Reference no: EM133045088

EG7010 Engineering Design Case Study - University of Leicester

Simulink / Simscape exercises

Question 1 Spring and dampers: series and parallel Construction

Create a mass-spring-damper system in Simscape. Start with a 1kg weight attached to a fixed reference point. Between these two elements and in series with them, should be a subsystem consisting of a spring of spring constant
k = 100 N/m in parallel with a damper of coefficient c = 1N/(m/s).
A translational motion sensor should track the position of the mass, using a scope and an ideal force source should provide an additional force of 5N, switched on using a Simulink step block. Like all Simscape models, it should have a Solver Configuration block.

Experiments

a. Run the model and from the graph produced, verify in your log-book that the expected displacement is seen. Use Tools/Measurements to place horizontal cursor lines on this graph at the level of maximum displacement and at the level which is half way between this level and the final resting level. Choose File/Print from the Scope window to save this as a Matlab figure or a pdf for future reference. Show this figure in your report.

b. Copy and paste (Ctrl-C and Ctrl-V) the spring and place the new spring in series with the old one, so they are still in parallel with the damper, as shown in figure 1.1.
Run the simulation again and comment on how the graph has changed qualitatively. In particular, notice the final resting level and the period of oscillation. There is no need to show the figure in your report.

c. Now erase the extra spring and instead, put it in parallel with the spring-damper subsystem, as shown in Figure 1.2.
Run the simulation again and comment on how the graph has changed qualitatively.

Figure 1.1

Figure 1.2

d. Now erase the extra spring and do the same two exercises (b and c) with the damper. For these, you should comment on the following question:

‘Compared to exercise 1a, are there more or fewer oscillations between the times when the system reaches its maximum displacement (horizontal cursor 1) and when the amplitude of the oscillation has died away to half of that value (horizontal cursor 2)?' Comment on your result.

e. Now copy and paste the entire spring-damper subsystem and put the new spring-damper subsystem in series with the original one. Let's call this output 1.

Now, going back to the original system with one spring and one damper in parallel, change the values of k and c until you get the same graphical output as output 1.

State what values of k and c you used.

Question 2 The logistic equation Background
The logistic equation is one of the simplest examples of feedback. It is represented algebraically as:

x_n+1 = rx_n(1 - x_n)

,where r is a coefficient that we will set and {x_n} represents a series of numbers that shrink or grow, depending on r. It has been used to model population dynamics in a finite region. For example, the number of fish in a pond in the next generation (x_n+1) depends on how many fish are in the current generation (x_n), but also on the finite food supply, represented by the factor (1-x_n). The point is that it is such a simple equation that it may be encountered in many systems from diverse fields of science.

The coefficient, r should be applied using a Simulink gain block with the variable ‘r' input into its dialogue box as the gain. The numerical value of the gain can then be altered quickly by typing, for example: ‘r=2' into the Matlab command window.

Construction
Arrange the blocks shown as in the figure. The step and edge detector together act as a way to input a value of x₁ = 1/2 into the system. The delay block ensures that the equation is evaluated in n steps, rather than all at once. The scope will show the value of x as it evolves.

Experiments
a. Set r equal to 0.5 and run the model and make sure the Stop Time is large enough so that convergence occurs. What is the eventual value?
b. Now set r to 0.999. Does it converge to the same value or a different one? Does the convergence occur more quickly or more slowly?
c. Now try several different values of r between 1 and 2. How does the eventual value depend on r?
d. Now try r=2.5, 2.8, 2.9, 2.99 and 2.999. What do you notice about the convergence?
e. Now try r=3 and change the Stop Time to view the eventual value. What do you notice?
f. Now try various different values of r between 3 and 3.4. Describe what happens to the eventual values.
g. What happens at r=3.56, 3.58, 3.6 and r>4?

Question 3 The hardening spring
This exercise models the effect of a small deviation from Hooke's law in a spring. For small displacements, springs have a restoring force which is proportional to displacement, but If a spring is stretched beyond its elastic limit, the restoring force will increase.

Construction
First make a Simulink model of a 1 kg weight suspended by a linear spring with spring constant, k = 900 N/m. It should also have a small frictional force equal to 1 Ns/m. Now drag the mouse over the whole assembly and copy it. Keep the copy in the same model, so that both outputs can be seen on one graph.

Edit the copied spring system so that its restoring force is equal to

F = 900x + 1700x³

, where x is the displacement from its rest length. Set the initial displacement 50 cm. Add a scope to show displacement as a function of time and alter it so that it can receive two inputs. Set the timer to five seconds and run the model. Show the scope's output with an appropriate legend.

Questions

a. Does the non-linear spring have a higher or lower natural frequency than the linear spring?

b. By counting the number of oscillations over the first five seconds for each spring, estimate by what percentage the frequency has changed.

c. Change the initial displacement of each spring. Does the frequency of each of them depend on initial displacement?

d. When the initial displacement is 2m, which of the springs is more quickly dampened. Can you explain this?

Question 4 Signal multiplexing Background
In communications, data or voice signals from many users are gathered together and put on the same link in a process known as multiplexing. For example, all the mobile phone conversations between Leicester and Birmingham can be sent on one high-speed link, so there is no need to have hundreds of wires or hundreds of radio transmissions between the two cities. Frequency-division multiplexing and Time-division multiplexing are two different methods of doing this. In this exercise, we'll multiplex two binary signals together using the fact that sine and cosine functions are orthogonal -i.e. they can each carry information without interfering with one another.

Construction
You will need the following Simulink blocks:
• String Constant
• String to ASCII
• Deserializer (a serial to parallel interface)
• Integer to bit converter
• BPSK Modulator (Baseband) (This takes the ‘0's and ‘1's and represents them as different phases of the carrier wave)
• BPSK demodulator (Baseband) (This converts them back)
• Bit to integer converter
• ASCII to string
• Display
Arrange the blocks in the order of the list above and connect them. Type any character string up to a maximum of 10 characters, enclosed in quotes into the dialog box of the String Constant block.
The String to ASCII block should have its ‘Output vector size' set to ‘10'.
The Integer to bit converter and its counterpart should be set to 7 bits per digit, as the ASCII code contains 128 symbols. The other blocks can be left in their original state.

Experiments

a. Run the model and show the its final state in your report. The text that you typed into the String Constant block should appear in the Display block at the other end of the model. There is no need to show each of these steps in the log-book, as long as you submit a working .slx file.

b. The message has been turned into a string of the symbols ‘+1' and ‘-1'. Add another Display block to the wire connecting the Modulator and Demodulator. Run the model again. The data stream should appear as these symbols.

c. Now copy the whole assembly and paste it in the same model, so you have two copies of it.
Alter the Modulator and Demodulator of one copy so that their phase offset is ‘pi/2'. Run the model again and you'll see that the symbols used to send this are now ±i.

d. Now delete the connection between Modulator and Demodulator in one copy. Add the output of the Modulator to the line connecting the other Modulator /Demodulator pair, so that both signals are going down one line (see Figure 4.1).
Type different text strings into the two String

Constant blocks and run the model. Each of the two separate texts should appear in the respective Display block.

e. Figure 4.1 Figure 4.1

f. Now connect a Constellation Diagram block to the line connecting modulators and demodulators. Run the model again and it should show four yellow points. This shows the 2x2 = 4 different combinations of symbol that are being sent down the line on an Argand diagram.

g. Now add a Phase Noise block and in its dialogue box set the frequency offset to 100Hz. Alter the Phase noise level and show its effect on the Constellation Diagram and on the integrity of the signal sent.

h. What is the highest value of phase noise that the signal can tolerate and remain intact?

i. Explain what effect phase noise has on the signal.

Attachment:- Simulink and Simscape Exercises.rar