Torsion Syste Introduction This lab experient studies dynaics of a torsional syste with single and ultiple degrees of freedo. The effects of various control configurations are studied in later part of the lab. (J 3) Encoder #3 ( 3 ) Third encoder/disk for Model 205a only k2 = L2 GJ Make certain upper disk is ounted below shaft clap (J 2 ) k1 = L1 GJ Encoder #2 ( ) 2 (J 1) 4 Movable asses each disk Brushless Servo Motor Encoder #1 ( 1) Rigid belt drive Hardware Figure 1: ECP Torsion Experient The syste we use in this experient is Model 205. Figure 1 shows the Model 205 Torsion Experient consisting of three disks supported by a torsionally flexible shaft that is suspended vertically on anti-friction ball bearings. The shaft is driven by a brushless servo otor connected via a rigid belt (negligible tensile flexibility) and pulley syste with a 3:1 speed reduction ratio. An encoder located on the base of the shaft easures the angular displaceent, 1 of the first disk, J 1. Two additional encoders easure the displaceents of the other two disks as shown. The torsional echanis represents any physical plants including rigid bodies; flexibility in drive shafts, gearing and belts; and coupled discrete vibration with actuator at the drive input and sensor collocated or at flexibly coupled output (non-collocated). 1
Safety - Ensure the asses are firly secured by screws. - Be sure to stay clear of the echanis when turning on the controller. Selecting Ipleent Algorith iediately ipleents the specified controller. If there is an instability or large control signal, iediately abort the control. If the syste appears stable after ipleenting the controller, first displace the disk with a light, non-sharp object (e.g. a plastic ruler) to verify stability prior to touching plant. Hardware/Software Equipent Check Before starting the lab, ake sure the equipent is working by following the steps below: Step 1: Enter the ECP progra by double clicking on its icon. You should see the Background Screen. Gently rotate the drive or load disk by hand. You should observe soe following errors and changes in encoder counts. The Control Loop Status should indicate "OPEN" and the Controller Status should indicate OK. Step 2: Make sure that you can rotate the disks freely. Now press the black "ON" button to turn on the power to the Control Box. You should notice the green power indicator LED lit, but the otor should reain in a disabled state. Do not touch the disks whenever power is applied to the Control Box since there is a potential for uncontrolled otion of the disks unless the controller has been safety checked. 2
Experient 1: Syste Identification In practice, the syste paraeters of a piece of equipent, such as the inertia, spring constant, and daping ratios are often unknown. In this section of the lab, these unknown paraeters will be deterined using a process called syste identification. These sae paraeters will be used later to ipleent various controllers. Procedure: Figure 1 Configuration For Plant Identification 1. Secure four 500g asses on the upper and lower disks as shown in the figure. Verify that the asses are secured and that each is at a center distance of 9.0 c fro the shaft center-line. 2. Clap the center disk to put the echanis in the configuration shown in Figure 1-1a using the 1/4" bolt, square nut, and clap spacer. Please use the washer in between the spacer and the disk to avoid bending and possibly breaking the apparatus. 3. With the controller powered up, enter the Control Algorith box via the Set-up enu and set T s = 0.00442 and Continuous Tie. Enter the Coand enu, go to Trajectory and select Step. Select Open Loop Step and input a step size of 0 (zero), a duration of 3000 s and 1 repetition. Exit to the Background Screen by consecutively selecting OK. This puts the controller board in a ode for acquiring 6 sec of data on coand but without driving the actuator. This procedure ay be repeated and the duration adjusted to vary the data acquisition period. 3
4. Go to Set up Data Acquisition in the Data enu and select Encoder #3 as data to acquire and specify data sapling every 2 (two) servo cycles, i.e. every 2 T s 's. Select OK to exit. Select Zero Position fro the Utility enu to zero the encoder positions. 5. Manually displace the upper disk approxiately 20 deg. Select Execute fro the Coand enu. With the upper disk displaced approxiately 20 deg ( 1000 encoder counts as read on the Background Screen display) in either direction, release the disk and click run iediately after. The disk will oscillate and slowly attenuate while encoder data is collected to record this response. Select OK after data is uploaded. Plot the data on screen to see the response 6. Export the data to save in a file using export raw data in the data enu. Use the atlab progra, plotdata., on the class website to plot Encoder 3 position vs. tie (plot key iplot=1). Clearly label the plots. 7. Find the natural frequency of the syste using the following steps: Use atlab data curser tool to choose several consecutive cycles (say 5 to 10) in the aplitude range between 100 and 1000 counts. Divide the nuber of cycles by the tie taken to coplete the. Convert the resulting frequency in Hz to radians/sec. This daped frequency, d, approxiates the natural frequency, n, according to: (Equation 1-2) dd 31 nd 31 2 1 d 31 dd 31 where the "d31" subscript denotes disk #3, trial #1. 8. The next experient is done with the four asses reoved fro the third (upper) disk. Repeat Steps 5 through 7 to obtain nd32 for the unloaded disk. Plot and label the results as you did in trial #1 and then calculate nd32. 9. Find the daping ratio of the syste in Test trial #2. Measure the reduction fro the initial cycle aplitude X o to the last cycle aplitude X n for n cycles easured in Step 8. Use the following logarithic decreent (for sall d 32 ) to calculate the daping ratio: d 32 1 X 0 d 32 ln (Equation 1-2) 2 1 2 n X n d 32 10. Perfor the sae experients for the loaded and unloaded lower disk part of the syste siilar to what you did for the upper part. Go to Set up Data Acquisition in the Data enu and select Encoder #1 as data to acquire. Deterine nd11, nd12 and d12. How does this daping ratio copare with that for the upper disk? Why ight it be different? 4
11. Use parallel axis theore to calculate the total inertia of the four weights about the center of the disk, J. The ass and size of the weight are: Brass Mass (incl. bolt & nut) = 0.5kg Diaeter of Brass Mass = 0.05 12. Deterine the disk inertia without the weights, J d3, and upper torsional shaft spring stiffness, k d3 using the following eqs: k d3 /(J +J d3 ) = ( nd31 ) 2 (Equation 1-3) k d3 /J d3 = ( nd32 ) 2 (Equation 1-4) 13. Find the daping coefficient c d3 by equating the first order ters in the following equation, where c is c d3 and J is J d3 for the given proble: s 2 + 2 n s + 2 n = s 2 + c/js + k/j (Equation 1-5) 14. Repeat the above calculations for the lower part of the syste to deterine J d1, k d1 and c d1, which include the effects associated with otor, belt, and pulleys. Syste basic property paraeters have now been deterined. Values for J 1 and J 2 for any configuration of asses ay be found by adding the calculated inertia contribution of the asses to that of the unloaded disk. The final report is expected to include: Four (4) MATLAB plots, with two (2) data cursor points on each plot, along with titles, labels and legends if necessary to show which plot corresponds to which situation. - Disk 3 Trial 1 - Disk 3 Trial 2 - Disk 1 Trial 1 - Disk 1 Trial 2 Calculations showing how you found the following values, along with units for every quantity: - 4 Natural frequencies ( nd31 nd32 nd11 nd12) - 2 Daping ratios ( d32 d12) - Inertia of the 4 asses (J) - Inertia of Disk 1 (Jd1), including otor etc. - Inertia of Disk 3 (Jd3) 5
- Spring constant for upper shaft (kd3 ) - Spring constant for lower shaft (kd1) - Daping constant of upper part (cd3 ) - Daping constant of lower part (cd1 ) For all the questions highlighted, the questions should be copied and pasted into your lab report and answered iediately thereafter. Experient 2.a: Rigid Body P, PD and PID Control In this part of the lab, a proportional controller will be ipleented, so that the syste will act like a specific frequency spring. Effects of various control configurations and control paraeters on the syste perforance will be studied. Note: You will need this value: k hw = 17.4 N-/rad Procedure: Proportional Control Action 1. Using the results fro Experient 1 to construct a odel of the plant with two ass pieces at 9.0 c radial center distance on the botto disk both other disks reoved. 2. Set-up the plant in the syste configuration described in Step 1. 3. Use Eq. 2-1 to deterine the value of k p so that the syste behaves like a 2 rad/s (or 1.0 Hz) spring-inertia oscillator. n n = k pk hw J (Equation 2-1) 4. Set up a closed-loop step of 0 (zero) counts, dwell tie = 3000 s, and 1 (one) rep Trajectory in the Coand enu. Set up to collect Encoder 1 and Coand Position via the Data Acquisition box in the Data enu. 5. Now, set up the controller. Enter the Control Algorith box, set Ts=0.00442 s and select Continuous Tie Control. Select PI + Velocity Feedback, which is the return path derivative for control. Enter k p value deterined above for 1 Hz or n 2 rad/s oscillation (k d & k i = 0) and select OK. Do not input value greater than k p = 0.20 Select Ipleent Algorith, then OK. 6
6. Run the experient: Manually rotate the lower disk roughly 60 deg and hold (not for too long). Select Execute under Coand, Release disk and click Run iediately after. 7 Export the data to MATLAB. Plot the encoder 1 position and coand position vs. tie. Calculate to confir the frequency using the Data Cursor Tool in the MATLAB Figure. Be sure to show the calculations and units. If the calculated k p does not give you the right n, talk to TA or experientally deterine a k p which does. The final report of this part is expected to include: One MATLAB Plot with titles, labels and two data curses used to deterine natural frequency. - Plot of the free response Calculations showing how you found the following values, along with units for every quantity. - Inertia of the Syste J for this experiental setup - Calculation for kp in Step 3 - Frequency of the response. Experient 2.b PD Control Design & Step Response This part is a continuation of Experient 2.a with the sae experiental setup. Control values of kp and kd are deterined to results in a underdaped, critically daped and overdaped syste as studied in e370 and e450 9. Use Eqs. 2-1 to deterine a value of kp to result in a syste natural frequency of n 4 rad/s. Then use Eq. 2-2 below to deterine three values of kd to result in a syste with (1) = 0.1 (underdaped), (2) = 1.0 (critically daped) and (3) = 2.0 (overdaped). = k d k hw k = d k hw 2J n 2 Jk p k hw (Equation 2-2) 10. Ipleent the underdaped controller (via PI + Velocity Feedback) and set up a trajectory for a 2500 count closed-loop Step with 4000 s dwell tie and 1 rep. 7
11. Execute this trajectory. Plot on screen both the coand and response on the sae vertical axis so that there is no graphical bias. Make sure your results indeed show an underdaped step response before you proceed to export data to atlab. 12. Repeat Step 11 for the critically daped and over-daped cases. Make sure they are indeed largely daped with no oscillations in the responses. 13. Export the data to your atlab folder. Then plot all three daping cases in one figure using progra plotdata3., coparing and coenting on the daping characteristics. 14. Now, for the underdaped design, reduce the value of kp to 50% and 10% of the current value and run the two experients. Observe the response on onitor. Increase the runtie if it does not reach a steady state before the coand is dropped off. Export the data to atlab and plot the responses for all three kp values in one figure. Use atlab data curser to deterine the following fro the response curves: 1) Percentage steady-state error, ess %, as a function of kp, which is the difference between the coand level and the response divided by the coand. It easures accuracy of the response. 2) Settling tie, tss, which is the tie taken for the response to reach within 2% to steady state. It easures how fast to reach target. 3) Percentage overshoot, Mp %, which is the difference between the peak value of the response and its steady-state level divided by the steady-state value. It easures response stability characteristics. 4) Peak tie, tp, which is the tie taken to reach the peak response. It easures response speed. Fill the results in Table 1 below in increasing order of kp: Table 1. Response characteristics as functions of control gain kp ess % tss Mp % tp 8
Briefly discuss the effect of the control gain on the syste steady-state and transient perforance in ters of these results. Which of the three control gains produces the best results? The final report of this part is expected to include: Two atlab plots along with titles and labels: - Plot for all three daping cases - Plot for all three kp cases including the calculation data cursers Calculations showing how you found the following values along with units. - kp in Step 9 - Under-daped kd - Critically daped kd - Over-daped kd - ess % and Mp % in Table 1 Experient 2.c Adding Integral Action In this part of the lab, a full PID controller will be ipleented. By adding integral action to the controller, the settling tie and overshoot of the response will be ipacted. 15. Deterine k i such that k i k hw = 3 N-/(rad-sec). Ipleent a controller with this value of k i and the critically daped k p & k d paraeters fro Step 9. Do not input k i >0.4. 16. Execute a 2500 count closed-loop step of 4000 s duration (1 rep). 17. Plot the encoder 1 response and coand position in MATLAB 18. Experientally deterine a value of ki that visibly gives you a better response judged by the overshoot and steady-state error in coparison to the previous run. This ki ay be saller than the one used in Step 15. 19. Plot responses and coand position for three cases of ki = 0 (fro Experient 2.b), the calculated ki and the better ki. Briefly describe the effects of adding integral action to the controller. The final report is expected to include: One MATLAB Plot, along with title, label and legend. Calculated ki with calculation equation. Experientally deterined better ki. 9
Experient 2.d Frequency Response In this portion of the lab, the input will now be a series of increasing frequency sine waves, used to deterine the frequency perforance of the syste. Part (1) Single DOF syste 20. Ipleent the underdaped syste configuration fro Step 9. 21. Set up a trajectory for a 400 count closed-loop Sine Sweep fro 0.1 Hz to 10 Hz of 60 seconds duration with Logarithic Sweep checked. Select Encoder 1 only as data to acquire in Data Acquisition enu. Execute the trajectory 22. Plot Encoder 1 position in Log frequency and Db aplitude. Observe the response peaks at a given frequency, which would be the syste natural frequency. 23. Export the data to atlab and plot the response using plot key, iplot=0, in the plotdata. progra. The final report is expected to include: One Matlab plot fro which to identify syste natural frequency with Data Curser tool. Coent on whether it is what you expected based on your syste design in Step 9. Part (2) 2DOF syste Mount all three disks to the device. Add a pair of weights to both lower and iddle disks at 9.0 c radius. Fasten the top disk to the frae so that it can t ove. The syste now has two degrees of freedo at the two oving disks, and the shaft segents connecting the three disks act as torsional springs. Set up a sine sweep action siilar to what you did in Step 21 except this tie setting the trajectory to sweep frequencies fro 0.5 Hz to 10Hz. Set the data acquisition to acquire position data for both disk 1 and disk 2 (encoder 1 and encoder 2). Execute the sine sweep and observe the responses fro both the disks. They are not in phase but you should see the responses increase, decrease, increase again and decrease again as the sweep frequency increases, suggesting two natural frequencies. Plot Encoder 1 position in Log frequency and Db aplitude, fro which to estiate the captured two natural frequencies (the peaks). Repeat it for Encoder 2 position to see if the captured peaks are the sae frequencies. Export the data to atlab to generate the plots. You need to slightly change the plot coand in the plotdada. when you plot the Encode 2 data (ask TA). The final report is expected to include: 10
Two (2) Plots, along with titles, labels and legends if necessary to show which plot corresponds to which situation. Use the Data Cursor to indicate the two natural frequency peaks in each plot. - Plot of Encoder 1 - Plot of Encoder 2 - The two natural frequencies fro Encoder 1 - The two natural frequencies fro Encoder 2 Copare and coent on the two sets of results on syste natural frequencies. 11