Physics of Oscillations
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Parametric Excitation of a Linear Torsion Pendulum

(A Virtual Lab for Undergraduate Students)

A computer model of a mechanical linear torsion oscillator excited by a square-wave modulation of its parameter (moment of inertia of the rotor) is presented on this page. The Java-applet requires some time to load, so please be patient while it is starting. After the applet loaded, you can switch to off-line mode. Java applets are run by web browsers (with Java plugin installed) under security restrictions to protect the user. In case you have Java 7 or Java 8 installed on your machine, trying to run Java applications generates a message:

  • Java applications blocked by your security settings.
As a workaround, you can use the Exception Site List feature of your operating system to run the applications blocked by security settings. Adding the URL of the blocked application (applet) to the Exception Site List allows the applet to run with some warnings.

Instead of running the applet, you can open (or download) the executable jar file (Java archive) OscillationsE.jar in which several simulations are packaged. When you start it, the list of available simulations appears. Select the simulation Parametric Excitation of the Torsion Spring Pendulum from this list, and start it together with the presently opened web page, which gives the description of the simulated physical system.

The model of a linear oscillator which is used in the simulation program is a balanced rotor (flywheel) with two equal weights, so that its center of mass lies on the axis of rotation. A spiral spring (with the other end fixed) flexes when the wheel turns. The spring creates the restoring torque that tends to return the rotor into the equilibrium position. The torque is assumed to be proportional to the angle of deflection (Hooke's law). To provide modulation of a system parameter, we assume that the weights can be shifted simultaneously along the rod in opposite directions into other symmetrical positions so that the rotor as a whole remains balanced, but its moment of inertia J is changed. Periodic modulation of the moment of inertia by such mass redistribution can cause, under certain conditions, a growth of (initially small) natural rotary oscillations.

The square-wave variation of a parameter can produce considerable oscillation of the rotor if the period of modulation is chosen properly. For example, suppose that the weights are drawn closer to each other at the instant at which the rotor passes through the equilibrium position, when its angular velocity is almost maximal. While the weights are being moved, the angular momentum of the system remains constant since no torque is needed to effect this displacement. Thus the resulting reduction in the moment of inertia is accompanied by an increment in the angular velocity, and the rotor acquires additional energy. The greater the angular velocity, the greater the increment in energy. This additional energy is supplied by the source that moves the weights along the rod.

On the other hand, if the weights are instantly moved apart along the rotating rod, the angular velocity and the energy of the rotor diminish. The decrease in energy is transmitted back to the source. In order that increments in energy occur regularly and exceed the amounts of energy returned, i.e., in order that, as a whole, the modulation of the moment of inertia regularly feed the oscillator with energy, the period of modulation must satisfy certain conditions.

For instance, let the weights be drawn closer to and moved apart from each other twice during one mean period of the natural oscillation. Furthermore, let the weights be drawn closer at the instant of maximal angular velocity. Then they are moved apart almost at the instant of extreme deflection, when the angular velocity is nearly zero. The angular velocity increases at the moment the weights come together, and vice versa. But if the angular momentum is zero at the moment the weights move apart, this particular motion causes no change in the angular velocity or kinetic energy of the rotor. Thus modulating the moment of inertia at a frequency twice the mean natural frequency of the oscillator generates the greatest growth of the amplitude, provided that the phase of the modulation is chosen in the way described above.

The model allows you to observe the motion of the pendulum on the screen. You can make a pause in the simulation and resume it by clicking on the button "Start/Pause" on the control panel located on the left-hand side of the applet window. You can vary the time scale by moving the slider (named "Delay" – down on the panel to the left side of the pendulum) for convenient observation. The model allows you also to display the graphs of time dependence of the angle and angular velocity (by checking the corresponding check-boxes on the same panel), as well as the graphs of energy transformations and phase diagram. You can resize and drag with the mouse the panels with graphs and the phase trajectory to any convenient place on the screen.

At first acquaintance with the program you can open the list of predefined examples (by using the check-box under the image of the system). These examples illustrate the most typical kinds of motion of the simulated system. Choosing an example from the list, you need not enter the values of parameters required for the illustrated mode – they will be assigned automatically.

The panel on the right-hand side of the pendulum allows you to vary parameters of the pendulum (absence or presence of viscous friction, and quality factor Q ), the period and depth of modulation, and also to change initial conditions (initial angle of deflection and initial angular velocity). You can change the values either by dragging the sliders, or by typing the desired values from the keyboard (editing the corresponding number fields). In the last case you should press "Enter" key after editing. The model will accept the new values after you press the button "Accept new values".


Principal goals of the lab:

  • To get acquainted with the physical nature of parametric excitation by using an obvious example of a mechanical torsion spring oscillator whose moment of inertia is modulated periodically in piecewise constant manner.
  • To explain peculiarities of parametric resonance by treating parametric oscillations at piecewise constant modulation of the inertia moment as natural oscillations with alternating periods.
  • To understand general principles of parametric excitation in linear oscillatory systems.
  • To calculate theoretically and prove in the simulation experiments the frequency intervals of parametric instability and the threshold of parametric excitation for the principal resonance and resonances of high orders.
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