Page 173 - Pipelines and Risers
P. 173
146 Chapter IO
where:
,...,
~(i)=cos(wt+~),n=1.2,3
and
=
~(x) c, cosh(s, x) + c, sinh(s,x) + c, cos(s,x) + c, si@, x)
The boundary conditions for a beam with end springs may be expressed as:
BC 1: EI-=~,- dv(0)
&(O)
dx dx
dZW)
BC 2: EIT=-k,- dW)
dx dx
BC 3: T=-EI-=k,,y1(0) d3m)
dx dx’
BC4: T--EI-=-~,~~(o
d’W)
dW)
dx dx’
where:
translational spring stiffness, left end of beam
k,,
translational spring stiffness, right end of beam
k,%
rotational spring stiffness, left end of beam
k,,
rotational spring stiffness, right end of beam
k,
I Length of pipe
Applying the boundary conditions in the general solutions, 4 linear equations are obtained
from which o is solved as frequency determinant. When ois known the four coefficients
except for an arbitrary factor can be determined.
The frequency determinant may be derived as:
