ME240 F23 HW11

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Jan 9, 2024

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ME 240, Fall 2023 Koller & Shorter ME 240: Introduction to Dynamics & Vibrations Homework #11 – Due Friday, November 17 th by 11:59 PM on Canvas Module 5 – Particle Kinetics Module 5 Learning Objectives (LOs) Please review the learning objectives for module 5 reiterated below. Each of the following problems maps back to these specific learning objectives and are indicated accordingly in the column to the right. Got it! Do you feel like you’ve learned this concept? Problems Learning objectives covered in each problem 1 2 3 4 5 6 7 * LO1: Develop an accurate free body diagram ☐ ☑ ☑ ☑ ☑ ☑ ☑ ☑ LO2: Select appropriate coordinate systems and be able to apply them correctly ☐ ☑ ☑ ☑ ☑ ☑ ☑ ☑ LO3: Define forces, moments, and kinematic variables using a vector notation ☐ ☑ ☑ ☑ ☑ ☑ ☑ ☑ LO4: Apply Newton’s & Euler’s laws ☐ ☑ ☑ ☑ ☑ ☑ ☑ ☑ LO5: Solve systems of linear equations ☐ ☑ ☑ ☑ ☑ ☑ ☐ ☑ LO6: Analyze two state problems using energy and momentum methods ☐ ☐ ☐ ☐ ☐ ☐ ☐ ☐ LO7: Construct equations of motion ☐ ☐ ☐ ☐ ☐ ☐ ☑ ☐ LO8: Identify the role of differential equations in rigid body kinetics ☐ ☐ ☐ ☐ ☐ ☐ ☑ ☐ LO9: Solve a differential equation using numerical methods and analyze the results (i.e. using MATLAB) ☐ ☐ ☐ ☐ ☐ ☐ ☑ ☐ *Indicates optional problems All problems have been adopted from “Engineering Mechanics: Dynamics (7 th Edition) by Meriam & Kraige unless noted otherwise. Section 1: Graded Problems The following problems will be graded in your homework submission. Problem 1 (6/8) This problem maps back to learning objectives 1-5.

ME 240, Fall 2023 Koller & Shorter Problem 2 (6/56) This problem maps back to learning objectives 1-5. Problem 3 (6/73) This problem maps back to learning objectives 1-5.

ME 240, Fall 2023 Koller & Shorter Problem 4 (6/78) This problem maps back to learning objectives 1-5. Hint: To solve this problem you are going to need to determine if it is slipping or not. To do so, you can first assume it is not slipping and solve for the force of friction. Once you have that magnitude, what criteria will tell you if the cylinder is slipping or not? Problem 5 (6/92) This problem maps back to learning objectives 1-5.

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ME 240, Fall 2023 Koller & Shorter Problem 6 (MATLAB) This problem maps back to learning objectives 1-4 & 7-9. We are going to revisit the simple pendulum example that was considered in Homework 6. However, in Homework 6 we modeled the pendulum as a point mass, here we are going to consider it as a rigid body. What you are tasked with doing is modeling this pendulum and simulating it in MATLAB using ode45 . The real system and a simple model representation of it are shown below:

ME 240, Fall 2023 Koller & Shorter The pendulum arm has a length of 17.8 cm from pivot to the center of the bulb that was tracked and a mass of 37.9 grams. You can assume that the center of mass is located halfway down the length of the arm. a) Derive an equation of motion of the pendulum by assuming a rigid body. Note that the pendulum has somewhat of an odd shape so modeling it as a slender rod may not be very accurate. Rather, we have experimentally approximated its radius of gyration about the pivot point for you and found it to be 0.119 meters. That is to say that 𝐼 𝑂 = 𝑚𝑘 2 where point 𝑂 is the pivot point and 𝑘 is the provided radius of gyration (our textbook’s description of the radius of gyration is copied to the end of this problem). Be sure to include a free body diagram in your solution. b) Simulate your equation of motion for three oscillation (roughly 2.8 seconds) and plot the simulated angle versus time in the same plot as the provided tracked data. There is a provided MATLAB live script with gaps in the code for you to fill in to aid this process. c) Calculate the horizontal and vertical components of the acceleration of the center of mass now that you have simulated the equation of motion. There is a provided MATLAB live script with gaps in the code for you to fill in to aid this process. d) Calculate the horizontal and vertical reaction forces at the pin. Plot these values versus time. There is a provided MATLAB live script with gaps in the code for you to fill in to aid this process. Your final reaction force plot should look like:

ME 240, Fall 2023 Koller & Shorter Section 2: Optional Problems These problems are ungraded and provide additional practice. We will post solutions to these problems with the homework solutions. Problem 7 (6/86) This problem maps back to learning objectives 1-5.

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