When a large number of different systems of the same generic form are to be designed, or when general characteristics of a set of systems are to be investigated it is desirable to plot the results in terms of dimensionless parameters if possible. For example, transforming the equation of motion is shown for the investigated case. It has been shown that the result may be generalized by plotting the non-dimensionless quantiting “M” and the phase angle “φ” against the ratio “n” for various values of “δ”.
When the energy loss is proportional to the square of the amplitude of strain it is known as linear hysteric damping. It may be presented as a result of the short forth part of this chapter.
In many practical situations the exciting force has a more complex waveform than the simple harmonic function. The investigations of this waveform are described in the next part. A reciprocating masses driven by slider-crank mechanisms, torque generated by a reciprocating engine and force generated by electromagnetic vibrators are known as examples of this phenomenon. It has been stressed in the article that many periodic forcing functioning which arise in engineering practice may be represented by the Fourier series. Fortunately this does not make the vibration analysis of such systems much more complicated as they may generally be considered as linear, and therefore the principle of superposition applies. It has been shown there that the motion of the system is the sum, in their correct phase relationships, of the motions arising from each of the harmonics applied separately. This is an important point in understanding the vibratory phenomena. The responses of a system to a complex cyclic exciting force are given in the article. They illustrate the sum of the harmonics which has been mentioned above.
A force which causes disturbances to be propagated through the ground, which could interfere with proper functioning of delicate instruments or machines, or, if large, cause damage to nearby buildings is a central task of such very small part of the work as the sixth one.
A case of excitation by a non-periodic force is considered in this chapter too. When the exciting force is non-periodic, such as that arising from an earthquake or due to the blast from an explosion, another method for calculating the response of a system is required. There is no difficulty in representing any varying force as a series of impulses. This is known as a standard method for this case. Therefore, the total response of the system to a non-periodic force may be obtained approximately by superimposing the responses to the discrete impulses which approximately represent the applied force. Special attention is paid to the integral, which is known in mathematical texts as the convolution integral or Duhamel’s integral. This equation (or integral) is the result of investigations carried out there.
Eventually, the last part of the first chapter is devoted to the variation of amplitude with time. The manner in which the steady state amplitude has been achieved hasn’t been previously concerned. This is reasonable because for most practically important systems the steady amplitude is achieved quickly and the rate at which it is approached is not of great interest. However, many mechanical systems have one or more resonant frequencies, or critical speeds, below their normal operating speeds and it is common knowledge that a system may be run through critical speeds up to its normal operating speed, without deleterious effects. Thus the time variation of amplitude may be of major interest when running through a critical speed but unfortunately great difficulty is experienced in finding a mathematical expression for it. Therefore, the approach to examining an easier case, the build-up of amplitude at a constant exciting force frequency equal to the natural frequency of the system has been described. So when excited at its natural frequency, the amplitude of an undamped system grows linearly with time. It may be considered as a result of carried out investigation.
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