190
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Arbitrarily choose ( — 0.1. Using the constants in équation (7.101), Mathematica®
(1999) gave the following resuit: crv = 0.135 ft. Note that this resuit is about
one-fourth the value calculated by a different theory in Example Problern 7.3. A
doser agreement between the two results can be obtained by choosing Ç ~ 0.2,
for which
= 0.196 ft. A least squares fit of the constants â, £, w based on
équation (7.102) may lead to even doser agreement between the results for
PROBLEMS
7.1 If y(t) is Gaussian with zéro mean and has a variance of
found in a statistics reference book.
7.2 Suppose that y(t) is Gaussian, exists only over a narrow band of frequencies, and is a smooth function of time. Also assume that each cycle crosses
the mean level j/(t) = 0 so that the maxima always occur for y(t') >0 and the
minima always occur for y(t) < 0.
(a) Sketch a function y(t) which behaves as defined.
(b) The probability distribution for the peaks of y(t) so defined is given by
équation (7.7), the well-known Rayleigh distribution. Sketch p(a) as a function
of the amplitude a of y(t). Prove that the maximum value of p(a) occurs when
the amplitude is equal to the standard déviation, or a =
2
/
a2 \
A *>'i“='»p(-^)
7.3 A digital time history of ÿ(t) is available in the form of points J/(tn)
even incréments of time, n = 1,2,... , n. Outline a computer-aided method that
will generate a probability density function p(y) from these data. How would
you check p(y) to see if it was Gaussian?
7.4 A schematic diagram of an instrument called a spectrum analyzer
is shown in Figure 7.7. The wave height input r?(t) is assumed to be a stationary ergodic random process. This input is filtered by a filter whose harmonie response function H(w) is a constant Ho in the narrow frequency band
(ixzo — Au>/2) < lu < (uiq +
The filter output y(t) is squared, and its time
average z(t) is calculated from
_
1
/*to/2
z(t) = E[y2(t)] =
y2{t)dt
T0 J-t0/2
The mean level of z(t), estimated from the output meter for sufficiently long
time penods tq, is
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Arbitrarily choose ( — 0.1. Using the constants in équation (7.101), Mathematica®
(1999) gave the following resuit: crv = 0.135 ft. Note that this resuit is about
one-fourth the value calculated by a different theory in Example Problern 7.3. A
doser agreement between the two results can be obtained by choosing Ç ~ 0.2,
for which
= 0.196 ft. A least squares fit of the constants â, £, w based on
équation (7.102) may lead to even doser agreement between the results for
PROBLEMS
7.1 If y(t) is Gaussian with zéro mean and has a variance of
found in a statistics reference book.
7.2 Suppose that y(t) is Gaussian, exists only over a narrow band of frequencies, and is a smooth function of time. Also assume that each cycle crosses
the mean level j/(t) = 0 so that the maxima always occur for y(t') >0 and the
minima always occur for y(t) < 0.
(a) Sketch a function y(t) which behaves as defined.
(b) The probability distribution for the peaks of y(t) so defined is given by
équation (7.7), the well-known Rayleigh distribution. Sketch p(a) as a function
of the amplitude a of y(t). Prove that the maximum value of p(a) occurs when
the amplitude is equal to the standard déviation, or a =
2
/
a2 \
A *>'i“='»p(-^)
7.3 A digital time history of ÿ(t) is available in the form of points J/(tn)
even incréments of time, n = 1,2,... , n. Outline a computer-aided method that
will generate a probability density function p(y) from these data. How would
you check p(y) to see if it was Gaussian?
7.4 A schematic diagram of an instrument called a spectrum analyzer
is shown in Figure 7.7. The wave height input r?(t) is assumed to be a stationary ergodic random process. This input is filtered by a filter whose harmonie response function H(w) is a constant Ho in the narrow frequency band
(ixzo — Au>/2) < lu < (uiq +
The filter output y(t) is squared, and its time
average z(t) is calculated from
_
1
/*to/2
z(t) = E[y2(t)] =
y2{t)dt
T0 J-t0/2
The mean level of z(t), estimated from the output meter for sufficiently long
time penods tq, is