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Some features of the forced vibrations modelling

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

187 apt., 38 bldg., Prospect Gagarina, Kharkov 61140 Ukraine

Phone: 38 (0572); e-mail: selflab@go.com , selflab@mail.ru

23 February, 2002

This is the third leading paper of a large cycle devoted to our new method to obtain the exact analytical solutions for vibrant elastic lumped and distributed systems. Its initial version was published in "Materials. Technologies. Tools", the journal of National Academy of Sciences of Belarus, 5 (2000), 3, pp.14-19 (in Russian). In the present version we retained generally the original part of the published paper, so the English-language readers can consider it as the elaborated version of the published paper. We have essentially widen the conventional methods analysis; in our view, it presents more brightly the advantages which our method adds to the calculation scope and which we embody in our investigations, both published and being about publishing in the nearest future.

Abstract

In this paper we survey the conventional methods to calculate the systems modelling vibrant 1D elastic lumped lines, in comparison with the new non-matrix method to obtain the exact analytical solutions for such systems. We consider the features arising when an external force acts on an interior element of such system. We analyse the conditions of the limiting process to the distributed lines and derive the conditions, when a lumped line can be modelled by the methods conventional for distributed lines.

Keywords: Mathematical physics; Wave physics; Theory of many-body systems; ODE systems; Finite deformation; Oscillation theory; Dynamical systems

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

We know the multiple types of vibrations, though they are caused by a small number of parameters: such as the system elements masses, the constraints stiffness, the type of dissipative forces and some others. The dependence of vibrations on the model structure conditions it and makes necessary to use a broad gamut of the basic models describing these processes. Even the most general classification divides the models into 1D, 2D and 3D, lumped and distributed, homogeneous and heterogeneous, dissipative and not, having one or few degrees of freedom, forced and free vibrations etc. For each of the mentioned types of basic models a special approach has been developed, and for most of them even several different approaches whose results often do not match. The more, most of the methods have quite narrow specialisation; it essentially hinders the general studying of the vibration process regularities. As we will show below, the methods used for distributed systems cannot give the exact analytical solutions for the entire complex of lumped models; the methods used for infinite models are inapplicable to those finite, and so on.

Many methods which one conventionally thinks analytical can be actually classified as numerical by their essence. They cannot give a complete pattern of all interconnections of a dynamical processes. To move over this limitation, we have developed the original non-matrix method to obtain exact analytical solutions. In [1], [2] and [3] we presented some results which we have obtained for infinite and finite elastic lumped lines. Stating them, we proceeded, on one hand, from the mind that it will be convenient to present the new method, beginning with the simple models easy comparable with the known results in some particular cases. On the other hand, ''in the engineering viewpoint, the particular case is important when the oscillators are put in series so that the nth oscillator is connected only with the previous (n- 1)th and next (n+1)th ones. As an example we can take a shaft with the disks fitted on it. The disks behave as the vibrant masses, and the between-the-disks shaft sections serve as the elastic constraints'' [4, p.277]. The integrating feature of the models considered in the mentioned sources is that the external force acted always on the start of line. But practically we often meet a case when the external force acts on the interior elements of a line. This difference essentially effects both on the pattern of solution and on the pattern of forced vibrations produced in the line.

In this paper we will analyse the features of vibrations caused by the mentioned reason. In this way we will develop the solutions obtained by the new method, extending them to the more complicated models of elastic systems.