# Design Options to Improve the Dynamic Behavior and the Control of Small H-Darrieus VAWTs
## 2 Background  
### Inertial Properties of the Rotors  
Generally speaking, the momentum of inertia of wind turbines rotors is much larger than that of the other components of the power train, so the contribution of the latter can be neglected for a first step dynamic analysis approach. The moment of inertia of the rotor consists essentially of the sum of that of the blades and that due to the hub (or to the spokes, in the case of a VAWT). A schematic and general representation of the geometric configuration of the blades of horizontal and vertical axis turbines, useful for an approximate but sufficiently accurate calculation of the moment of inertia, is given in Figure 1. Note that, in general, \(L_{B}=k_{b}\cdot H\), where \(0\leq k_{b}\leq 1\), depending on the arm to the blade connection point. The moment of inertia with respect to the \(O^{\prime}\) axis is given by the following:  
\[I^{\prime}=I+md^{2} \tag{1}\]  
Figure 1: Simplified schemes of wind turbine rotors for HAWT with rectangular (**a**) and triangular (**b**) blade planform, as well as for H-VAWT (**c**). where \(I^{\prime}\) is the moment of inertia concerning the generic axis \(O^{\prime}\), \(I\) is the moment of inertia concerning the \(O\) axis (parallel to \(O^{\prime}\)) that crosses the blade center of gravity, and \(m\) is the mass of the body. Typically, the blade of a HAWT has a morphology falling in between a tip tapered and constant chord that can be assimilated to a rectangle and a triangle, respectively (Figure 1a,b). The moment of inertia of a rectangular shaped blade \(I^{\prime}_{R}\) of mass \(m_{R}\) is is equal to the following:  
\[I^{\prime}_{R}=I_{R}+m_{R}\bigg{(}\frac{R}{2}\bigg{)}^{2}=\frac{1}{12}m_{R}R^ {2}+m_{R}\bigg{(}\frac{R}{2}\bigg{)}^{2}=\frac{1}{3}m_{R}R^{2}\]  
On the other hand, the moment of inertia of a triangular shaped blade \(I^{\prime}_{T}\) with mass \(m_{T}\) is equal to the following:  
\[I^{\prime}_{T}=I_{T}+m_{T}\bigg{(}\frac{R}{3}\bigg{)}^{2}=\frac{1}{18}m_{T}R^ {2}+m_{T}\bigg{(}\frac{R}{3}\bigg{)}^{2}=\frac{1}{6}m_{T}R^{2}\]  
Assuming that the density of the blade material is the same, the moment of inertia of a constant-chord blade is double that of a variable-tapered, triangle-shaped one. Commercial blades with a length less than 10 m, where the width of the spar-cap box and the skin do not vary along the blade, show the following main relationships for the inertia, mass, and blade length [32]:  
\[I^{\prime}=k_{I}m_{B}R^{2}\]  
where \(k_{I}\) ranges from 0.21 to 0.22 [33] (\(\approx 1/5\) to \(2/9\)), figures that well agree with the outcome of the above analytical model. Focusing on H-Darrieus VAWTs, the moment of inertia is equal to the sum of that of the blades support arms \(I_{S}\) (spokes) and one of the blades \(I_{B}\) in the following formula (see Figure 1c):  
\[I^{\prime}_{H}=I_{S}+I_{B}\]  
Assuming both blades and support arms have a constant chord \(c\), they can be assimilated to rectangular bodies, and the moment of inertia results in the following:  
\[I^{\prime}_{H}=\frac{1}{3}m_{S}R^{2}+\frac{1}{12}m_{B}c^{2}+m_{B}(R)^{2}\]  
Introducing the definition of rotor solidity \(\sigma\)[34],  
\[\sigma=\frac{N_{B}cH}{2 RH}=\frac{N_{B}c}{2R}\]  
and substituting the value of \(c\) from the previous relation (assuming \(m_{S}=m_{B}\)), the following holds:  
\[I^{\prime}_{H}=\bigg{(}\frac{4}{3}+\frac{1}{3}\frac{\sigma^{2}}{N_{B}^{2}} \bigg{)}m_{B}R^{2}\]  
The term \(\frac{1}{3}\frac{\sigma^{2}}{N_{B}^{2}}\) is much smaller than 1, and thus, we have the following:  
\[I^{\prime}=I+md^{2} \tag{1}\]0  
It should be noted that the mass of a VAWT blade is, for structural reasons, equal to \(5\div 10\) times that of a HAWT of the same radius. This means that, at least when smaller sizes are considered, the inertia of an H-VAWT (with the same number of blades and swept area, and considering rotors having different aspect ratios) is from ten (for relatively large swept areas) to hundred times (for small swept areas) higher than the HAWTs counterparts.