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The features of inclined force acting

The features of inclined force affection on 1D homogeneous

elastic lumped line and related modernisation

of the equations of wave physics

Sergey B. Karavashkin

Special Laboratory for Fundamental Elaboration SELF

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

 Abstract

We will analyse the exact analytical solutions for 1D elastic lumped lines affected by external force inclined to the line axis. We will show that in this case an inclined wave described by an implicit function propagates along the line. We will extend this conclusion both to free vibrations and to distributed lines. We will prove that the presented solution in the form of implicit function is a generalising for the wave equation.

When taken into consideration exactly, the pattern of dynamical processes leads to the conclusion that the divergence of a vector in dynamical fields is not zero but proportional to the scalar product of the partial derivative of the given vector with respect to time into the vector of wave propagation direction.

Keywords: Mathematical physics, Wave physics, Dynamics, Elastic lumped lines, Inclined force action, General solution of the wave equation, Vector algebra, Divergence of vector in dynamical fields, ODE systems

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 the previous papers of this cycle, [1]- [3], we considered longitudinal vibrations in a 1D elastic lumped line. As well, the vibrations of more general form are possible in such line, if we attribute the concept of one-dimensionality only to the general shape of a line, not to the degree of freedom of the line elements vibration. In this paper we will study the features of exact analytical solutions for this class of problems.

We will also show some results developing the vector algebra tool for dynamic fields, as well as some important improvements of general solution of the wave equation. Though these results seem to be simple, they will considerably improve our understanding of basic models of dynamic systems; it will be worthy to formulate them. This is why, having a target to formulate most completely the basis of the exact analytical solutions for dynamical many-body systems, we will not emphasise the complicacy either simplicity of considered problems. We will draw our attention to the importance of such or other aspect in general context of the studied phenomenon.

2. Vibrations in a semi-finite elastic line with an inclined external force affecting the line start

In the view of stated above, supposing an elastic line elements having two degrees of vibration freedom, we can present this model as shown in Fig. 1.

Fig. 1. The model of inclined vibrations in a 1D elastic lumped line

In case of small vibration amplitude (linear vibrations), this model can be described by two systems of equations – in x- and y-projections of the external force affection relatively:

and

where _i is the momentary longitudinal shift of the kth element of the line, y_i is the shift of the ith element of elastic line in the vertical plane, m is the mass of line element, s is the coefficient of line stiffness, F (t) is the external force affecting the line, and is the angle of an external force inclination to the line axis.