statement of this problem in tectonic and seismic processes.