Page 205 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 187

Making allowance for the shorter tube at the free end, the kinetic energy of the pipe is

~(" 4q ) +clp [2
~~
,
T-L ,ml3 {I 3 ~ '2 + 4 2 ~ ' 4 ) ~ 4 (3.157)
)
1
3
-
'
p= 1 q=o q=o
where
2
3
clp = 1 + (e - 1)6,~. ~2~ = 1 + (e - 1)6,~, ~3~ = 1 + (e - 1)6pN,
(3.158)
and SP~ Kronecker's delta, while the kinetic energy of the fluid is
is

(3.159)

both correct up to the quadratic terms. The potential energy of the system is

The equations of motion are derived via Hamilton's principle, equation (3.10). In this
case, R = wLk - cLi, t = @Nk + i, where
N N
(3.161)
p= 1 p= 1

The equations of motion, in dimensionless form, are
2 "
~C3r4r + Nyc2r4r + 2N2U2C1r(@~ - 4r)

r= 1,2,3 ..... N, (3.162)