The Chatic Oscillator混沌振子.docVIP

The Chaotic Oscillator Introduction In this lab, you will explore a simple mechanical system that exhibits some rather complicated behavior. At the heart of this system is the spring-disk system illustrated below. By attaching a mass on the edge of the disk, we create a physical pendulum. (Fig. 1) The torque on the pendulum due to the asymmetrically placed edge mass is (1) where R is the radius of the disk and the torque is positive if the vector points out of the paper. Note that this relationship assumes that ??= 0 radians when the edge mass is at its lowest point and ? is positive in the counterclockwise direction. Figure 1. The physical pendulum.1 The pendulum is attached to a drive shaft about which a string is wound and attached on either side to a spring of spring constant k. (Fig. 2) If the springs are equally stretched to a total length L at equilibrium (??= 0), the torque on the disk due to the combined springs is (2) where r is the radius of the drive wheel, not the disk. (Fig. 2) Figure 2. The springs which exert forces on the drive wheel when the pendulum is displaced from equilibrium.1 In addition, there is an adjustable magnetic damper that provides a velocity-dependent damping force giving the torque . (3) Figure 3. The magnetic damper.1 Finally, the pendulum is driven by a motor that provides additional stretch and contraction to the springs. The driver torque is (4) where A is the amplitude of the motor’s driving arm displacement and ? is the angular frequency of the arm’s rotation. So, the net torque on the damped-driven physical pendulum system is: (5) This is equal to the rotational inertia of the pendulum, I, times the angular acceleration, ; this is the rotational version of Fnet = ma. The goal of mechanics is to solve such an equation of motion for ??(t), for any choice of driving force frequency, amplitude and phase. We will now see that this is quite challenging for the full equation of motion that w