Linear Systems And Control Discussion
1. Determine the transfer function of the system where force F is the input and displacement x is the output variable.
2. Draw the block diagram of the system indicating clearly all variables.
3. If the input force F is applied in a form of an impulse of magnitude 𝐹0 derive the time-domain expression for output x. Plot x(t). Then, as a measure of the slamming effect, calculate the value of kinetic energy at the time when the door closes.
4. A more realistic form of the input force would be a couple (a positive step function, followed by a negative step of the same magnitude) as shown by Figure 3. Derive x(t), plot it and calculate the value of kinetic energy at the time when the door closes.
5. The first design requirement for the mechanism is that kinetic energy is equal to zero at the point of impact. If the values of M and k are the same as originally defined, determine the range of values of C for which this requirement is satisfied.
6. The second design requirement is that the door closes as quickly as possible. Find the value of C* (M and k having their original values) for which both requirements are satisfied. What are the values of time constants in this case?
7. Assume you are a newly employed engineer and your (non-engineer) manager wants all doors in the building to have the hydraulic part replaced with an electronic device. You find out that the device is in fact a control loop that consists of a displacement sensor in the feedback branch and a controller in the main branch (Figure 4). The transfer function of the sensor is H(s)=1, while the transfer function of the controller is
A is an adjustable parameter that can take values from 0 to very high. The stiffness of the spring is the same as previously defined while the change in mass of the door is negligible.
a) Plot the root locus of the new system (make sure you define K for this). Show clearly the starting points (if any), end points (if any), centroids and asymptotes (if any), break-away and break-in points (if any) as well the intersections with the imaginary axis (if any). Provide the numerical values of s and K in all these cases.
b) What values of K (if any) satisfy both requirements defined in Parts 5 and 6?
c) By comparing the values of the time constants for the open loop (Part 6) and closed loop (Part 7) systems, which system has more desirable behaviour? Explain in up to 100 words.