Page 432 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

13

Random Vibrations

The types of functions we have considered up to now can be classified as
deterministic, i.e., mathematical expressions can be written that will determine
their instantaneous values at any time t. There are, however, a number of physical
phenomena that results in nondeterministic data for which future instantaneous
values cannot be predicted in a deterministic sense. As examples, we can mention
the noise of a jet engine, the heights of waves in a choppy sea, ground motion
during an earthquake, and pressure gusts encountered by an airplane in flight.
These phenomena all have one thing in common: the unpredictability of their
instantaneous value at any future time. Nondeterministic data of this type are
referred to as random time functions.

13.1 RANDOM PHENOMENA

A sample of a typical random time function is shown in Fig. 13.1-1. In spite of the
irregular character of the function, many random phenomena exhibit some degree
of statistical regularity, and certain averaging procedures can be applied to
establish gross characteristics useful in engineering design.
In any statistical method, a large amount of data is necessary to establish
reliability. For example, to establish the statistics of the pressure fluctuation due to
air turbulence over a certain air route, an airplane may collect hundreds of records
of the type shown in Fig. 13.1-2.
Each record is called a sample, and the total collection of samples is called
the ensemble. We can compute the ensemble average of the instantaneous pressure
at time t^. We can also multiply the instantaneous pressures in each sample at
times and + r, atid average these results for the ensemble. If such averages
do not differ as we choose different values of t^, then the random process
described by this ensemble is said to be stationary.

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