1. Draw a free-body diagram showing all the forces acting on the mass m shown in Figure 2.
2. From the earlier description, diagrams and the laws of Physics, show that the motion of the system in Figure 2 can be described by the LCCDE (linear constant-coefficient differential equation) below:

3. Using the Laplace transform of the equation above, find an expression for  , the system transfer function.
The mass-spring-damper system is a damped second order system. It is common to express the homogenous second order DE for such a damped system as

4. From equations (1) and (2), determine expressions for  (the damping ratio) and  (the natural frequency) in terms of the parameters m, k and C 5. Determine the characteristic equation and eigenvalues (characteristic values) for this system based on equation (2) above (in terms of  and  ).
6. From the answer to part 5, determine the full mathematical expression (in terms of  and  ) for the natural response of the system for the following cases:

a.  = 0
b. 0 <  < 1
c.  = 1
d.  > 1

Consider a suspension system with the following parameters:  = 380 kg = 15,000 N/m
7. Determine  (in rad/s) for this suspension system and the corresponding value for  (in Hz).
8. Calculate the required value of  in order to achieve  = 1

9. Plot the impulse response and step response of the system (for 2 seconds duration and time ‘step size’ of 1 millisecond) using the impulse and step functions. Include all plots (properly labelled) in your submission.
10. Determine the frequency response from 0 to 200 rad/s using the freqs command. Plot the magnitude and phase response over this frequency range. Hint: Use frequency ‘step size’ of 0.1 rad/s.
Hint 1: You can plot all 4 graphs in one go using a 2 x 2 matrix of plots using subplot(22n), where n determines which of the 4 subplots gets used.
Hint 2: In order to clearly see variations over a range of frequencies, it is best to use a log scale for the frequency and magnitude (phase would still be displayed using linear scale). The functions loglog (for magnitude) and semilogx (for phase) can be used instead of plot.
11. Determine the magnitude response at . Determine the frequency of the -3dB point (magnitude = 1⁄√2 of passband). Hint: Use the ‘data cursor’ tool on the plot of the magnitude response. It shows the x and y values of the plot as you move along the curve.
12. Discuss the response of the system. Why do the impulse and step responses have that particular shape? How well will this system fulfil its purpose of a vehicle suspension?

DETAILED ASSIGNMENT

20201008053439assignment_signals_and_systems___matlab