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Physical Pendulum without Small Angle Approximation

Specialized MATLAB material developed by Gautam Vemuri and Andy Gavrin - Published November 24, 2020

The simple pendulum usually studied by analytic methods invokes the small angle approximation (SAA) so that one can easily reduce the equation of motion to Hooke’s law and thereby obtain the period of the pendulum and other associated quantities. If the approximation is relaxed, the problem becomes analytically intractable and one must resort to computational methods. In this exercise, the pendulum with and without the SAA are compared to allow students to discover what happens to the temporal behavior for larger angles of displacement. The students will also be able to obtain quantitative estimates of what a “small angle” means, and the limits of validity of the SAA. In the computations, students will learn that the second order differential equation that describes the motion of the pendulum can be reduced to two coupled first order differential equations, which can then be solved by the Euler-Cromer algorithm. In this specialized exercise set, students will also learn to use the ODE45 package in MATLAB to solve differential equations and its advantages over Euler-Cromer method, as well as ‘findpeaks’ command in MATLAB.
Subject Area Mechanics First Year and Beyond the First Year MATLAB • Convert 2nd order ordinary differential equation (ODE) to a pair of coupled 1st order ODEs for numerical implementation (Exercise 1). • Solve the equation of motion for a pendulum, both with and without the small angle approximation, by using Euler-Cromer integration (Exercise 1). • Determine the period of an oscillatory function numerically, e.g., by counting peaks (Exercise 2). • Solve the equation of motion for a damped pendulum, both with and without the small angle approximation, by using MATLAB’s ODE45 differential equation solver (Exercise 3). • Make and interpret phase plots (Exercise 3). • Use MATLAB’s ODE45 differential equation solver to solve the damped, driven oscillator, and to explore the properties of the solution (Exercise 4). • Use Euler-Cromer integration to find the motion of a nonlinear oscillator with restoring force given by F = -kx^3, and show that its period is inversely proportional to its amplitude. (Exercise 5). • Evaluate the range of validity of an approximation. • Use a numerical solution to qualitatively explore the behavior of a complex system. • Use a numerical solution as an experimental platform from which they can extract and analyze data. 120 min
Exercise 1: Using the Euler Cromer method, solve $\ddot{\theta}=-{\omega}^{2}sin{\theta}$ and plot position, $\theta$, vs time, up to a total time of 10 periods, for a simple pendulum with SAA (i.e. $sin{\theta}={\theta}$) and without SAA for initial angles of 5, 15, 30, 45 and 60 degrees (minimal set: 5, 30 and 60 deg). Take $\omega=2\pi$, initial velocity zero, and integration step size of 0.01. Put both plots on same graph and discuss qualitatively how the time behavior of the pendulum’s oscillations change if the SAA is not used. (Students can use PICUP article by K. Roos, Simple Hanging Harmonic Oscillator, as reference for Euler-Cromer method, as well as for behavior of pendulum with SAA) Exercise 2: Extend the code for Exercise 1 to determine the smallest angle (integer multiple of 5 will suffice) for which the average period of the pendulum without SAA exceeds the period of the pendulum with SAA by 5%. You can start with an angle of 5 degrees and increment by 5 degrees. Students can use findpeaks in MATLAB to determine peak amplitudes and then determine period. (Will require students to think about how to get average period since numerical results can have some variations and uncertainties – one way, as shown in sample code, is to use the command mean(diff()) in MATLAB. Another way, depending on student and instructor interest is to use fast Fourier transform (fft) to get mean frequency of oscillations). Exercise 3: Now introduce a damping coefficient and solve $\ddot{\theta}+k{\dot{\theta}}+\omega^{2}sin\theta=0$ by using ODE45 in MATLB. First set $k = 0$ and $\theta = 5$ and make sure that you get correct results for an undamped oscillator by comparing to results of Exercise 1. Then set $k = 0.05$ and compute and plot position vs time for initial angles of 5 and 60, initial velocity zero and for $\omega=2\pi$. Make phase plots of $\dot{\theta}$ vs $\theta$ for damped ($k = 0.05$) oscillator with amplitude of 60 degrees and interpret the plot. ((minimal set: can skip phase plots; Students can consult PICUP article by D. O'Neil, Energy and Phase Space of Damped Oscillator for reference) Exercise 4: Now, using ODE45, model the forced oscillator, given by $\ddot{\theta}+k{\dot{\theta}}+\omega^{2}sin\theta=Fcos(\omega_Ft)$ by choosing suitable values of $k$, $\omega$, $F$ and $\omega_F$. $k$ is the damping rate, $\omega$ is the natural frequency of the oscillator, $\omega_F$ is the frequency of the forcing term and $F$ is the amplitude of the forcing term. (Requires students to think of what values to use). Plot amplitude vs time for a few different parameters. Exercise 5: Consider an oscillator described by $\ddot{\theta}+{\alpha}{\theta}^{3}=0$, where you can take $\alpha = 1$. (Nonlinear oscillator). Using Euler-Cromer method, compute and plot a graph of the period of the oscillator vs 1/amplitude. Take amplitudes from 5 to 85 degrees and initial velocity zero. Does the period vary with amplitude, and how does the variation compare with the simple harmonic oscillator with SAA?