f19MCO

Mechanical Chaotic Oscillator

Abstract: Four parameters (damping coefficient, constant frictional torque, magnetic dipole moment, and driving frequency) in a damped driven oscillator are compared to those obtained in simpler conditions. We collect position and time data from the mechanical oscillator and fit it to the model to obtain the parameters. As a result, three out of four parameters are within six uncertainties deviation under different conditions and the model for damped driven oscillator has a reduced chi-squared of about 2 across three trials.

About the project: We originally wanted to investigate the chaotic regime, but it turned out there are many obstacles before we even thought about fitting it. We will primarily focus on the obstacles encountered, and the potential directions one can take regarding the MCO device.

Apparatus:

Here is a top-down view of the device. It is essentially a harmonic oscillator. In a harmonic oscillator, one must have a restoring torque (because the thing rotating is better analyzed in rotational terms, the moment of inertia will replace mass, and torque replace force.) The restoring torque will ensure the rotor (magnetic dipole) has an equilibrium position. To complicate things, there are intrinsically at least two other terms, damping torque and constant kinetic frictional torque. These two torques impede the rotor from rotating, as common sense will agree. The damping torque comes from changing magnetic flux in the flywheel and coils. The kinetic friction is everywhere. Last but not least, the driving torque can be added to drive the rotor.

(A top-down view of the mechanical chaotic oscillator device. The central rectangle is a magnetic dipole set up parallel to the ground. The green ellipses represent a set of DC Helmholtz coils which generate a constant magnetic field to induce the restoring force on the dipole. The red ellipses are a set of AC Helmholtz coils to induce the periodic driving force on the dipole.)

(A photo taken from the side of the device. The flywheel, the shaft, and the magnetic dipole comprise the rotor which rotates horizontally under the influence of the magnetic fields generated by two sets of coils: larger DC coils and smaller AC coils. There was originally an adjustable magnetic on top of the device to change the damping coefficient, but we decided to remove it to better see how the device runs without more complexity.)

As one may see from the top-down view, the non-linearity is introduced from the orthogonal fields generated by AC and DC Helmholtz coils. This allows the potential for chaos.

Chaos: sensitive dependence on initial conditions (as well as parameters); trajectory is topologically dense (think about rational numbers, they are dense in real numbers, meaning for any non-rational number, you can find a rational number that is infinitely close to that chosen number).

What chaos entails is that we need to have very high accuracy on the initial conditions as well as the parameters to be able to predict the trajectory.

As it turned out, the "high accuracy" part needs a lot more work before one can delve into the chaotic regime.

Data Pre-processing:

The data of the position and angular velocity are acquired using an optical encoder disk and HEDS module[3] with a 12-bit resolution (meaning the position is accurate up to 0.0015 rad). The encoder disk is located at the top of the constant field coils, and it is connected to a HEDS optical encoder module mounted next to the shaft. The module then gives a quadrature output signal processed by a micro-controller[3]. The micro-controller resolves the quadrature signal from the HEDS module into positional data. The controller also tracks time, therefore the angular velocity can be calculated[3]. Through the serial protocol, we used LabVIEW to directly control the driving frequency as well as the driving amplitude. With the same program, we collect the data of position. Setting up the time it takes to collect 256 data points, we derive the time step between each position datum. Given the resolution, we encounter digitization noise in the interpolated angular velocity data. To get a smoother interpolated velocity, we use the Savitzky Golay filter to pre-process the position data in MATLAB with a window size of 121 and an order of four to reduce the digitization noise. The larger the window size, the more sensitive the result is regarding the distribution of data points across the window. With larger polynomial order, the filtered result is smoother in higher derivatives, such as angular velocity. As for the uncertainty in the filtered position, we use the lower bound which is the uncertainty aforementioned. Because the uncertainty cannot be lowered by functions operating on the data, taking the lower bound is an underestimate of the uncertainty on filtered data.

What's Done: we fit the model to the device where we powered up both sets of coils, but the driving amplitude is small and the initial angle from equilibrium is small too.

The model:

With solution:

The solution has an under-damped term (first line), which was empirically determined when we disconnected the AC coils.

The second line is the steady-state solution sharing the frequency of that in the equation of motion. The third line is related to the constant friction.

We first determined the moment of inertia and DC B field. Using these two, we estimated the damping coefficient, constant frictional torque, as well as magnetic dipole moment under two regimes: damped oscillation and damped, driven oscillation. We then compare the parameters to see how well the fit was on describing the damped driven oscillation. We also look at goodness of fit, such as reduced chi-square (~2 across 3 trials) and the residual plot. The time scale is only 4 seconds so the weighted residuals can be examined pretty clearly (compared to ~100 seconds or so in past MCO project.)

(Left: The time trajectory of the position of the last of the three trials under damped, driven condition with constant frictional term (Eq. 5). Across all three trials the trajectories look very similar so only one is chosen.)

(Right: The residual plot of the last of the three trials under damped, driven condition with constant frictional term (Eq. 5). Across all three trials the residuals look very similar so only one is chosen. In the residual plot, the overall region the plot occupies is [-4;3]. resulting in a relatively small but still not ideal (one is ideal) reduced chi-squared ~3.)

Possible improvements / directions for the future on this project:

1) the uncertainty of the filtered position (using a Savitzky Golay filter) can be estimated to reduce the underestimation of the uncertainty in our current results.

2) the magnetic field of the DC coils at the center can be determined directly without inferring using a second device, for more accurate determination of the parameter.

3) the device variable friction behavior should be investigated, potentially by rebuilding the rotating part of the device and by observing the time trajectory of the position using pure rotation.

4) the flywheel can be better controlled by using a mechanical trigger than a human hand, to achieve better systematic control over the initial conditions.

5) magnetic dipole moment can be directly measured instead of being fitted under a damped oscillation model.

The gist is to be as accurate and direct as possible about the parameters. Only when we are very confident about our parameters (the value, the variable or constant nature, and the uncertainty), should we start thinking about chaotic regime.

Reference:

[1]G. D. Davidson, “The damped driven pendulum: Bifurcation analysis of experimental data,” 2011.

[2]N. Macdonald, “Coupled oscillators in chaotic modes,” Nature, vol. 274, no. 5674, p. 847–847, 1978.

[3]B. M. Thacker and E. Ayars, “Mechanical chaotic oscillator,” 2015.