Geology Reference
In-Depth Information
modeling shocks and events induced by general longitudinal transients. Such a
formulation, of course, will lead to harmonic results if such results in fact exist.
4.3.7.1 Pentadiagonal difference equations.
Unlike the axial wave equation treated previously, which contains
second-
order
spatial derivatives, the beam equation is
fourth-order
. This means that the
simple
tridiagonal equations
we obtained earlier for longitudinal vibrations no
longer apply. However, the idea behind
central differencing
, which preserves
left-and-right symmetry, still applies, since a symmetrically constrained beam
acting under symmetric axial stresses, must respond symmetrically about its
midpoint. If we apply our earlier central difference formulas recursively, it is
possible to show that
4
u/ x
4
= ( U
i-2
- 4U
i-1
+ 6 U
i
- 4U
i+1
+ U
i+2
)/(
x)
4
+O{ (
x)
2
}
(4.3.61)
applies at the point x
i
. Thus, Equation 4.3.40 can be discretized or finite
differenced as follows,
EI (v
i-2,n
-4v
i-1,n
+6 v
i,n
-4v
i+1,n
+v
i+2,n
)/(
x)
4
+ N
i,n
(v
i-1,n
-2v
i,n
+v
i+1,n
)/(
x)
2
+ { ( N
i+1,n
- N
i-1,n
)/2
x}{(v
i+1,n
-v
i-1,n
)/2
x}
+ kv
i,n
+
(v
i,n
-v
i,n-2
)/2
t
+
A(v
i,n
-2v
i,n-1
+v
i,n-2
)/(
t)
2
= q
(v)
i,n-1
(4.3.62)
with
N
i,n
= - AE { u
i+1,n
- u
i-1,n
}/2 x (4.3.63)
where, as in axial vibrations, the time derivatives are approximated using
backward differences by virtue of
causality
requirements. Now, we can rewrite
Equation 4.3.62 in a more enlightening form which is immediately suggestive of
a stable computational algorithm or recipe. We recast our difference
approximation in the form
{EI/( x)
4
} v
i-2,n
+ {-4EI/(
x)
4
+ N
i,n
/(
x)
2
- (N
i+1,n
-N
i-1,n
)/(4(
x)
2
)} v
i-1,n
+ { 6 EI/ (
x)
4
-2N
i,n
/(
x)
2
+ k + /(2
t) + A/(
t)
2
} v
i,n
x)
4
+ N
i,n
/(
+ {-4EI/(
x)
2
+ ( N
i+1,n
-N
i-1,n
)/(4(
x)
2
)} v
i+1,n
+ { EI/ ( x)
4
} v
i+2,n
=q
(v)
i,n-1
+ (
/(2
t))v
i,n-2
+ { 2
A/(
t)
2
}v
i,n-1
- { A/(
t)
2
}v
i,n-2
(4.3.64)
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