
8.17. Consider a uniform bar in bending, simply supported at both ends and subjected to the excitation,
f(x, t) = F (t)δ(x − L/2),
where F (t) is an ergodic random process with ideal white noise power spectral density, and δ(x−L/2) is
a spatial Dirac delta function. Derive expressions for the cross-correlation function between the response
at the points x = L/4 and x = 3L/4 and for the mean-square value of the response at x = L/4. Assume
constant m, c, and EI along the beam.
Solution: The equation of motion and boundary conditions are given by
EI
∂
4
y
∂x
4
+ c
∂y
∂t
+ m
∂
2
y
∂t
2
= F (t) δ (x − L/2) for 0 < x < L
y (0, t) = 0
EI
∂
2
y
∂x
2
0,t
= 0
EI
∂
2
y
∂x
2
L,t
=