physics notes

Velocity-Dependent Forces III: Damping and Resonance

The damped, driven oscillator is a well-known analytic problem that occurs in many guises. The damping force is linearly dependent on velocity, and Newton's law can be written:
In the analytic treatment the equation is usually rewritten to introduce the angular frequency of the undamped oscillator and a dimensionless 'quality' factor that is inversely proportional to the strength of the damping force.
The equation simplifies further if we use a dimensionless time variable. The numerical solution is straightforward: start from the initial conditions and use the Feynman algorithm for velocity-dependent forces. In the first example there is no applied force and the results of the calculation are shown on a plot of v versus x rather than x versus t. This axis system is known as phase space. The initial conditions define a point in phase space, and the motion of the system produces a phase path that terminates at the origin (for finite Q). A set of phase paths from various points in phase space is known as the phase diagram for the system. For Q = 1/2 the system is said to be critically damped. For higher values the system is underdamped, and for lower values it is overdamped. The program uses a set of initial points on a circle to generate phase diagrams for all three cases. In the underdamped case the displacement changes sign during the motion. In the other cases it does not change sign for initial points in the first and third quadrants. In the second and fourth quadrants the situation is more complicated. Physically this is easy to understand. No matter what the damping, if the particle starts out moving toward the origin fast enough, the displacement will change sign. Hence there will be a region near the v-axis in quadrants 2 and 4 for which phase paths cross the v-axis on their way to the origin. A green line (determined analytically) shows the boundary of this region. For critically-damped motion the phase paths approach the origin tangent to this boundary, whereas for overdamped motion they approach along a common line at a different angle.

Liberty Basic Source   Java Source

There is a story behind the above example. 'Once upon a time' the text I was using (Marion's Classical Dynamics) had a figure showing the phase diagram for overdamped motion (page 108 in the 2nd edition). I assigned a problem asking for the equation of the region boundary (the green line), and a student came to ask me about it. He said he had no trouble solving the problem, but there had to be something wrong with the figure in the text. After thinking about it, I agreed with him and asked him to come up with a corrected figure. He returned the next day with a figure similar to the overdamped plot in the example. (I don't remember whether he generated the plot numerically or analytically.) The figure in the text makes the mistake of showing the phase paths approaching the origin along the region boundary.
The second example deals with the driven oscillator and presents its results directly on an x versus t plot. It allows one to vary the quality factor Q, the applied force/undamped oscillator frequency ratio r=f/fo, and the shape (sine wave or square wave) of the applied force.

Liberty Basic Source   Java Source

The program allows the analytic solution for a sinusoidal applied force to be superimposed on the numerical solution (excluding the undamped oscillator at resonance). The net effect of the superposition is to change the color of the response curve from blue to magenta. The analytic solution follows the usual procedure for solving linear differential equations with constant coefficients. The solution is written as the sum of two parts: the complete solution (with two arbitrary constants) of the equation with the driving force set equal to zero, and a particular solution of the equation with the driving force applied. The initial conditions are used in the final stage of the process to determine values for the two constants. With the applied force zero, an exponential trial solution reduces the differential equation to an algebraic equation.
If Q<1/2, the two solutions are negative exponentials, and the system is said to be overdamped. If Q>1/2 the exponent is complex, and the solution is a damped sinusoidal oscillation with amplitude and phase as the two constants. (The example has four Q values, one below 1/2 and three above.)
With a sinusoidal force applied, the particular solution will be a sinusoidal displacement with amplitude and phase determined by direct substitution. The result is a bit messy, but it involves only Q, r, and the static displacement F/k produced by a force of the same amplitude.
The main advantage of the analytic solution is that it provides expressions for the frequency dependence of the amplitude and phase of the steady-state displacement.
There is a third way of solving the damped driven oscillator problem that is better in many ways than either the numerical or analytic approaches. Connect a capacitor, inductor, and variable resistor in series across a waveform generator. Connect the generator output (the force) to one channel of an oscilloscope, and the voltage across the capacitor (the displacement) to the second channel. The waveform generator applies a sine or square wave of variable frequency, and varying the resistance varies Q. Only the steady-state solution can be shown for a sine wave, but a low-frequency square wave demonstrates the transient solution.

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