## Jo MoG

where JO, 0, and MO are, respectively, the mass moment of inertia about point O, the angle of rotation, and the external moment about O.

Figure G.1: Single degree of freedom system consisting of a rigid body of length L connected to the ground by means of aj, pin joint.

According to the parallel-axis theorem, the mass moment of inertia about point O is related to the mass moment of inertia about its center of mass, Jc , as follows

where m the mass of the rigid body, and yb2j is the distance from the center of mass to point O (the so-called bodytojoint vector in SD/FAST®'s system description file). Substituting the expression for the bending moment in Eq. (G.1) gives

where L the length of the rigid body, and c the torsional spring stiffness. Taking the Laplace transform on each side of Eq. (G.3), and assuming zero initial conditions results in the following equation

JO s26(s) = F(s)L — c6(s) ^ ( Jo s2 + c) 6(s) = F(s)L (G.4)

The ratio of transformed response 6(s)L (output) to the transformed excitation F(s) (input) can be expressed as

which is known as the system transfer function. The transfer function describes the system under analysis in terms of poles and residues, and gives the dynamic response of a system under any type of excitation, including periodic and harmonic ones. Evaluating the transfer function only in the frequency domain, i.e. along the imaginary axis, gives

which is called the system's frequency response function (FRF). Notice that the FRF is just a particular case of the transfer function, and that it gives the dynamic response of a system under sinusoidal excitation only. In practice, however, it may replace the transfer function without loss of useful information [126].

Notice that Eq. (G.3) is an ordinary second order mass-spring equation. This means that we need two state variables to transform this differential equation to state-space form. We choose 6 and 6 as state variables, or define

The output equation is y = Ld!

In vector-matrix form, Eq. (G.7) and (G.8) can be written as

 " 9i ' h 0 1 " " 9i " 9 + 0 = I-i °J L 92 92 The above equation is in standard state-space form

## 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 non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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