Nonlinear Oscillations
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Pendulum with a Square-wave Modulated Length

(A Virtual Lab for Undergraduate Students)

A computer model of an ordinary rigid planar pendulum excited by a square-wave (piecewise-constant) modulation of its length is presented on this page. The simulation is implemented as a Java applet. It requires some time to load, so please be patient while it is starting.

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.

Steps to Add URLs to the Exception Site list:

  • Go to the Java Control Panel (On Windows click Start, then Control Panel, and find Java there).
  • Click on the Security tab of the Java Control Panel.
  • Click on the Edit Site List button.
  • Click Add in the Exception Site List window.
  • Click in the empty field under the Location field to enter the URL.
  • Type in (or paste) the required URL that hosts the applet, namely
  • Click OK to save the URL that you entered.
  • Continue on the Security Warning dialog.
  • Reload the web page with the applet.
The operating system may ask your permission to start the Java applet. Say "yes" to this request --- this is absolutely safe. After the applet is loaded, you can switch to the off-line mode.

Instead of running the applet, you can open (or download) the executable jar file (Java archive) PendParSquare.jar in which the simulation is packaged. You can start it while reading the present web page, which gives the description of the simulated physical system.

The model of a pendulum that is used in the simulation program is a massless rod with a massive bob on its end. The force of gravity creates a restoring torque that tends to return the pendulum into the equilibrium position. The torque is proportional to sine of the deflection angle. To provide modulation of a system parameter, we assume that the bob can be shifted up and down along the rod. Periodic modulation of the pendulum length by such mass reconfiguration can cause, under certain conditions, a growth of (initially small) natural oscillations of the pendulum.

The periodic square-wave variation of a parameter can produce considerable oscillation of the pendulum provided the period and phase of modulation are chosen properly. For example, suppose that the bob is shifted up at the instant at which the pendulum passes through the equilibrium position. At this instant the angular velocity of the pendulum has its maximum value. The angular momentum of the pendulum with respect to the pivot remains constant when the bob is shifted along the rod, because no torque is needed for this displacement. Thus the resulting reduction in the moment of inertia is accompanied by an increment in the angular velocity, and the pendulum gets additional kinetic energy. The greater the angular velocity, the greater the increment in energy. This additional energy is supplied by the source that moves the bob along the rod.

On the other hand, if the bob is instantly moved down along the rotating rod, the angular velocity and kinetic energy of the pendulum 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 pendulum length regularly feed the pendulum with energy, the period and phase of modulation must satisfy certain conditions.

For instance, let the bob be shifted both up and down twice during one mean period of the natural oscillation. Moreover, let the bob be shifted up at the instant at which the angular velocity reaches its maximum. Then the bob will be shifted down almost at the instant of extreme deflection, when the angular velocity is nearly zero. The angular velocity increases at the moment when the bob is shifted up, and vice versa. But if the angular momentum is zero at the moment the bob is shifted down, this particular motion causes no change in the angular velocity or kinetic energy of the pendulum. Thus modulating the length of the pendulum at a frequency twice the mean natural frequency generates the greatest growth of the amplitude, provided that the phase of the modulation is chosen in the way described above.

The theoretical background for understanding the phenomenon of parametric excitation can be found in the paper «Pendulum with a Square-wave Modulated Length». We recommend you to read this paper before proceeding to work with the simulation program. The paper presents a summary of the relevant theory.

The simulation program (Java applet) 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 the button "Start/Pause" on the control panel which is located on the left-hand side of the applet window. You can vary the time scale of the modulation for convenient observation by moving the slider named "Anim. delay" (down on the panel to the left side of the pendulum). The model allows you also to display the graphs of time dependence of the angle and angular velocity as well as the graphs of energy transformations and the phase diagram. The graphs are displayed in separate windows, if you by check the corresponding check-boxes on the control panel. You can resize and drag with the mouse the panels with graphs and the phase trajectory to any convenient place on the screen. (If these windows occur covered by other panels, simply double click on the corresponding check-boxes.)

Instead of using the applet in your browser, you can download the Java executable archive file PendParSquare.jar on your machine, launch it and comfortably play with the computer model of the pendulum out of the browser window. In particular, you can drag and resize the panel that displays the pendulum with your mouse for convenient observation, open additional windows with different graphs and with the phase trajectory, vary parameters of the system, and much more.

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). When you type in new values, the corresponding field becomes yellow. In the last case you should press "Enter" key after editing. If the field becomes red, your input is inadmissible. Please make corrections and press "Enter" once more. The field with admissible new values should be gray, as initially. The model will accept the new values after you click at 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 system --- rigid planar pendulum --- whose length (and hence the moment of inertia) is modulated periodically in a piecewise constant manner.
  • To explain peculiarities of parametric resonance by treating parametric oscillations at piecewise constant modulation of the pendulum length as natural (free) oscillations with alternating periods.
  • To understand the role of nonlinear properties of the pendulum (namely, dependence of the natural period on the amplitude) in restricting the growth of the amplitude in conditions of parametric resonance.
  • To explore the parametric autoresonance phenomenon, the phase locking between the modulation and oscillations of the pendulum.
  • To observe bifurcations of the symmetry breaking, and bifurcations of period doubling in a simple and familiar nonlinear system.
  • To understand general principles of parametric excitation in 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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