Steady State Angle of a Pendulum with Wind

[Imurphy]

I'm having trouble solving this problem. I'm trying to find the steady state angle of a rod with wind /fluid drag. Assume Cd, L, U, etc. are known. The viscous and form drag forces are a function of the angle θ so I don't have enough equations to solve the problem. Maybe there's some type of Lagrangian approach to this that I'm not thinking about. It seems like a simple problem but I'm missing something. See image.

Thanks,
imurphy

 

[Simon Bridge]

Draw a FBD showing all the forces as functions of the angle.
How many forces are acting on the rod?

 

[Imurphy]

I added the FBD. To clarify, the system is a rigid pendulum with an evenly distributed mass. The fluid is water, so there is also a significant buoyant force (Fb). The weight and buoyant force aren't a function of the angle in this model. In the real system, some of the fluid is water, and some is air, so in that case, the buoyant force only is considered below the water line.

Knowns: L, U, aerodynamic properties, mass/inertia properties, CG location, density of the rod, density of the fluid.

 

[Simon Bridge]

Ah yeah - so buoyancy will eventually depend on angle (as the proportion of the rod immersed).
FBD looks good enough to talk about.
Do you know the details of those function of theta-ss in the diagram?

After that - have you tried proceeding how you normally would for a FBD?
Resolve the forces into components perpendicular and parallel to the tension.
Sum the forces in each direction etc.

 

[Imurphy]

Yes, I have all of those functions written out. Just didn't want to write too much or I thought no one would read it/answer. I was trying to simplify the problem to get to the root of the issue. I've tried solving it by summing the Y forces and X forces against the reaction forces Fx and Fy (components of tension) and it is not solvable. After you mentioned the FBD, I think the solution may be summing the moments about the pivot axis. Its really just a statics problem. My last resort would be trying to do some type of equilibrium calculation using energy methods but that would be the most time consuming method.

The drag calculations are done using a quasi-steady state formulation, but with separate terms for viscous drag and form drag:
Fdv=1/2 ρ Cdv U^2 Av
Fdf=1/2 ρ Cdf U^2 Af

Where
Av=D*L*sin(θss)
Af=D*L*cos(θss)

For the case where there are two fluids, the L above is replaced by Lw(θss), the length of the rope under water as a function of steady state angle in the Fluid area calculations Av and Af.

 

[Simon Bridge]

Summing the moments would work too.

If you have got as far as two equations and two unknowns, and they appear intractable, that is quite a different issue.

 

[Imurphy]

Essentially, that it what happened. I believe the equations become coupled when they became a function of theta. It looks like right now there is 5 unknowns and 4 uncoupled equations, therefore plugging it into a solver or solving by hand didn't work. I think summing the moments should give me that last, uncoupled, equation. I haven't had a chance to sit down and work it out yet.

 

[Simon Bridge]

From what you've written... I just get two equations and the two unknowns are the tension and the angle - in fact, one of the equations has the angle and stuff you know in it. So what do you get and what are the unknowns?

 

[Imurphy]

Ok, perhaps there isn't a problem after all. I just can't solve a symbolic solution. If I sum the moments for the simple case with a single fluid, I get 1 equation 1 unknown. I tried solving it symbolically and I get "no explicit solution found", but if I take out all the constants except theta I can get MATLAB to find a numerical solution. If there are 5 other constants its not able to solve the equation.

 

[Simon Bridge]

Then you want to post that last equation then and we'll have a look at it?

It is difficult to see what you expect from the forums since you do not like to post any details that would help us help you.

 

[Imurphy]

Sure, I haven't given too many details. It looks like I'm getting numerical solutions by summing the moments around the axis of rotation. There's just some intellectual property concerns with posting all of my equations online.

Thanks Simon, your replies have helped me rethink the problem.

 

[Simon Bridge]

IP covers copyright, patents, trademark, and trade secrets.
Nothing we have been talking about looks to be commercially sensitive since it is all obvious to anyone in the field, so it is puzzling that you'd be concerned. Besides, publication usually helps with IP conflicts more than it hurts. That's why we tend to be concerned when people claiming to have made a great discovery refuse to publish for eg.

What you are asking about is so basic I suspect it has already been patented many times. Have you checked?

You certainly should be up-front about asking for free-of-charge assistance with a project you expect to restrict via IP sometime. Of course you will be making a contribution to PF should your efforts bear fruit ;)

Cheers. I look forward to developments.