Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 83

Numerous textbooks and papers on beam theory are available. Among the
better known are Timoshenko and Goodier (1934), Love (1944), Clark and
Reissner (1951), Den Hartog (1952) and Sechler (1952); also, Timoshenko
(1958), Abramson, Plass, and Ripperger (1958), Timoshenko and Gere (1961),
Clough and Penzien (1975) and Graff (1975). These references provide a range
of analytical and numerical solution techniques that have proven useful in
engineering calculations. In this example, we wish to introduce basic analysis
concepts and discuss difficulties associated with different approaches.
Let us attempt a traveling wave solution of the form v(x,t) = f(x-ct), where
the speed c is constant for all waves, as we did for the classical wave equation.
We find that Ac 2 f " + EI f "" = 0, which is obviously not true universally
(since the function f does not cancel); thus, different waves are likely to
propagate with different speeds. Now, the substitution v(x,t) = sin (kx - t)
leads to - A 2 + EI k 4 = 0 or = (EI/ A) k 2 . Since the phase velocity /k
= (EI/ A) k depends on the wavenumber k, beam disturbances are
“dispersive,” with the exact propagation velocities depending on the form of the
initial disturbance. As an alternative to “f(x-ct),” we now separate variables,
taking
v(x,t) = q(t)r(x) (3.2)

Substitution in Equation 3.1 and division by Aq(t)r(x) leads to q"(t)/q(t)
= - (EI/ A)r""(x)/r(x). Since the left side is a function of time and the right
depends only on space, each must equal the same constant. Thus,

q"(t)/q(t) = - (EI/ A)r""(x)/r(x) = -

2

(3.3)

where the right side is a negative constant (positive or zero values lead to trivial
solutions). Hence, we have the two ordinary differential equations

r""(x) - 4 r(x) = 0 (3.4)

q"(t) +

2 q(t) = 0

(3.5)

where

4 = A 2 /EI

(3.6a)

= 2 (EI/ A) (3.6b)

The general solutions to Equations 3.4 and 3.5 are

r(x) = A sin x + B cos x +C sinh x +D cosh x (3.7)

q(t) = E sin t + F cos t (3.8)

In order to explore the consequences of Equation 3.1 more fully, we consider a
finite length beam in 0 < x < L and two boundary condition models.