Use the files below for the question

Use the files below for the question. The initial equations for each question must be clearly specified if using software, and the final answers must be clearly stated. Include the input and output for any MATLAB code if used. Please provide a brief explanation for each part. Generous tip will be given for correct answer.

You are tasked with designing a control system for a voltage-controlled pin ball launcher. The launcher utilizes a rubber cylinder that ensures static friction when the ball comes in contact. The walls are frictionless. The ball absorbs energy while in contact with the rubber and is propelled with velocity, v. The design is shown below. Pinball Exits with velocity, v D2 D rubber Top View Rubber W Shaft Rotating shaft and rubber cylinder rubber Motor Pinball Enters with zero velocity Front View JE Mrubber D, Mshas Dz? 12 Mrubber = 0.0006kg, Mshaft = 0.003kg 8 8 Mball = 0.08kg, D2 = 0.005m, D2 = 0.01m, k, = 12 Nom.s rad ku = 10 N·ms ke = 15 Nm rad A Where ki is due to the internal friction loss of the motor, ku is due to the losses from the rubber cylinder, Mis the mass of the marble, ku is the constant that relates the torque output to the input current. The equivalent circuit for the motor is presented below. R: + o w 000 R, «R, R = 101, L = 10mH, Vems = kew, k, = 0.1 Vis rad Ωs) a) (7 points) Derive a function that relates the voltage input to motor power output. (Hint: T(S) find the transfer function for and then multiply them together to obtain the Vin(s) Vin (S) power) b) (7 points) Derive a transfer function that relate the voltage input to the output angle of the motor shaft. c) (7 points) Assuming the marble is in contact with the rubber cylinder for 1/3 radians. Estimate how long the marble is in contact with the rubber cylinder if the step input is V(s)=Vin/s. (Hint: check the coefficients in the denominator - some of them may be small enough to ignore when compared to the others) d) (4 points) Using the results from (a) and (c) estimate the velocity of the pin ball. You can assume all energy is converted into translational motion and Vin=5V. You can assume the motor has been operating at steady state when the ball is launched.