beam responses
273
Define
c
m
= 2CnWn
(10.108)
With the last équation and the natural frequencies .. „ given by équation (10.74)
équation (10.107) becomes
ÿn T 2^ncunjfn 4- ujnyn — _ g,(|f ।
(10.109)
It is observed that équation (10.109) is in the form of équation (9.62) for
a finite degree of freedom System. Assume now that the nth modal response
yn(t) is statistically indépendant of the mth modal response Vm(f). the same
assumption that previously led to the spectral density of the response given by
équation (9.70). By analogy the response spectral density Sv(cu) for the beam
is given by superimposing modal responses according to équation (10.105) and
equating its cross-spectral density terms to zéro. That is,
Y
s^) = \ —(10.110)
Here
is the harmonie response function for the nth mode, derived from
équation (10.109) by letting
Un
mu/'
(10.111)
qn =
(10.112)
where go is an arbitrary constant. The results are
En(w) =
(10.113)
(10.114)
l^nMI2 =
The last ingrédient needed in équation (10.110) is 5'9n(^’)> the spectral den
sity of the generalized force. To dérivé this term, first observe that the beam
loading g(x,i) is related to the surface wave height
by
g(x, t) = | (10.115)
where the transfer function G(u>) is calculated for the chosen waye theory and
flow régime as discussed in Chapter 4. Rewrite équation (10.115) y rep ac g
? — q(x, t) with its expanded form given by équation (10.106). Now mu.tip y