Environmental Engineering Reference
In-Depth Information
5. Derive the following equation for the base overturning moment, M
h
starting
with (
10.6
) for a constant U:
h
2
:
M
h
¼
M
0
þ
T
max
h
þ
1
d
0
þ
d
1
h
3
4
qU
2
C
d
6. Derive F
wind
and M
h
for the case of U(y) as given by Eqs.
1.14
and
1.15
.
7. Recast (
10.16
) for the second derivative of deflection into two first order
ODEs and check that the formulation in
oct_tower.m
is correct. What are the
boundary conditions to be used in solving the ODEs? What numerical
methods are available to solve these equations?
8. Using Eqs.
10.6
and
10.16
, show that the maximum deflection, d
max
,ofa
monopole tower of constant diameter and wall thickness with no turbine
moment (so a
0
= 0) is given by
:
d
max
¼
h
3
EI
a
1
3
þ
a
2
h
4
and explain why a
3
does not appear.
9. Use
tapered_oct_tower.m
to estimate the natural frequency of the example
monopole tower without a turbine and so confirm the value of 1.07 Hz in
Table
10.2
.
10. The 10.66 m tapered tower for the Skystream turbine shown in Fig.
1.2
has a
circular cross-section made from 10 gauge steel (3.42 mm). d
0
= 15.67 cm
and d
h
= 26.56 cm. Show that the tower mass is nearly 250 kg.
11. The Skystream 2.4 kW turbine has a mass of 77 kg. Estimate the fundamental
natural frequency with and without the turbine.
12. Use
tapered_oct_tower.m
to find the stress distribution, tower top deflection,
and natural frequency for the example tower if T
max
is replaced by 2T
max
in
accordance with the IEC SLM safety factors.
13. Compare the exact solution from Exercise 10.8 with your results from the
program
tapered_oct_tower.m
. Would you expect the 4th-5th order Runge-
Kutta method (ode45 in Matlab) to give the exact answer in this case?
14. For a constant diameter tower with constant thickness, no turbine, but a
constant wind loading, use the result of Exercise 10.8 to determine the
effective stiffness and show that the Method A from the text leads to the
following approximation for the natural frequency
1
=
2
1
ph
3
=
2
EI
m
t
n
1
¼
15. For a constant diameter tower with constant thickness with turbine but no
wind loading, determine the effective stiffness and show that the resulting
approximation for the natural frequency is
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