ME250 W23 Final Exam

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University of Michigan *

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250

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Mechanical Engineering

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Feb 20, 2024

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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?

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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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Component Direc±on of Rota±on (CW, CCW, or FIXED) S1 (IN) CW A1/S2 A2 FIXED R1 FIXED R2 (OUT) P1 P2 P3 P4 P5 P6 B.) (5 Points) Chris instead decides to use the following transmission from what he learned in ME250, assuming that it will meet the overall eﬃciency requirement: (Not drawn to scale) The input (motor) sha³ A is connected to gear 1, and the output (roller) sha³ E is connected to gear 7. All gears are radially constrained via bronze bushings. Gear 1 has a diametral pitch of 8, pressure angle of 20°, and 72 teeth. Gear 2 has a diameter of 5 inches. Pulley 3 has a circular pitch of 0.5 and diameter of 9.549 inches. Gear 7 has a circular pitch of 0.2618, pressure angle of 14.5°, and diameter of 3 inches. i.) What is the pressure angle of gear 2? ii.) What is the diametral pitch of gear 5?

iii.) The ra±o of the pitch diameter between gear 5 and gear 7 is 3:1. What diameter would pulley 4 need to be, to meet Sam’s minimum required gear ra±o ? ? ??????𝑥 C.) (6 Points) Use your answer(s) from (B) to help Chris determine the following pertaining to the gearbox. Assume each gear or pulley interface equally contributes to the gear train eﬃciency, and assume the gearbox is not backdrivable. NOTE: 1 inch = 2.54 cm i.) When the motor is running at its no-load speed, what is the maximum linear velocity of the belt in inches/second? ii.) When the motor is running at its stall torque, what is the maximum tension the belt must withstand? D.) (3 Points) Chris thinks there must be ways to increase the eﬃciency of the gearbox shown in (B). Describe three methods to achieve this.

Question 3 : Static Analysis, Free Body Diagrams (18 Pts) Uh oh! It looks like Cam had a rough day on his farm, and got his tractor stuck in a ditch! Cam needs your help to figure out if he can get his tractor out of the ditch without it tipping over, so he asks you to run some calculations on the situation. Cam has a team of Oxen to help pull the tractor out, and some rope which will be tied to the center of the uphill-facing wheel. Cam provides you with the following table of values: Parameter Value L wheelbase 3.0 [m] D wheel 2.0 [m] θ 15.0° μ 0.15 F oxen 20,000 [N] M tractor 3,000 [kg]

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Assume the center of gravity acts at the center of the tractor and in line with the axle, there is friction at the surface of the incline, and the oxen pull with a force that is parallel to the horizontal ground. If you make any other assumptions, please state them. A. Label the following box with forces to create a free body diagram for the slip/tip of the tractor. You may treat the entire tractor as one rigid body (i.e., a box on an inclined plane). The normal force N and the point of rotation ‘o’ have been provided for you. (4 pts)

B. Write the SYMBOLIC equilibrium equations for the free body diagram, but do NOT solve. Report your answers in the boxes provided. (6 pts) C. Using the values provided in the table, and g = 10 m/s 2 , determine whether the tractor will tip or slip. Defend your answer with justification. (6 pts)

Question 4 (20 points) After seeing the tractor changes implemented by Chris and Sam and the challenges that Cam faces on his farm, Dom decides that it would be beneficial to make a production line to mass produce pink tractors. The first station in the production line is body production, where the body of the tractor is formed (Production Rate = 65 parts/hr) . After body production, the line splits into two types of axle manufacturing: - Light-duty axles are manufactured on the light-duty axle station (Production Rate = 75 parts/hr) - Heavy-duty axles are manufactured on the heavy-duty axle station station (Production Rate = 80 parts/hr) . The two lines reunite at the wheel assembly station (Production Rate = 100 parts/hr) where the tractor wheels are put together. The line then splits again for two types of transmission manufacturing. - The standard transmissions are assembled at the standard transmission station (Production Rate = 500 parts/hr). - Special NASCAR transmissions are assembled at the NASCAR transmission station (Production Rate = 150 parts/hr). These lines are brought back together at the engine assembly station (Production Rate = UNKNOWN parts/hr). Once the engine is assembled, everything is put together at the final assembly station (Production Rate = 50 parts/hr). A). (4 Points) Draw a functional diagram of the flow line for tractor production. Identify known and unknown production rates for each block.

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B.) (2 Points) What is the smallest production rate for the engine assembly station so that it isn’t a bottleneck of the tractor production system? C.) (4pts) All of the tractor axles got broken when Cam’s tractor got stuck in the ditch, so Dom decides that she has to make custom axles for the tractor during the axle manufacturing station of the production line. She decides to do this by turning 3” of the length down the ends of a 3” round piece of stock to fit into a 2.75” ID wheel bearing. However, the material supplier gave her four different material options: Cast Iron, 304 Stainless Steel, 4140 Steel, and 1215 Carbon Steel . She wants to be able to make the axles as fast as possible. Help her select which material to use for the tractor axles. (See the question on the next page.)

C.) Continued Given the original stock diameter , final stock diameter , find the maximum spindle 𝐷 ? = 3" 𝐷 ? = 2. 875" speed [rev/min] for turning stock down on the lathe when using Carbide tooling for each of the three given materials. D.) (4pts) Given a feed rate [in/rev], find the maximum material removal rate, MRR, for each of the ? = 0. 003 three given materials [in^3/min]. E.) (4 pts) Find the machining time [min] for each of the three given materials. 𝑇 ? F.) (2 pts) Based on machining time, which material should Dom choose? List one other factor that she could take into account when she’s selecting the axle material.

Question 5: (13 pts) Prof. Austin-Breneman wants to upgrade his tractor to film a shot-for-shot remake of Fast and Furious V for a class video. The following pictures are a series of accessories he is adding to the tractor which were made using one of the manufacturing processes discussed in this course. Indicate which primary shaping process(es) were used to make the products and justify your answer. A. Lightweight plastic sunshade canopy available in multiple colors(3 feet x 4 ft) (2 pts) B. Metal rear brake drum (2 pts) C. Metal hood (2 pts)

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D. Prof. Austin-Breneman is looking to upgrade the seat for a more comfortable ride. He finds two options, the red metal one on the left, and the black plastic one on the right. What are the manufacturing processes used to create the two seats? Describe one manufacturing trade-off between the two different approaches? (4 pts) E. Prof. Austin-Breneman would like to create some custom tractor wheel rims like the red part in the photo shown below. He is considering sand casting and die casting as possible process choices. Suggest one process and give two reasons for your selection. (3 pts) Honor Code: I have neither given nor received aid on this examination, nor have I concealed any violation of the honor code. Signature:___________________________