Question

In this problem, we will try to illustrate the following: a waggon is linked to a wall with a spring and a damper. The position of the waggon is given by x(t). The waggon can move without any friction on the ground. The mass is M. (35 pnts) The differential equation that describes the system is the following: d2x M + b + cx(t) = F dt2 Where: M=5 [kg] mass of the waggon b= 1 [Ns/m] damper constant C= 2 [N/m] spring constant F-force acting on the waggon, zero from the beginning and 2[N] after 5 [sec]. X(t)- position dx(t)/dt-velocity ssume velocity and position are zero from the start. Decide using SIMULINK the stationary sition of the waggon after a long time by reading from the Scope in your model. Determine the sition, velocity, and acceleration of the wood block after a long time and display the results in notations next to the scope listing these 3 values. Create comment block in SIMULINK model explain your model and rationale.

Answer 1

The Simulink model uses signal connections, which define how data flows from one block to another.

The Simscape model uses physical connections, which permit a bidirectional flow of energy between components. Physical connections make it possible to add further stages to the mass-spring-damper simply by using copy and paste. Input/output connections require rederiving and reimplementing the equations.

The initial deflection for the spring is 0 meter. This is shown in the block annotations for the Spring and one of the Integrator blocks.

Simulink Model Simscape Model V_sl V SC 1/m -14 Mass V 1/Mass xD Damper W5 Spring b Damping k Stiffness Mass-Spring-Damper in Simulink and Simscape 1. Plot spring deflection (see code) V_SC 2. Explore simulation results using sscexplore 3. Learn more about this example V_sl 4. Open model of double mass-spring-damper 5. Learn more about modeling physical networks Simscape H Simulink Velocity of Mass

Entering State-Space Models into MATLAB

Now we will demonstrate how to enter the equations derived above into an m-file for MATLAB. Let's assign the following numerical values to each of the variables.

m                   mass                                    5.0 kg
k                   spring constant                         2.0 N/m
b                   damping constant                        1.0 Ns/m
F                   input force                             2.0 N

Create a new m-file and enter the following commands.

m = 5;
k = 2;
b = 1;
F = 2;

A = [0 1; -k/m -b/m];
B = [0 1/m]';
C = [1 0];
D = [0];

sys = ss(A,B,C,D)

sys =
 
  A = 
         x1    x2
   x1     0     1
   x2    -0.4  -0.2
 
  B = 
       u1
   x1   0
   x2   0.2
 
  C = 
       x1  x2
   y1   1   0
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

The Laplace transform for this system assuming zero initial conditions is

Position of Mass Simscape Simulink 0.8 0.6 0.4 0.2 Position (m) um -0.2 -0.4 -0.6 -0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (s)