Solutions for finite elastic lumped lines

Volume 4 (1999), No 4, pp. 5-13	5
Solutions for finite elastic lumped lines

NATIONAL ACADEMY OF SCIENCES OF BELARUS STATE COMMITTEE ON SCIENCE AND TECHNOLOGIES OF REPUBLIC OF BELARUS ASSOTIATION “NOMATECH” INSTITUTE OF RESOURCE SAVING PROBLEMS V.A.BELYI METAL-POLYMER RESEARCH INSTITUTE OF NASB MATERIALS. TECHNOLOGIES. TOOLS 1999, #4, pp.5-13

MECHANICS OF STRESSEDLY-DEFORMED STATE

Exact analytical solution for 1D elastic homogeneous finite lumped line vibration S.B. Karavashkin Special Laboratory for Fundamental Elaboration SELF 187 apt., 38 bldg., Prospect Gagarina, Kharkov 61140, Ukraine Phone: +38 (0572) 276624; e-mail: sbkarav@altavista.com Abstract We will analyse in this paper the principal demerits of conventional approaches to the problem of vibrant 1D homogeneous finite lumped line and present the exact analytical solutions for forced and free vibrations in finite lines with free ends and with the free end and fixed start. We will analyse these solutions and their distinctions from the conventional concept on the vibration pattern in such lines. We will give the check of presented solutions proving them to be complete and exact Keywords: Mathematical physics, Wave physics, Dynamics, Finite elastic lumped lines, ODE systems, Microwave vibrations in elastic lines Classification by MSC 2000: 30E25; 70E55; 70J35; 70J60; 70K40; 70F40 Classification by PASC 2001: 02.60.Lj; 05.10.-a; 05.45.-a; 45.30.+s; 46.15.-x; 46.25.Cc; 46.40.-f; 46.40.Fr 1. Introduction In [1] we investigated the exact analytical solutions for infinite 1D elastic lumped lines and showed that on distinct of conventional solutions, the exact analytical solutions have generally three vibration regimes: periodical, aperiodical and critical. With it the vibration pattern essentially changes dependently on the line type (semi-finite or infinite) and on the vibration conditions, i.e., are they forced or free. Generally, finite lines have the same distinctions, but also some others are inherent in them. In this paper we will study these features. 2. The drawbacks of conventional methods Two basic approaches are currently applied to find exact solutions for vibrations in 1D finite elastic lumped lines. They are the allowed modes method (see, e.g., [2]) and Krylov method (see, e.g., [3], [4] or [5]). To see their drawbacks, analyse each briefly.

In the allowed modes method the standard system of equations
	(1)
is taken as the basis. We will study it after Pain [2] relatively to the transversal vibrations (see Fig. 1); the essence of this technique for those longitudinal is the same. The solution of this system of equations is sought in the following form:
	(2)
with the following boundary conditions for a fixed-ends finite line:
	(3)
Substituting y_r to the initial system (1), we yield a system of algebraic equations of a following type:
	(4)
which can be naturally presented as the system of ratios constant for given line parameters and frequency :