Homework 2 - F23

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School

University of Michigan *

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Course

250

Subject

Mechanical Engineering

Date

Dec 6, 2023

Type

pdf

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Uploaded by CaptainPartridgePerson774

ME250 | F23 | University of Michigan HW2: Lectures 5, 6, 7, and 8 (63 pts) Due Friday, October 6th by 11:59 pm on Canvas This is an individual assignment, and your solution must be entirely prepared by you. Homework assignments must be completed on your own (unless they are team assignments), however you are encouraged to discuss the problems with your classmates. Upload a PDF of your solution to the Assignments tab on Canvas. Problem 1: Lecture 5 Catapult Data Analysis (15 pts) Using the data you collected from the in-person Lecture 5 activity, answer the following questions. Note that your team should have collected 40 data points. If not, pool the data you collected during the in-person activity across your section. Each student must answer the following questions individually and submit their own assignment via Canvas. A. Use the equation provided to determine the best angle of release for the catapult. d thr = ? 2 ?𝑖?(2θ ??? ) ? B. Randomly allocate 10 of the 40 data points to the validation set. Use the remaining 30 data points as training data. Create a chart of distance thrown (y-axis) vs the release angle (x-axis) which includes the training data and include linear and quadratic lines of best fit to the training data (Note: it is OK to include 2 graphs, one showing linear best fit and one showing quadratic best fit). C. Please fill out the following table using your validation set and predictions based on your empirical models. Experimental validation data set Theoretical predictions of distance thrown at this release angle based on the linear and quadratic models derived in part (B) Release angle Distance thrown Linear model prediction Quadratic model prediction Data point 1 Data point 2 Data point 3 Data point 4 Data point 5 Data point 6 Data point 7 Data point 8 Data point 9

Data point 10 D. Subsequently, calculate the RMSE validation error for the linear and quadratic models. Please use google sheets instead of excel. Empirical model [f(x)] formed using the training data set (30 data points) RMSE Validation Error Linear model f(x) Quadratic model f(x) E. Comparing your first principles model to your empirical models, what would you recommend as the best angle of release for the catapult and why? (You are not limited to the three release angles tested in class) Problem 2: First Principles Analysis and Empirical Testing (15 pts) Pendulums have been widely used as the most precise timekeepers throughout the 17 th and 18 th century due to their key property of isochronism. For small deflections from the vertical position, F=ma can be used to approximate the pendulum’s period of oscillation as: 𝑇 = 2π ? ? , Where L is the length of the pendulum, and g is the acceleration due to gravity. A. Use a first principle model to calculate the required length (m) of a pendulum that keeps its time period equal to 1.5 sec. B. Using the data below, estimate a quadratic empirical relation between string length and the time period. Provide a plot of this empirical relationship and the data below. Write the obtained equation. C. Using the empirical relation obtained in part B, calculate the required length of the pendulum to keep its time period equal to 1.5 sec. Give your answer to the closest 0.001 increment. Time period (sec) Length (m) Time period (sec) Length (m) 0.31 0.02 1.15 0.34 0.45 0.05 1.38 0.46 0.74 0.12 1.58 0.59 0.92 0.23 1.95 0.97

Problem 3: Gears and Transmissions (16 pts) 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.) Sam selects a gearbox with overall gear ratio, ratio, , and overall efficiency, ? ??????? = 32 . His tractor has mass and a driven roller diameter η ??????? = 0. 85 ? ??????? = 30, 000?? . The tractor is pulling a trailer with mass . Assume that ? ??𝑖??? ?????? = 0. 8? ? ???𝑖??? = 90, 000?? 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]?

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