Massachusetts Institute of Technology Department of Mechanical Engineering 2.003J/1.053J Dynamics & Control I Fall 2007 Homework 8 Issued: Nov. 9. 2007 Due: Nov.30. 2007 Instructions (please read carefully) : We have three nonlinear dynamics problems posted in this homework but please choose and solve only ONE that interests you. The hardcopy report should contain your written answer for each sub-question and graphs (no m-files). Supporting materials which are not included in your report (such as m-files with appropriate comments) should be also submitted on the MIT Server site. Without the m-files, your answers will not be accepted. There may be several places where you will have questions or get stuck. Please ask the TAs for any guidelines/ideas/clarifications. Problem 8.1 : Nonlinear parametric pendulum The equation of motion governing an undamped simple pendulum is given by: d 2θ dt2 +ω02 sinθ= 0 (1) where ω0 = g is the natural frequency of the pendulum, g the acceleration due to gravity, L L the length of the pendulum and θ the angle made with the vertical. Now, we can add a damping force to the equation by including a term that is proportional to the velocity of the pendulum. The equation of motion now becomes: d 2θ+ 2γ dθ +ω2 sin θ= 0 (2) dt2 dt 0 where γ is the damping coefficient. In this problem, we are interested in studying the nonlinear parametric pendulum, whose equation of motion is given by: Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].d 2θ dθ dt2 + 2γ dt +ω02 ⎡⎣1+h cos (2ω0t)⎤⎦sin θ= 0 (3) The above is a simple model for a child playing on a swing. The forcing term models the periodic pumping of the child's legs at approximately twice the natural frequency of the swing. h is the amplitude of forcing. Let us see if we can answer the following question in this problem: starting near the equilibrium point θθ = 0=, can the child get the swing going by pumping his/her legs this way, or does he/she need a push ? For this problem, choose ω0 =1. Unless otherwise specified, use γ=0.1. Also, remember that θ physically varies between 0 and 2π, and any value greater than 2π in the solution should be brought back to the physical range using the function mod. i) For the unforced and undamped pendulum ( h =γ= 0 ), find the time series θ(t)vs. t with the initial conditions: a) θ()0 = 0, θ 0 ()= 0.01 b) θ()0 = 0, θ 0  ()= 3 Show that the trajectory in a) is sinusoidal, whereas the trajectory in b) is not. This happens because, in part b), θ is so large thatsinθ≈θ is not a good approximation, and hence nonlinear effects kick in. Also, show that the time period of oscillations is dependent on the initial conditions for large θ (and θ ), and is independent of the initial conditions for small θ (and θ ). ii) The rest state ==of the pendulum is known to be unstable whenθθ 0 γ2 <(hω0 )2 /16 . Verify the prediction of the critical value of h necessary for sustained oscillations of the pendulum. (Get the time series for two values of h , one that is just below and the other that is just above the critical value of h given by the instability condition.) iii) Plot the time series ()(for any initial condition) for h =1.50, 1.80, 2.00, 2.05, 2.062.θ t Calculate the time periods and comment on the pattern you observe. (Remember to let the initial transients die away before you estimate the time period.) Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].iv) The motion of the pendulum for h =2.2 is on a "strange attractor." An attractor (a set of points to which solutions get attracted) is informally described as strange if the dynamics on the attractor are chaotic. When a system gets into the chaotic regime, the behavior of the solution will seem random even though we have a deterministic system that governs the motion of the pendulum. For h =2.2, plot the trajectory on a θ −θ plane, and observe the chaotic (random) nature of the "strange attractor"! (Use any non-zero initial condition; you can also plot the time series of θ or θ and observe that the behavior is random and there is no clear time period in the signal.) Again, remember to let the initial transients die away before you make your plots. Unpredictability is an essential feature of chaos. If a system is in the chaotic regime, it is impossible to predict the exact position and velocity of the pendulum at a future time even if we are given the current position and velocity with maximum accuracy. This property of unpredictability arises from what we call "sensitive dependence on initial conditions." We may start the system at (0.1,0.1) and observe a specific trajectory; but, starting at (0.1,0.10000001) may result in a completely different trajectory if the behavior is chaotic. Show that there is sensitive dependence on initial conditions for h =2.2. (For showing sensitive dependence on initial conditions, consider two initial conditions that are different only by a very small quantity (something like δθ = 0.001 radians should do the job), and show that the solutions are very different after some time when plotted on top of each other.) v) Finally, make a quick comparison of the time series of θ for h =1.8, which is well below the critical value for chaos, and h =2.2 when the system is chaotic. Use the same initial conditions for the two cases. Problem 8.2 : The growth/decay of population of animal species A popular model for the growth/decay of population of animal species is given by the logistic map. It takes into account the two basic factors that contribute to the birth or death of any animal species: a) Reproduction means the population will increase at a rate proportional to the current population. This can be stated as "the population at a future time is proportional to the population at the current time," i.e. xn+1 ∝ xn (4) Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].where xn+1 is the population at time n +1 given that the population at time