s19ChaoticRotor

Mechanical Chaotic Oscillator

Matthew Tate and Ruiren Shi

University of Minnesota - Methods of Experimental Physics II

Abstract

The chaotic nature of the Mechanical Chaotic Oscillator as developed by Eric Ayars is investigated. The apparatus is designed to be a nonlinear driven damped oscillator and is interfaced with a software-driven control panel to set system parameters of the motion. A proposed modeling equation is computed and compared against motion data taken from the device to analyze the devices control-ability. Parameter values were obtained and model equation phase-space data (φ and φ˙) are matched to experimental phase-space data for the case of simple harmonic motion.

Introduction:

In order to understand what Chaos theory is, it is necessary to understand Newtonian Determinism and how Chaos Theory diverges from it. Newtonian Determinism, in the physics implementation, is the belief that with the knowledge of all initial states and properties of the bodies within a system, we are able to accurately predict the systems time-evolution for all times in the future. Therefore if we are given enough initial information about the system. we shall be able to predict its behavior in its entirety. Chaos Theory diverges from this approach in that the system’s behavior evolves throughout time in such a manner that it enters regions of Chaos and diverges from any trajectory we expect for its behavior. It's not quite the same as quantum mechanics, in that we need to apply a statistical approach to understand what is happening within the system. Instead, Chaos is better to be thought of as having a few defining characteristics to its motion and thought of as a type of motion too complex to map entirely through analytical methods.

Chaos is typically described with the following three criteria[9] to its motion:

(1) Topological mixing - trajectories in phase space that pass through any region A in that space will eventually pass through another region B given enough time (regions within phase space will always overlap, noting that we may choose any regions A,B in the phase space).

(2) Dense periodic orbits - any points within the attractor of the system are arbitrarily close (within an epsilon neighborhood for epsilon greater than zero) to a periodic orbit.

(3) Sensitivity to initial conditions - means that a small change in the initial conditions can results in a big difference in the trajectories (This popularly known as the ”Butterfly Effect”). Thus in the simplest terms, chaotic behavior is a space-filling type of motion (All points within phase-space will eventually be overlapped by some trajectory)

Utilizing an apparatus designed by Eric Ayars [1] known as the ”Mechanical Chaotic Oscillator” (MCO), we seek to investigate the predictability of this system both inside of and outside of chaotic regions of the systems phase space. Due to limited time and various obstacles encountered throughout this pursuit, this goal was truncated strictly discussing the simple harmonic motion case. This case proves to be vital for future understanding of the experimental apparatus, as it provides a familiar/simplified system to which we may determine system parameter values. The MCO is significant due to its intentional design being that of a controllable chaos capable dynamical 2 system. A device that should allow researchers and students alike to investigate the unusual world of Chaos.

Again, prefacing with the restriction of our experimental goal to a determination of the MCO’s defining parameters and analysis of the agreement between experimental data and simulated values. This goal is accomplished with tentative confidence due to important inaccuracies that shall be analyzed at depth within the analysis of this final report.

Apparatus:

Figure 1: Eric Ayar’s experimental apparatus of the Mechanical Chaotic Oscillator labeled for clarity [1].

The apparatus is constructed as shown in Figure 1, from nonmagnetic material and its labeled components. The apparatus consists of a magnetic dipole (labeled ”3”) centered within two pairs of mutually orthogonal Helmholtz coils, which serves as the system oscillator. The dipole is constructed from two neodymium magnets fitted into a spacing housing and constrained to rotate in a single plane of rotation (XY-plane as shown in Figure 2). The angle φ is taken to be a measure of the dipoles angular position with respect to φ = 0 being defined at alignment with the constant magnetic field (B_field) coil pair (labeled ”5”). The constant magnet field Helmholtz coil pair is configured such that it is mutually orthogonal to a driven time-varying magnetic field coil pair (labeled ”7”). The driven Helmholtz coil pair is controlled by an Arduino-based microprocessor board (Teensy micro-controller: labeled ”6”) interfaced with a Windows PC running Lab-View to communicate coil driving amplitude (B_drive) and frequency (ω_d).

An inertia disk (labeled ”4”) is mounted to the top of the shaft to impart a larger moment of inertia to the dipole shaft system and facilitate inductive braking through the use of a magnet 3 mounted to a micrometer screw. The inductive braking occurs through the creation of eddy currents within the rotating disk (the dampening magnetic moves relative to the rotating disk) and thereby a dissipation of its kinetic energy.

The dipole’s angular position φ is recorded through the use of a shaft mounted quadrature optical incremental encoder (labeled ”2”) and exported to various data analytic software through a ”Lab-View .lvm” file structure. The optical encoder takes data with a precision of 4096 cts/rev , and provides positional data as well as directional data. This data is fed to the "Teensy microcontroller" and then sent as phase-space information with values of time data (t), angular position data (φ), angular velocity data (φ˙ - calculated by ∆φ/ ∆t ). This system offers a data point collection period of 1 /(256∗ωd) .

Theory:

The apparatus as previously described is diagrammed in Figure 2.

Figure 2: Theoretical diagram of the apparatus. B(t)drive (Green vectors) corresponds to the alternating magnetic field lines that is driven alternating current sent through the vertically labeled Driving Helmholtz Coils. B_field (Blue vectors) corresponds to the constant field lines that are powered by a constant current flow through the Constant Helmholtz Coils (supplied via current locked power supply)[1].

Analyzing the torque on the magnetic dipole within the mutually orthogonal B_field and B_drive magnetic fields results in the model equation we wish to compare to the experimental data:

Where B_field is the magnitude of constant magnetic field, B_drive is the magnitude of the driven magnetic field, mu is the dipole strength, I is the inertia of the dipole mechanical assembly (approximated to be that of the aluminum disk), Beta is the dampening coefficient.

To simplify the equation, we cut down the number of parameters by making mu*B_field/I as A, mu*B_drive/I as B and beta/I as C, and thus the equation now becomes

Understanding where this model came from is a matter of simple physics, however, understanding how this system evolves into chaos is a rather difficult task. For this experiment, it is our focus not to investigate and/or prove the system is chaotic but to calibrate the device parameters such that we are able to find agreement between our simulated data and the experimental data. These device parameters being B_field, Bdrive, ωd, µ, beta, and the system’s initial conditions (φ_0, (dφ/dt)_0).

Data:

Table values for Figure 4.

A linear fit is created to determine parameter A as a function of I. We know this is the case due to the substitution made for A is such that it contains B_field, which dependent on I.

Figure 4: Linear fit for parameter values for A, used to determine a relationship between A and I.

Figure 5: Residual plot for 0.813 ± 0.001A. Truncated: t0 = 8s

Figure 6: Residual plot for 0.817 ± 0.001A. Truncated: t0 = 8s

Figure 7: Residual plot for 0.822 ± 0.001A. Truncated: t0 = 8s

Figure 8: Residual plot for 0.825 ± 0.001A.

Figure 9: Residual plot for 0.829 ± 0.001A. Truncated: t0 = 8s

Figure 10: φ vs t plot, for 0.813 ± 0.001A. Truncated: t = [8, 18]s

Figure 11: φ vs t plot, for 0.817 ± 0.001A. Truncated: t = [8, 18]s

Figure 12: φ vs t plot, for 0.822 ± 0.001A. Truncated: t = [8, 18]s

Figure 13: φ vs t plot, for 0.825 ± 0.001A. Truncated: t = [8, 18]s

Figure 14: φ vs t plot, for 0.829 ± 0.001A. Truncated: t = [8, 18]s

Analysis:

The values determined for system parameters are displayed in the last table of the data section. Note that these parameter values do not have any sensible error values as they are fitted values.

The results for the parameter values face a few problems. Firstly, A appears to have an offset must larger than its slope ∗ I term. This is problematic as it states that the value of this parameter A is dominated by the value of this offset (−32 1/s^2 ). To investigate this I took data at small currents (with DVM in series in current-mode) and tried to fit this data using this offset value. Data for one of these small current values is shown in Figure 15.

Figure 15: Phase-space (φ˙ vs φ) plot for 0.002 ± 0.001A.

Now to compare this to a phase space diagram from a larger current value (Figure 16):

By comparing these two we see that there is evident unaccounted for phenomena occurring within the 0.002 ± 0.001A. This phenomena is causing the frequency to oscillate between values instead of holding hitting some resonant frequency and stabilizing. As a suggested phenomena that could be causing this frequency oscillation: magnetic field produced by the Helmholtz coil could be lagging behind the current sent through the driving coil. This oscillation in frequency is more clearly depicted by the Figure 17.

Figure 17: Poincare Plot for 0.002±0.001A data set cycle colors and corresponding numbers denoted in the following table (Note: 256 data points per cycle).

Another issue with the parameter values is that when we took rough measurements initially B_field differed in value from B_drive by only one order of magnitude. In the result, we find that in order to get the simulation data points to match the experimental data, now B is suppressed by three orders of magnitude. It should be noted that as B approaches A the amplitude of the simulated data is simply modulated until a threshold point, at which the fitting technique fails to produce similar behavior to that of the experimental data. Due to lack of time, this phenomena is largely still unexplored. With the little time available we were able to notice a dependency of this threshold value on the value for ωd (I believe it has to do with the peak number of the modulation term of the model equation i.e. ”cos(ωdt)” that a change in ωd causes).

Conclusion:

Further investigation into the offset value of −32 1/s^2 for parameter A must be done. This could possibly involve a different fitting technique, as well as more, investigate for the omega_d value. The offset behavior is expected to be determinable from low-current values being passed through the constant field coils, however, this data as shown in Figure 15, this data is undergoing some unaccounted for phenomena that current skews fitting to that data. In theory, a value for B should be determinable from this data set; due to the oscillatory behavior of the data set’s frequency makes this determination impossible.

References:

[1] Thacker, Brandon. Ayars, Eric PhD. ”Mechanical Chaotic Oscillator” California State University at Chico August 5, 2015.

[2] Durbin, Samantha. ”What are weather models, exactly, and how do they work?”: Washington Post May 18, 2018.

[3] Encyclopedia.com. ”Chaos Theory and Meteorological Predictions”: www.encyclopedia.com 2008.

[4] Lorentz, Edward N. ”Deterministic Nonperiodic Flow”: Massachusetts Institute of Technology January 7, 1963.

[5] Pecora, Louis M. Carol, Thomas L. ”Synchronization in chaotic systems”: Code 6341, Naval Research Laboratory, Washington, D.C.

[6] Vaidya, Priya G. He, Rong. ”Transfer of Information between Synchronized Chaotic Systems”: Washington State University, November 18, 1993.

[7] Cuomo, Kevin. Oppenheim, Alan. ”Circuit Implementation of Synchronized Chaos with Applications to Communications”: Research Laboratory of Electronics, Massachusetts Institute of Technology January 21, 1993.

[8] Oestreicher, Christian. ”A history of chaos theory”: Department of Public Education, State of Geneva, Switzerland September 2007.

[9] Chu-Carrol, Mark. ”Good Math/Bad Math: Category Archives: Chaos”: Good Math/Bad Math Febuary 7, 2010.

[10] Lagarias, J.C. Reeds, J.A. Wright, M.H. Wright, P.E. ”Convergence Properties of NelderMead Simplex Method in Low Dimensions”: SIAM Journal of Optimization Vol 9, Number 1, 1998.