modeling beams and cables
255
2. Clamped support, or zéro displacement and zéro slope at the end:
dv
Ve = 0;
= 0
(10.26)
3. No support, or zéro moment and zéro transverse shear at the end:
d2ve
d^Ve
EI—^=0-,
EI^-=0
(10.27)
axz
oxJ
Let X = X(æ) dénoté the mode shape and w the frequency parameter. For
harmonie beam vibrations of the form given by équation (10.14) applied to
équation (10.24), the characteristic beam équation becomes
A'"" - a2X = 0
(10.28)
where the frequency parameter is
w = £a/5
(10-29)
V m
and where a has yet to be calculated. The boundary conditions associated
with équation (10.28) are recast using équation (10.14). Letting X(e) designate
either X(0) or X((), équations (10.25)-(10.27) become, respectively:
X(e) = X"(e) = 0,
simple support
(10.30)
X(e) = X'(e) = 0,
clamped end
(10.31)
X"(e) = X"'(ë) = 0,
free end
(10.32)
The general solution to équation (10.28) is given by
X(z) = Dr sïnax + D2cosaz + r>3sinhaz + D.coshai
(10.33)
where /), D2, D3, and D4 are constants. This solution, together with the appropriate end conditions lead to the calculation of a, the free vibration frequencies,
and the mode shapes. This procedure is now illustrated.
Example Problem 10.1. Compute the range of undamped, free vibration
frequencies for a submerged, uniform cross brace of a jacket temp ate p
,
as shown in Figure 10.4a. Because each end of this brace is we
to a eg,
ends are not simple supports, but because of leg and joint flexibihty, these en
are not fully clamped either. Since the true end fodty is somewhere between
these extremes, the actual frequencies will lie between those ca eu a
or
extreme end fixity.
255
2. Clamped support, or zéro displacement and zéro slope at the end:
dv
Ve = 0;
= 0
(10.26)
3. No support, or zéro moment and zéro transverse shear at the end:
d2ve
d^Ve
EI—^=0-,
EI^-=0
(10.27)
axz
oxJ
Let X = X(æ) dénoté the mode shape and w the frequency parameter. For
harmonie beam vibrations of the form given by équation (10.14) applied to
équation (10.24), the characteristic beam équation becomes
A'"" - a2X = 0
(10.28)
where the frequency parameter is
w = £a/5
(10-29)
V m
and where a has yet to be calculated. The boundary conditions associated
with équation (10.28) are recast using équation (10.14). Letting X(e) designate
either X(0) or X((), équations (10.25)-(10.27) become, respectively:
X(e) = X"(e) = 0,
simple support
(10.30)
X(e) = X'(e) = 0,
clamped end
(10.31)
X"(e) = X"'(ë) = 0,
free end
(10.32)
The general solution to équation (10.28) is given by
X(z) = Dr sïnax + D2cosaz + r>3sinhaz + D.coshai
(10.33)
where /), D2, D3, and D4 are constants. This solution, together with the appropriate end conditions lead to the calculation of a, the free vibration frequencies,
and the mode shapes. This procedure is now illustrated.
Example Problem 10.1. Compute the range of undamped, free vibration
frequencies for a submerged, uniform cross brace of a jacket temp ate p
,
as shown in Figure 10.4a. Because each end of this brace is we
to a eg,
ends are not simple supports, but because of leg and joint flexibihty, these en
are not fully clamped either. Since the true end fodty is somewhere between
these extremes, the actual frequencies will lie between those ca eu a
or
extreme end fixity.