V.2 No 1

71

Some features of the forced vibrations modelling

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

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

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

We will survey the conventional methods to calculate the systems that model vibrant 1D elastic lumped lines, in comparison with the new non-matrix method to yield the exact analytical solutions for such systems. We will consider the features arising when the external force affects an interior element of such system. We will analyse the conditions of the limiting process to the related distributed lines and derive the conditions at which a lumped line can be modelled by a distributed line.

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

A salient feature of vibration processes is that they are very diversiform, while not many parameters affect them - such as the mass of elements of the system, the constraints stiffness, the type of dissipative forces and some others. To a definite extent this is because vibrations depend on the structure of model that 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 these 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 each other. The more, most methods have quite narrow specialisation; it essentially hinders the general studying of vibration process regularities. As we will show below, the methods used for distributed systems cannot give the exact analytical solutions for the whole complex of lumped models; the methods used for infinite models are inapplicable to those finite, and so on.

Many methods conventionally thought analytical can be rather classified as numerical by their essence. They cannot give a complete pattern of dynamical processes in all their interconnections. To move this limitation over, we have developed the original non-matrix method allowing to yield exact analytical solutions. In [1], [2] and [3] we presented some results which we yielded for infinite and finite elastic lumped lines. Studying them, we proceeded, on one hand, from the idea that it will be convenient to present the new method, beginning with simple models easily comparable with the results known in some particular cases. On the other hand, “in the engineering view, 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 between-the-disks shaft sections serve as the elastic constraints” [4, p. 277]. The integring feature of models considered in the above references is that the external force affected always the start of line, while practically we often meet a case when the external force affects the interior elements of a line. This difference has a great effect both on the pattern of solution and on the pattern of forced vibrations produced in the line.

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