Page 297 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

284 Vibration of Continuous Systems Chap. 9

Substituting these boundary conditions in the general solution, we obtain
(y) v = (i + C = 0, A = - C

( ^ ) "" Pi ^ Px -\- B cosh f3x - C sin ¡3x + D cos fSx] , . „ = 0

I3[B + D] = 0, B = -D
d^y ==l^^[A cosh pi ^ B sinh pi —C cos pi —D sin pi] = i)

dx^ x = /
/i(cosh pi cos pi) + 5(sinh pi + sin pi) = 0
= p^[A sinh pi + B cosh pi + C sin pi - D cos pi] = 0
dx^
^(sinh p i - sin p i ) + 5(cosh p i + cos p i ) = 0
From the last two equations, v/e obtain
cosh pi + cos p i _ sinh p i + sin p i
sinh pi - sin pi cosh pi + cos pi
which reduces to
cosh pi cos ^/ + 1 = 0

This last equation is satisfied by a number of values of pL corresponding to each
normal mode of oscillation, which for the first and second modes arc 1.875 and 4.695,
respectively. The natural frequency for the first mode is hence given by
(1.875)^ 3.515

Example 9.5-2
Figure 9.5-2 shows a satellite boom in the process of deloyment. The coiled portion,
which is stored, is rotated and deployed out through straight guides to extend 100 ft
or more.
This particular boom has the following properties:
Deployed diameter = 12.50 in.
Bay length = 7.277 in.
Boom weight = 0.0274 Ib/in. of length

Bending stiffness, E7 = 15.03 X 10 lb ■in. about the neutral axis

Torsional stiffness, GA = 5.50 X 10'" lb • in.“
Determine the natural frequencies in bending and in its free unloaded state if its
length is 20 ft. The boom can be represented as a uniform beam.
Solution: The natural frequencies in bending can be found from the equation
. , El
t> i