Home :: Programs :: N-Mass Pendulum
This simulation above is a simple extension of the double pendulum to 20 masses. The bottom mass is given an initial angular velocity, which corresponds to a "flick" to the left. Notice how the wave propagates up the "chain" and attempts to reflect from the top. When it does, however, the wave begins to appear choppy and unnatural. I presume the reason why this is the case lies in a few major factors:
As noted in the derivation (pdf) to the double pendulum simulation, the first factor is a significant one. A real string in a macroscopic environment can only transfer a tensile force, not a compressive one. The rigid nature of the rods connecting the masses, however, means that compressive forces (including the internal forces that bring about the beautiful wave pattern seen above) can be, and indeed are transferred. This phenomenon manifests itself clearly in the simulation as a down-up motion of the chain. Look closely at the bottom mass when the wave reflects off the ceiling. Notice that it, along with the entire chain, dips down and then up (several times). This would not happen if the rods were not rigid.
A solution is to model the string in accordance with Hooke's law - by connecting the masses with springs, rather than rods. All strings stretch slightly under tension, as should our string. A Hooke's law-abiding string would still be able to transfer compressive forces, and the effect would still be visible, but not as much. Increasing the number of masses should make the effect less and less pronounced. I am sure that the effect still exists to some degree in real strings, although not necessarily for long ones - as dissipative effects become important. Even now, while the connecting media are rigid rods, increasing the number of masses should help the problem. Most importantly, the modification corresponds to true life, as the atomic forces that keep the string together can always be approximated closely to spring-like forces.
The second factor can be easily remedied by an easy solution and a hard solution. The easy solution is to extend the simulation to include a number of masses that makes it virtually indistinguishable from that of a continuous string. In the Processing code given, this is simply a matter of changing a constant and the initial conditions. Indeed, the only possibly intractable necessity is computing power and time. The hard solution would be to scratch the entire "discrete mass" view of a string and approach the problem from the standpoint of continuum systems.
The third factor applies to the later portions of the simulation in particular. The chain exhibits chaotic motion, and even though real strings do as well, the simulation is more true to life if (especially) small-scale motions are damped away, to eventually leave the string in an perpetual equilibrium "hanging" position.