ME250 W23 Final Exam

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University of Michigan *

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250

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Mechanical Engineering

Date

Feb 20, 2024

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pdf

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16

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ME250 W23 FINAL EXAM This is an open-book exam. You may NOT consult any resources outside of your ME250 lecture notes, your HW, and any other resources posted on the ME250 W23 Canvas site. You should have paper (or a tablet computer) for writing, pencil, calculator, and a way to upload your solutions. The exam time slot includes time for upload. Make sure to start uploading your exam to Gradescope with at least 5 minutes left. The Gradescope exam assignment will let you upload your exam up to 1 hour late, so do NOT email your late submission. Exams uploaded late WILL face stiff penalties, including potential non-acceptance of the submission (0% score), unless proof you completed the work on time is provided in extenuating technical circumstances. You can provide proof of extenuating circumstances via an email to me250W23instructors@umich.edu in a real time emergency. The Honor Code is on the last page of this exam.
Question 1 (16 points) After watching Clarkson's Farm, Sam is motivated to electrify the agricultural industry. He decides to make an electric quad-track tractor. He starts with a rear-track-drive concept that uses a single DC electric motor and a gearbox. The motor is powered by two large batteries connected in series for an operating voltage of 1600V. The DC electric motor has the following torque/speed and power/efficiency curves: A.) (5 points) Sam selects a gearbox with overall gear ratio, ratio, , and overall efficiency, ? ??????𝑥 = 32 . His tractor has mass and a driven roller diameter η ??????𝑥 = 0. 85 ? ??????? = 30, 000𝑘? . The tractor is pulling a trailer with mass . Assume that ? ??𝑖𝑣?? ?????? = 0. 8? ? ???𝑖??? = 90, 000𝑘? gravity = 9.81 m/s^2 and there is no efficiency loss between the roller wheels, treads, and ground. If his tractor is pulling the trailer up a hill with a slope of 10 degrees, what is his tractor's maximum speed [m/s]?
B.) (4 points) Sam wants to size the gearbox such that his tractor's motor operates at peak efficiency when pulling a 90,000kg trailer up a 10 degree incline. The gearbox's overall efficiency, . η ??????𝑥 = 0. 85 His tractor has mass and a driven roller diameter . The ? ??????? = 30, 000𝑘? ? ??𝑖𝑣?? ?????? = 0. 8? tractor is pulling a trailer with mass . Assume that gravity = 9.81 m/s^2 and ? ???𝑖??? = 90, 000𝑘? there is no efficiency loss between the roller wheels, treads, and ground. What speed should the motor be running at to achieve peak efficiency? What gear ratio should Sam choose?
C.) (4 points) Sam decides to crank the power up by adding an electric motor to drive the front track. The front and rear motors are identical and operate on the following torque/speed and power/efficiency curves at 1600V: Sam chooses to use the following motor voltages and gearboxes: Rear Motor (unchanged): 2 batteries in series producing 1600V operating voltage Gearbox: Gear Ratio , and overall efficiency . ? ??????𝑥 = 32 η ??????𝑥 = 0. 85 Driven roller diameter D = 0.8m Front Motor ( changes are italicized ): 1 battery producing 800V operating voltage Gearbox: Gear ratio 16, and overall efficiency 0.95. ? ??????𝑥 = η ??????𝑥 = Driven roller diameter D = 0.8m What is Sam's tractor's maximum pulling force on a flat surface [N]?
D.) (3 points) If Sam uses the configuration above, at what speed in rpm, , is the rear motor most ? ???? efficient? At what speed in rpm, , is the front motor most efficient? At what speed in m/s, ? ????? , is the tractor most efficient? 𝑣 ???????
Ques±on 2 (18 Points) To minimize cost, Sam ul±mately decides that his electrified tractor will be powered by a single rear motor with the following specifica±ons: Rear Stall torque, τ ? = 100𝑁? No load speed, ω 𝑁𝐿 = 1800𝑟𝑝? Minimum gear ra±o, ? ??????𝑥 = 32 Overall efficiency, η ??????𝑥 = 0. 85 He then asks Chris, a highly experienced transmission engineer, to develop the gearboxes. A.) (4 Points) Because of the high motor torque and low availability of space, Chris’s first thought is to stack two stages of planetary gearboxes together for the rear wheels to get the desired gear ra±o. Complete the table on the next page with the rota±on direc±on of each component. Assume the first stage arm (A1) is rigidly a²ached (i.e. rotates in the same direc±on) to the second stage sun gear (S2); the first stage ring gear (R1) is fixed; the second stage arm (A2) is fixed; the first stage sun gear (S1) is the motor input and turns clockwise (CW); and the second stage ring gear (R2) is the gearbox output.
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