Massachusetts Institute of Technology Department of Mechanical Engineering 2.003 Modeling Dynamics and Control I Spring 2005 Prelab 1, Feb/02/05 Your prelab is due at the start of your lab section. We suggest you make a copy of your prelab, so you can use your results during lab. Note: No late prelabs will be accepted. Come to lab on time! Required Reading • Class Notes Chapter 1 MATLAB You will be using MATLAB extensively throughout this course. Problem 4 at the end of this prelab includes a short plotting exercise. This exercise demonstrates some basic MATLAB commands for creating and plotting vec-tors and matrices of data. You can find more information on using MATLAB in several of the handouts on page of the course 2.003 on the MIT server.the toolsPrelab 1 Feb/02/2005lab, we measure the time constant of a first-order spring-damperand then use this value to estimate the magnitude of the damping.shows the spring-damper system, and Figure gives an idealizedwhich we will use here to predict its behavior.2.003 In this system Figure model (L = wh(stiffness k is a constant)Cantilever SpringL(camera)x(t)(damping b can be adjusted with knob)Airpot Dashpot0. w = 0. h = 0.x(t) k bIntroduction Figure 1: Picture of spring and dashpot system. 205 m, 0127 m, 00127 m) Figure 2: Idealized spring-damp er system. We will model the spring-steel cantilever as a linear spring; that is, we as-sume that a plot of force vs. displacement is a straight line. The constitutive relationship for a linear spring has the form fk (t) = kx(t) (1) where fk (t) = force applied to spring in x direction (N) k = the spring stiffness (N/m) x(t) = the displacement across the spring (m) 2 122.003 Prelab 1 Feb/02/2005 The stiffness k in the x direction (see Fig. 1) as seen at a position L along a uniform cantilever beam can be derived from the beam bending equations as 3EI k = L3 (2) where E = Young’s modulus (210 GPa for spring steel) I = wh3/12 w = width of the beam’s cross section h = height (thickness) of the beam’s cross section L = length of the beam An Airpot 1dashpot attached to the end of the cantilever adds damping to the system. The Airpot is an air piston-cylinder arrangement with a control-lable ”leak”. The Airpot—with simplifying assumptions—can be modeled as a pure damper where the damping force is proportional to velocity, so we write it’s constitutive relationship as fb(t) = bx˙(t) (3) where fb(t) = force applied to damper in the x direction (N) b = the damping coefficient x˙(t) = the velocity difference across the damper From a free body diagram the spring and dashpot forces sum at the node joining them (be careful about signs as you go through this step!). Using the constitutive relationships for the spring and damper given by Equations ( ) and ( ), we can write a simple first-order model of the homogeneous re-sponse of our cantilever and Airpot system as 0 = bx˙(t) + kx(t). (4) This is the equation of motion of the system. The solution to this differential equation has the form Ae−t/τx(t) = (5) (You are asked to derive the solution in Problem 3.) In words, the free (or “natural”) response of this model to an initial displacement decays exponentially. The exponential response of a first-order system is characterized by its time constant, τ. The time constant is the time required for the magnitude of the output to “decay away” by about 63%. (See Prof. Trumper’s supplementary notes on first-order system response for more details.) 1http://www.airpot.com/ 3 31x(t)Model Ak bx(t)Model Bkbx(t)Model Ck b2.003 Prelab 1 Feb/02/2005 Problems 1. A model for the first-order spring-damper system is shown in Figure . The same image is labelled as “Model A” in Figure below. Which of the other two models below (B or C) also model the same dynamic system? (Explain briefly.) Figure 3: Idealized spring-damp er systems. 2. Determine the numerical value of the spring constant k of the steel beam shown in Figure , at the position L where the Airpot is at-tached. 3. Derive the solution to Equation ( ) if the system is given an initial displacement a0 and released. The solution should be of the form Ae−t/τx(t) = (6) where A is a constant to be determined from initial conditions, and τ is the time constant. 4. Suppose that we have measured the following output data from a spring-dashpot system, and that the dimensions of the spring steel cantilever beam are as sketched in Figure . time (s) x(t) (m) 0.00 1.57 0.05 1.17 0.10 0.89 0.15 0.68 0.20 0.52 0.25 0.40 0.30 0.30 0.35 0.23 0.40 0.17 0.45 0.13 0.50 0.11 4 321152.003 Prelab 1 Feb/02/2005 (a) Plot the data using MATLAB and include this plot in your prelab. You may use the MATLAB code on the next page to create the plot. (b) Use your plot to estimate the time constant. (Just get a rough estimate by “eyeballing” it.) Estimate A and τ from the data, and compare your estimate with the Matlab plot done on the MAtlab M-file excercise. (c) Determine the damping constant b that best fits this data, given the spring constant you calculated in Problem 2. 52.003 Prelab 1 Feb/02/2005 MATLAB M-file exercise on plotting You may use the code below to create your MATLAB plot of the data in Problem 4 if you wish. We are asking you to create a simple MATLAB plot in part to insure you have some (recent) experience in plotting within the MATLAB environment, since you will be asked to make similar plots during the laboratory sessions for 2.003. When using MATLAB, you may often find it convenient to create an “m- file” to write out a sequence of MATLAB commands so you can more easily repeat the same set of commands (on multiple sets of data, for instance). An m-file can be