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542 CHAPTER 15 Special Functions and Eigenfunction Expansions

If we consult a table or use MAPLE to call up these zeros (see Example 15.7), we find that
ω 1 ≈ 2.40483,ω 2 ≈ 5.52008,ω 3 ≈ 8.65373,
and so on. Using these we can obtain approximate numerical values for the frequencies of the
first three normal modes of vibration, given the length of the chain.

15.3.5 Critical Length of a Rod
We will analyze the critical bending length of a rod. Suppose we have a thin rod of constant
weight w per unit length, length L and circular cross section of radius a. The rod is clamped
in a vertical position. If the rod is “long enough,” and the upper end is displaced and held
in position until the rod is at rest, the rod will remain bent. By contrast, if the rod is “short
enough”, it will return to its vertical position after the end has been displaced slightly. The crit-
ical length L C is the transition between these two states. If L ≥ L C , the rod remains bent, but
if L < L C , it returns to the vertical after a small displacement. We want to derive an expression
for L C .
Let E be the Young’s modulus for the material of the rod. This is the ratio of stress to
strain for a linear compression. Figure 15.8 shows the rod after a small displacement. The x-axis
is vertical, along the original position of the rod. Downward is positive and the origin O is
at the upper end. P : (x, y) and Q : (ξ,η) are points on the bent rod, as shown. The moment
about P of the weight of an element w x at Q is w x[y(ξ) − y(x)].Byintegratingthiswe
obtain the moment about P of the weight of the rod above P. From the theory of elasticity, this
moment about P equals EI y (x). Assuming that the part of the rod above P is in equilibrium,

then
x

EI y (x) = w[y(ξ) − y(x)]dξ.
0
Differentiate this equation with respect to x:
x
(3)

EI y (x) = w[y(x) − y(x)]− wy (x)dξ =−wxy (x).
0
(0, 0)

Q
x
P: (x, y)

FIGURE 15.8 Displacement of
a rod.

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October 14, 2010 15:20 THM/NEIL Page-542 27410_15_ch15_p505-562