Comparison

The exact analytical solution of Section C.1 is used to evaluate the three approximations. In order to do so, the relative frequency errors are computed. This error is defined as:

Approximated eigenfrequency

Exact eigenfrequency

 Mode Number of superelements N„, with k = 1 2 3 4 5 1 3.599 3.636 3.636 3.636 3.636 2 36.86 22.12 22.70 22.75 22.75 3 - 73.88 60.90 63.18 63.46 4 - 159.7 140.0 117.8 122.9

Table C.4: The first four eigenfrequencies in radians per second of the superelement approximation as function of the number of superelements.

The relative errors for the first four eigenfrequencies of both the finite element and superelement approximation are plotted as function of the number of degrees of freedom in Fig. C.3. Notice that the number of degrees of freedom Ndof are related to the number of superelements Nse as follows:

Ndof = 2Nse

Finite element method

Finite element method

15 20 25

Degrees of freedom [-]

15 20 25

Degrees of freedom [-]

Figure C.3: The relative errors for the first four eigenfrequencies of the finite element and superelement approximation as function of the number of degrees of freedom. Solid line: first mode, dashed line: second mode, dashed-dotted line: third mode, dotted line: fourth mode, and dashed-dotted horizontal lines: + 1 % and - 1 % error bound respectively.

The finite element errors are all positive (i.e. the eigenfrequencies are overestimated by the finite element model implying that the model is stiffer than the real system), and decrease monotonically with an increasing number of finite elements.

In the superelement approximation, some eigenfrequencies are smaller, and some are larger than in reality, while the errors reach the indicated 1 % error bound rather

The relative errors for the first four eigenfrequencies of the both the lumped-mass and superelement approximation are plotted as function of the number of degrees of freedom in Fig. C.4. Note that, of course, each lumped-mass has one degree of freedom.

Figure C.4: The relative errors for the first four eigenfrequencies of the lumped-mass and superelement approximation as function of the number of degrees of freedom. Solid line: first mode, dashed line: second mode, dashed-dotted line: third mode, dotted line: fourth mode, and dashed-dotted horizontal lines: + 1 % and - 1 % error bound respectively.

Figure C.4 shows that for an Euler-Bernoulli beam both the lumped-mass and the superelement modeling approach represent a consistent approximation to the continuum model (in the sense that it represents a discretization of the continuum model with an approximation accuracy that increases with an increasing number of lumped-masses/superelements).

It can be concluded that the superelement approach is particularly useful for approximating the first number of eigenfrequencies with a limited number of superelements (i.e. degrees of freedom). In general, these lowest frequency modes have the largest amplitude and are the most important to be approximated well both for time-domain simulation and control system design.

Appendix D

Renewable Energy Eco Friendly

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.

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