Name: Partner: Date: LAB 1: Dynamic Equations of a Spring-Mass-Damper System Objectives : Physical setup Building the model with Simulink Analysis and explanation. Physical setup Newton's laws of motion form the basis for analyzing mechanical systems. Newton’s second law, Equation (1), states that the sum of the forces acting on a body equals the product of its mass and acceleration. Newton's third law, for our purposes, states that if two bodies are in contact, then they experience the same magnitude contact force, just acting in opposite directions. (1) The free-body diagram for this system is shown below. The spring force is proportional to the displacement of the mass, , and the viscous damping force is proportional to the velocity of the mass, . Both forces oppose the motion of 1 the mass and are, therefore, shown in the negative X-direction. Note also that corresponds to the position of the mass when the spring is upstretched. Now we proceed by summing the forces and applying Newton’s second law, Equation (1), in each direction. In this case, there are no forces acting in the Ydirection; however, in the X-direction we have: (2) This equation, known as the governing equation, completely characterizes the dynamic state of the system. Later, we will see how to use this to calculate the response of the system to any external input properties such as stability and performance. , as well as to analyze system Building the model with Simulink To build the simulation model, open Simulink and open a new model window. Then follow the steps listed below. 1. Insert an Integrator block from the Simulink/Continuous library and draw lines to and from its input and output terminals. 2. Label the input line "Xddot" and the output line "Xdot". To add such a label, double-click in the empty space just below the line. 3. Insert another Integrator block above the previous one and draw lines to and from its input and output terminals. 4. Etc… For the Second Order Differential Equation (Single Degree of Freedom SystemSDOF) model the initial conditions for both integrator blocks should be set to zero. As for the gain blocks, use the following values m = 2, b = 2, and k = 4. 2 Where (xddot) is the acceleration of the mass m, (xdot) is the velocity, x is the displacement, f(t) is the forcing function (input force), b is the damping coefficient, and k is the spring constant. To model this system we start by looking at the terms in the bracket. There are three input signal lines: f(t), xdot, and x. We also notice that both xdot and x are multiplied by constants. Analysis: 1. Provide an analytic explanation of the giving formula 2? 2. Provide the final Simulink design of the Spring-Mass-Damper System? 3. Run the simulation using the given parameter? 4. Draw the function: Force= f (time) using an appropriate scale? 3