# Basics Of The Theory Of Vibratory Immersion and Extraction

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The theory of vibrational and impact-vibrational immersion is connected with the study of essentially non-linear systems, in which the nonlinearity is determined both by the forces of resistance (for example, when studying longitudinal-rotational vibrations) and by the dynamics of the vibration exciter itself, for example, when considering its dynamic model.

In a theoretical study of the processes of vibration and impact-vibration immersion (extraction,) the movement of the systems under study is taken with a period equal to the time of one or several revolutions of the eccentrics. In the mathematical model of the process, this period is represented as successive stages of movement, characterized by a certain set of acting forces and moments. Each stage is described either by dynamic equations, which are second-order differential equations (system motion,) or by static equations, which are first-order trigonometric equations (system stop.)

The composition of the possible stages of movement in the considered period (cycle) is determined by the accepted calculation scheme of the problem being solved and is reflected in the algorithm.

The algorithm sets the conditions for the end of one or another stage of movement–the transition to another set of acting forces and moments, as well as the conditions for the completion of the cycle. However, the sequence of stages and even their number in a cycle are often unknown and are revealed only in the process of solution, which makes it difficult to use approximate calculation methods, since their application requires data to determine the nature of the movement.

In those cases where it is possible to use approximate methods, with their help it is possible to obtain a solution in an analytical form, which makes it possible to directly calculate the desired characteristics of the process, as, for example, was done in §5 (vibrational extraction during longitudinal vibrations) and in §6 (vibrational extraction during rotational, helical and longitudinal-rotational oscillations.)

Of the known exact methods–the method of phased integration, the inverse method and the method of point transformations for the problems under consideration–it is most expedient to use the first method, which corresponds to the features of calculations on a computer and does not require an analysis of the stability of motion (inverse method) in the process of solving and the involvement of geometric images and interpretation motion (method of point transformations.)

This is the introduction to Part II, which contains the following chapters:

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