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Square-wave Excitation of a Linear Torsion Pendulum

(A Virtual Lab for Undergraduate Students)

A computer model of a mechanical linear oscillator driven by a square-wave force 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 Torsion Spring Pendulum under the Square-Wave Torque 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). The other end of the spring is attached to the rod (exciter) which is forced to turn abruptly back and forth in equal time intervals about its middle position. Therefore a square-wave external torque is applied to the flywheel. This is an example of kinematical excitation of forced oscillations by a non-sinusoidal external force.

The behavior of oscillatory systems under periodic external forces is one of the most important topics in the theory of oscillations. A noteworthy distinctive characteristic of forced oscillations is the phenomenon of resonance, in which a small periodic disturbing force can produce an extraordinarily large response in the oscillator. Resonance is found everywhere in physics and so a basic understanding of this fundamental problem has wide and various applications.

An external torque whose shape is that of a periodic square-wave can be realized by abruptly displacing the driving rod alternately in opposite directions through the same angle in equal time intervals. This is an example of the kinematic excitation of forced oscillations. If the displacements of the rod and thus of the equilibrium position of the flywheel occur very quickly, there is no significant change in either the angular displacement or velocity of the flywheel during the displacement of the rod. In the mathematical model of the system these forced displacements of the driving rod are set to occur instantaneously.

The model allows you to observe the motion of the pendulum. 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 amplitude of the external driving torque, 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 study forced oscillations of a linear mechanical oscillator (torsion pendulum) under the external torque with non-sinusoidal time dependence.
  • To investigate peculiarities of forced oscillations excited by a non-sinusoidal external force.
  • To understand how the steady-state forced oscillations excited by the piecewise constant torque can be represented as a succession of free oscillations about alternating displaced equilibrium positions.
  • To study the energy transformations in the course of steady-state forced oscillations and transient processes under a non-sinusoidal excitation.
  • To study transient processes at different periods of excitation – at resonances (when the period of excitation equals an odd number of natural periods), when the period is much longer or shorter than the natural period.
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