double pendulum - 1 Chaotic Double Pendulum Jusuk Lee,...

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1 Chaotic Double Pendulum Jusuk Lee, Brandon Sim, Max Sun Ms. Jensvold – Physics C AP Period 5 22 September 2009 Investigation How does the motion of double pendulum differ based on different angles and height? In this lab, we explore the motion of double pendulums. By differing initial conditions, we see that the pendulum can exhibit either chaotic or nonchaotic behavior. By doing this, the different characteristics of chaotic and nonchaotic motion are observed. The purpose of this lab is to explore the motions of the double pendulum system and to make a hypothesis about the mathematical nature of the system. Through techniques of classical mechanics, notably the analysis of the system’s Lagrangian, we are able to find that the system is in fact chaotic; however, it can be modeled precisely for a given set of initial conditions. In addition, with the data as well as additional analysis, we give a set of boundaries for the chaotic motion of the system based on its initial parameters. Method Materials Needed: • Double pendulum kit from CIPT lending library (wheels, screw, bolts, washers, clamps, etc.) • Protractor • Photogates with software like VernierPro (Although many parts of the demonstration require only the pendulum kit). Experimental Work Section 1 Question 1: Two characteristics of the double pendulum’s chaotic motion are the initial angles Θ 1 and Θ 2 as well as the period of the top pendulum. Question 2: The behavior of the double pendulum is chaotic for certain energies (which is dependent on the initial angles of the upper and lower pendulums, respectively, Θ 1 and Θ 2 .) Above certain energy, the behavior of the double pendulum is erratic and chaotic, depending largely on the initial conditions Θ 1 and Θ 2. Under this threshold energy, however, the behavior of a double pendulum is the same as that of a normal pendulum. Section 2 Question 3 : Pendulums do follow Newton’s Laws of motion. In fact, the chaotic motion of the double pendulum can be modeled by numerically integrating (RK4 or other methods) several ODE’s (ordinary differential equations). However, with just classical Lagrangian mechanics, the
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double pendulum - 1 Chaotic Double Pendulum Jusuk Lee,...

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