Driven Mass-Spring System with Damping

1

58:080 Experimental Engineering

Lab 2c

Driven Mass-Spring System with Damping

Objective

Warning: though the experiment has educational objectives (to learn about boiling heat transfer, etc.), these should not be included in your report.

-To measure and investigate the dynamic characteristics of a driven spring-mass-damper system.

-To understand the behavior of vibration systems and theory.

Equipment

Name / Model / S/N
ECP Rectilinear Control System / 210 A
Computer
Control Box

Figure1: Model 210A Rectilinear Plant

The Driving Mass-Spring workstation includes an ECP Model 210A rectilinear control system that is connected to a PC containing the required ECP software. The rectilinear control system consists of three mass carriages, three encoders, two dashpot dampers, a control box, and a mechanical actuator. The three mass carriages have an associated encoder that measured the carriage displacement and reported their measured values to the computer. The second and third carriages have the possibility of adding a dashpot damper. The magnitude of damping is controlled by an adjustable screw. The system control box contained the actuator that moved the carriages. A control algorithm is downloaded to the control box which would execute the trajectory on the carriages.

required reading

References [1] and [2] for theory and Appendix A and B for Safety and Precautions for ECP Systemand Demonstration of ECP Executive Program.

prelab questions(10% of the total grade of the lab, 2.5% each)

1-What is the Resonance Frequency?

2-What is the Damping Ratio?

3-When a system is said to be critically or over or under damped?

4-Read carefully the instructions in this write-up and describe how you will perform your measurements to obtain errors on the system parameters (spring constants and damping ratios), as well as natural frequencies of the systems.

Procedure

The procedure to follow for Day 1 of this lab can be found in Chapter 6 of the ECP manual available online. Before beginning, read Appendix A and B to familiarize yourself with the safety features and description of the equipment for this lab. The goal of day 1 will be to take measurements needed to determine the system parameters spring constants, damping coefficient, and the effective masses of the carriages.

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1.System Identification

This section gives a procedure for identifying the plant parameters: masses of the system’s carriages, spring constant and damping coefficient. The approach will be to indirectly measure the mass, spring, and damping parameters by making measurements of the plant while set up in a pair of classical spring-mass configurations.

Procedure:

1.Clamp the second mass to put the mechanism in the configuration shown in Figure 2a using a shim (e.g. 1/4 inch nut) between the stop tab and stop bumper so as not to engage the limit switch (see Section 2.2). Verify that the low stiffness spring (nominally 400 N/m (2.25 lb/in.)) is connecting the first and second mass carriages.

2.Identify four brass masses to use. These are nominally 500 g. Weigh them and determine a more accurate mass for each. Making sure you take enough measurements to perform proper uncertainty analysis. Secure the four 500g masses on the first mass carriage.

3.With the controller powered up, enter the Control Algorithm box via the Set-up menu and set Ts = 0.00442. Enter the Command menu, go to Trajectory and select Step, Set-up. Select Open Loop Step and input a step size of 0 (zero), a duration of 3000 ms and 1 repetition. Exit to the background screen by consecutively selecting OK. This puts the controller in a mode for acquiring 6 sec of data on command but without driving the actuator. This procedure may be repeated and the duration adjusted to vary the data acquisition period.

Figure 2: Configurations For Plant Identification
(Model 210a shown. Four 500 g. weights on each active carriage.)

4.Go to Set up Data Acquisition in the Data menu and select Encoder #1 and Encoder #2 as data to acquire and specify data sampling every 2 (two) servo cycles (i.e. every 2 Ts's). Select OK to exit. Select Zero Position from the Utility menu to zero the encoder positions.

5.Select Execute from the Command menu. Prepare to manually displace the first mass carriage approximately 2.5 cm. Exercise caution in displacing the carriage so as not to engage the travel limit switch, or else the data acquisition will cease. With the first mass displaced approximately 2.5 cm in either direction, select Run from the Execute box and release the mass approximately 1 second later. The mass will oscillate and attenuate while encoder data is collected to record this response. Select OK after data is uploaded.

6.Select Set-up Plot from the Plotting menu and choose Encoder #1 Position then select Plot Data from the Plotting menu. You will see the first mass time response.

7.Choose several consecutive cycles (say ~5, also make sure you have enough data to obtain a precision error for the results) in the amplitude range between 5500 and 1000 counts (This is representative of oscillation amplitudes during later closed loop control maneuvers. Much smaller amplitude responses become dominated by nonlinear friction effects and do not reflect the salient system dynamics). Divide the number of cycles by the time taken to complete them being sure to take beginning and end times from the same phase of the respective cycles.[1] Convert the resulting frequency in Hz to radians/sec. This damped frequency, d, approximates the natural frequency, n, according to:

(1)

where the "m11" subscript denotes mass #1, trial #1. (Close the graph window by clicking on the left button in the upper right hand corner of the graph. This will collapse the graph to icon form where it may later be brought back up by double-clicking on it.)

8.Remove the four masses from the first mass carriage and repeat Steps 5 through 7 to obtain nm12 for the unloaded carriage. If necessary, repeat Step 3 to reduce the execution (data sampling only in this case) duration.

9.Measure the initial cycle amplitude Xo and the last cycle amplitude Xn for the n cycles measured in Step 8. Using relationships associated with the logarithmic decrement:

(2)

find the damping ratio m12 and show that for this small value the approximations of Eq's (1, -2) are valid.

10.Repeat Steps 5 through 9 for the second mass carriage. Here in Step 6 you will need to remove Encoder #1 position and add Encoder #2 position to the plot set-up. Hence obtain nm21 , nm22 and m22. How does this damping ratio compare with that for the first mass? Be sure to save this plotted data as it will be used in the next experiment.

11.Connect the mass carriage extension bracket and dashpot to the second mass as shown in Figure 2c. Open the damping (air flow) adjustment knobapproximately one turn from the fully opened position; or remove the knob and screw it in carefully about one turn. Repeat Steps 5, 6, and 9 with four 500 g masses on the second carriage and using only amplitudes ≥ 500 counts in your damping ratio calculation. Hence obtain d where the "d" subscript denotes "dashpot".

12.Each brass weight has a mass of 500 ± 10 g. Hopefully, this will be verified by your measurements. Use the more precise mass measurements in the analysis below. Calling the mass of the four weights combined mw, use the following relationships (2 equations and 2 unknowns) to solve for the unloaded carriage mass mc2, and spring constant k.[2]

k/(mw+mc2) = (nm21)2(3)

k/mc2 = (nm22)2(4)

Find the damping coefficient cm2 by equating the first order terms in the equation form:

(5)

Repeat the above for the first mass carriage, spring and damping mc1, cm1and k respectively.[3] [4]

Calculate the damping coefficient of the dashpot, cd.

1. Remove the carriage extension bracket and dashpot from the second mass carriage, replace the low stiffness spring with a high stiffness spring (800 N/m nominally), and repeat Steps 5 and 6 to obtain the resulting natural frequency m23. Calling the value of stiffness obtained in Step 12 above klow stiffness, calculate khigh stiffness and klow stiffness from the frequency measurements of this step.
2. Note any differences between nominal masses and spring constants and what you measure and calculate.

Now all dynamic parameters have been identified! Values for m1 and m2 for any configuration of masses may be found by adding the calculated mass contribution of the weights to that of the unloaded carriages[5].

Analysis

Discuss and calculate the properties and parameters mentioned in the procedure for the mass/spring/damper system. MAKE SURE YOU HAVE ENOUGH DATA TO PERFORM A HIGH-QUALITY UNCERTAINTY ANALYSIS ON ANY OF THE PARAMETERS MEASURED!!

Day 2 Procedure

1)Read and follow the safety precautions in Attachment A at the end of this write-up.Note: All users must read and understand Attachment A (Ecp Manual Section 2.3, Safety) before performing any procedures described herein.

2)Familiarize yourself with the operation of the ECP Executive program by completing the demonstration of the ECP Executive program. Work through steps 1 through 4 of the ECP Executive Program Demonstration in Appendix Bat the end of this write-up. The complete section 3.2 of the manual is provided in Appendix B for completeness; but again, complete only through step 4 to familiarize yourself with the operation. However, instead of using the “default.cfg” in the handout use the one created for this lab, “lab2c_default.cfg”. Go through the procedures in Appendix Bfollowing the instructions for the Model 210. Do not use the third carriage and encoder. This will setup the ECP hardware as a single degree of freedom system, using only mass carriage 2, and encoder 2, while the driving input to the system will be sensed at encoder 1. Note also the control settings may be different than the Appendix Bvalues (the gains discussed in step 2 in Appendix B); record these in your logbooks. Disregard any discussion for the third encoder, since we are not using it. At the end of the demo you should have a good understanding of: loading configuration files, using the trajectory settings, executing a trajectory, and plotting data.

3)Use the "Export Raw Data" function under the "Data" menu to export acquired data, and then bring the data into the Excel program. This will be useful in storing data if you need it. Learn how to export and plot data. Consult the TA or manual for assistance if needed.

4)Abort the control algorithm. Add two masses to the encoder 2 carriage, and place one mass on carriage 1. Disconnect the damper. Note which spring you are using to connect the two carriages. You should know these masses and the carriage mass from your data collected and analyzed on day 1 of the lab. Using what you have learned in steps 2) and 3) above, load the configuration file "lab2c_step.cfg". Go to the algorithm settings and record them. Go to the trajectory settings menu and record them. Return to the Setup Control Algorithm menu and implement the algorithm. Setup a plot of the encoder 1 and 2 position output. Execute the trajectory. Observe the plot of the data and do not delete this plot. Export the data to a file and determine the natural frequency. Drive the system using the configuration file “lab2c_step.cfg”. Plot the data from the encoders 1 and 2 and determine the frequency of the oscillations. Thendetermine the spring constant from the measured natural frequency from  = (k/m)1/2. Compare this with the spring constant parameter value determined from the data taken on day 1 for the spring between encoders 1 and 2. Repeat this several times and estimate the precision errors, for one spring. Discuss the comparison, and possible reasons for differences. Repeat this step for two other springs.

5)Press the "Abort Control" button. Add one more massto the encoder 2 carriage. Load the configuration file "lab2c_step.cfg". Implement the algorithm and execute the trajectory again. Export the data to a file and determine the natural frequency. Again, using the encoder 2 carriage massand measured masses on encoder2, determine the spring constant from the measured natural frequency. Repeat this several times and estimate a repeatability error.

6)Report on and discuss the effect of mass on natural frequency. Plot the natural frequency versus mass. Is the damping Coulomb- or viscous-type damping?

7)Attach the damper as you did in day one, by removing the cap and then carefully screwing it in one turn. Load the configuration file "lab2c_step.cfg" with the masses present on the carriage from step 5. Run the trajectory and store the step function data with the damper attached.

8)Press the "Abort Control" button. Load the configuration file "lab2c_transmission_band.cfg". Record the control and trajectory settings. Implement the algorithm and execute the sine input trajectory. Observe the data plot of the two encoder positions. Is the system over- or under-damped? Record the amplitude and frequency data, and note the steady state time-delay between the two encoders. This time delay can be used to determine the phase shift.

9)Repeat step 8) by loading, and running the configuration files: "lab2c_resonance_band.cfg" and "lab2c_filter_band.cfg". Record the amplitude and frequency data for each file run, and note the steady state time-delay between the two encoders for each case. This time delay can be used to determine the phase shift in the next step that you will complete outside the lab.

10)Load again the configuration "lab2c_resonance_band.cfg". Run the apparatus using this setting while turning the damper clockwise to increase damping until you find the setting for critical damping ratio (approximate to the best of your ability) for the system. How many turns is this from fully open (or damper screw removed)?

11)Outside the lab, make your own frequency response plots like Figure 3 and 4magnitude ratio and Phase shift, respectively (see plots repeated below). The plots should have one theoretical curve corresponding the damping ratio you estimated your data was run at in steps 9) and 10), and plot the three experimental data points corresponding to the frequency ratios (ω/ωn) executed in steps 8) and 9).

Figure 3: Second-order system frequency response: amplitude.

Figure 4: Second-order system frequency response: phase shift.

Day 3

In this part of the experiment you will be setting-up a 2DOF (two degrees of freedom) system. As you did in the previous days, you will first determine the system parameters, and with them you will perform comparisons of the theoretical and measured transfer functions trough the corresponding Bode diagrams. In this section you will perform an interesting way of obtaining the system Bode diagrams through the use of a sine sweep excitation input.

The idea is to analyze a symmetric system composed of two carriages with the same mass, two equal springs and two equal dampers. Of course, this is not going to be possible in the real experiment, but we will try to make the setup as symmetric as possible.

You will be setting up the experiment with three masses on carriages two and three, you will be using the springs with constants of 400 N/m, and with the dashpots you will fully extract the adjustment screws.

The 2DOF setup should resemble Figure 5c.

The equations of motion for this system are briefly described in Appendix D. In this appendix the carriages’ transfer functions are obtained, leading to the expressions

Remember from the previous sections that was the transfer function corresponding to the 1DOF system.

The magnitude of these transfer functions when gives you the relative amplitude between carriage displacements and input displacements, and the phase between these is obtained as the phase of the transfer function, i.e.

Procedure:

First, as you did in the first day, you will determine the system characteristics. You will find the parameters,,andfrom these set of measurements. You will use the two softer springs of approximately 400 N/m.

1)Setup the system as shown in Figure 5-a. That is, connect carriages one and two with one of the springs, connect a dashpot to the second carriage, and put three weights on it. Clamp the first carriage. You will determine , and .

2)Remove completely the adjustment screw from the dashpot.

3)Follow the procedures given in the first day for obtaining natural frequency, damping constant and spring constant. Remember that for obtaining the damping constant what you do is perform one free oscillation run and you infer it from the equation . For obtaining the carriage mass and spring constant you perform two runs, one with the three masses and one without masses. Then you solve a system of two equations for the spring constant and carriage mass.