Examples from Classical Mechanics - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 84

3.1.1 Example 3-1a: Hinged ends.
The boundary conditions r(0) = 0, r"(0) = 0, r(L) = 0 and r"(L) = 0 assume
the same zero displacement and moment at each end. Following the detailed
analysis described in Example 3-1b, we find that trivial solutions are obtained
unless sin L = 0. This requires L= , 2 , 3 , ... , that is, L = n where n = 1,
2, 3, ... or = n /L. Equation 3.7, which reduces to r(x) = sin x here, implies
the eigenfunction r n (x) = sin n x/L, for which the corresponding time
amplitude takes a functional ( E n sin {(n /L) 2 (EI/ A)}t + F n cos
{(n /L) 2 (EI/ A)}t ) form . Hence, the general solution is
v(x,t) = ( E n sin {(n /L) 2 (EI/ A)}t
+ F n cos {(n /L) 2 (EI/ A)}t ) sin n x/L (3.9)

where the summation index n varies from 1 to .
The coefficients E n and F n are still undetermined, but following the
procedure in Example 1-3, they are fixed once the initial transverse
displacement and speed v(x,0) and v t (x,0) are specified. Separation of variables,
in this and other elementary examples, leads to “trouble-free” results.

3.1.2 Example 3-1b: Clamped end, other end free.
Here we introduce difficulties that highlight the weaknesses of analytical
methods. The conditions r(0) = 0 and r'(0) = 0 imply zero displacement and
slope, while r"(L) = 0 and r"'(L) = 0 call for zero shear force and moment. From
the first two conditions, we have B + D = 0 and A + C = 0, and from the last
two, we obtain

C {sin L + sinh L} + D{cos L + cosh L} = 0 (3.10)

C {cos L + cosh L} + D{sinh L - sin L} = 0 (3.11)

For nontrivial solutions, the determinant formed by the curly-bracketed
coefficients of C and D must vanish, leading to the “eigenvalue relation”

cos L cosh L = -1 (3.12)

This “transcendental equation” has an infinite number of eigenvalue solutions,
each of which require intensive numerical solution. In fact, the first six for

1 ,

2 , ..., 6 are determined from

L = 1.875, 4.694, 7.855, 10.996, 14.137, 17.279 (3.13)

The corresponding eigenfunction is no longer the “r n (x) = sin n x/L”
found for Example 3-1a, but the unwieldy trigonometric-hyperbolic function