Lab 2: Time and Frequency 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 and frequency domain analysis. The steps we will go through in this lab are very commonly used in system identification (determining the transfer function) of unknown systems. We will utilize these models in later labs so do a good job in this lab, your results in later labs will be affected by how well you perform in this lab. You will be assigned to model either rectilinear or torsional systems. Background A one degree of freedom rectilinear mass-spring-damper system cmkk1211F(t)x (t)1 or a rotational mass-spring-damper system can be modeled as 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 files from the folder basic_files into this folder. Part A: One degree of Freedom Rectilinear Systems You will need to go through the following steps for each configuration (different masses and/or springs). You need to fill out the data sheet indicating each configuration on the last page of the lab and turn it in! It needs to be signed before you start the second 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 connecting the first cart to the second cart (you may also have an additional spring between the motor and the first cart) and at least two masses on the cart. Do not use the damper. If you use two springs, the stiffer of the two springs should be between the first and second cart! Be sure you write down all of the information you need to duplicate this configuration. Step 2: Log Decrement (Time Domain) 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 determine the static gain. 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.) • 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 the cart to be analyzed (cart one in this case) • Select Load IC (initial condition) Response (the variables time and x1 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. • 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 put this figure in your memo. Step 3: Time Domain Estimate of the Static Gain You will go through the following steps: • Reset the system using ECPDSPresetmdl.mdl. • Modify Model210_Openloop.mdl so the input is a step. To make any changes to Model210_Openloop.mdl, the mode must be Normal. • Set the amplitude initially to something small, like 0.01 or 0.02 cm. • Compile Model210_Openloop.mdl • 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, the system will tell you if you have to. • You only need to run the system until it comes to steady state, then stop it. • Estimate the static gain as ssxKA= wheressxis the steady state value of the cart position, and Ais the input amplitude. You should do this in Matlab, don't use the X-Y Graph. The variables x1 and time should be in your workspace. You need to increase the value of the input amplitude until the cart is moving a maximum of about 2 cm or so. Use the static gain values associated with this input amplitude as the static gain. Step 4: Fitting the Estimated Frequency Response to the Measured Frequency Response We will be constructing the magnitude portion of the Bode plot and fitting this measured frequency response to the frequency response of the expected transfer function to determineK, ζ, and nω. For each frequency 2fωπ= we have as input () cos( )ut A tω= where, for our systems, Ais measured in centimeters. After a transition period, the steady state output will be1() cos( )xt B tωθ=+ , whereBis also measured in cm. Since we will be looking only at the magnitude portion of the Bode plot, we will ignore the phase angleθ. You will go through the following steps For frequencies Hz 0.5,1,1.5...7.5f =• Modify Model210_Openloop.mdl so the input is a sinusoid. You may have to set the mode to Normal. • Set the frequency and amplitude of the sinusoid. Try a small amplitude to start, like 0.01 cm. Generally this amplitude should be as large as you can make it without the system hitting a limit. This amplitude will probably vary with each frequency. • Compile Model210_Openloop.mdl, if necessary. (Assume it is not necessary. The system will let you know if it is 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