MAS212 Assignment 2: The Damped Driven Pendulum Dr. Sam Dolan ()
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MAS212 Assignment 2: The Damped Driven Pendulum Dr. Sam Dolan ()
MAS212 Assignment 2: The Damped Driven Pendulum Dr. Sam Dolan ([email protected]) Introduction: In this assignment you will use Python to investigate a nonlinear differential equation which models the motion of a damped, driven pendulum rod. Part 1 focusses on the linear case, describing small-amplitude oscillations. Part 2 focusses on the non-linear case, which exhibits some interesting behaviour. The Submission: A completed assignment will comprise a PDF of your report on Parts 1 & 2 (maximum 6 sides, plus appendices) and a Python script and/or a IPython Notebook. The report will carry more credit than the code. Please ensure that every section below is addressed in your report. The Deadline: The deadline for submission is found on the course website. Files should be submitted at http://mas212.group.shef.ac.uk/uploads. Part 1: The linear damped pendulum Consider a mass attached to the end of a light rod. The other end of the rod is fixed at P , such that the rod can rotate freely in a plane. Let θ be the angle between the rod, and the vertical line beneath P . For small-amplitude motion, θ 1, the system may be described by the differential equation θ¨ + 2γ θ˙ + ω 2 θ = 0, (1) 2 where θ = θ(t) and θ˙ ≡ dθ , θ¨ ≡ ddt2θ . The parameters γ and ω are determined dt by the air resistance and the mass; and the strength of gravity and the length of the rod, respectively. ˙ Rewrite (1) as (a) First-order form. Introduce a new variable η = θ. two first-order ODEs. (b) Numerical integration. Use scipy.integrate.odeint to find a numerical solution for the case of initial conditions θ0 = 0.1, η0 = 0; and parameters ω = 1, (i) γ = 0, (ii) γ = 0.5, (iii) γ = 1.0 and (iv) γ = 2.0. (c) Figures. By plotting the cases (i)–(iv) as five lines on a plot, with labels, make: (1) a time-domain plot θ(t) as a function of t, and (2) a phase plot, η versus θ. (d) Closed-form solution. Derive a general solution for (1), in terms of two arbitrary constants A and B. Derive specific solutions for cases (i), (iii) and (v). Comment on these solutions with reference to critical damping. In your own words, describe in one paragraph how a general closed-form solution is superior to/more illuminating than specific numerical solutions. Part 2: The damped driven pendulum Consider the non-linear equation 1 θ¨ + θ˙ + sin θ = α cos(Ωt) q (2) with parameters q, the quality factor, and α and Ω, the driving amplitude and frequency, respectively. Note that the pendulum can now swing ‘all the way around’ by passing over point P . Hence, we make the association θ + 2πk ≡ θ for k ∈ Z. (Note also that Eq. (2) reduces to Eq. (1) when sin θ ≈ θ and α = 0, after a rescaling of the time coordinate.) Our aim here is to address a key question: over the long term, does a periodic driving force (the right-hand side) produce a periodic response in θ? (a) First-order form. Show that Eq. (2) can be written as three firstorder equations: η φ˙ = Ω. (3) θ˙ = η, η˙ = − − sin θ + α cos φ, q (b) Numerical solution. Write a function solve ddp(ts, params, ics) to numerically solve the first-order equations. Here ts represents a numpy array of values of the independent variable, t; params represents the values of q, α and Ω; and ics represents the initial conditions. The function should return the values of variables θ, η and φ at the times specified in ts. (c) Plots. Make phase plots of η versus θ for 100 < t < 1000 and q = 2, Ω = 0.6667 with the following values of α: (i) α = 0.9, (ii) α = 1.07, (iii) α = 1.15 and (iv) α = 1.35. (Use initial conditions θ0 = η0 = φ0 = 0.) Comment on what your plots imply about the behaviour of the pendulum. (d) Extensions. Investigate this system thoroughly using Python code. How does its behaviour change as the driving amplitude α is modified? Attempt to describe the long-term behaviour of the pendulum in words, possibly discussing concepts such as deterministic chaos; ergodicity; self-similarity; and ‘strange attractors’. Illustrate your discussion with, for example, a Poincar´e section and/or a bifurcation diagram.