MATH 1063 - Mathematical Methods for Engineers 1 Team Project: A “Plunger” Mechanism Note: This project is designed to help develop Graduate Qualities 2, 3, 4, and 6, namely lifelong learning, effective problem solving, working autonomously and collaboratively, and communi- cating effectively. It should also provide opportunity for varying student Learning Styles. C P x O θ φ y A OC = 20 cm CA = 20 cm AB = 12 cm 8 cm B Consider the mechanical device illustrated, in which the piston AB (of length 12 cm) slides back and forth so that at time t sec, t ≥ 0, the distance x is given by x(t) = 8 + 2 cospit cm. This means that each cycle has a period of 2 sec and x varies from a maximum of 10 cm initially to a minimum of 6 cm halfway through each cycle. The point O (the ori- gin) is fixed, and so as the piston moves back and forth the two driven arms AC, OC rotate according to the angles θ, φ respec- tively. The point C clearly travels along an arc which is a small part of a circle centred at O with radius 20 cm. 1. Produce the following commands in a Matlab m-file named simulate.m, and then run it. % This program simulates the mechanical system defined in device.m figure(1); for t=0:0.02:4; x=8+2*cos(pi*t); alpha=atan(8/x); beta=acos(sqrt(x^2+64)/40); % find coordinates of A Xa=-x; Ya=8; % find coordinates of B Xb=-x-12; Yb=8; % find coordinates of C Xc=-20*cos(alpha+beta); Yc=20*sin(alpha+beta); axis([-23 14 0 28]); axis(’equal’); axis manual; %(*) hold on; plot([Xb Xa],[Yb Ya],’b’); plot([Xa Xc],[Ya Yc],’r’); plot([0 Xc],[0 Yc],’g’); hold off; pause(0.1); clf; %(*) end; 1 2. Describe the meaning and purpose of every non-comment line in the M-file simulate.m. In particular, explain why it is necessary to include the lines ending with %(*). 3. Use a diagram to illustrate what are represented by alpha (α) and beta (β) in the above M-file and explain how they are related to θ, φ. Hence derive the expressions in the above M-file for the coordinates of C relative to the origin O. 4. Briefly explain why it is obvious that the maximum values of θ(t) and φ(t) both occur when t = 0, whereas the minimum values occur at t = 1. When is the velocity of the piston a maximum? 5. Write an M-file named device.m which • creates symbolic expressions for the angles θ(t), φ(t) and hence the angular velocities θ′(t), φ′(t). • plots the angles θ(t), φ(t) [radians] on common axes with suitable legends over two cycles starting at t = 0, and also evaluates the maximum and minimum values each angle can reach. • determines the first time when the angle φ = 60◦. • plots the angular velocities θ′(t), φ′(t) [radians/sec] on common axes over two cycles starting at t = 0, and determines the maximum value of each (these do not occur at the same instant when the piston has its maximum speed!). • plots the path (using equal axes) travelled by the point P (on the driven arm AC) exactly midway between the points A and C. Submission of Reports and Assessment A team report is to be submitted by each team by the due date. Presentation should be accord- ing to the guidelines given in the handout Writing Mathematics Reports. Only A4 size papers should be used. The names and UniSA network user names of each participating team member should appear on the cover page of the report. The assessment will take into account both presentation, and your mathematical and computa- tional analysis of the problem, including your Matlab codes, figures and outputs. Each team as a result will obtain a team mark for its report. Then the individual marks will be allocated based on the Individual Peer Assessment Form handed in, in confidence, by each student to their practical supervisor. For each absence from the practical class, there will be a reduction, worth 20% of the full mark, from the individual mark. No joint work is allowed between different teams. 2