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Probabilistic Models for Dynamical Systems, 2nd Edition
book

Probabilistic Models for Dynamical Systems, 2nd Edition

by Haym Benaroya, Seon Mi Han, Mark Nagurka
May 2013
Intermediate to advanced content levelIntermediate to advanced
764 pages
6h 43m
English
CRC Press
Content preview from Probabilistic Models for Dynamical Systems, 2nd Edition
4.25. Reconsider Example 4.14 and assume that the projectile speed and angle are correlated with
ρ
V
0
a
= 0.5. Compare the results with those of the uncorrelated case. Draw conclusions.
Solution: Recall
R =
υ
2
g
sin 2φ
µ
φ
= 30
=
π
6
δ
φ
= 0.10
= σ
φ
= µ
φ
δ
φ
=
π
6
(0.10)
µ
υ
= 300 m/s
σ
υ
= 35 m/s.
For the general case of Y = g(X
1
, X
2
) where in the Taylor expansion terms to second-derivative are retained,
we have
E {Y } = g
µ
X
1
, µ
X
2
+
1
2
2
g
∂X
2
1
V arX
1
+
2
g
∂X
1
∂X
2
ρ
12
σ
X
1
σ
X
2
+
1
2
2
g
∂X
2
2
V arX
2
V arY =
∂g
∂X
1
2
V arX
1
+ 2
∂g
∂X
1
∂g
∂X
2
ρ
12
σ
X
1
σ
X
2
+
∂g
∂X
2
2
V arX
2
,
where all derivatives are evaluated at the respective mean values. Therefore,
E(R) =
µ
2
υ
g
sin 2µ
φ
+
1
2
2
g
sin 2µ
φ
·σ
2
υ
+
4µ
υ
g
cos 2µ
φ
· ρ
φυ
σ
φ
σ
υ
+
1
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Publisher Resources

ISBN: 9781439849897