5. The Squeeze
As mentioned previously, the Savonius wind turbine’s blade is geometrically squeezed as it is twisted (when using the geometrically pure linear trace-based model). This phenomenon can be analyzed by observing the squeeze in Geometry Expressions in both the top and side views, then modeling the squeeze in a computer algebra system. In Gemonetry Expressions, the top view model is consulted for a geometrically accurate representation of the squeeze and the side view model (with only one section) is consulted for a contextual/visual representation – see the interactive figure at right. By moving the value of s, a proportional point on the line segment, the figure moves up and down through all of the turbine’s horizontal cross sections. The locus of that point (in green/blue) in terms of t is always a semi-circle (ellipse in the side view) but gets larger and smaller. The top view version, animated below/left, shows the mathematically accurate version of this same concept: moving up and down the turbine blade.
One important question that comes out of this squeeze is at what point the squeeze is the most extreme (or where is it most squeezed) in terms of s? It turns out that this happens where the distance between the two intersection points of the circle is equal to the diameter of the locus. This is because this locus always goes through two points, the intersections of the circles. How can this point be located, in terms of s, though? It is important to keep in mind that the point s being in the same place as one circle intersection doesn’t mean the locus is at its minimum size. It is best to eyeball where s is when the circle is at its minimum size and then replace s with that estimate. This estimation can be confirmed by calculating the locus’ symbolic equation then drawing a circle and constraining it to have the same equation as the locus. Then, calculating the circle’s radius and the distance between the circle intersections, and finding the radius to be half the distance means the correct value for s has been found. Fortunately, the minimum radius is found at 0.5s, or halfway between the top and the bottom. This means that the squeeze is vertically symmetrical. Now the question is what effect theta (the twist) has on the amount of squeeze. The figure below illustrates how the angle of twist, with s constrained at the maximum squeeze value of 0.5, effects the amount of squeeze. Remember, the smaller the locus, the more the squeeze.
Modeling the Squeeze with CAS
First, Geometry Expressions’ calculation for the symbolic radius of the locus of the point constrained at s is copied as a Maple input and pasted into Maple (or whichever computer algebra system you use). Then, the constant r is set at 1 and theta is replaced with pi. Plotting the result gives the amount of squeeze moving up/down the turbine when it is twisted pi radians (note that Maple works almost exclusively in radians). Going back to the theta equation, a three-dimensional plot of the squeeze can also be created. In this plot, the unmarked axis is the radius of the locus and the other two dimensions represent theta and s. Finally, there is a two-dimensional plot of how the twist affects the radius at the minimum radius and a calculation of the radius at a specific theta value. The entire Maple worksheet is below.
Maple Worksheet:
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