Author's home page Physics of Oscillations – Contents
Torsion Pendulum with Dry and Viscous Friction
(A Virtual Lab for Students)
A computer model of an ordinary torsion spring pendulum with viscous and dry friction is presented on this page. The pendulum can execute free (unforced) rotary oscillations and oscillations under sinusoidal driving torque. Mechanical vibration systems with combined viscous and dry (Coulomb) friction are of considerable importance in numerous applications of dynamics in engineering. When friction is viscous, the spring oscillatory systems are described by linear differential equations. This case allows an exhaustive explicit analytical solution. On the contrary, dry friction results in a nonlinearity.
The simulation of the system is implemented here as a Java applet. 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:
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Instead of using the applet in your browser, you can download the Java executable archive file ForcedDryFriction.jar in which the simulation is packaged. You can start it on your machine, 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 the same time you can read the present web page, which gives the description of the simulated physical system.
About the program.
The simulation program 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 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.)
The theoretical background for understanding peculiarities in behavior of an oscillator with dry friction can be found in the paper «Torsion Spring Oscillator with Dry Friction». We recommend you to look through this paper before proceeding to work with the simulation program. The paper presents a summary of the relevant theory.
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 (the width of the dead zone d, absence or presence of viscous friction, and quality factor Q ), the frequency and amplitude of the external torque, and also to change the 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" (the lowest button on Parameters panel).
The simulated physical system.
The rotating component of the torsion spring oscillator is a balanced flywheel whose center of mass lies on the axis of rotation, similar to devices used in mechanical watches. A spiral spring with one end attached to the flywheel flexes when the flywheel is turned. The other end of the spring is attached to the driving rod, which can be turned by an external force about the axis common with the flywheel. The spring is described by Hooke's law, that is, the spring provides a restoring torque whose magnitude is proportional to the angular displacement of the flywheel from that of the driving rod. In other words, the flywheel is in equilibrium (the spring is unstrained) when the rod of the flywheel is parallel to the driving rod.
In this model of an oscillatory system, free or natural oscillations of the flywheel occur when the driving rod is fixed immovable. Forced oscillations are excited when the driving rod rotates back and forth through some angle sinusoidally about its middle position. This mode differs from the dynamical mode considered usually in textbooks, according to which oscillations are excited by a given external force exerted on the system. Our mode can be called kinematical, because in this mode oscillations are excited by forcing one part of the system (the driving rod) to execute a given motion (in our case a simple harmonic motion). This kinematical mode is especially convenient for visualization in computer simulations, because the motion of the exciter is displayed on the screen simultaneously with oscillations of the flywheel, so that the phase shift between the exciter and the flywheel is obvious.
An idealized mathematical model of dry friction described by the so-called z-characteristic is assumed in the simulation. In this model, the force of kinetic friction does not depend on speed and equals the limiting force of static friction.
There is a range of values of the angular displacement called the stagnation interval or dead zone in which static friction can balance the restoring elastic torque of the strained spring. At any point within this interval the system can be at rest in a state of neutral equilibrium, in contrast to a single position of stable equilibrium provided by the spring in the case of viscous friction. If the angular velocity of the flywheel becomes zero at some point within the dead zone, the system remains at rest there.
The stagnation interval extends equally to either side of the point at which the spring is unstrained. The stronger the dry friction in the system, the more extended the stagnation interval. The boundaries d and –d of the interval are determined by the limiting torque of static friction. These boundaries (corresponding to the middle position of the driving rod) are shown by two arrows on the image of the oscillator on the screen.
While the flywheel is rotating in one direction, the torque of kinetic dry friction is directed against rotation and is constant in magnitude. This rotation is described by the differential equation of a linear oscillator with additional constant term. In other words, this constant torque causes only the displacement of the middle point of the flywheel oscillations, occurring under the torque of the elastic spring, to the opposite boundary of the dead zone. At the moment when the direction of rotation reverses, the middle point of oscillation abruptly jumps to the other boundary of the dead zone. Whenever the sign of the angular velocity changes, the pertinent equation of motion also changes. The nonlinear character of the problem reveals itself in alternate transitions from one of the linear equations to the other.
Below the most typical examples of motion and peculiarities in behavior of the oscillator with dry friction are discussed.
If damping of free oscillations occurs solely due to viscous friction, the amplitude decreases exponentially with time. That is, the consecutive maximal deflections of the oscillator from its equilibrium position form a diminishing geometric progression because their ratio is constant. Such oscillations continue indefinitely, their amplitude asymptotically approaching zero. The exponential character of damping caused by viscous friction follows from the proportionality of friction to velocity. Some other relationship between friction and velocity produces damping with different characteristics. Click here in order to observe the process of damping of natural oscillations under dry friction. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
Let the flywheel be displaced to the right (clockwise) from the equilibrium position, and then released without a push. In the phase plane this initial state is represented by the point which lies to the extreme right on the horizontal axis. If this displacement exceeds the boundary of the stagnation interval, the flywheel begins moving to the left and the graph of its motion is a segment of a cosine curve whose middle point is displaced to the right-hand boundary of the dead zone. The displacement d of the midpoint from zero is caused by the constant torque of kinetic friction. This torque is directed to the right (clockwise) while the flywheel is moving to the left. The corresponding portion of the phase trajectory lyes below the horizontal axis. This curve is the lower half of an ellipse (or of a circle if the scales have been chosen appropriately) whose center is at the point d on the horizontal axis. This point corresponds to the right-hand boundary of the stagnation interval.
The subsequent motion is again a half-cycle of harmonic oscillation with the same frequency but with the midpoint –d displaced to the left, i.e., with the midpoint at the left-hand boundary of the stagnation interval. This displacement is caused by the constant torque of kinetic friction, whose direction was reversed when the direction of motion was reversed. In the phase plane this stage of the motion is represented by half an ellipse lying above the abscissa axis. The center of this second semi-ellipse is at the point –d on the axis. Thus, the flywheel executes harmonic oscillations about the midpoints alternately located at d and –d. The frequency of each cycle is the natural frequency, and so the duration of each full cycle equals the period T0 of free oscillations in the absence of friction. The complete phase trajectory is formed by such increasingly smaller semi-ellipses, alternately centered at d and –d. The diameters of these consecutive semi-ellipses decrease each half-cycle by 2d. The loops of the phase curve are equidistant. The phase trajectory terminates on the abscissa axis at the point at which the curve meets the axis inside the dead zone.
An important feature of free oscillations damped by dry friction is that the motion completely ceases after a finite number of cycles. As the system oscillates, the sign of its velocity changes periodically, and each subsequent change occurs at a smaller displacement from the mid-point of the stagnation interval. Eventually the turning point of the motion occurs within the stagnation interval, where static friction can balance the restoring torque of the spring, and so the motion abruptly stops. At which point of the interval this event occurs, depends on the initial conditions, which may vary from one situation to the next.
In the suggested example the dead zone d equals 12 degrees, which means that after each cycle the maximum elongation decreases by 4d = 48 degrees. The initial angular displacement equals 175 degrees to the right-hand side. Therefore the subsequent maximum displacements to this side are 175 – 48 = 127 degrees, 127 – 48 = 79 degrees, 79 – 48 = 31 degrees. During the next half-cycle the flywheel moves to the left and stops dead at 31 – 24 = 7 degrees within the dead zone after crossing the zero point (at –7 degrees).
Click here in order to observe this process of damping of natural oscillations under dry friction. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
Generally at large enough dry friction sticking may occur: the flywheel remains at rest for a finite time after the velocity reaches zero. However, if the amplitude of excitation exceeds some threshold value, the motion of the flywheel can be purely sliding (non-sticking). In conditions of exact tuning to resonance in the absence of viscous friction the amplitude of oscillations grows indefinitely. Let for simplicity the initial deflection of the flywheel coincide with the left boundary of the dead zone, and initial angular velocity zero. Such initial conditions provide the sliding (non-sticking) motion from the very beginning with two turnarounds during each cycle of excitation. Click here in order to observe the process of resonant growth of oscillations in the presence dry friction. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
If the amplitude of the exciter is smaller than the critical value, the flywheel, depending on the initial conditions, either remains immovable from the very beginning, or makes several movements with sticking and then finally stops at some point of the dead zone. For given width d of the dead zone (for given dry friction) there exists the critical (minimal) value of the drive amplitude which provides non-sticking resonant forced oscillations of the flywheel after a rather short transient (see the paper «Torsion Spring Oscillator with Dry Friction» for a detailed theory). During the transient, depending on the initial conditions, sticking is possible. After the transient is over, at the initial moment of each in turn cycle of excitation, angular velocity of the flywheel is zero. Since the increment in the elongation is the same for each cycle, the curls of the phase curve are equidistant.
In the suggested example the drive amplitude equals 25 degrees, the dead zone equals 15 degrees. The initial position of the flywheel (–15 degrees) coincides with the left boundary of the dead zone, and the initial velocity equals zero. According to the theory (see the paper «Torsion Spring Oscillator with Dry Friction»), the increment in the amplitude during each cycle of excitation in this case should be 18.54 degrees from the very beginning. Therefore the next maximum displacement to the left side should be 15 + 18.54 = 33.54 degrees. Click here in order to observe the process of resonant growth of oscillations in the presence dry friction. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
The amplitude of steady-state forced oscillations at the threshold depends on the initial conditions. If the initial velocity is zero, steady-state oscillations occur from the very beginning, without any transient, in case the initial displacement is negative and lies beyond the dead zone. The amplitude of oscillations equals the initial displacement in magnitude. This mode of oscillations is unstable with respect to variations in parameters of the system: a slight increment in the drive amplitude or decrement in the dead zone width causes an indefinite growth of the amplitude; otherwise, the amplitude gradually diminishes and the oscillations damp out.
In the suggested example the dead zone equals 15 degrees, therefore the threshold drive amplitude is 19.1 degrees (more exactly, 19.0986 degrees). For initial displacement –25 degrees and initial velocity zero, steady-state oscillations occur from the very beginning, and the amplitude equals 25 degrees.
Click here in order to observe stationary periodic oscillations at the threshold conditions. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
The resonant growth of amplitude over the threshold is restricted if some amount of viscous friction is present in the system. In a dual-damped system steady-state oscillations with a constant amplitude eventually establish for arbitrary initial conditions. In the suggested example (drive amplitude 15 degrees, dead zone 5 degrees, quality factor 10) the theoretical value of the steady-state amplitude equals 86.3 degrees. Click here in order to observe how the resonant growth of the amplitude slows down, and a steady-state regime is established due to viscous friction. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
In conditions of exact tuning to resonance the energy is transferred to the oscillator from the external source (from the exciter) with maximal efficiency, if at the beginning of each excitation cycle the flywheel occurs at an extreme elongation to the left-hand side. Indeed, in this case the sinusoidally varying external torque exerted on the flywheel by the exciter acts during the whole cycle in the direction of the flywheel rotation, and over the threshold overcomes the torque of dry friction: the amplitude grows linearly increasing during a cycle by the same amount. On the contrary, if at the beginning of the excitation cycle the flywheel occurs at an extreme elongation to the right-hand side, the external torque of the spring during the whole cycle is directed against the flywheel's angular velocity together with the frictional torque. In this case the amplitude reduces during each cycle. After the amplitude reduces to zero, the phase relations between the flywheel and exciter change to the opposite and become favorable for the transfer of energy to the oscillator: the amplitude begins to grow. Click here in order to observe how at resonant conditions over the threshold the amplitude dinishes at the first stage if initially the flywheel is displaced to the right-hand side, and after turning to zero the amplitude begins to grow. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
In the case of exact tuning to resonance, in contrast to the oscillator with viscous damping, dry friction alone is unable to restrict the growth of the amplitude of forced oscillations over the threshold. In non-resonant cases of harmonic excitation, after a transient of a finite duration, steady-state oscillations of constant amplitude can establish due to dry friction even in the absence of viscous friction. The periodic motion of such non-resonant oscillations consists of two non-sticking symmetric phases of equal duration T/2. In the suggested example (drive frequency 0.7 natural frequency, drive amplitude 45 degrees, dead zone 20 degrees) the theoretical value (see the paper «Torsion Spring Oscillator with Dry Friction») of the steady-state amplitude is 80.6 degrees. Click here in order to observe how due to dry friction a steady-state regime eventually establishes after a transient when the driving frequency differs from the natural one. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
Generally characteristics of forced steady-state behavior of the oscillator with dry friction, as well as of the oscillator with viscous friction, are uniquely defined by the system parameters, and by the frequency and amplitude of the excitation. Certain exceptions are revealed if the frequency of sinusoidal excitation coincides with one of subharmonics of the natural frequency. Occurrence of special motions can be expected in these cases. Such motions correspond to asymmetric oscillations: the angular excursion to one side is greater than to the other. In contrast with the general case of forced oscillations, for which the steady-state regime is described by the unique solution, a continuum of asymmetric non-sticking solutions exists at sub-resonant frequencies of excitation, each of which is a limit cycle (attractor) that corresponds to initial conditions from a certain basin of attraction. The steady-state regime occurs from the very beginning (that is, without any transient) if the initial conditions are chosen properly, that is, at a quite definite value of the initial velocity whose value depends on the drive amplitude, but is independent of the dead zone width, and at the initial displacement from some interval. There exists a continuum of different steady-state non-sticking motions with the same total angular excursion, proportional to the drive amplitude. This is a manifestation of multistability –– a typical feature of nonlinear systems. The character of steady-state oscillations vary from one of the most asymmetric cases (at the initial displacement on the boundary of the above mentioned interval) through the symmetric case to the other most asymmetric case. If the initial displacement lies beyond this interval, one of the limit cycles from the same continuum is eventually established after a transient process, during which oscillations with sticking for finite time intervals take place.
Next examples show three different limit cycles from the continuum of non-sticking oscillations, occurring at the same values of the system parameters (driving frequency equals one half of the natural frequency, drive amplitude 60 degrees, dead zone 4 degrees). At these parameters steady-state oscillations occur from the very beginning at initial velocity 0.698 natural units (see the paper «Torsion Spring Oscillator with Dry Friction»). Total angular excursion between the extreme elongations equals (8/3)60 = 160 degrees. Click here in order to observe one of the most asymmetric stady-state regimes (–96 degrees to the left and 64 to the right) which takes place at the initial displacement 8 degrees. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
Click here in order to observe the unique symmetric stady-state regime (80 degrees to both sides) from the above mentioned continuum which takes place at the initial displacement –8 degrees. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
Click here in order to observe the other most asymmetric stady-state regime (–64 degrees to the left and 96 to the right) which takes place at the initial displacement –24 degrees. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
The next example shows the most asymmetric limit cycle from the other continuum of non-sticking oscillations, occurring at the same values of the system parameters (driving frequency equals 1/4 the natural frequency, drive amplitude 90 degrees, dead zone 3 degrees). At these parameters steady-state oscillations occur from the very beginning at initial velocity 0.4189 natural units. Total angular excursion between the extreme elongations equals (32/15)90 = 192 degrees. Click here in order to observe one of the most asymmetric stady-state regimes (–93 degrees to the left and 99 to the right) which takes place at the initial displacement 3 degrees. Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
At sub-resonant drive frequencies of odd orders non-sticking solutions for an oscillator with dry friction do not exist: at least twice during each cycle of the steady-state motion velocity turns to zero for finite time intervals. Click here in order to observe an example of such steady-state oscillations (driving frequency equals 1/3 the natural frequency, drive amplitude 90 degrees, dead zone 12 degrees, maximum elongation 87.7 degrees). Click also here to put the applet on the upper part of the screen, and the button "Start" to initiate the simulation.
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Nonlinear Oscillations – a Virtual Lab