110
SINGLE DEGREE OF FREEDOM STRUCTURES
leads to
2
dx
max
(5.34)
3. Using the saine function
calculate the maximum kinetic energy
based on the sum of équations (5.25) and (5.26). The resuit is
K max
2
dx
max
+lmd
max
X—t
(5.35)
4. Compute the natural frequency by equating the energies of Steps 2 and
3, and cast the resuit in the following form:
(3tt2E//£2 - mdg) tt2/(8£)
3ml' + 3mol" + m-d
Here, the length parameters are defined by
3d l . 7rd
l
l = 'à ~ ô- Sln
+ 77T sin
8
2tt
Z
16tt
2ird
~T
(5.36)
(5.37)
3
l
r = -0-d) + -Sin
O
Z7T
itd
T
l . 2ttcZ
Ï6tt Sm ~T
(5.38)
This same problem but without the dead weight effect of the deck was previously modeled in Example Problem 2.8 by équation (2.47), from which the
équation for undamped, free vibrations is deduced as
\3mfid + 3mo(£ — d)fi + md] v + 36——v = 0
(5.39)
XV hen équation (5.2) is used with the last équation, the resulting frequency is
________ 36EI/13_________
3mfad + 3mo(£ —
+ md
(5.40)
ln the last two équations, m is the virtual mass per unit length of each submerged
leg. which is the sum of its actual mass per unit length mo and it5 added mass
per unit length, or
m = mo + CAP^D2
(5.41)