Geology Reference
In-Depth Information
3
Examples from Classical Mechanics
We will introduce a range of wave propagation problems showing the
diversity of model equations taking forms
other
than
2
u/ t
2
- c
2 2
u/ x
2
= 0 .
We also illustrate the concepts discussed in Chapters 1 and 2 with examples
found in structural, fluid-dynamic and acoustical applications. Building on these
problems, the more challenging issues encountered in drillstring vibrations, in
which axial, torsional and lateral modes interact, are addressed in Chapter 4,
where complicated effects like rock-bit interaction, bit bounce and rate-of-
penetration are studied. The remainder of the topic then deals with problems in
geophysics, electromagnetic logging, multiphase reservoir flow and Stoneley
wave approaches to permeability determination.
3.1 Example 3-1. Lateral Vibration of Simple Beams
Consider the lateral bending vibrations of a uniform bar with mass density
, cross-sectional area A, Young' s modulus E and moment of inertia I (see
Figure 3.1). The transverse displacement v(x,t) satisfies
A
2
v/ t
2
+ EI
4
v/ x
4
= 0
(3.1)
Figure 3.1.
A simple beam.
82
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