Homework 3

pdf

School

Pennsylvania State University *

*We aren’t endorsed by this school

Course

370

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

pdf

Pages

2

Uploaded by CountInternet11042

Report
ME 370 Homework 3 Problem 3.1 Watch the following video about debugging Matlab code: https://youtu.be/PdNY9n8lV1Y . Based on the video, give two examples of things you can do to find and eliminate errors in Matlab code. Problem 3.2 For the system on the right, the moment in the spring is given by 𝑘𝑘 tan −1 𝜃𝜃 where 𝑘𝑘 is spring constant and 𝜃𝜃 is the stretch in the spring. The moment in the damper is given by 𝑐𝑐𝜃𝜃 ̇ 5 where 𝑐𝑐 is the damping constant. Write the EOM as a first-order system. Problem 3.3 Just after a gust of wind, a 40 lb traffic light has a position of -1 in and a velocity of -6 in/s relative to its rest position. Suppose the equivalent stiffness is 6.3 lb/in. a) Find the response of the system analytically. b) Suppose a student were to calculate the phase shift ϕ using their calculator. Explain why they should or should not add π to the value from the calculator. Partial answer: a) 𝑥𝑥 = (1.26 in) sin(7.80 𝑡𝑡 -2.23) Problem 3.4 Download the files prob3_4.m and trafficLightEOM.m from Canvas. These files will be used to find the response of the system in Problem 3.3 numerically (i.e., using ode45). There are several errors in the files. For all problems that involve Matlab, submit the code you used/wrote as well as the output. Matlab has a publish option that may be useful: https://www.mathworks.com/help/matlab/matlab_prog/publishing- matlab-code.html a) Find and fix all of the errors. Also briefly explain why the error is an error (as if you were explaining it to your friend). Circle, highlight, or otherwise clearly indicate where you modified the code. b) Plot of the position of the traffic light with respect to time for both the analytical and numerical responses. Does your plot seem correct? Why or why not? Hint: When you download the files, make sure you can see them in the Current Folder window in Matlab. If you cannot, either move the files to your current folder or change the working folder. See the third video in Module 3E for additional details. Partial answer: b)
Problem 3.5 A 1000 lb machine is attached to a wall mount with four hollow Neoprene (Modulus of elasticity 𝐸𝐸 = 1,500 psi, Yield strength 2,000 psi) pads of outer diameter 4 in, inner diameter 2 in, and length 3 in. What is the maximum initial speed of the cart so that vibratory stress (the stress caused by the vibration) in each leg is less than 10% of the yield stress? Assume the initial position of the cart is 0. Hint: The stress caused by an axial load is given by 𝜎𝜎 = 𝐹𝐹 / 𝐴𝐴 where 𝐹𝐹 is the axial force in the bar and 𝐴𝐴 is the cross-sectional area. Think about what vibratory system element the legs are and how you can calculate the force in that element. Also think about how amplitude is related to displacement. Answer: 34.1 in/s Problem 3.6 To properly control airplanes, the inertia needs to be known. [Data and picture from Soule, H., Miller, M., 1934. The Experimental Determination of the Moments of Inertia of Airplanes.] a) Draw the free body diagram of the swinging gear-plus- airplane (pendulum). Write the equation of motion. The total weight of the pendulum is 2,591 lb. The distance from the oscillation axis to the pendulum center of gravity is 9.050 ft. b) Using your equation of motion, find the equivalent stiffness of the pendulum. c) Find the mass moment of inertia of the pendulum about the oscillation axis. When the pendulum is gently swung, its period of oscillation is 3.759 sec. d) Find the mass moment of inertia of the airplane about the airplane’s center of mass. The inertia of the airplane about the oscillation axis is 𝐼𝐼 𝑂𝑂 = 𝐼𝐼 1 − 𝐼𝐼 2 where 𝐼𝐼 1 is the inertia of the pendulum about the oscillation axis and 𝐼𝐼 2 = 638.1 slug ft 2 is the inertia of the frame about the oscillation axis. The weight of the airplane is 2,208 lb and the distance from the oscillation axis to the airplane’s center of mass is 9.513 ft. Hints: Derive the equation of motion and use the approximation that sin 𝜃𝜃 ≈ 𝜃𝜃 for small angles to find the equivalent stiffness for the pendulum. The parallel axis theorem will be useful for finding the inertia of the airplane about the center of mass. Answer: b) 23,450 lb ft/rad c) 8,393 slug ft 2 d) 1,549 slug ft 2
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
  • Access to all documents
  • Unlimited textbook solutions
  • 24/7 expert homework help