538
15 Random Vibrations
The ensemble mean square is
E[X
2
(t)] = A
2
π
0
sin
2
(ωt + α) p(α)dα
=
A
2
π
π
0
1 − cos 2(ωt + α)
2
dα =
A
2
2
The temporal mean is
X (t) = lim
T →∞
1
T
T
0
A sin(ωt + α)dt = 0
The temporal mean square is
X
2
(t)
= lim
T →∞
1
T
T
0
A
2 sin
2
(ωt + α)dt =
A
2
2
The temporal autocorrelation is
φ(τ ) = lim
T →∞
1
T
T
0
A sin(ωt + α)A sin(ωt + α + τ )dt
=
A
2
2
cos ωτ
The ensemble autocorrelation is
R(τ ) =
π
0
A sin(ωt + α)A sin(ωt + ωτ + α)
1
π
dα
=
2 A
π
cos ωt cos ωτ
Comparing ensemble and temporal averages of the problem, it is clear that the
random process described is not stationary.