STRUCTURAL RESPONSE STATISTICS: PART I
173
transform and the inverse Fourier transform from digitized data. In practice,
however, it is usually more efficient to calculate Sy(w) directly from experimental
time historiés (for rj(11. for instance) by means of an electronic instrument called
a frequency analyzer such as described in Problem 7.4, or by means of computeraided Fast Fourier Transform (FFT) methods to evaluate the Fourier coefficients
of équation (6.2) for use in équations (6.5). Newland (1975) elaborated on these
méthodologies, and Cooley and Tukey (1965) presented efficient algorithms for
calculating the Fourier coefficients.
Example Problem 7.2. A linear, single degree of freedom flexible stucture
is subjected to a total wave force pi(t), in line with the motion of the structure.
Assume a distribution of simple, linear waves for which the wave height rj(t)
is stationary, ergodic, and Gaussian, with a zéro mean and with a spectral
density ^(w). Starting from basic définitions, relate the spectral density of the
wave load Spi(cj) to Sp(cj) through a known transfer function G(w) defined by
équation (4.24).
First, rewrite the structural load-wave height relationship as
p1(t)=r?(t)|G(^)|
(7.23)
Then rewrite équation (7.20) twice using équation (7.19), first substituting r/ for
y and then pi for y. The results for the respective spectral densities are
i r°°
Sr,M - - I
Efo(t) -Ht + T)]e-^dr
(7.24)
J-oo
1
f°°
Spl(u>) = —
£[pi(«)Pi(t + T)]e ^Tdr
(7.25)
2tt J ...
When équation (7.23) is substituted into équation (7.25) and this resuit is compared to équation (7.24), the required relationship is deduced as
Spl(u>) = |G(w)|2Sp(o;)
(7.26)
7.4 STRUCTURAL RESPONSE STATISTICS: PART I
The dérivation of response statistics that follow are based on a linear model
in the form of équation (2.43), written in terms of the displacement coordinate
v. However, similar results can be obtained based on a linear model in the
form of équation (2.81), written in terms of the rotational coordinate 9. After
going through the following dérivation for the displacement model, the reader
is encouraged to then dérivé the response statistics for the rotational model.
Given the linear structural model
mv + cjv + k-pu — pi(f)
(7.27)
173
transform and the inverse Fourier transform from digitized data. In practice,
however, it is usually more efficient to calculate Sy(w) directly from experimental
time historiés (for rj(11. for instance) by means of an electronic instrument called
a frequency analyzer such as described in Problem 7.4, or by means of computeraided Fast Fourier Transform (FFT) methods to evaluate the Fourier coefficients
of équation (6.2) for use in équations (6.5). Newland (1975) elaborated on these
méthodologies, and Cooley and Tukey (1965) presented efficient algorithms for
calculating the Fourier coefficients.
Example Problem 7.2. A linear, single degree of freedom flexible stucture
is subjected to a total wave force pi(t), in line with the motion of the structure.
Assume a distribution of simple, linear waves for which the wave height rj(t)
is stationary, ergodic, and Gaussian, with a zéro mean and with a spectral
density ^(w). Starting from basic définitions, relate the spectral density of the
wave load Spi(cj) to Sp(cj) through a known transfer function G(w) defined by
équation (4.24).
First, rewrite the structural load-wave height relationship as
p1(t)=r?(t)|G(^)|
(7.23)
Then rewrite équation (7.20) twice using équation (7.19), first substituting r/ for
y and then pi for y. The results for the respective spectral densities are
i r°°
Sr,M - - I
Efo(t) -Ht + T)]e-^dr
(7.24)
J-oo
1
f°°
Spl(u>) = —
£[pi(«)Pi(t + T)]e ^Tdr
(7.25)
2tt J ...
When équation (7.23) is substituted into équation (7.25) and this resuit is compared to équation (7.24), the required relationship is deduced as
Spl(u>) = |G(w)|2Sp(o;)
(7.26)
7.4 STRUCTURAL RESPONSE STATISTICS: PART I
The dérivation of response statistics that follow are based on a linear model
in the form of équation (2.43), written in terms of the displacement coordinate
v. However, similar results can be obtained based on a linear model in the
form of équation (2.81), written in terms of the rotational coordinate 9. After
going through the following dérivation for the displacement model, the reader
is encouraged to then dérivé the response statistics for the rotational model.
Given the linear structural model
mv + cjv + k-pu — pi(f)
(7.27)