# Why is the theoretical formulation not explaining why the graph of the square of the velocity and angular displacement is nonlinear (experimentally)?

So I did a wind tunnel experiment where I put a rope restrained (1 degree of freedom, so it can wave in the direction of the wind) at one end so that it displaces according to the variation in wind speed. I have a NON-LINEAR graph when I plot the wind speed and the angular displacement and I'm trying to explain why that's happening.

Here's a schematic of the problem that I've made. As we know, wind load has a linear relationship with the square of wind speed and a nonlinear relationship with wind speed. The opposite applies here. Why please? How can we consider friction in the rotation point? And we are considering only pure horizontal wind.

Plot of tangent of theta (Y-axis) against the square of the wind speed (X-axis) just according to the mechanical analysis.

below is the experimental setup. How can we analyze the effect of friction at the rotation point O?

• CAn you share the plot of the angular displacement with the wind speed or square of wind speed? Sep 8 at 10:25
• Also can you clarify whether the "rope" is rigid or its able to oscillate? Sep 8 at 10:38
• @NMech I don't know how to reply with an image so I just edited the main post. please check it out. Sep 8 at 10:39
• It is rigid and able to oscillate in the direction of the horizontal wind. It maintains it's straight shape throughout the displacement process (it's not flexible). Sep 8 at 10:41
• @AJN by the intended design, I want the rope angle to vary according to the variation in wind speed. So angular displacement is not an error. Sep 8 at 12:54

From what I see the angles that you have are greater than 45 degrees. More specifically, $$atan(2.25)=66^o$$.

(Although probably not the only reason), the $$C_L$$, $$C_D$$ will change significantly between 90 and 30 degrees angle of attack.

figure: variation angle of attack (source researchgate)

Both these coefficient have an effect on the torsional moment that raises the rope, therefore, since they have a non linear behavior, the angle will also have a linear relationship.

• Thank you for your comments. Of course these your points are valid, but I need specific convincing reasons that actually explain the phenomena. Thank you again Sep 8 at 12:56
• Thinking about it even more, I now suspect friction in the restrained end Any ideas on how I can theoretically add this please? I'll add a schematic of the experimental setup to the question. Sep 9 at 0:01