Geology Reference
In-Depth Information
Sign conventions.
Observe that the axial force N(x,t) in Equations 4.5.1
and 4.5.2 satisfies N > 0 for compression and N < 0 for tension, and that
N = - AE u/ x (4.5.3)
This expression is consistent with our axial stress convention, which assumes
AE u
x
< 0 for compression and AE u
x
> 0 for tension. Also note that in the
limit of vanishing stiffness EI, torque T, spring constant k, damping factor ,
and external loading q, Equations 4.5.1 and 4.5.2 reduce to classical wave
equations for transverse string vibrations. We further point to the obvious sign
differences in front of the torsional terms in Equations 4.5.1 and 4.5.2 and their
physical implications. These equations require that any slope in one direction
causes curvature in the other, consistently with the well known arguments of
Den Hartog (1952). The sign of the axial force N, of course, remains the same
regardless of lateral mode.
4.5.4.2 Finite differencing the coupled bending equations.
As in our discussion for lateral vibrations without torque, our coupled
equations for v(x,t) and w(x,t) are each characterized by
fourth-order
spatial
derivative terms. Thus the approximating difference equations will take on a
banded pentadiagonal structure. They are no longer tridiagonal as for axial and
torsional vibrations. In addition, we now have
third-order
spatial derivative
terms related to nonvanishing torque, e.g., (T
2
w/ x
2
)/ x = T
3
w/ x
3
+
( T/ x)(
2
w/ x
2
). These must be modeled with central differences throughout
in order to ensure midpoint symmetry, in the limit of a symmetrically
constrained uniform beam (the use of backward or forward differences will lead
to incorrectly biased asymmetric results for symmetric problem formulations).
Algebraic manipulations similar to those used earlier lead to the formula
3
u/ x
3
= ( - U
i-2
+ 2 U
i-1
- 2U
i+1
+ U
i+2
)/{2( x)
3
} + O( x)
2
and, as we had
before,
4
u/ x
4
= ( U
i-2
- 4U
i-1
+ 6 U
i
- 4U
i+1
+ U
i+2
)/( x)
4
+O( x)
2
. These
provide the required spatial symmetry about the index i, and offer second-order
accuracy in addition to stability benefits. Equations 4.5.1 to 4.5.3 become,
respectively,
EI (v
i-2,n
-4v
i-1,n
+6 v
i,n
-4v
i+1,n
+v
i+2,n
)/(
x)
4
(4.5.4)
+ 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
-T
i,n
(-w
i-2,n-1
+2 w
i-1,n-1
-2w
i+1,n-1
+w
i+2,n-1
)/2(
x)
3
- {(T
i+1,n
-T
i-1,n
)/2
x}(w
i-1,n-1
-2w
i,n-1
+w
i+1,n-1
)/(
x)
2
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