Page 369 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

356 Mode-Summation Procedures for Continuous Systems Chap. 11

A

|y///
X
+

TT7TT
W Figure 11.3-3.

Solution: The problem is approached in a manner similar to that of the direct problem in
which, in place of and co^, we use and il,. We now relieve the constraints at the
supports by introducing opposing forces -F(a) and -M(a) equal to kyia) and
KyXal
To carry out this problem in greater detail, we start with the equation

Q i

which replaces Eq. (11.3-8). Letting Djico) = M ,n f[l - (co/O,)^], the displacement
at jc = is

-F{a)^f(a) - M{a)^'(a)^,{a)
y{a) = («)'?, = E
D,{(o)
We now replace -F(a) and -M(a) with ky(a) and Ky'(a) and write
ky{a)^f{a) + Ky'{a)<t>'^(a)^^(a)
y(a) = E
i D,(a>)
fcy(a)«I>;(a)<l>,(«) + Ky'{a)^',^(a)
y'{a) = E
D,(a>)
These equations can now be rearranged as

y(a) 1 - ^ E = y '(« )^ E A(o>)

i - ^ E
r D,{co)
The frequency equation then becomes

i - ^ E 1 - /iE - kK = 0
r r A ( ^ ) D,{w)