Page 26 - Mechanical Engineers Reference Book

Vibrations 111 5
Further reading
Johnston, E. R. and Beer, F. P., Mechanicsfor Engineers, Volume
1, Statics; Volume 2, Dynamics, McGraw-Hill, New York (1987)
Meriam, J. i. and Kraige, L. G., Engineering Mechanics, Volume
1, Statics, second edition, Wiley, Chichester (1987)
Gorman, D. J., Free vibration Analysis of Beams and Shafts,
Wiley, Chichester (1975)
Nestorides, E. J., A Handbook of Torsional Vibration, Cambridge
University Press, Cambridge (1958)
Harker, IR., Generalised Methods of Vibration Analysis, Wiley,
Chichester (1983)
Tse, F. S., Morse, I. E. and Hinkle, R. T., Mechanical Vibrations:
Theoq and Applicationr: second edition, Allyn and Bacon, New
York (1979)
Hatter, D., Matrix Computer Methods of Vibration Analysis,
Butterworths, London (1973)
Nikravesh, P. E., Computer Aided Analysis of Mechanical Systems,
Prentice-Hall, Englewood CEffs, NJ (1988)

British Standards Figure 1.26 Ensemble of a random process
BS 3318: Locating the centre of gravity of earth moving equipment
and heavy objects the value of a particular parameter, x, and that value is found
BS 3851: 1982 Glossary of terms used in mechanical balancing of to vary in an unpredictable way that is not a function of any
rotary machines other parameter, then x is a random variable.
BS 3852: 1986: Dynamic balancing machines
BS 4675: 1986: Mechanical vibrations in rotating and reciprocating
machinery 1.4.3.2 Probability distribution
BS 6414: 1983: Methods for specifymg characteristics of vibration
and shock absorbers If n experimental values of a variable x are xl, x2, x3, . , . x,,
the probability that the value of x will be less than x' is n'ln,
where n' is the number of x values which are less than or equal
1.4.3 Random vibrations to x'. That is,
1.4.3.1 Introduction Prob(x < x') = n'/n
If the vibration response parameters of a dynamic system are When n approaches 0: this expression is the probability
accurately known as functions of time, the vibration is said to distribution function of x, denoted by P(x), so that
be deterministic. However, in many systems and processes
responses cannot be accurately predicted; these are called
random processes. Examples of a random process are turbu-
lence, fatigue, the meshing of imperfect gears, surface irregu-
larities, the motion of a car running along a rough road and The typical variation of P(x) wi:h x is shown in Figure 1.27.
building vibration excited by an eaxthquake (Figure 1.25). Since x(t) denotes a physical quantity,
A collection of sample functions xl(t), x2(t), x3(t), . . . ,xn(t)
which make up the random process x(t) is called an ensemble Prob(x < -0:) = 0, and Prob(x < +%) = 1
(Figure 1.26). These functions may comprise, for example, The probability density function is the derivative of P(x)
records of pressure fluctuations or vibration levels, taken with respect to x and this is denoted by p(x). That is,
under the same conditions but at different times.
Any quantity which cannot be precisely predicted is non- WX)
deterministic and is known as a random variable or aprobabil- P(4 =
istic quantity. That is, if 3 series of tests are conducted to find
= Lt [P(x + Ax) - P(q
Ax+ 0

<

1 X
x
%+Ax
'
I
0
Fiaure 1.27
Figure 1.25 Example random process variable as f(t) -I-- Probabilitv distribution function as f(x)
~