Video-Captured Dynamics of a Double Pendulum

Transcription

1 WJP, PHY81 (010) Wabash Journal of Physics v.0, p.1 Video-Captured Dynamics of a Double Pendulum Scott Pond, Thomas Warn, M. J. Madsen, and J. rown Department of Physics, Wabash College, Crawfordsville, IN 479 (Dated: March 0, 010) Analyzing the dynamics of a double pendulum to determine the chaoticness of its motion can be done by looking at its Lyapunov exponents. Using a particle-tracking program, we tracked the motion of three different colored dots on a double pendulum to track the motion of the entire system. First introduced in 1890 by Henri Poincaré, the study of chaos has expanded greatly. Poincaré stated that A small error in the [initial conditions] will produce an enormous error in the [final phenomena] [? ]. Chaos theory characterizes the time-evolution dynamics that a system exhibits due to its hypersensitivity to its initial conditions [? ]. utterfly effect is a term that refers to the idea that over time the wind created from a butterfly s wings could eventually lead to a hurricane on the other side of the planet; this is the premise of chaos theory [? ]. Determining if a system is truly chaotic requires an infinite amount of time, however, as this is impossible to experimentally achieve, we can use Lyapunov exponents to show that a system is sufficiently chaotic. Classical chaos is simple to observe and demonstrate with a double pendulum [??? ]. Tracking the motion of the double pendulum has been done by analyzing Polaroid photos of the pendulum, but our goal was to use a video of the motion and track the motion of the the system as it evolves. Last semester rad Vest and Thomas Warn built the double pendulum, which we will analyze [? ]. For this project we used a high speed camera to capture the motion of the double pendulum. Using the tracking software, we can see how the system evolves with time. The program locates and tracks blobs of specific colors, red for our case. To ensure the camera can differentiate between the dots and the metal pendulum and frame, we spray painted the front face of the pendulum black and used a large garbage bag to cover the frame. We placed three red dots on the pendulum: one the top axle, one just above the lower pendulum, and one near the bottom of the lower arm. The two dots that are relevant to the tracking are the lower two. The dot on the top axle allows us to determine the angle

2 WJP, PHY81 (010) Wabash Journal of Physics v.0, p. the lower arm makes with the upper arm. It was observed last semester that the table would shake while the pendulum was moving, interfering with the results. For our experiment, we will secure the double pendulum to the floor using steel and aluminum struts. The double pendulum that we will use is made primarily from aluminum. FIG?? shows the dimensions and materials that were used to construct the double pendulum, and FIG?? shows an exploded view of the assembled pendulum. Quantity: Quantity: 1 Quantity: 1 Quantity: Holes 1,, and centered 5/8 from edges / / 5/16 1 7/8 Steel Axels Quantity: 1 1/4 /8 5/16 Plastic spacer 10 /4 8 1/ 5/16 1/ Quantity: /4 1/ Aluminum 1 1 1/ 1/ ID: Centered FIG. 1: The drawings of the pieces made by rad Vest and Thomas Warn [? ]. These pieces were then assembled for the double pendulum. ecause proving chaos requires an infinite amount of time, we need to find some characteristic that describes the chaoticness of the system, unless the system s motion is random. Lyapunov exponents are one such characteristic. Calculating Lyapunov exponents is made simple by selecting two points, initially infinitesimally close on a phase-space plot, then tracking the divergence of the points as the system evolves over time. A positive Lyapunov exponent describes a chaotic system; a negative exponent describes a dampened system, and if the exponent equals zero, then the system is harmonic. FIG?? shows that, by using the initial separation between the two points ɛ, the final separation

3 WJP, PHY81 (010) Wabash Journal of Physics v.0, p. A: Fiberglass spacers : earings A A FIG. : Exploded view of the assembly of the double pendulum from Vest and Warn [? ] between the two points d n, and the number of iterations n, the Lyapunov exponent is given by λ = ln dn ɛ n. (1) efore we could use Mathematica to analyze the data, we first had to determine the angle the arms made with the respect to the vertical FIG??. x 1 = L 1 sin θ 1 () y 1 = L 1 cos θ 1 () x = x 1 + L sin θ (4) y = y 1 L cos θ (5) L 1 = (x 1 x 0 ) + (y 1 y 0 ) (6)

4 WJP, PHY81 (010) Wabash Journal of Physics v.0, p.4 n 1 ε d n FIG. : Calculating Lyapunov exponents requires using the initial separation between the two points ɛ, the final separation between the two points d n, and the number of iterations n L = (x x 1 ) + (y y 1 ) (7) θ 1 = arcsin( x 1 L 1 ) = arccos( y 1 L 1 ) (8) y 1 = L 1 cos θ 1 (9) Using these simple relationships, we can find θ 1 and θ are given by x 1 θ 1 = arcsin( (x1 x 0 ) + (y 1 y 0 ) ) = arccos( y 1 (x1 x 0 ) + (y 1 y 0 ) ) (10) and x x 1 θ = arcsin( (x1 x 0 ) + (y 1 y 0 ) ) = arccos( y + y 1 ). (11) (x1 x 0 ) + (y 1 y 0 ) Using these equations we created an Excel spreadsheet of the collected data to determine the values of L 1, L, θ 1, and θ. These equations give θ 1 and θ, but we also need θ 1 and θ these values come from an the derivate of an interpolated function of each θ. After determining θ 1, θ, θ 1, and θ, we used the Mathematica notebook and same procedure used by Vest and Warn to analyze the dynamics of the double pendulum.

5 WJP, PHY81 (010) Wabash Journal of Physics v.0, p.5 θ1 θ FIG. 4: θ 1 and θ are the angles each makes with respect to it s vertical. Determining the equations of motion for the double pendulum is easily done by finding the Lagrangian. The potential energy for this system is given by U = m ( 1l 1 cos θ 1 m l 1 cos 1 + l ) cos 1 The kinetic energy is a little more tricky. The kinetic energy for the upper arm is given by (1) K 1 = m 1l 1 θ 1 + m l θ, (1) and the kinetic energy of the lower arm is given by K = 1 m [ l θ 1 + l 1 l θ1 θ (sin θ 1 cos θ + sin θ 1 sin θ ) + l θ ]. (14)

6 WJP, PHY81 (010) Wabash Journal of Physics v.0, p.6 Adding the kinetic energies give the total kinetic energy. So the Lagrangian L = K U is given by L = m 1l1 θ 1 + m l θ + 1 [ m l θ 1 + l 1 l θ1 θ (sin θ 1 cos θ + sin θ 1 sin θ ) + l θ ] + m 1l 1 cos θ 1 + m ( l 1 cos 1 + l cos 1 Our tracking software data randomly skips large chunks of data points, making it very difficult to accurately obtain θ 1 andθ. ). While these problems prevented us from getting viable results, the spread sheet, Lagrangian and Mathematica file will be benefical for the next group once they create or find a working software. [] Poincaré, Henri. Sur le problme des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt. Acta Mathematica 1, 1-70 (1890). [] DeSerio, Robert. American Journal of Physics. 71, (00). [] Levien, R.., Tan, S. M.American Journal of Physics. 61, (19). [] Rafat, M. Z., Whealand, M. S., edding,t. R. American Journal of Physics. 77, (009). [] Shinbrot, Troy; Grebogi, Celso; Wisdom, Jack; Yorke, James A. American Journal of Physics. 60, (199) 60, 6 (199). [] Hilborn, Robert C. American Journal of Physics. 7, (004). [] Vest, rad C, Warn, Thomas. Wabash Journal of Physics. 81, 1-10 (009).