Superelement approach
In the superelement approach, a (part of a) flexible body is approximated with a number of socalled superelements. Each symmetric superelement consists of 3 rigid bodies connected by joints (marked o in Fig. 3.15) containing ideal torsional springs that model the elastic properties in bending direction. The attractive feature of modeling the flexibility by joint springs and dampers is that the spring and damper forces are readily incorporated into the standard (rigid) multibody body packages (e.g. SD/FAST® [109]).
It should be noted that the centre body of a superelement can be divided in two parts of equal length to include axial deflection, and torsion deformation as well. In this thesis we limit ourselves (initially) to bending since the first torsional mode and the first two (nonrotating) bending modes of the Lagerwey LW50/750 wind turbine are sufficient apart (see Table A.2 on page 223). That is, for the Lagerwey LW50/750 wind turbine it is not necessary to take the torsional mode of vibration into account.
Figure 3.15: Deflections and slopes of a superelement with bending stifness EI, and length L. Each (symmetric) superelement consists of three rigid bodies (with lengths kL, (1 — 2k)L, and kL) connected by joints (O).
The main question is "What should the values of the spring constants be in order to produce a comprehensive and accurate dynamic model of a flexible body?"
Figure 3.15: Deflections and slopes of a superelement with bending stifness EI, and length L. Each (symmetric) superelement consists of three rigid bodies (with lengths kL, (1 — 2k)L, and kL) connected by joints (O).
Accurate in the sense that i) the elastic deformations of the superelement under a static load should be equal to those of a flexible beam, ii) the superelement should have the same mass and inertia properties as a rigid beam with identical dimensions, and iii) the eigenfrequencies of the superelement should be as close as possible to those of a continuous beam. Next, we will derive the spring constants in bending direction required in the superelement approach. The interested reader is referred to Molenaar [189] for the determination of the spring constants representing the axial deflection and the torsion deformation.
Continuous bending expressions
If it is assumed that both the shear deformation and rotational inertia of the flexible body crosssections are negligible if compared with bending deformation and translational inertia, the spring constants can be derived from the differential equation of the deflection curve of a prismatic beam (i.e. beam with constant cross section throughout its length). This equation is given in Gere & Timoshenko [76] as d2v _ M
with
v 
Transverse displacement (v ± y) 
[m] 
y 
Distance from the origin 
[m] 
M 
Bending moment 
[Nm] 
E 
Modulus of elasticity 
[Pa] 
I 
Area moment of inertia 
[m4] 
It should be noted that Eq. (3.57) is valid only when Hooke's law applies for the material, and when the slope of the deflection curve is very small. Also, since effects of shear deformations are disregarded, the equation describes only deformations due to pure bending.
For a tapered beam, the presented relationship gives satisfactory results provided that the angle of taper is small (i.e. < 10°). In that case, Eq. (3.57) has to be written in the following form d2v M
in which I(y) is the area moment of inertia of the crosssection at distance y from the origin.
Determination parameters superelement
The superelement parameters (i.e. the torsional spring constants cz1, cz2, and cz3 see Fig. 3.15) are found by comparing the deflection and the angle of rotation at the free end of a EulerBernoulli beam (i.e. a prismatic beam with length L, crosssection area A, constant flexural rigidity EIz, and uniformly distributed mass per unit length p = m/L, where m is the total mass of the beam) subjected to a load F and couple M at the free end of the beam (see Fig. 3.16). Since this is a case of pure bending, we may use Eq. (3.57) to determine the total deflection S and the total angle of rotation 0 at the free end [76].
Substituting the expression for the bending moment, the differential equation becomes
EIzv" = M = FL  Fy with v" = d—v. The first integration of this equation gives
The constant of integration C\ can be found from the condition that the slope of the beam is zero at the support; thus v'(0) = 0, which results in Ci = 0. Therefore
Integration of this equation yields
EIz v
FLy2 Fy3
The boundary condition on the deflection at the support is v(0) = 0, which shows that C2 =0. Thus, the equation of the deflection curve is
The angle of rotation 0F and the deflection SF at the free end of the beam loaded by a force F are readily found by substituting y = L into Eqs. (3.59) and (3.60) respectively.
The equation of the deflection curve for an EulerBernoulli beam loaded by a couple M at the end of the beam (see Fig. 3.16) can be determined analogously. The results for both cases are summarized in the following equation
6 EIz
Inversion of this equation results in
2L3 3L2 3L2 6L
F 
EIz 
12 
6L 
' S '  
M 
L3 
6L 
4L2 
0 
From Fig. 3.15 it can be easily derived that
A7i A72
E,I= constant
E,I= constant
Figure 3.16: Deflections and slopes of an EulerBernoulli beam with bending stiffness EI, and length L.
and that the following relation holds
■ Ô ' 
' L(1 — k) kL  
0 
with k the partitioning coefficient. Substituting Eq. (3.64) in Eq. (3.62) and backsubstituting the result in Eq. (3.63) gives The spring coefficients cz1, cz2, and cz3 are found by comparing the elements in the above equation. The resulting spring coefficients are 6EIz 
Renewable Energy 101
Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of nonrenewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a nonrenewable supply, the nonrenewable energy sources release emissions into the air, which has an adverse effect on the environment.
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