Volume 4 (1999), No 3, pp. 1523 
15 
Exact analytic solution for 1D infinite vibrant elastic lumped line 

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 METALPOLYMER RESEARCH INSTITUTE OF NASB MATERIALS TECHNOLOGIES TOOLS SCIENTIFICENGINEERING JOURNAL COMEL  MPRI NASB  1999, Vol. 4 (1999), No.3, pp. 513 MECHANICS OF STRESSEDLYDEFORMED STATE 
Exact analytic solution for 1D infinite vibrant elastic lumped line S.B. Karavashkin Special Laboratory for Fundamental Elaboration SELF email: selftrans@yandex.ru , selflab@mail.ru We will analyse the most important drawbacks of conventional solutions for the problem of vibrant infinite 1D elastic lines with lumped parameters. We will present the exact analytic solutions for forced and free vibrations of semifinite and infinite homogeneous elastic lines. We will analyse these solutions, examine their physical meaning and verify them, to perform their exactness and completeness. 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. The problems of conventional base models and their solutions Conventionally all problems of wave physics are modelled, basing on two types of problems that study vibration of systems having distributed and lumped masses. On the whole, the solution for distributed systems is known. But most of real vibrant systems cannot be considered through this model, we have to model them as lumped systems. This necessity grows year to year with the increasing frequencies of vibrations that violate the principle of continuity of the studied bodies at the range of these frequencies. We would point the endeavours of numerous authors (see, e.g., [1, 2]) using the exact solution for distributed lines that tried to compose some compound systems of distributed and lumped masses, in order at least to estimate the parameters of lumped vibrations. But if we can study the exact pattern of processes in space and time, this is always preferable, because even most successful estimations and perfect algorithms cannot serve instead exact solutions as in the amount of information as in the range of possible analysis in modelling.

Fig. 1. The typical mechanical model of an infinite chain of elastically constrained lumped masses (after H.G. Pain [3])

Consider the most widespread methods to solve the problem of lumped vibrations. In Fig. 1 the typical mechanical model of an infinite chain of elastically constrained lumped masses is presented after H.J. Pain [3]. The differential system of equations describing this model has, after Pain too, the following form: 
(1) 
where m is the mass of line element, _{n} is the momentary longitudinal shift of the nth element of line and s is the stiffness coefficent of line. The expression 
(2) 
where f is the vibration frequency in the line, is the circular frequency of affecting force and a is the betweenelements distance, is thought to be the solution of this system. But if we try to substitute (2) to the initial system (1), we will see this expression to be not a solution. The more, if in an infinite system there is a forced vibration, one more expression has to appear in (1), to account the external force affection as 
(3) 
where F (t) is the external force affecting the line. Naturally, (2) does not satisfy this equation all the more. Besides, as we will see further, the given system has one, not three essentially different solutions for forced vibrations and the same for free ones. Thus, we can unambiguously conclude that the exact solution for the problem of lumped systems does not exist for models of infinite line. In this paper we will consider some results of study that we undertook in order to improve this situation. 