AVERAGES AND PROBABILITES
167
From the viewpoint of structural dynamics, équation (7.1) implies that the
fluctuations are about a static, zéro mean. For such cases the variance
defined as the mean square value of i/(t), is often used as a measure of these
fluctuations. That is,
1 rTo/2
1 = ~
y2(.t)dt = a?
T0 J-to/2
(7-2)
When équation (7.1) is satisfied, the standard déviation or the root-mean-square
(rms) of y(t) is defined as the positive square root of équation (7.2), or ay.
Once ay is calculated, its value needs to be interpreted in a statistical sense
to be useful to the structural design engineer. For instance, if the rms value
of deflection y = v = v(t) is av = 2 m, a practical question is: What are the
chances that v will be smaller or larger than 2 m? Also, if the structure will
fail statically for v > 5 m, what are the chances that 2.5
density fonction, p(v), expressed in terms of av, are defined to answer these
important questions.
Stripped of mathematical elegance, the incrément of the probability distribution function, AP, is defined symbolically at y = yo as
AP = P[y0 < y < (y0 + At/)]
(7-3)
This is the probability that y is between a fixed value yo and (yo + △?/)• Specifically, AP is defined as the fraction of the total time Tq that y(t) is in this
bandwidth At/. The vertical strips of Figure 7.2 show the typical time incréments Atj, At2,... within this bandwidth. Thus
AP — —(Ati + At2 + • • • )
To
The probability density function is defined as a limiting process, or
AP
dP
p(y)=lim(Ny
0.1 — =
(7.4)
(7-5)
where yo is replaced by y for generality. Thus it is possible to obtain a plot
of p(y) versus y by dividing a sample trace y(t) of duration tq into a sufficient
number of levels y = yo, measuring the time spent at each level in each time
band, and forming the ratio NP/'Ny - p(y) where AP is given by équation
(7.4). With automated digital sampling (see Problem 7.3), this is accomplished
in an efficient way.
In the statistical response calculations involving structures, p(y) is rarely
calculated. Instead it is assumed a priori that p(y) has a Gaussian or normal
distribution of zéro mean, given in terms of the variance
167
From the viewpoint of structural dynamics, équation (7.1) implies that the
fluctuations are about a static, zéro mean. For such cases the variance
defined as the mean square value of i/(t), is often used as a measure of these
fluctuations. That is,
1 rTo/2
1 = ~
y2(.t)dt = a?
T0 J-to/2
(7-2)
When équation (7.1) is satisfied, the standard déviation or the root-mean-square
(rms) of y(t) is defined as the positive square root of équation (7.2), or ay.
Once ay is calculated, its value needs to be interpreted in a statistical sense
to be useful to the structural design engineer. For instance, if the rms value
of deflection y = v = v(t) is av = 2 m, a practical question is: What are the
chances that v will be smaller or larger than 2 m? Also, if the structure will
fail statically for v > 5 m, what are the chances that 2.5
density fonction, p(v), expressed in terms of av, are defined to answer these
important questions.
Stripped of mathematical elegance, the incrément of the probability distribution function, AP, is defined symbolically at y = yo as
AP = P[y0 < y < (y0 + At/)]
(7-3)
This is the probability that y is between a fixed value yo and (yo + △?/)• Specifically, AP is defined as the fraction of the total time Tq that y(t) is in this
bandwidth At/. The vertical strips of Figure 7.2 show the typical time incréments Atj, At2,... within this bandwidth. Thus
AP — —(Ati + At2 + • • • )
To
The probability density function is defined as a limiting process, or
AP
dP
p(y)=lim(Ny
0.1 — =
(7.4)
(7-5)
where yo is replaced by y for generality. Thus it is possible to obtain a plot
of p(y) versus y by dividing a sample trace y(t) of duration tq into a sufficient
number of levels y = yo, measuring the time spent at each level in each time
band, and forming the ratio NP/'Ny - p(y) where AP is given by équation
(7.4). With automated digital sampling (see Problem 7.3), this is accomplished
in an efficient way.
In the statistical response calculations involving structures, p(y) is rarely
calculated. Instead it is assumed a priori that p(y) has a Gaussian or normal
distribution of zéro mean, given in terms of the variance