Page 284 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.2 Longitudinal Vibration of Rods 271

ment can be written as
00 nirx
> ' ( - ^ - 0 = E ( C „ s in c o s w „ i ) s i n ^ (9.1-14)
n=-\
nirc
" /
By fitting this equation to the initial conditions of y(jc,0) and y(jc,0), the and
can be evaluated.
Example 9.1-1
A uniform string of length / is fixed at the ends and stretched under tension T. If the
string is displaced into an arbitrary shape y(.r,0) and released, determine C„ and
of Eq. (9.1-14).
Solution: At r = 0, the displacement and velocity are
00
mrx
y(^.o) = L si sm /
« = 1
y(j:,0) = (o„C„ sin = 0
/
n = 1
If we multiply each equation by sin kirx/l and integrate from x = 0 to x = /, all the
terms on the right side will be zero, except the term n = k. Thus, we arrive at the
result
. krrx
2 L ri . . Krr.
dx
Q = 0, k= 1,2,3,.

9.2 LONGITUDINAL VIBRATION OF RODS

The rod considered in this section is assumed to be thin and uniform along its
length. Due to axial forces, there will be displacements u along the rod that will be
a function of both position x and time t. Because the rod has an infinite number of
natural modes of vibration, the distribution of the displacements will differ with
each mode.
Let us consider an element of this rod of length dx (Fig. 9.2-1). If u is the
displacement at jc, the displacement at jc + ¿¿c will be u { d u / d x ) dx. It is

Pm----
1— ^ — d x\^ ^ ----;
1 ^

Figure 9.2-1. Displacement of rod
element.