# Analysis of motion of a body moving on a string?

I was wondering about something I observed yesterday. To give some background, one of my hobbies is slacklining. This is essentially like tight-rope walking but with a one inch piece of (in this case polyester) webbing. Usually the set-up is you have an anchor around a tree, a pulley system, the webbing, then another anchor around a second tree. Thus a 'static' end and a non-static end. You tension the line to an appropriate amount and the goal is to walk from one end to the other, working on balance and what not.

Anyways, yesterday there was a little bit of wind and when I stood on top of the webbing, the line, that was initially just barely whipping in the wind, started to oscillate up and down, and as I continued to stand on the line, the amplitude seemed to build on itself and get to be larger and larger oscillations. Once I got off the line it came back to just being in the air.

My question is this, what is going on and what is the best way to model this motion? Is it some kind of driven oscillatory motion? I'm not bouncing up and down, just standing on top. Do the waves travel through the line, into the tree and back building on itself?

How accurate of an analysis is possible? What method should be used? Statics, Dynamics, Lagrangian? Any input would be very welcome.

• Sounds like the position you were standing on it and the tension induced in the line changed its resonant frequency to match the driven oscillations caused by the wind. I imagine that had you moved to a different spot on the line, the oscillations would have abated a bit. Very cool that you picked the right spot. Reminds me of the Tacoma Narrows bridge – Jim May 6 '15 at 15:33
• @ACuriousJim It does seem very similar! I was standing on one end, essentially touching the tree. When you are walking closer to the middle taking steps cause the line to move, but not to the same degree if that makes sense. – Jordan Simon May 6 '15 at 16:14

Yes the energy input into the string comes from the wind. Essentially, the flat cross section acts like a wing producing lift when moving upwards, and downforce when moving downwards. On each cycle there is more and more energy added. This energy travels to the ends and some of it dissipates and some of it bounces back only to add to the motion.

When you get on the string the tension goes up by a lot. The natural frequency of the string goes up with tension according to $f \propto \sqrt{T}$. My guess is that the natural frequency came close the to the excitation frequency of the wind. Empirically this is $f \propto v_{wind}$.

A lot of the physics are similar to overhead power line vibration (called Aeolian Vibration). See http://www.tdee.ulg.ac.be/userfiles/file/Vibrations_eoliennes_intro.pdf for some of the basics.

Aeolian vibration frequency is related to the Strouhal number (empirically found to be $S_t = 0.185$) $$f= S_t \frac{v_{wind}}{d}$$

A related concept is that of mechanical impedance (and matching). See http://www.bksv.com/doc/17-179.pdf for an overview.

The mechanical impedance of the strung cable is $$Z =\sqrt{ \frac{T}{\lambda} }$$ where $T$ is tension and $\lambda=\rho A$ is the linear mass density (as in mass/length).

The cable share between supports without you on it is a catenary shape $$y = a \left( \cosh(x/a)-1 \right)-d$$ where $a=T/(g \lambda)$ is the catenary constant, and $d$ is the sag amount in order for the ends to be at $y=0$. When you walk on the string, it forms two catenary shapes that meet where you stand, which do not share slope. The slope discontinuity is such as to support your weight. This creates two loops for the waves to go back and forth (before and after you) and you need to fit integer number of wavelengths into each segment to achieve resonance.