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Forced Oscillations of a Linear Torsion Pendulum

(A Virtual Lab for Undergraduate Students)

A computer model of a mechanical linear oscillator driven by a sinusoidal 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 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 Forced oscillations 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 (in this position the needle of the rotor is vertical and points to the zero of the dial). 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 move periodically back and forth (turning about the axis common with the axis of the flywheel) with a sinusiodal time dependence.

In the conventional classification of oscillations by their mode of excitation, oscillations are called forced if an oscillator is subjected to an external periodic influence whose effect on the system can be expressed by a separate term, a periodic function of the time, in the differential equation of motion. We are interested in the response of the system to the periodic 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.

The phenomenon of resonance depends upon the whole functional form of the driving force and occurs over an extended interval of time rather than at some particular instant.

A mechanical system such as this one is ideal for the study of resonance because it is possible to see directly what is happening. When the driving rod (the exciter) is turned through a given angle, the equilibrium position of the flywheel is displaced through the same angle, alongside the rod. The flywheel can execute free damped oscillations about this displaced position. For weak and for moderate friction the angular frequency of these oscillations is close to the natural frequency of the flywheel.

If the rod is forced to execute a periodic oscillatory motion, the flywheel is subjected to the action of a periodic external torque. This action is an example of the kinematic excitation of forced oscillations. This method of excitation is characterized by a given periodic motion of some part of the system. The kinematic mode of excitation is chosen here for the computer simulations of forced oscillations because the motion of the exciting rod can be displayed directly on the computer screen. Computer experiments with the system can show clearly, among other things, the phase shift between the exciter and the flywheel, and the ratio of their amplitudes.

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 frequency 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 sinusoidal time dependence.
  • To investigate the resonance character of the oscillator steady-state amplitude-frequency response.
  • To understand the energy transformations in the course of steady-state forced oscillations and transient processes.
  • To study transient processes at different frequencies (at resonance, below and over resonance).

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