ECE-320 Lab 2 Time Domain Modeling of One Degree of Freedom Systems Overview In this lab you will be modeling two one degree of freedom systems using time-domain analysis. The goal is to review the log-decrement method and step response analysis for modeling 2nd order systems. If your station has both a rectilinear (model 210) and torsional (model 205) you should make one model of each type of system. Otherwise make two different systems (and corresponding models) using either the model 210 or model 205. Background A one degree of freedom rectilinear mass-spring-damper system can be modeled as By drawing a free body diagram and balancing forces, we get the equation of motion: cmkk1211F(t)x (t)1 11 11 1 2 1() () ( ) () ()mxt cxt k k xt Ft+++ =  A one degree of freedom rotational mass-spring-damper system can be modeled as 1By drawing a free body diagram and balancing torques, we get the equation of motion () () () ()Jt ct kt Ttθθθ++ =  Despite the fact that the systems appear quite different, the transfer functions for both of the one degree of freedom systems can be put into the standard form 22()121nnKGsssζωω=++ Here Kis the static gain, nωis the natural frequency, and ζis the damping ratio. These are the parameters we need to determine for these models. You will need to set up a folder for Lab 2 and copy all of the files from the basic_files folder into this new folder. Part A: One degree of Freedom Rectilinear Systems You will need to go through the following steps for two different configurations (different masses and/or springs) unless you are going to use one torsional and one rectilinear system. Be sure to load the correct controller personality file for the ECP system (and reset the controller)!!! Step 1: Set Up the System. Only the first cart should move, all other carts should be fixed. You need to have at least one spring connected to the cart and at least one mass on the cart. Do not use the damper. Be sure you write down all of the information you need to duplicate this configuration. Step 2: Log Decrement Estimate of ζand nω As you recall, the log decrement method is a way of estimating the natural frequency nωand damping ratio ζof a second order system. However, this method does not detethe static gain. rmine You will go through the following steps: • Reset the system using ECPDSPresetmdl.mdl. • Modify Model210_Openloop.mdl so the input has zero amplitude. • Compile Model210_Openloop.mdl if necessary. • Connect Model210_Openloop.mdl to the ECP system. (The mode should be External.) 2• Displace the first mass, and hold it. • Start (play) Model210_Openloop.mdl and let the mass go. • Run the m-file log_dec.m. This should be in the same directory as Model210_Openloop.mdl and log_dec.fig. This routine assumes the position of the first cart is labeled x1 and the time is labeled time. (These are the defaults in Model210_Openloop.mdl.) The program log_dec comes up with the following GUI: You need to • Select Cart 1 • Select Load IC (initial condition) Response (the variables time and y will be loaded from the workspace). At this point some initial estimates will be made. • Set/modify the Final Time • Select Plot IC Response to plot the initial condition response • Choose to identify the positive peaks (Locate + Peaks) or negative peaks (Locate - Peaks) . If the peaks are not numbered consecutively, you need to decrease the Samples Between Peaks and try again until all peaks have been identified. 3• Choose the initial peak (Peak x(n)) and final peak (Peak x(n+N)) to use in the log-decrement analysis. These should be fairly close to the beginning of the initial condition response. Don't try and use more than a few peaks. • Select Estimate Parameters to get the initial estimates of ζand nω • Select Make Log-Decrement Figure to get a plot and summary of the results. You need to include this figure in your memo. Step 3: Estimating the Static Gain K You will go through the following steps: • Reset the system using ECPDSPresetmdl.mdl. • Modify Model210_Openloop.mdl so the input is a step. You may have to set the mode to Normal. • Set the amplitude to something small, like 0.01 or 0.02 cm. • Compile Model210_Openloop.mdl, if necessary. • Connect Model210_Openloop.mdl to the ECP system. (The mode should be External.) • Run Model210_Openloop.mdl. If the cart does not seem to move much, increase the amplitude of the step. If the cart moves too much, decrease the amplitude of the step. You may have to recompile after the change. • You only need to run the system until it comes to steady state, then stop it. Estimate the static gain as ssyKA= wheressyis the steady state value of the cart position, and Ais the input amplitude. You should determine the value of ssyin Matlab, don't use the X-Y Graph. The variables y and time should be in your workspace. You should use three different input amplitudes and produce three different estimates for the static gain. Try and make your systems move so the steady state values are between 0.5 and 1.0 cm. Average your three estimates to produce a final estimate of the static gain. Your static gain should generally be between 10 and 30. If yours is not, it is likely you did not use the same units for A and ssy. Step 4: Second Order Step Response We will now try to identify the required system parameters by determining the step response of the real (ECP) system, simulating an ideal second order system, and comparing the two responses. If you cannot make the ideal system match the real system, make the ideal system match the real system as well as possible at the beginning of the response. 4This part of the lab should be done independently of Parts 2 and 3. That is, do not just use the values for the natural frequency nω, damping ratio ζ, and static gain K you obtained previously. Part of the reason for doing this is so you can develop a better feel for how changing these parameters will change the step response. You might want to have one partner do Parts 2 and 3 and the other do this part for the same system, and then switch when you do the other system. You will go through the following steps: • Reset the system using ECPDSPresetmdl.mdl. • Modify Model210_Openloop.mdl so the input is a step. You may have to set the mode to Normal. • Set the amplitude to something small, like 0.01 or 0.02 cm.