PolhodePolhode MotionMotion ofof thethe GrGravityavity Probe-BProbe-B GyroscopesGyroscopes Michael Dolphin, Daniel DeBra, Michael Salomon, Alex Silbergleit, Paul Worden Euler’s Equations Rotor Asymmetry Observations & Explanation Dissipation Model (1) Dissipation Model (2) The Gravity Probe B gyroscopes are the roundest objects ever Given E and L, the shape of the polhode path is determined by a To account for energy dissipation, a term is added to the Euler Model Fit manufactured. They operate in a near torque-free single asymmetry parameter Q2 (I is the moment of inertia about ith equations which satisfies 3 fundamental assumptions: i The predicted environment. As round as they are, they still have principal axis) 1) Angular Momentum is conserved. polhode period from asymmetry, and exhibit polhode motion as they spin. 2) Energy is never added. the dissipation model Angular Momentum: Kinetic Energy: 3) Energy dissipation goes to zero fits extremely well to ONLY when spin axis aligned with principal the measured General Euler Equations (no external axes of gyro. polhode period. torque): * The polhode period of each gyro was observed to be changing over time. Each of the four Even when the Polhode Motion exhibited similar asymptotic behavior. The only explanation that agrees with all the Spin axis moves along polhode path, observation was that of energy dissipation in the rotor body. Dissipation moves the spin axis baseline exponential which is determined by the intersection towards the maximum inertia axis where energy is minimum. * Modified Euler Equations introduced by Alex Silbergleit. is removed, the of two ellipsoids: constant angular Algorithm to find Q2: momentum (L) and constant kinetic • Chose a long time stretch of data and set Q2 (and some other parameters). dissipation model energy (E). 2 2 • Compute initial energy (E) from polhode period (T (t)). captures the residual Q = .10 Q = .50 With kinetic energy on the order of 1 J, the dissipated power needed to move the spin axis all i p i • Integrate energy equation, and find predicted Epr(t). Conservation of energy and angular momentum the way from min to max inertia axis over one year is ~10-13 W. behavior impressively •Convert Epr(t) to polhode period (Tppr(t)). whose expressions are: Even if there is no energy dissipation, knowledge of Q2 is needed While we can assume constant energy over one polhode period, we need to account for the • Fit to polhode period data using quadratic cost function. well. to accurately determine the polhode path. dissipation of energy over the course of the mission. • Repeat process to find minimum cost. Vary Q2 over desired range, and find best Q2 by analyzing the cost function. GSS Position Model (1) GSS Position Model (2) GSS Position Model (3) Cost Analysis of Q2 Results And Applications To match the signal, a relatively 2 B D simple model of the gyro surface Two versions of the dissipation By combining all our results, we get an estimate for Q in 1st and 3rd spherical harmonics model and the GSS position for each of the four gyros. c*Sinλ was employed to account for slight differences between two model for gyro 1. Results shown C sets of orthogonal in terms of fit cost function vs Q2. measurements. Gyro 1 Gyro 2 Gyro 3 Gyro 4 2π The resulting model matches the The alternate dissipation model method is ω data throughout the entire A A based on a 3rd order exponential mission. 0.33 0.72 0.32 interpolation of the measured polhode 0.36 The best fit model was found for period data points: Q2 (0.29 – (0.56 – (0.30 – Mass unbalance creates a signal at spin frequency in the position as measured by a range of Q2 values. Each best fit (0.14 - 0.43) Even if the asymmetry parameter (Q2) was equal to zero, and thus the the Gyro Suspension System (GSS). The amplitude of the signal is a function of was evaluated by a least squares 0.38) 0.75) 0.40) polhode paths were circular, we would expect “M” shaped variations at both the length of the mass offset, and its angle to the spin axis. cost function. polhode frequency in the amplitude of the position signal at spin frequency. The importance of knowing Q2 is that it is used in computation of Gyro 1 is shown here with The above dissipation model allows for theoretical expressions of the fit polhode phase and polhode angle for each gyroscope for the entire The GSS position is The “M” shape curves indeed its surface model as found coefficients C as functions of Q2. By mission, which is needed for science data analysis. are visible for some times of the using this method. n measured with RMS matching those theoretical expressions mission. This example is for gyro error of 0.45 nm. The Also, some of the polhode with the found values provides the Accurate knowledge of the polhode enables us to account for the 1, relatively early in the mission paths used in the fitting sought number for the asymmetry polhode frequencies in a the signals in which it appears. Some of the signal due to mass (October, 2004). process are shown. parameter (the only approach to measurements which include polhode frequencies are: GSS position, Spin frequency unbalance is of order 1 The period of this signal accommodate gyros 3 and 4). Triangles denote locations GSS suspension force, vehicle attitude and control system, vehicle matches known polhode to 10 nm, and is of the spin axis at each data orbit, LF science SQUID signal, HF SQUID trapped flux signal, measured frequency. clearly visible in the point. All three methods suggested spin rate and spin phase. The knowledge of polhode phase is critical to frequency analysis of This also provides an estimate of Q2 for gyro 1 is in the region the Trapped Flux Mapping efforts (see Trapped Flux Mapping poster). the signal. the mass unbalance vector. Q2 of .29 to .39 This research supported by NASA W. W. Hansen Experimental Physics Laboratory • Stanford University, Stanford, CA 94305-4085 • http://einstein.stanford.edu on contract NAS8-39225.