Environmental Engineering Reference
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accuracy and reliability of more computationally expensive fi nite element analy-
ses of wind turbine tower. Recent studies using the fi nite element technique for
free vibration analyses of structures in wind engineering include Bazeos et al. [ 6 ]
and Dutta et al. [ 7 ]. Murtagh et al. [8] derived an expression in closed form to
yield the eigenvalues and eigenvectors of a tower-nacelle system comprising of a
prismatic cantilever beam with a rigid mass at its free end.
2.2 Rotating blades
The free vibration properties of realistic wind turbine blades are computationally
more diffi cult to obtain, and models are usually mathematically complicated due
to the complex geometry of the blade and the effects of blade rotation. Baumgart
[9] used a combination of fi nite elements and virtual work, accounting for the
complex geometry of the blade to obtain the modal parameters. Naguleswaran [10]
proposed an approach to determine the free vibration characteristics of a spanwise
rotating beam subjected to centrifugal stiffening. This model [10] can be used in
many industrial fi elds, such as wind turbine blades, aircraft rotor blades and tur-
bine rotor blades. Naguleswaran [10] and Banerjee [11] both used the Frobenius
method to obtain the natural frequencies of spanwise rotating uniform beams for
several cases of boundary conditions. Chung and Yoo [12] used the fi nite element
method to obtain the dynamic properties of a rotating cantilever, whereas Lee
et al. [13] carried out experimental studies on the same. All studies indicate that
the natural frequencies rise as the rotational frequency of the blade increases. Vari-
ous software codes have been developed by engineers to dynamically analyse
the various components of a wind turbine tower. Buhl [14] presented guidelines
for the use of the software code ADAMS in free and forced vibrations of wind
turbine towers.
Under the action of rotation, the free vibration parameters of the blades are
affected by two axial phenomena. The fi rst is centrifugal stiffening and the second
is blade gravity (self weight) effects. In order to fi nd the free vibration properties
of the blades, each blade can be discretized into a lumped parameter system com-
prising of ' n ' degrees of freedom. The eigenvalues of a blade undergoing fl apping
motion may be obtained from the eigenvalue analysis:
2
[
K
]
w
[
M
] =0
(1 )
B
B
B
where
represents the modifi ed stiffness matrix due to the geo-
metric stiffness matrix [ K BG ], accounting for the effect of axial load, w B is the
natural frequency, [ K B ] is the fl exural stiffness matrix and [ M B ] is the mass matrix.
The mass matrix may be formulated as a diagonal matrix with the mass m i at each
discrete node i .
The geometric stiffness matrix contains force contributions due to blade rota-
tion which are always tensile, and contributions from the self weight of the blade,
which may be either tensile or compressive, depending on blade position. The
geometric stiffness matrix is
[
K=K K
]
[
+
]
B
B
BG
 
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