AA Pneumatic Chaotic Pendulum
Devlin GualtieriTikalon LLC, Ledgewood, New Jersey ([email protected])
I present a chaotic pendulum based on the repulsive force between a random array of point sources of air flow and the conical tip of a rigid pendulum. Source code is provided for generation of random aperture arrays. The chaotic motion was analyzed using machine vision techniques. A computer simulation of this system is also presented, as are the results of some example simulations.
IntroductionThere are numerous examples of chaotic pendulums based on repulsive forces between an array of permanent magnets and a rigid rod tipped with a permanent magnet.[1-2] Air flow will also provide a repulsive lateral force on a pendulum when acting on a cone. A pneumatic chaotic pendulum was constructed using an air plenum drilled with an array of randomly-sized holes in random positions. A diagram of its construction can be viewed as fig. 1. Fig. 1. The pneumatic chaotic pendulum.The 9-inch by 9-inch air plenum is a frame that’s capped at the top and bottom by plexiglas polycarbonate sheets. The sheet transparency allows the tip of the pendulum to be viewed from below. The aperture array is constrained to a 4-inch by 4-inch patch in the center of the top sheet.Fine wires are wrapped around the ceramic pendulum rod to supply electrical current to the LED at the cone tip.Air PlenumThe air plenum is constructed as a 9-inch by 9-inch frame that’s capped at the top and bottom by transparent plexiglas polycarbonate sheets. The aperture array of pnematic point sources is constrained to a 4-inch by 4-inch patch in the center of the top sheet. Holes were drilled through this sheet in a range of sizes from 0.0980-inch to 0.1495-inch (ANSI drill sizes 40 to 25). A computer program (source code in Appendix II) produced a drilling template for twenty holes of random size in random positions, as shown in fig. 2. Fig. 2. Drilling template for the top sheet of the air plenum. The numbers are the ANSI drill sizes, and the red dot marks the center of the array. PendulumSince air flow was limited (a Porter Cable Model C2002 Portable Electric Pancake Air Compressor with an airflow of about 3 cubic feet per minute was used), it was necessary to fabricate a pendulum of small weight. The cone was hollow, constructed from paper, and it was mounted at the end of a thin, 0.125 inch diameter, ceramic rod manufactured as a feed-through for a fine gauge thermocouple. Alternatively, a hollow but rigid plastic tube can be used. This might be constructed from several plastic drinking straws coupled together.A number of cone angles was tried, and maximum motion was obtained with a cone of about 60-degree tip angle. A small light-emitting diode was affixed to the tip of the cone to assist in the machine vision data collection, as described below. The combined eight of the rod, cone, LED, and LED wiring was 9.0 grams, and the rigid rod was attached to the support with a short cotton thread. It was important to adjust the rate of airflow with the valve, since too high a flow resulted in the cone’s orbiting the entire array.Machine Vision Data CollectionTo facilitate data collection, the motion of the tip of the pendulum was recorded on video for analysis. This process was aided by fitting the tip of the cone with a small light-emitting diode operating at low power and collecting data in a darkened room. A small HD resolution video camera with a USB interface was used with the free and open source video capture program, cheese . Although the camera was capable of color video at 1920 x1080 resolution, the data collection was done using grayscale at 640 x 480.The video stream, captured as a webm video file, was processed by the free and open source video editing program, ffmpeg , to create sequential grayscale pgm images at half-second intervals. An image analysis program extracted the (x,y) coordinates of the tip as a function of time. Fig. 3 shows the (x,y) position as pixels from the pendulum rest position at half-second intervals for twelve minutes of operation after an initial 30-second period to eliminate any transient effects. Note the unexplained region at larger x and y values that might relate to the placement and sizes of the apertures.Fig. 3. The pendulum tip position, as pixels from the pendulum rest position, at half-second intervals for twelve minutes of operation. Data for the initial 30-seconds were eliminated to exclude potential transient effects.Note the unexplained region at larger x and y values that might relate to the placement and sizes of the apertures, as would the shape of the region of allowed values.ata AnaysisThe radial deviation from the pendulum rest position was calculated from the (x,y) coordinates, and a middle portion of the movement is shown as fig. 4. Fig. 4. Radial deviation from the pendulum rest position as a function of time. Shown are data from 200 to 400 seconds.Summary of ExperimentThis proof-of-principle experiment opens an opportunity to explore the affect of the aperture number, sizes, and positions on the pendulum motion. The mechanics of the system are simple, so a computer simulation was undertaken (Source code in Appendix III). This simulation reproduces the action of this mechanical system, and it allowed a more thorough analysis of its performance. The simulation illustrates a sensitivity to initial conditions, in this case the starting position of the pendulum tip, that’s common in chaotic systems.Physical ModelA diagram of the forces acting upon the pendulum’s conical tip is shown as fig. 5. The pendulum is always acted upon by a restoring force to its rest position. Its conical tip can also be acted upon by multiple air flow sources depending on its position relative to the apertures drilled into the air plenum at random positions.he acceleration and direction of the pendulum are calculated from the sum of the resolved force components of these forces in the x- and y-directions, and the pendulum position is calculated from these accelerations after each time tick, δt. Some simplification allowed an easier computation. Air flow through the plenum apertures was considered to be laminar, and the resultant force was considered to be acting at a point. The resolved force components depend on the angle of air flow with respect to the cone. This angle is principally the angle of the cone, but it will change with the pendulum position. In these calculations, just the cone angle was used. Fig. 5. Forces acting on the conical tip of the pneumatic chaotic pendulum.The air plenum is drilled with random-sized holes in random positions. The pendulum is acted upon by a restoring force to its rest position F r and also forces from air flow F a from the plenum apertures.Pendulum acceleration and direction are calculated from the resolved force components in the x- and y-directions, and the pendulum position is calculated from these accelerations after each time tick, δt. Simulation ProgramA synopsis of the simulation program follows:1) An array of randomly-sized holes at random position is generated in the air plenum. - A minimum spacing is required to exist between the holes.2) A random initial pendulum position (x,y) is selected within the aperture array.3) The force vectors acting on the conical tip are calculated:- The pendulum restoring force to its resting position.- The air forces from all apertures within the cone’s projected cross-section. For simplification, a circular cross-section is used instead of the slightly elliptical one that would be present.4) The x- and y- components of the forces are summed.5) The resulting acceleration is calculated and summed with the last acceleration.6) A time tick δt is set.7) The resultant displacement vector is calculated from δs = ½ a δt .) A new pendulum position (x,y) is calculated9) Loop to step 3.The program logs the initial conditions, and the pendulum position at each clock tick. There is also provision for automatic generation of an (x,y) plot using GNUPlot (Appendix IV).[3]Simulation ResultsFig. 6 illustrates the results of the simulation for various aperture arrays and pendulum initial positions. In some cases, the pendulum’s initial position causes it to swing freely without any influence from the aperture array.Fig. 6. Examples of the pendulum motion for different aperture arrays (shown in red) and initial pendulum positions. In (a), the pendulum swings without influence of any of the air sources, while (b) through (d) are the usual response.ig. 7 illustrates the temporal progression of the pendulum cone radial distance from the pendulum resting position, as sampled at one second intervals over a 1000 second period, for one set of initial conditions. Fig. 8 shows a histogram of the difference in the radial positions at 0.05 second intervals for the same initial conditions that produced the pendulum motion shown in fig. 3. The line in fig. 8 is a fit of the data to a normal distribution.Fig. 7. Radial distance of the pendulum’s conical tip from the pendulum resting position, sampled at one second intervals. Fig. 8. Histogram of the difference in radial positions at 0.05 second intervals from the pendulum resting point for the pneumatic chaotic pendulum.The line is a fit of the data to a normal distribution, and it indicates a diminished probability of circular orbital arcs around the pendulum resting position.ig. 9 shows the histograms of the pendulum x- and y- values sampled at 50 second intervals for a representative simulation of 100,000 seconds duration. The lines are fits to normal distributions.Fig. 9. Left, histogram of x-positions, and, right, histogram of y-positions, taken at 50 second intervals over the course of a representative 100,000 second simulation. The lines are fits to normal distributions.Sensitivity to Initial Pendulum PositionSensitivity of initial condition, the so-called butterfly effect . ppendix III – Program Source Code (chaotic_pendulum_simulation.c) ppendix IV – GNUPlot Script (graph.gnu) set terminal pngcairo enhanced font "arial,16" fontscale 1.0 size 750, 750 set output 'graph.png'set grid nopolarset grid xtics nomxtics noytics nomytics noztics nomztics \ nox2tics nomx2tics y2tics nomy2tics nocbtics nomcbticsset grid layerdefault linetype -1 linecolor rgb "gray" linewidth 0.200unset keyset autoscale xset autoscale yset title "Pneumatic Chaotic Pendulum" set xlabel "X" set ylabel "Y" plot 'output.txt' skip 17 index 1 u 1:2 w lines lt rgb "blue", 'output.txt' skip 17 index 0 u 1:2 pt 7 lc rgb "red" ps 1 ppendix V – Initial Conditionsppendix V – Initial Conditions