ProcedureDEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF MINNESOTA EE 4237 State Space Control Laboratory Experiment 7: Gyroscope Dynamics: Nutation and Precession Objective: 1. To study the Gyroscope system 2. To study Nutation and Precession References: 1. ECP systems manual (Model 750) Apparatus: 1. Gyroscope Model 750 2. PC 3. Control Box Prelab Report: 1. What is a Gyroscope? What it is comprised of? Give a brief summary of what the experiment is all about. 2. What are the application areas where gyroscope is important? Postlab Report: • Answer the questions asked within or at the end of the procedure.ecpChapter 6. Experiments 816.2 Gyroscopic Dynamics: Nutation & Precession This section measures the nutation and precession of the gyroscope as a function of wheel speed. It also demonstrates the damping of nutation through rate feedback at Gimbal #2, and finally, gives another way of interpreting precession in terms of conservation of angular momentum. All tests in this section are performed with the apparatus in the configuration of Figure 6.2-1. Important Notice: From this point on in the instructions, no specific reminders to perform the required safety procedures shall be given. The user must follow the safety the Safety guidelines of Section 2.3 at all times when operating this equipment. EncoderBrakeCapstanPulleyGimbal Angles: q2o = 0, q3o = 0Axis 4 Brake: OFFAxis 3 Brake: ONAxis 2 Virtual Brake: OFF Figure 6.2-1. Configuration For All Tests In This Section 6.2.1 Nutation: Frequency & Mode Shapes Procedure 1. Setup the mechanism as shown in Figure 6.2-1. 2. Write a simple real-time algorithm to activate Motor #2 (i.e. put control effort values on the DAC) with a Control Effort equal to the (Commanded Position)/32.1 Use the global real-time variables “control_effort2” and “cmd1_pos” for this purpose. 3. Go to Trajectory 1 Configuration. Enter Impulse and specify an Amplitude of 16000 counts, a Pulse Width of 50 ms, a Dwell Time of 4000 ms, and 1 repetition (this prepares the controller board to input a 16000 count positive-going impulse followed immediately by 4 seconds of zero input during which data is collected. 4. Setup Data Acquisition (Setup menu). Specify Commanded Position 1, Sensor 2 Position, Sensor 4 Position, and Control Effort 2 as data to be acquired with a Sample Period of 4 servo cycles. 1 The “/32” factor accounts for a firmware gain that multiplies all commanded position and encoder signals by 32 for increased internal resolution.ecpChapter 6. Experiments 825. Enter the ECP Multivariable Executive program. Implement your algorithm from Step 2 above with sampling period set to Ts = 0.00442 seconds. 6. Initialize Rotor Speed to 200 RPM and zero the encoder positions (Utility menu). Execute the maneuver selecting Normal Data Sampling and Execute Trajectory 1 Only . 7. Plot the Encoder 2 and Encoder 4 Position data and subsequently the velocity data. Note the frequency of the oscillations and the relative amplitude and phase of the Encoder 4 response verses the Encoder 2 response. Save your plots. 8. Disable the rotor speed loop (Command menu). You may also want to turn off the Control Box to more rapidly decelerate the rotor. Wait for the rotor to stop (if you turned off the Control Box, turn it back on at this point). Repeat Steps 6 and 7 for a rotor speed of 400 RPM. 9. Repeat Step 8 for a rotor speed of 800 RPM. 6.2.2 Precession Procedure 10. Repeat Steps 1 through 6 of the Section 6.2.1 except in Step 3 setup the Impulse trajectory for an Amplitude of 6000 counts, a Pulse Width of 8000 ms, a Dwell Time of 0 ms, and 1 repetition (this prepares the controller board to input a 6000 count constant input for 8 seconds). The first maneuver should be at 200 RPM and should result in a initial transient series of attenuating nutation oscillations followed by a steady state response. Plot the position data for Encoders 2 and 4 and also their velocity data. Save your plots. 11. Repeat Step 10 for the 400 and 800 RPM cases. Note the change in steady state velocity for Encoder 4 with rotor speed. 6.2.3 Nutation Damping Procedure 12. Augment your algorithm from Step 1 of Section 6.2.1 to add rate feedback damping at Axis 2. I.e. add a term u2damp of the form u2damp = -kv q2 s where kv is the rate feedback gain. You may use the backwards difference transformation to implement discrete time differentiation according to s - 1-z-1Ts (6.2-8) where Ts is the sampling period. Use Ts =0.00884 s. in this experiment. Have your laboratory supervisor review and approve your algorithms before proceeding.ecpChapter 6. Experiments 8313. Set the rotor speed to 400 RPM. Beginning with a value of kv = 0.005 Implement your algorithm with Ts =0.00884 seconds. 14. Repeat Step 10 except maintain the rotor speed at 400 RPM rotor speed. Do you see a reduction in the nutation mode amplitude? 15. Repeat Steps 13 and 14 for various increasing values of kv. Do not exceed kv=0.10, higher values could lead to excessive numerical noise and damage to the system! How are the nutation oscillations affected by increased rate feedback gain? Save your plot of a case where the nutation is well damped. Exercises A. From the plotted data of Section 6.2.1, measure the frequency of the nutation mode for the three rotor speeds. (Hint: divide the number of cycles considered [typically between 2 and 5] by the time taken to complete them. Zoom the plot if necessary or export the raw numerical data to get precise readings and make sure that you start and end the evaluation period at the same phase in the respective cycles.) Compare your result with that predicted by the theory (i.e. the eigenvalues of Eq.(5.4-13) or equivalently the characteristic roots of Eq’s (5.4-7 & -8)). Assuming the measured and provided moments of inertia are within 10% of their actual values, are your results in agreement with the theory? What is the relationship between rotor speed and nutation frequency? B. Consider the position plots versus those of velocity for the data of Section 6.2.1. For a given test, are the oscillation frequencies the same? Are the relative amplitudes of the outputs at encoders 2 and 4 the same? Are the steady state values the same? Explain your answers (you may neglect the effects of friction). C. Solve for the eigenvectors in the system matrix of Eq.(5.4-13). Solve for