Superelement approximation
In this case, the EulerBernoulli beam has been modeled using a number of superelements. Again, the beam is built in at the base. The torsional spring constants for each superelement are determined as follows:
Lse with E the modulus of elasticity, Iz the area moment of inertia, and Lse the length of the superelement which is, in turn, defined as
Lse = Nse with L the length of the EulerBernoulli beam and Nse the number of superelements the beam is subdivided in.
The first four eigenfrequencies of the superelement approximation as function of the number of elements are listed in Table 4.3. The pattern is clear: dividing the beam into more superelements produces more eigenfrequencies (of which only the first four are shown), and improves the accuracy (compare columns with exact values listed in Table 4.1). The limiting case being an infinite number of superelements of which the eigenfrequencies equal to those of the exact solution.
Mode 
Number of superelements jv„, with yfc = 1(1 L)  
1 
2 
3 
4 
5 
6 
7 
8 
9  
1 
3.599 
3.636 
3.636 
3.636 
3.636 
3.636 
3.636 
3.636 
3.636 
2 
36.86 
22.12 
22.70 
22.75 
22.75 
22.76 
22.76 
22.76 
22.76 
3 
 
73.88 
60.90 
63.18 
63.46 
63.53 
63.55 
63.56 
63.57 
4 
 
159.7 
140.0 
117.8 
122.9 
123.7 
124.0 
124.1 
124.1 
Table 4.3: The first four eigenfrequencies in radians per second of the superelement approximation as function of the number of superelements.
Table 4.3: The first four eigenfrequencies in radians per second of the superelement approximation as function of the number of superelements.
The exact analytical solution is used to evaluate the superelement approximation. In order to do so, the relative frequency errors are computed. This error is defined as:
Approximated eigenfrequency
Exact eigenfrequency
The relative errors for the first four eigenfrequencies of the superelement approximation are plotted in Fig. 4.3 as function of the number of superelements. Some eigenfrequencies are smaller, and some are larger than in reality, while the errors reach the indicated 1 % error bound rather fast.
It can be concluded that the superelement modeling method used to discritize the EulerBernoulli beam represents a consistent approximation to the exact model with an approximation accuracy that increases with an increasing number of superelements.
1 1  
1 v• ________________  
i 
 1% 
Figure 4.3: The relative errors for the first four eigenfrequencies of the superelement approximation as function of the number of superelements Nse with k = 1 (1 — ). Dasheddotted lines: + 1 % and  1 % error bound respectively.
In addition, the mode shapes become also better defined with an increasing number of superelements, since information on more locations along the beam is available. Fig. 4.4 compares the first four analytical undamped mode shapes of an EulerBernoulli beam with those of the superelement approximation. The mode shapes are plotted from x = 0 to the full beam length of L = 50 m. The comparison shows that at least n superelements are required to accurately approximate the first n analytical mode shapes of an EulerBernoulli beam.
Recall that the form of the analytical mode shapes is given by Eq. (4.7). The mode shapes of the superelement approximation are computed by extracting the statespace matrices A: n x n state or system matrix, B: n x r input matrix, C: p x n output matrix, and D: p x r direct feedthrough matrix (with n number of states, r number of inputs, and p number of outputs) from the simulation model (i.e. Beamlsd, Beam2sd, Beam3sd and Beam4sd). For example, for Beamlsd n = 4, r =1, and p =1. Subsequently, the matrix eigenvalue problem is solved producing a diagonal matrix of generalized eigenvalues and a full matrix whose columns are the corresponding eigenvectors.
Each column of the eigenvector matrix contains the stacked vector of local joint position states (i.e. the angle of rotation of the 1 degree of freedom rotational joint expressed in the bodylocal frame of reference) and local joint velocity states. The modes are obtained by repeatedly picking an eigenvalue and selecting the associated vector of local position states out of the eigenvector matrix. Subsequently, the vector of local position states is converted to global displacements using the pin joint locations expressed in the Newtonian reference frame. Finally, the mode shapes are
ro 5
se scaled such that the displacement at the free end of the beam is identical to the displacement of the exact mode shape. Remember that a mode shape is just a measure of the motion of the beam and consequently can be scaled arbitrarily.
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 x [m] x [m] x [m] x [m]
Figure 4.4: Comparison of the first four mode shapes of an EulerBernoulli beam. Thin lines: analytical mode shapes (with C4 = 1 for each mode), thick lines: superelement approximation for Nse = 1 to Nse = 4, with o: 1 degree of freedom rotational joints (i.e. pin joints).
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 x [m] x [m] x [m] x [m]
Figure 4.4: Comparison of the first four mode shapes of an EulerBernoulli beam. Thin lines: analytical mode shapes (with C4 = 1 for each mode), thick lines: superelement approximation for Nse = 1 to Nse = 4, with o: 1 degree of freedom rotational joints (i.e. pin joints).
Finally, it is shown in Fig. 4.5 that the accuracy of approximating the centrifugal stiffening using the superelement approximation is nearperfect. In this figure the influence of centrifugal stiffening on the first four modes of an EulerBernoulli beam is plotted as function of the rotational speed. It is important to stress that the superelement modeling approach automatically acounts for centrifugal stiffening effects. After all, as the length of the rigid bodies within each superelement is constant, it follows that deformation of the blade automatically produces axial deformations and thereby automatically produces centrifugal stiffening.
It should be noted that the practical importance of centrifugal stiffening is considerable in the analysis of wind turbine rotor blades, since the eigenfrequencies of a spinning roter blade raise due to an additional stiffening term caused by the component of the centrifugal acceleration field along the blade.
4 Mode rd
3 Mode
2 Mode st
1 Mode
Figure 4.5: The influence of centrifugal stiffening on the first four modes of an EulerBernoulli beam. Solid line: exact, dasheddotted line: Nse = 3, dashed line: Nse = 9, and dasheddotted vertical line: maximum rotor speed nmax = 36 [r.p.m] (see Table A.4 on page 226).
Renewable Energy 101
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