EGB211 - Assessment Item No 3: Computer lab assignment Due date: Monday 22nd May 2017 Only groups of 3 or 4 students are allowed (4 is preferred), unless prior group-specific arrangements have been agreed with the lecturer. Introduction In this problem we will utilise MATLAB to describe projectile motion, we will do this by discretising our standard equations of motion. The simplest version of projectile motion is analytically solvable and this analytical solution will be included in this assignment as a reference for the assessment of the numerical results. In fact, when we include complex forces such as drag the numerical solution becomes necessary to overcome the complexity of a non-linear differential equation. This document will serve as both assignment sheet and template to develop and report your solutions. Please answer all the questions in all the four parts of this assignment. A total of 25 points are split among the four parts as specified in the titles. Also remember to write your group composition at the bottom of the page. Student Name Student Number % Contribution EGB211 Computer Lab Assignment, Page 1/12 Signature Data Consider the following problem parameters: Property Projectile (spherical) Mass Quantity Unit m 10 kg Diameter D 0.1 m Initial speed v0 500 m/s 25 degrees y0 30 m air 1.225 kg/m3 Cd 0.5 Initial angle of trajectory with horizontal axis Initial altitude Drag Air density Drag coefficient Symbol The drag force has Cartesian components F x= 1 2 C d A air v x sign ( v x ) 2 F y= 1 2 C d A air v y sign ( v y ) 2 where A is the cross-sectional area of the projectile. EGB211 Computer Lab Assignment, Page 2/12 ... Figure 1. Schematic qualitative diagram representing the problem (not to scale) 1. Theoretical part (5 points) 1.1 Free body diagram and equation of motion Draw the free body diagram and obtain the non-linear equation of motion (including air resistance). Use the following box. EGB211 Computer Lab Assignment, Page 3/12 EGB211 Computer Lab Assignment, Page 4/12 1.2 Analytical solution neglecting air resistance Analytically obtain the expression of x(t ) and y (t ) , horizontal and vertical position components of the projectile position in time. To do this, neglect the air resistance (drag) term of the equation of motion. Also, analytically calculate the landing position and time of landing. Use the following box (the size can be adapted to the space you require). EGB211 Computer Lab Assignment, Page 5/12 1.3 Finite difference approximation Obtain the Finite difference approximation of the non-linear differential equation characterising this problem. Use the following box (the size can be adapted to the space you require). EGB211 Computer Lab Assignment, Page 6/12 2. Synthesis of the algorithm (7 points) 1.4 Provide your Matlab code use to simulate and display the projectile motion Please copy paste from Matlab in the following box (you can adapt the size of the box to your code). Ensure your code is well commented. EGB211 Computer Lab Assignment, Page 7/12 1.5 Discuss your initial simulation parameter choices Discuss your initial choices of the simulation parameters (i.e. time step, total simulation time), based on what you know before running the actual simulation. Use the following box, keeping your answer within the limits of the current box size. EGB211 Computer Lab Assignment, Page 8/12 3. Numerical results (8 points) 1.6 Numerical results without air resistance Express the following quantities in the table Quantity Value x position of the impact Time of impact Unit m s Show the following plots. For comparison, show both numerical and analytical solutions on each plot. Horizontal position component in time Vertical position component in time Trajectory of the projectile until it hits the ground Magnitude of velocity in time Magnitude of acceleration in time Ensure your plots are neat, and well presented. EGB211 Computer Lab Assignment, Page 9/12 1.7 Numerical results with air resistance Express the following quantities in the table Quantity Value x position of the impact Time of impact Unit m s Show the following plots. For comparison, show both numerical and analytical solution on each plot (analytical solution without drag). Horizontal position component in time Vertical position component in time Trajectory of the projectile until it hits the ground Magnitude of velocity in time Magnitude of acceleration in time Ensure your plots are neat, and well presented. EGB211 Computer Lab Assignment, Page 10/12 1.8 Convergence Show how the solution changes with the choice of different time steps. To do that, plot: the landing position as a function of the time step and the computation time as a function of the time step. Obtain these plots for both the numerical simulations with and without air resistance. Show for a variety of timesteps (at least five), select time steps to appropriate show the relationships. EGB211 Computer Lab Assignment, Page 11/12 4. Critical analysis and discussion (5 points) 1.9 Discussion without air resistance Discuss the quality of your numerical solution without air resistance in comparison with the analytical solution and justify the choice of the time step considering the convergence effects seen in the previous part. Use the following box, keeping your discussion within the limits of box using this font. 1.10 Discussion with air resistance Add air resistance to your considerations: Is it relevant in the motion? How? How does it affect convergence and computational time? Use the following box, keeping your discussion within the limits of box using this font. EGB211 Computer Lab Assignment, Page 12/12 EGB211: Dynamics. Computer Lab Assignment Student ID Weight 100% Name Criteria/Grade 1-2 3 4 5 6 7 Approach to problem solving, completeness and correctness of the solution, including FBDs and other graphical results Less than 40% of the questions completed with correct procedure and values. Solution procedures and code lack coherence and work is clumsy. More than 40% of the questions completed with correct procedure and values. Solution procedure and code lack a systematic approach and work is not neat. More than 50% of the questions completed with correct procedure and values. Work and solution procedure and code are somewhat neat and understandable. More than 65% of the questions completed with correct procedure and values. Fairly neat work. Solution procedure and code are fairly clear and easy to understand. More than 75% of the questions completed with correct procedure and values. Largely neat and systematic work. Solution procedure and code are largely clear and easy to understand. More than 85% of the questions completed with correct procedure and values. Very neat and systematic work. Solution procedure and code are clear and easy to understand