MTE202 - Ordinary Differential Equations Fall 2020 Project Due date: Via crowdmark by Wednesday November 11th at 12 pm (EST). Instructions: a) Write a report which solves and discusses the ideas addressed in the problems below. A digitial copy, a pdf, of either a typed report or a scan of a neatly handwritten report needs to be submitted via crowmark. b) When asked to discuss something be sure to connect what is physically happening with the changes in the math. c) You can work in pairs and the report should include a cover page which includes a statement of whom is involved in the partnership. This includes a statement that you worked alone if this is the case. d) You can work with more than your parnter on solving the problems but each partnership should have a unique report which interperts and discusses the results. Any discovered attempts at plagerisim will be passed onto the department for disiplinary action. If you want to avoid any sense of impropriety a statement of whom you worked with (beyond your partner) to solve answers should also be included on the cover page. e) You will be graded on content and clarity. So this means how correct your answer is and how clear was your argument/discussion. f) Present your analysis and results in a clear and concise discussion. The max limit of the report is 15 pages (including appendices) not including a cover page. (a) The report should summarize and explain the problems encountered, the ways to set up the mathematical models for the physical problems given, the techniques used to solve the mathematical models, and the exact solving procedures for the final solution. (b) The report should discuss the results obtained in terms of how to understand the system behaviour, what the role of each component is in said behaviour. For example, what the damper is doing in the system, etc ... (c) Keep your discussion short and to the point. (d) Try to avoid rounding, but when a numerical answer is required a maximum of 4 decimal places. (e) Remember plots are supporting evidence for the claims/statements you make in your discussion. This means you should make them easy to interpret. Problem 1 A 2 kg mass is attached to a spring with stiffness k = 8 kg/s2 . At a time t = 0, the spring is compressed 4 m to the left of the equilibrium position and the mass is given an initial velocity of 6 m/s. Note: Set the direction towards the right of equilibrium as the positive direction. a) Draw a diagram for the system showing all forces applied to the mass. b) Suppose that damping is negligible and there is no external forcing. Find the ODE describing the displacement of the mass. c) Find the displacement of the mass using the appropriate method, be sure to justify your choice of solution technique. d) Show that, in general, the displacement of the mass can be written as A sin(ωt + φ), where A is the amplitude, ω is the frequency, and φ is the phase angle of the displacement. Then determine the amplitude, frequency, and phase for the solution found in part 1c). Note: Do not use phasor notation! e) Plot the displacement of the mass for 0 ≤ t ≤ 9. f) Discuss how the solution and the behaviour of the system changes as the stiffness of the spring increases, what changes are expected in the displacement, how are the amplitude, frequency, and phase of the displacement affected? Use plots to aid in your discussion. Note: When calculating the phase, φ, remember that tan(x) = tan(x + π), and your solution must satisfy the physical meaning of the example i.e. must relate to something such as the initial condition. Problem 2 The system described in Problem 1 will likely result in vibration. To reduce the vibration, it is suggested to add a damper into the system. Assume all other conditions are the same (i.e. the same parameters and initial conditions). a) Find the ODE describing the displacement of the mass in the damped system. b) Determine the smallest value for the damping coefficient that avoids oscillations occurring in the displacement of the mass. Be sure to justify why this is the smallest value. Then solve the ODE using this value to find the displacement function of the mass. Determine the maximum displacement from equilibrium, the displacement at t = 2 s, and plot the solution, for 0 ≤ t ≤ 9, to verify that no oscillations occur. c) Discuss what happens to the solution when the damping value if increased above the value found in 2b). Specifically: i) Pick a value for a larger damping coefficient and find the solution based on this value. ii) Determine the maximum displacement from equilibrium and the displacement at t = 2. Compare these values with those found in 2b). iii) Plot the displacement of the mass, for 0 ≤ t ≤ 9, and compare it with the plot from 2b). Discuss which case has the stronger damping. d) Now consider what happens when the damping coefficient decreases from the value found in 2b). Specifically, i) Pick a damping coefficient of 2 kg/s and find the solution based on this value. ii) Determine the maximum displacement from equilibrium and the displacement at t = 2. Compare these values with those found in 2b) and 2c). iii) Plot the displacement of the mass, for 0 ≤ t ≤ 9, and compare it with the plot from 2b) and 2c). Discuss the differences in these three cases. Problem 3 In Problem 1 and Problem 2, no external forces act on the mass. However, in reality, external forces always exist. a) Consider the case when an external force, f (t) = 10e−t , acts on the damped mass described in 2d). Solve the non-homogeneous problem, plot the solution, compare it to your solution found in 2d) and discuss any differences in the solution and the physical behaviour of the system. b) Consider the case when an external force, f (t) = sin(2t), acts on the undamped system from Problem 1. Solve the non-homogeneous problem, and plot the solution, compare it to your solution found in 1b), and discuss any differences in the solution and the physical behaviour of the system. What is happening to the displacement of the mass over time? Is there a physical phenomena that corresponds to your results?