You can do this assignment on your own or in groups, as long as you hand in your own solutions and it is clear that you understand your solutions. Formulate your solutions step- by step.Marketing Research and Data Analysis

Printed circuit board (PCB) drilling is a procedure that involves crafting cavities, slots, or holes on an electronic circuit board before you mount components. You need an automated machine or PCB drill bit to execute the process. The PCB (green part) is carried by a table that is able to move in z-direction. The driller (black) is connected through a slider beam to the actuator (blue), actuating in the x-direction (generating force without adding stiffness to the motion direction). In this assignment, only x-direction actuation is considered, see Figure 1(b). The mass of the actuator and the driller are 𝑚₁ = 2 kg and 𝑚₂ = 0.5 kg, respectively.

1. Model the two-body system (actuator and driller) as connected rigidly through a beam of negligible mass (x1 =x2 = x). Design a proportional feedback control action to steer the driller’s position towards a desired position 𝑥_{𝑟𝑒𝑟𝑟}, in a known environment.

• In the working environment, the carrying table might vibrate in x-direction up to 4 Hz with the amplitude of 1mm which is regarded as a disturbance input d. The maximum allowed error of the hole’s position on the PCB board is 0.02 mm. Sketch a block diagram of the system and proportional controller, identifying how the two input signals r and d enter the feedback loop.

• Compute the transfer function from input d to output x. Identify what kind of filtering action is the transfer function performing (viz. low-pass, high-pass, etc.). What is the (undamped) natural frequency of the closed-loop transfer function computed which results in an attenuation of disturbance d compatible with the requirement specified in question B?

• Firstly, design the gain of the P-controller to realise the natural frequency found in Question C). Derive the closed loop transfer function X(𝑠)/X_{𝑟𝑒𝑟𝑟}(𝑠).
• Identify the equivalent mechanical system of the closed loop system determined in question C and give a mechanical interpretation of the P-controller.
• How can the controller be modified to include a damper? What does it happen if we consider the effect of the gravity on the bodies? How would you design the controller?

Now consider the beam connecting the two masses is not rigid. Investigate how the system is affected by this lack of rigidity. The actuator applies the force on 𝑚₁, a sensor measures the position of 𝑚₂. The dimension of the beam is 𝐿_𝑥 = 𝐿_𝑧 = 40mm and 𝐿_𝑦 = 200mm. The beam stiffness value is given by

1. Determine the model of the system including the effect of the flexible beam. Draw the Bode diagrams from the input to 𝑥₂.  Determine the resonance frequency if any.
2. Assume the presence of some damping in the beam connecting the two masses. Analyse how the frequency profile changes as the damping value changes. Study how the gain of a P-controller changes as the damping value changes to guarantee stability of the closed loop system.
3. Design a PD-controller for this system considering two different damping coefficients, to obtain the desired unity-gain cross-over frequency in point A. Analyse the obtained results. How would you change the controller to obtain the desired behaviour?
4. Analyse the performance of the achieved closed-loop system. Discuss the results of the analysis.