## Saturday, December 26, 2020

### A Tarzan Rope in the Wind (Problem)

With all of my grades entered, my mind can turn off for two weeks.  In my case, that means exploring my curiosity.  Today, that resulted in a fascinating mechanics problem.

My kids have a Tarzan rope in the backyard - a rope suspended vertically and hanging freely.  I noticed this morning that it had been displaced significantly by the wind: it was now draped over a swing that hangs nearby.  "That must have been some wind," I thought.  Rope has a small ratio of surface area to mass, which means it should not be overly affected by aerodynamic drag forces.  With some physics, I should be able to estimate the minimum speed of last night's wind.

To make the exercise worthwhile, I have no intention of simply solving a numerical problem: boring.  Instead, I will solve a generalized problem before specifying any parameters.  Before doing so, I will make some assumptions that will hopefully render the problem to one that can be solved without numerical software.

I will assume that the wind is lateral and constant.  This may lead to an overestimate of the wind, because it is possible that some sort of driving frequency was present in the wind, causing the fundamental mode of the rope to resonate somewhat.  Still, it is probably a fair assumption.  Also, the fact that the wind force is not time variant will reduce the governing dynamics from what could have been a partial differential equation to an ordinary one.  This is because the lateral displacement of the rope (y) varies along the vertical rope's length (x).  A time-varying displacement would make the solution vary according to y(x,t), a multivariable function.  Now, I can begin my search for the single variable function, y(x).  I will also assume what appears to be true: the rope is uniform in terms of its properties and cross-sectional geometry across its whole length.

At this point I could probably Google "steady-state lateral deformation of a vertically suspended uniform rope exposed to a uniform and constant lateral wind", but I strongly doubt anything useful will turn up.  So, because I can (I hope) solve this problem, I am diving into it head first.

Yup, it's boxing day, but instead of looking for deals on stuff, I am entertaining myself for free as my wife shakes her head (well, she doesn't, but that is only because she thinks I am up to something more important).

Before beginning this analysis, I must determine what approach to take.  It is clear that with a constant wind speed at all locations of the rope at all times, the rope will reach a steady state y(x).  So, I am in search of an equilibrium position.  This simplifies things considerably from a typical first principles analysis.  Rather than applying Newton's second law for all of the infinitesimally small segments of rope, dx, I can do this using the first law.  That is because acceleration has been removed from the scenario.

I could do a quick first pass using an assumed modes shortcut.  If I assume that the shape of the rope will follow a specific y(x), I could quickly establish a single algebraic equation in which wind speed is the only unknown.  Here, I would effectively be starting with an assumed solution, but if my guessed shape happens to be good, the answer it gives could be surprisingly accurate.  It is a good moment to pause and ask ourselves the following question: If we were forced to assume the shape that a constant wind would impose upon the freely suspended rope, what mathematical function might it follow?

Four possibilities immediately come to mind:

(1) Linear

(2) 1/4 sine wave

(3) Some sort of polynomial

(4) Some sort of exponential

I will not reveal here which of the above options my intuition leans towards.  I will leave that as a fun mental exercise for the reader.  I will dissect each of the above options in my solutions post.

Alternatively, I could use a more generalized approach - one that requires no shape assumption.  The downside here would be that an ordinary differential equation would require solving.  It would probably be solvable analytically, without the use of numerical tools.  The type of y(x) function that would result would either confirm or negate the choice of assumed function used in the shortcut approach described above.  I guess I should be rigorous, and explore both options.  I could then note the extent to which the quicker method is valid.

I suppose this is where I leave you, for now.  If you are a mechanics savvy reader and are up to the challenge, feel free to post your solution for y(x) in the comments (you can use any approach you like, but please state the one used).  To ensure we use the same variables, express your y(x) in terms of the following variables:

Surface gravity: g

Air density: p

Wind speed: v

Rope mass: M

Rope diameter: D

Rope length: L

Rope elastic modulus: E

Rope shear modulus: G

Rope shape (cylinder) coefficient of drag: Cd

It is likely than one or more of the above parameters will not appear in your solution for y(x).  It is also possible that some additional assumption on your part will be required, though at this time I cannot think of one.

Alright, now for the hard part, if I do not get overwhelmed by laziness: working out the detailed solutions.  See you on the other side of 2020.

Unknown said...

Treating the rope as a rigid body, we can use Euler's beam deflection equation to obtain the function y(x). If we model the rope as a vertical cantilever beam and the drag force as a uniformly distributed load, we can express the bending moment as a function of x. The first roadblock is encountered after solving for M(x). The ODE that stems from Euler's equation is a non linear second order differential equation, which is rigorous to solve analytically. To ease the analytical solution, it was assumed that y'(x) ≈ 0. The ODE then becomes linear and of second order, which can be easily solved by integrating twice. As such, y(x) is then found to be:

y(x)= (w/(24EI))*(4Lx^3-x^4-6(Lx)^2+Cx+K)

where:
w is the force per unit length applied via the drag force
I is the moment of inertia
E is stiffness
C and K are the boundary condition constants
L is length

We can also use superposition to find the deflection in x direction caused by the weight of the rope. However, this entire approach can cause an erroneous approximation for the following reasons:

Euler's equation is derived off of the existence of a neutral axis. Due to the fact that rope is not an isotropic material, the neutral axis might not necessarily be at the center, which can falsify the ODE. The y'(x) ≈ 0 assumption is garbage as soon as the angle of deflection becomes large. Nonetheless, FEA or CFD would come in handy here to support or disprove the solution.

The Engineer said...

Interesting approach! Rope is a tricky material, because it is not quite a string, which cannot support shear load, and simply follows the shape dictated by the forces and the axial tension. It is also not like a metallic beam - besides being non-isotropic, its modulus of elasticity is also much lower. When I solve this problem, I will treat it like a string. Thanks for contributing here :)