A 'torsional clock'
We often model the residual damping of our TO-as-oscillator by a fractinoal torque, τ ≈ -b dθ/dt. Given such a torque, oscillations are damped, with an exponentially-decaying envelope. But such decay can be avoided, provided the TO is actively driven by an external torque + b dθ/dt. The Auxiliary Electronics add-on for the TO contains all that's needed to form such a drive signal, out of the electronic signal proportional to θ(t) emerging from the TO. With this form of active feedback, the TO provides the timing of oscillations, while the AE provides the energy to keep them sustained. The combination is as much a clock as any driven-pendulum or quartz-oscillator system. Students will get to see just how such sustained oscillations can be controlled and stabilized, and they will also be able to measure just how steady is the behavior of the resulting electro-mechanical clock system.
Below we show the electronically-measured period of such a TO+AE clock, displayed a succession of single-period timings. The period is near 1.166 s, and is best written as 1,166,xx,μs. Note that the measurements #7 through #12 are anomalously high, by about 400μs - that's because during these periods, we deliberately added a little brass washer, of mass ≈.04 g, to the periphery of the rotor. That increased its rotational inertia, and hence its period. The effect, though small, is very easily detected.
There's another way to change the period of this electro-mechanical clock, and that's the addition of a static magnetic field Bh in a horizontal direction. The field component, acting on the rotor-mounted magnets, creates a torque which supplements at the Hooke's-Law restoring torque on the oscillating system, and hence changes its period of oscillation. The graph below shows how the average period, measured over four cycle of clock operation, depends on this added field Bh. From the data show, we see that each 1 μ T sensitivity represents the detection of field changes of just 1 milli-gauss, and this by a steampunk-looking wood-and-copper combination that students can understand and model fully.
Our manual for the Auxiliary Electronics shows students some general features of time-keeping including the ways that feedback imperfections can 'pull' a high-Q resonator away from its natural frequency. It also introduces to the Allan deviation as a suitable measure of clock performance.
A 'servo for torque'
The rotor of our torsional oscillator is a mechanical object having rotational inertia I about its (vertical) axis of rotation, and getting its restoring force from a Hooke's-Law torque τ = -κ θgenerated by twists in the music-wire fiber that supports the rotor. Here θ is the angular coordinate of the TO's rotation, and the κ is a torsion constant for the fiber. Additional torque on the rotor can be provided by damping, and by external torques - these include those exerted by any magnetic field that exerts a torque on permanent magnets mounted on the rotor.
For the TO itself, acting passively, a slowly-varying external torque will create (in a reasonably well-damped TO) a similar slowly-varying angular response, according to θ(t) ≈ τ(t)/κ. So in this sense, the TO itself is a 'torque meter', revealing the τ(t),and treat that as an 'error signal', which the AE is configured to drive back to zero. In this approach, the TO+AE acts as servomechanism, and the best measure of the external torque is to measure the electronic signal proportional to the counter-torque that the TO is applying to the rotor to keep it at the θ = 0 position.
Some results of this scheme are shown in the data below. On the left, the TO is acting passively without any feedback from the AE. The upper trace shows the timing of an external torque, taking on the values ±19 x 10⁻⁶ N∙m in a square-wave fashion, with a 10-s period. The lower trace shows the resulting angular displacement of the TO's rotor, converted into its angular-position output signal, indicating an angular displacement of ±0.33 milliradians. (Such motion is totally invisible to the eye - points on the rotor's rim move through ±20 micrometers of displacement - and even the electronic signal from the rotational motion is afflicted by noise.)
The box has a convenient collection of input/output connectors, and it also has six independent electronic modules, conveniently interconnected by the user via 'pin jacks' to configure just the sort of feedback desired. One of those modules is a voltage-to-current converter, designed expressly to connect to the 'drive coils' that allow a tailored torque to be applied to the oscillating TO system. The other modules can be used, in various configurations, to transform the angular-position output of the TO into a chosen torque-drive signal that gets sent back into the TO.
Built into the modules are analog integration and differentiation functions; these allow the creation of waveforms that are phase-shifted by 90 degrees relative to their inputs. Also included are a general-purpose scaling amplifier (gain 0 to ±1 or to ±10) and a special-purpose limiting amplifier (gain 1.00 for small signals, dropping toward differential gain 0.8 for larger signals). There is also a timing module, which can be used to produce precise time-zero-crossing digital pulse signals from the low-frequency sinusoidal signals typically emerging with the TO.
The Auxiliary Electronics all fit in a simple and modular box, which is powered by an AUX. OUT power-supply point built into our TO's.
TeachSpin's Torsional Oscillator is a first-class way to teach and learn about simple harmonic motion, and by itself, it permits dozens of experimental investigations. Buf from the beginning, we've realized that the addition of some rather simple external electronics allows the Torsional Oscillator to become part of an electro-mechanical system which is ideal for teaching physics students about the concept, and the uses, of feedback. So we have introduced the TO's Auxiliary Electronics, a simple package which extends the capabilities of the TO into new domains, and makes new kinds of teaching and learning possible.
What's special about the TO itself is that it produces a real-time analog output voltage proportional to the (angular) coordinate of a system undergoing simple harmonic motion, and that it accepts a real-time analog input current which creates a proportional torque on that system. So the Auxiliary Electronics permits students to take that TO output, to modify it electronically in the AE, and to feed it back into the TO as a drive. Better still, they can understand just how the feedback ought to affect the properties of the TO. Result: TO+AE combination can display behavior that's not available in the TO itself.
One of the most glamorous outcomes made possible by the TO+AE combination is the abolition of damping, so that the TO's oscillations are no longer damped sinusoids, but instead turn into undampled continuing oscillations of the TO, at its own natural frequency. The result is a 'torsional clock', operating as an electro-mechanical oscillator, in which the TO provides the timing, while the AE provides the energy needed to keep the oscillations from decaying due to residual damping. Such a clock is capable of remarkable timing performance, and is also an illustration of the general technique of applying feedback to a lightly damped system to turn the comination into an oscillator.
TeachSpin provides a manual with the new Auxiliary Electronics, taking users subsequently through the possibilitiees latent in the TO + AE combination. Here we illustrate two such possibilities:
On the right, the TO is active, part of the TO+AE system with feedback. The top two traces have the same significance as formerly; here we see the success that feedback has in its 'striving' to keep the angular-displacement signal near zero. The novelty is the bottom trace, which is a signal proportional to the drive-coil torque, generated by the AE, which provides the counter-torque required to null the effects of the external torque. The signal-to-noise advantage of this technique dramatically evident.