272
CONTINUOUS SYSTEMS
and so on. However, since natural damping always exists, this pipeline is stable
for ~ ivn under parametric excitation.
Statistical Wave Excitation
Most of the background analysis needed to calculate the statistical dynamic
responses of beams to stationary, ergodic wave excitation has already been developed. Outlined now are the basic assumptions, methodology, and results
of this statistical analysis. In this section, the reader is encouraged to carry
through the details of the mathematical manipulations. Further expositions on
this topic are given by Clough and Penzien (1993) and Gould and Abu-Sitta
(1980).
The mathematical model chosen for analysis is the uniform Bernoulli-Euler
beam without longitudinal loading (P = 0) but with light, linear viscous damping. From équation (10.8) this model is
d^v
_dv
d2v
_
E'ââ + râ<+”'âP = ‘'" "
(10.104)
where the loading per unit length q = q(x, t) is stationary, ergodic and Gaussian,
defined by the spectral density S9(cv).
Equation (10.104) is reduced to a familiar form by expanding both the displacement solution v = v(x, t) and the loading q = q(x, t) in terms of the
undamped normal modes Xn = Xn(x) and the generalized modal coordinates
ün = ?/n(t) and qn - qn(t). That is, let
OO
v(x,t) = ^xnyn
n=l
OO
g(z, t) = ^Xn qn
n=l
(10.105)
(10.106)
The procedure now is analogous to that used previously to calculate beam responses with deterministic loading where équation (10.88) was derived starting
with équation (10.82) and the undamped beam model given by équation (10.69).
That is, substitute équations (10.105) and (10.106) into équation (10.104); multiply each term by Xm = Xm(x); from équation (10.73) let X"" = a^X„; intégrale each term of the resulting équation over the interval (0,1)', interchange
the order of intégration and summation under the assumption that the sériés
are uniformly convergent; and apply the orthogonality condition of (10.70) and
(10.71) to the resuit, which is valid for any of the six sets of beam support conditions of (10.81). Thus ail terms in the sériés vanish except for m = n, which
leads to
thÿn + cÿn 4- EIa*yn = qn(t)
(10.107)
CONTINUOUS SYSTEMS
and so on. However, since natural damping always exists, this pipeline is stable
for ~ ivn under parametric excitation.
Statistical Wave Excitation
Most of the background analysis needed to calculate the statistical dynamic
responses of beams to stationary, ergodic wave excitation has already been developed. Outlined now are the basic assumptions, methodology, and results
of this statistical analysis. In this section, the reader is encouraged to carry
through the details of the mathematical manipulations. Further expositions on
this topic are given by Clough and Penzien (1993) and Gould and Abu-Sitta
(1980).
The mathematical model chosen for analysis is the uniform Bernoulli-Euler
beam without longitudinal loading (P = 0) but with light, linear viscous damping. From équation (10.8) this model is
d^v
_dv
d2v
_
E'ââ + râ<+”'âP = ‘'" "
(10.104)
where the loading per unit length q = q(x, t) is stationary, ergodic and Gaussian,
defined by the spectral density S9(cv).
Equation (10.104) is reduced to a familiar form by expanding both the displacement solution v = v(x, t) and the loading q = q(x, t) in terms of the
undamped normal modes Xn = Xn(x) and the generalized modal coordinates
ün = ?/n(t) and qn - qn(t). That is, let
OO
v(x,t) = ^xnyn
n=l
OO
g(z, t) = ^Xn qn
n=l
(10.105)
(10.106)
The procedure now is analogous to that used previously to calculate beam responses with deterministic loading where équation (10.88) was derived starting
with équation (10.82) and the undamped beam model given by équation (10.69).
That is, substitute équations (10.105) and (10.106) into équation (10.104); multiply each term by Xm = Xm(x); from équation (10.73) let X"" = a^X„; intégrale each term of the resulting équation over the interval (0,1)', interchange
the order of intégration and summation under the assumption that the sériés
are uniformly convergent; and apply the orthogonality condition of (10.70) and
(10.71) to the resuit, which is valid for any of the six sets of beam support conditions of (10.81). Thus ail terms in the sériés vanish except for m = n, which
leads to
thÿn + cÿn 4- EIa*yn = qn(t)
(10.107)