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Name: EE 179.1 Section: Course and Year: Laboratory Schedule: Activity 2: Mathematical Modeling of Physical Systems Abstract <100-300 words well-chosen words> 1 Objectives To grasp the important role of mathematical models of physical systems in the design and analysis of control systems. To learn how Scilab helps in solving such models. 2 List of equipment/software Personal Computer Scilab 3 Deliverables Scilab scripts and their results for all the assignments and exercises properly discussed and explained. Analytical conclusion for this lab activity. 4 Modeling Physical Systems 4.1 Mass-Spring System Model Consider the following Mass -Spring system shown in Figure 1 where K is the spring force, fv is the friction coefficient, x(t) is the displacement and f(t) is the applied force: Figure 1. Mass-Spring System and its Free-Body Diagram Name: EE 179.1 Section: Course and Year: Laboratory Schedule: Referring to the free-body diagram, we get 𝑀 𝑑2 𝑥(𝑡) 𝑑𝑡 2 + 𝑓𝑣 𝑑𝑥(𝑡) 𝑑𝑡 + 𝐾𝑥(𝑡) = 𝑓(𝑡) . (1) The second order linear differential equation (1) describes the relationship between the displacement and the applied force. The differential equation can then be used to study the time behavior of x(t) under various changes of the applied force. The objectives behind modeling the mass-damper system can be many and may include: Understanding the dynamics of such system. Studying the effect of each parameter on the system such as mass M, the friction coefficient B, and the elastic characteristic Kx(t). Designing a new component such as damper or spring. Reproducing a problem in order to suggest a solution. 4.2 Using Scilab in Solving Ordinary Differential Equations Scilab can help solve linear or nonlinear ordinary differential equations (ODE) usging ode tool. To show how you can solve ODE using Scilab, we will proceed in two ways. We first see how we can solve a first order ODE and then see how we can solve a second order ODE. 4. 3 First Order ODE: Speed Cruise Control Example Assume a zero spring force which means that K = 0. Equation (1) becomes 𝑀 𝑑2 𝑥(𝑡) 𝑑𝑡 2 + 𝑓𝑣 𝑑𝑥(𝑡) 𝑑𝑡 = 𝑓(𝑡) (2) or 𝑀 𝑑𝑣(𝑡) 𝑑𝑡 + 𝑓𝑣 𝑣(𝑡) = 𝑓(𝑡) since 𝑎(𝑡) = 𝑑𝑣(𝑡) 𝑑𝑡 = 𝑑2 𝑥(𝑡) 𝑑𝑡 2 , (3) Name: EE 179.1 Section: and Course and Year: Laboratory Schedule: 𝑣(𝑡) = 𝑑𝑥(𝑡) 𝑑𝑡 . Equation (3) is a first order linear ODE and we can use Scilab to solve for this differential equation. We can model and solve for equation (3) by writing the following script in Scilab. You can also input the codes directly into the Scilab Console. Code: //declare constant values M = 750 B = 30 Fa = 300 //the force f(t) //differential equation definition function dvdt = f(t,v) dvdt = (Fa-B*v)/M; endfunction //initial conditions @ t=0, v=0 t0 = 0 v0 = 0 t = 0:0.1:5; v = ode(v0, t0, t, f); clf; plot2d(t, v) //read references for ode tool Name: EE 179.1 Section: 4. 4 Course and Year: Laboratory Schedule: Second Order ODE: Mass-Spring System Example In reality, the spring force and/or friction force can have a more complicated expression or could be represented by a graph or data table. For instance, a nonlinear spring can be designed such that the elastic characteristic is 𝐾𝑥 𝑟 (𝑡) where r > 1. Figure 2 is an example of a nonlinear spring. Figure 2. Nonlinear Spring[1] In such case, equation (1) becomes 𝑀 𝑑2 𝑥(𝑡) 𝑑𝑡 2 + 𝑓𝑣 𝑑𝑥(𝑡) 𝑑𝑡 + 𝐾𝑥 𝑟 (𝑡) = 𝑓(𝑡) (4) Equation (3) represents another possible model that describes the dynamic behavior of the mass-damper system under external force. Unlike equation (1), this is said to be a nonlinear differential equation. We can also solve equation (3) using Scilab ode tool but this time, the second order differential equation has to be decomposed in a set of first order differential equations as follows: Let 𝑥(𝑡) = 𝑋1 , so 𝑑𝑋2 𝑑𝑡 so 𝑓 𝐾 𝑑𝑋1 𝑑𝑡 = 𝑋2 = − 𝑀𝑣 𝑋2 − 𝑀 𝑋1 𝑟 + 𝑓(𝑡) 𝑀 . and 𝑑𝑥(𝑡) 𝑑𝑡 = 𝑋2, Name: EE 179.1 Section: In vector form, let Course and Year: Laboratory Schedule: 𝑋 𝑥 𝑋 = [ 1] = [ ′] ; 𝑥 𝑋2 𝑑𝑋1 𝑑𝑋 𝑑𝑡 𝑑𝑡 = [𝑑𝑋 ] = [ 𝑥′ ] 2 𝑥′′ 𝑑𝑡 so we get 𝑑𝑋 𝑑𝑡 𝑋2 = [ 𝑓𝑣 𝐾 𝑓(𝑡)] . − 𝑀 𝑋2 − 𝑀 𝑋1 𝑟 + 𝑀 The Scilab ode tool can now be used: Code: //declare constant values M = 750 B = 30 Fa = 300 //the force f(t) K = 15 r = 1 //differential equation definition function xprime = f(t,x) xprime(1) = x(2); xprime(2) = -(B/M)*x(2)-(K/M)*((x(1))^r)+(Fa/M); endfunction t0 = 0 t = 0:0.1:5; x0 = 0; //set initial position xprime0 = 0; //set initial velocity x = ode([x0; xprime0], t0, t, f); clf; plot2d(t, x(1,:)) // “x(1,:)” plots the position vs time // “x(n, :)” to plot a solution in matrix form // with dimension n x T Name: EE 179.1 Section: 5 5.1 Course and Year: Laboratory Schedule: Assessment Assignment 1. Show and discuss the graphs obtained from the sample simulations. Name: EE 179.1 Section: 5.2 Course and Year: Laboratory Schedule: Exercise 1 Referring to the mass-spring system example: 1. Plot the position and the speed of the system both with respect to time in separate graphs. 2. Change the value of r to 2 and 3. Plot the position and speed both with respect to time. Discuss the results. 3. With r = 1, vary the value of K (multiply by 5 and 10) and discuss the results. Name: EE 179.1 Section: 5.3 Course and Year: Laboratory Schedule: Exercise 2 Consider the mechanical system depicted in the figure. The input is given by f(t), and the output is given by y(t). Determine the differential equation governing the system and using Scilab, write a script and plot the system response such that forcing function f(t) = 1. Let m = 10, k = 1, and b = 0.5. a. The peak displacement of the block is about ____. b. At time t = ____, the output reaches peak displacement. c. If k = 10, the peak value of y is ____. d. What happens to the system when b is changed? Name: EE 179.1 Section: Course and Year: Laboratory Schedule: 6 References CISE 302 Linear Control Systems Laboratory Manual. (2011, October). Systems Engineering Department, King Fahd University of Petroleum & Minerals. Scilab Enterprises . (2013). Scilab for very beginners. 78000 Versailles, France. Sengupta, A. (15, April 2010). Introduction to ODEs in Scilab. Indian Institute of Technology Bombay.