King fahd university of petroleum & minerals college of computer sciences & engineering
Experiment # 10: Simulation of systems represented by state variable model
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Experiment # 10: Simulation of systems represented by state variable modelINTRODUCTION In a linear translational system such as the two-mass system shown below in the figure, the state variables are the displacements and the velocities of the two masses. Let x1 and v1 denote the position and velocity of mass M1 with respect to fixed reference point and x2 and v2 denote the position and velocity of mass M2. The positive direction for both displacements is towards right. The two springs are neither stretched nor compressed when x1 = x2 =0. 
PROCEDURE: • Draw the free body diagram for the two-mass system • Write the modeling equation in state variable form describing the system from the free body diagram • Compare the equations with the equations given below 
• Draw the corresponding Simulink diagram. • Use the To Workspace blocks for t, fa(t), x1, and x2 in order to allow MATLAB to plot the desired responses. Set the save format to array in block parameters. EXERCISE 1: Plot the first 10 seconds of the response when the applied force fa(t) increases from 0 to 10 N at t= 1 sec. The parameter values are: M1 = M2 =10 kg, B1 = B2= B3= 20 N.s/m, and K1 = K2 = K3 =10 N/m. Plot both x1 and x2 on the same axes. PROCEDURE: • Use Step block to provide the input fa(t). • In the Step block, set the initial and final values and the time at which the step occurs. • Select the duration of the simulation to be 10 seconds from the Simulation > Parameters entry on the toolbar • Write an m-file setting the system parameters, • Include system execution commands in the m-file • Also include plotting statements in the same m-file. • Run the simulation in MATLAB command window and show your work.
• Compare the results of Exercise I and 2 and comment on its physical interpretation.
• Results of the programs • Comments on the results • Comparison of results with analytical values where possible SE 207 Lab # 12 Simulation of a Second Order System IntroductionMany practical system are well approximated by second order models. A general second order model; is given by 
where In this experiment we will study the effect of different parameters on the response of the system.
What are the values of the damping ratio that give under damped response? What are the values of the damping ratio that give damped response? .
Let the input be a unit step and the damping ratio  = 0.5. Simulate and plot the response of the system for the following natural frequencies {0.5, 1 , 2 , 10}.
What do you observe? Part 3:Simulate the system having = 0,  =0.5 for the following inputs,
Plot the responses and comment on them. SE 207-Lab # 13
Variations in the thickness of the finished product can be attributed to many factors> These include variations in the material and the hydraulic process that pushes the rollers. A feedback control system is used to control the thickness of the finished product by controlling the hydraulic pressure that push the rollers. A simplified sketch of the rolling process with the control system is shown in figure 1. A block diagram of the system is shown in figure 2. The process and controller parameters are  ,
Figure 1: Simplified Sketch of Rolling Control System 
Figure 1: Block diagram of Rolling Control System Transfer functions: The transfer function of a linear time-invariant system is defined as the ratio of Laplace transform of the output to the Laplace transform of the input. It is widely used to model of linear systems. SIMULINK offers a simple way of representing transfer functions. Procedure: Case 1: Simulate the system with the given controller assuming no disturbance plot the response y(t) and the error signal e(t) when yr is a step of size 0.1 m
plot y(t) Case 4: Repeat Case 3 with M=900 and plot(y) SE207: Modeling and Simulation Lab # 13
Simulation of the Response of a Car to Road Depressions. Objective: To simulate a simplified model of a car and investigates the effect of changes in the mass and spring constants as well as the shape of the road depression. Introduction: Models of the car suspension systems are important in the design of a car that satisfy the customer’s expectation of a nice ride. Practical car models are used to determine the forces on the deriver and his motion as the car moves on road’s surface. Cars (at least small cars) have four wheels and each one moves independently as the car moves. Both translational and rotational motion is present. A typical simplified lumped model involves seven degrees of freedom. Translational motion of the four wheels and rotational motion in three axes. In this lab experiment we use further simplification and consider the motion of the front right side of the car and ignore other sides and ignore rotational motion as well. 
Figure 1: Simplified model of the car (The right front wheel) Procedure: STEP 1: Verify that the mathematical model of the car is given by 
where
Typical output of interest are the displacements, velocities and accelerations of the masses. The model parameters: 
STEP 2: Draw simulation diagram STEP 3: Do the following cases. Case 1: Simulate the system with the parameters above to predict the vertical velocity and accelerations of the driver seat  as car moves over a step.
u(t) = is a step of magnitude 0.05 meters. Select a suitable time horizon. Case 2: Simulate the system with the parameters above to predict the vertical velocity and accelerations of the driver seat when the road surface is modeled by  .
Case 3: How does the behavior change when more passengers are added ? Repeat Case 2 with Mb = 350kg and 400 kg. Case 4: What happen if the tire’s spring constant is reduced to 160 000N/m? Repeat Case 2 with the new spring constant. Case 5: What happen if the shock absorber damping coefficient is reduced to 90% , and 80% of its original value? Repeat Case 2 with the new damping coefficient. What to submit?:
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