Basic Definitions of the Theory of Vibrations

 

The theory of vibrations represents a comprehensive section of physics covering a very wide range of problems in the field of mechanics, electrical engineering, radio engineering, optics, etc. The theory of vibrations is of special importance for applied problems encountered in engineering practice, among others in the design of machines and structures. There have been cases when an engineering structure designed for a large factor of safety to withstand static loading failed under the action of relativity small periodically acting forces.

In many cases stiff and very strong structures have proved unserviceable in the presence of varying forces whereas a similar lighter structures and not so strong at first glance, sustains these forces absolutely safely. Hence the problem of vibrations and, in general, the behavior of elastic systems under varying loads call for especial attention of the designer.

In the study of vibrations of elastic systems it is customary to distinguish them primarily by the number of degrees of freedom.

By the number of degrees of freedom is meant the number of independent coordinates defining the position of a system. For example, a rigid mass attached to a spring has one degree of freedom since its position is determined by only one coordinate measured from a certain point. Of course, this is true only so long as mass of the spring may be neglected in comparison with the mass of the vibrating weight. Otherwise, in order to specify the position of the system at any instant of time it would be necessary to introduce an infinite set of coordinates defining the position of all points of the elastic spring, and the system would have an infinite number of degrees of freedom.

If the mass of a shaft is neglected, it may be said that the system has two degrees of freedom it is necessary to have two angular coordinates defining the rotation of the rigid disks. A massive nut rotating and moving along a screw has one degree of freedom since its position in space is determined by only one parameter, for example, the angle of rotation. The movement along the axis depends on this angle. The number of degrees of freedom is virtually determined by the choice of a design scheme, i.e., by the degree of approximation to which we consider it necessary (or possible) to investigate a real object. One can often, therefore, hear the expressions: “We regard the beam as a system with two degrees of freedom”, or “the problem is solved on the assumption that the system has one degree of freedom”. This means that appropriate simplifications have been made in solving a specific problem. With a particular scheme available, it is not difficult to guess their nature.

In study of elastic vibrating systems a distinction is made between natural and forced vibrations, by a natural or free vibration is meant a vibratory motion that is performed by a system freed of external active force action and left to itself. An example of natural vibrations is provided by the vibrations of the prongs of a tuning fork. In this case the motion is induced by an initial impulse imparted to the system when it was struck. The natural vibrations continue until the energy is completely expended in doing work against the friction on air and the internal friction in the metal.

By a forced vibration is meant a motion of an elastic system produced by varying external forces called exciting forces. An example of forced vibrations is the motion performed by an elastic foundation if an imperfectly balanced motor is mounted on it. In this case the motor is a source of energy periodically delivered to the system and expended in overcoming frictional forces in the process of forced vibration. The force exerted by the motor on the elastic foundation is an exciting force.

As is known, the time interval between two successive maximum displacements of an elastic system from its equilibrium position is called the period of natural or forced vibration, as the case may be. The period of vibration is denoted by T. Its reciprocal is called the frequency of vibration.

V=1/T and represents the number of vibrations per unit of time. The frequency is measured in hertz, the hertz being the number of cycles per second.

 

Part 3. Speaking practice

1. Give the rest of the irregular verbs forms. Memorize them:

Withstand, said, made, meant, left, known, struck.