Physics of Oscillations
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Torsion Spring Pendulum with Dry Friction

(A Virtual Lab for Undergraduate Students)

A computer model of a mechanical oscillator subjected to both viscous and dry friction 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 the 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 Torsion Spring Oscillator with Dry and Viscous Friction 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 an 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). This angle is measured by the dial points. The whole device is similar to oscillators used in mechanical watches.

An oscillatory system with dry friction is characterized by some dead interval (stagnation interval) on both sides of the equilibrium position. Within this dead zone static dry friction can counterbalance the elastic force of spring tension. We assume a simplified mathematical model of dry friction, described by the so-called z-characteristic. The torque of kinetic friction is assumed to be independent of the velocity end equal to the limiting torque of static friction. The magnitude of this static friction is characterized by half-width of the dead zone (measured in degrees). We suppose that direction of the frictional torque changes abruptly when the direction of motion changes.

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 physical system modelled here shows the origin of accidental errors in reading some measuring instruments with a needle, like a moving-coil galvanometer. The needle of such an instrument shows not just to the point of the dial which corresponds to the value of the measured quantity, but rather to some point of the stagnation interval around this value. Position of the point within the dead zone depends on the initial conditions.

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.

The torsion pendulum is characterized by its moment of inertia I about the axis of rotation, and the spring constant D which is the coefficient of proportionality between the restoring torque and the angle of deflection from the equilibrium position. The strength of dry friction is indicated in terms of the half-size of the dead zone (in degrees), within which static friction can counterbalance the elastic force of the spring tension. The influence of viscous friction is characterized by dimensionless quality factor Q.

The panel on the right-hand side of the pendulum allows you to vary parameters of the pendulum (absence or presence of viscous friction, quality factor Q, the half-size of the dead zone), and also to change initial conditions of excitation (initial angle of deflection and initial angular velocity in units of the natural frequency). 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 the character of damping of natural oscillations under the influence of dry friction.
  • To compare peculiarities of damping caused by viscous and dry friction.
  • To investigate the energy transformations in the course of damping of natural oscillations caused by dry friction.
  • To understand the origin of accidental errors in reading some measuring instruments with a needle, like a moving-coil galvanometer.

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Physics of Oscillations – a Virtual Lab