	Dan Ludwigsen, Assistant Professor of Applied Physics
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One Degree-Of-Freedom

Oscillators

Lab #1 in pdf format (60k).

The ubiquitous one degree-of-freedom (1-DOF) oscillator is a mass, spring, and dashpot system. All of the mass of the system is lumped into an effective mass, the stiffness or compliance is modeled by the spring, and any damping or energy loss (especially if it's proportional to velocity) is captured in the dashpot -- a mechanical element similar to a shock absorber or piston in oil.

This hands-on demonstration in Acoustics II has two parts, and neither of them employ the typical mas-spring-dashpot. The first apparatus is a plane pendulum damped by significant air drag, and the second is a Helmholtz resonator. This acoustical oscillator foreshadows the work we do later in the course with mechanical/electrical/acoustic circuit theory, and it also lets students work with the spectrum analyzer.

The damped pendulum

Because gravity provides a restoring force, the pendulum exhibits oscillation about an equilibrium point. If we keep the displacement small, the restoring force is approximately linear to the displacement angle. The effective stiffness is thus not quite like a spring, and the mass is not involved in the pendulum's natural frequency as it is in the mass and spring system. These effects can be investigated with additional time in the lab.

Our plane pendulum exercise has two major points. The first is designed to emphasize that the frequency of a damped oscillator doesn't change as energy is lost and amplitude decreases. Students simply see this experimentally by measuring the period soon after releasing the pendulum, and comparing with the period many cycles later.

The second point is to calculate from experimental data the characterizing parameters of the system. student with data

For example, students may need to quantify the amount of damping in a system. By a technique employing the logarithmic decrement, students can derive the damping coefficient. (With the assumption that the damping is proportional to velocity, the amplitude envelope is fit by a decaying exponential function.

The Helmholtz oscillator

Our Helmholtz oscillator is simply a glass pop bottle. The volume of the bottle provides acoustical "springiness," while we think of the air in the neck of the bottle as a unified mass, oscillating in and out around an equilibrium displacement. Placing a microphone in the bottle allows us to measure the resulting oscillation in pressure. students with the analyzer The system is driven at a single frequency (which is then swept through a range) by a loudspeaker placed near the bottle. Varying amounts of loose cotton, placed in or on the bottle's opening, effectively change the damping in the system.

As the spectrum analyzer sweeps through the range of driving frequencies, its trace shows the response of the system. The spectrum clearly indicates the resonance frequency, and students can also measure the points on either side which have 3 dB less amplitude. From this data, they calculate the quality factor for the system with different amounts of damping from the cotton. The damping coefficient can also be determined from this data.

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