The Nonlinear Pendulum
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1 The Nonlinear Pendulum Evan Sheridan Feburary 18th 013 Abstract Both the non-linear linear pendulum are investigated compared using the pendulum.c program that utilizes the trapezoid method for solving differential equations. The damped pendulum was then analysed using the improved Runge-Kutta method after time it was found the non-linear pendulum approximated the Linear pendulum. Finally, the damped driven oscillator was analysed signs of chaotic behaviour were exhibited for A > 1.50m. 1
2 1 Aims To use the pendulum.c program to determine the motion of the simple pendulum To alter the pendulum.c program so that it uses the Runge-Kutta Method analyse the motion of the damped driven pendulum Use Gnuplot as an aid to interpret data. Backround Theory In general for a simple pendulum we have the equation of motion: d θ dt = g sinθ) 1.1) l If we make a small angle approximation, i.e sinθ) θ then we get : d θ dt = g l θ 1.) We call 1.) the equation of motion for the linear pendulum 1.1) the equation for the nonlinear pendulum. Now we wish to solve for the solution of these second order differential equations. In general we use a computational method to do this it can be done by 3 of the the following methods: The Simple Euler Method, The Trapezoid Method The Runge-Kutta Method. The Simple Euler Method In this method we take a second order differential equation reduce the problem to solving to first order differential equations. We do this by introducing the a new variable. In the specific case of the non-linear pendulum we have the following: ωt) = dθ dt fθ, ω, t) = dω dt Now if we wish to solve these equations numerically we need to begin with our initial values θ 0, ω 0 use a Taylor Expansion in order to derive θ 1,...,n, ω 1,...,n. The Taylor Expansion is given by: θt + t) = θt) + dθt) dt t + d θt) t d + O t 3 ) t! ωt + t) = ωt) + dω dt t + d ω d t which when writing in iterative notation becomes: t! + O t 3 ) θ n+1 = θ n + ω n t + O t ) 1
3 ω n+1 = ω n θ n t + O t ) Thus, now using these iterative functions we having the initial values we can solve for the motion trajectory. Obviously, in order to get any idea of what the motion looks like we would have to do quite a few iterations this is why computational methods are used for it would be far too tedious to this by h. The Simple Euler Method lays the foundation the next two methods are improvements on it. The Trapezoid Method This method makes a better approximation for the iterative functions. Instead we find that the area under a trapezoid for a θt) curve is gives the following result: θ n+1 θ n + t dθt) dθt + t + dt dt By explicitly writing out the Taylor Expansions for both θ n, ω n we get the following: k1a + k b θ n+1 = θ n + k1b+ k b ω n+1 = ω n + with k 1a = ω n t; k a = ω n + fθ n, ω n, t) t) t; k b = tfθ n+1, ω n+1 + k 1b, t n+1 ) All in all, the trapezoid method is a better approximation for solving the second order differential equation than the Simple Euler Method. The Runge-Kutta Method The equation of the Non-Linear Damped Driven Pendulum is given by: fθ, ω, t) = β sinθ) kω + A cosωt) In order to solve for the motion trajectory in this instance we use the 4th Order Runge-Kutta Method. Using the Runge-Kutta method we get the follwing: k1a + k a + k 3a + k 4a θt + h) = θt) + 6 k1b + k b + k 3b + k 4b ωt + h) = ωt) + 6 with k a = h ω + k ) 1b k 3a = h ω + k ) b k 1a = hω k 1b = hfθ, ω, t) k b = hf k 3b = hf θ + k 1a, ω + k 1b, t + h ) θ + k a, ω + k b, t + h ) k 4a = hω + k 3b ) k 4b = hfθ + k 3a, ω + k 3b, t + h)
4 The Experiment Exercises 1 1i) In this instance we analysed equation 1.) for a set of different starting values. We got the following plots: In each case the pendulum is simply oscillating about the it s equilibrium position as expected. 1ii) In this instance we analyses equation 1.1) against 1.) plotted θt) got the following results: As we can see for θ = 0 at the beginning the Non-Linear Linear case are quite similar as we expect because of the small angle approximation. However, as θ gets larger the trajectories differ as expected. Similar behaviour for θ = 0.. The dramatic difference occurs for θ = 3.14 which is evident from the plot. Essentially, this says that we must be very careful when applying the small angle approximation the approximation must really be quite small. 3
5 Out of the curiosity the Phase Portraits of the Non-Linear Undamped Pendulum were plotted: The red phase path corresponding to θ = 3.14) is identified as the separatrix. It is the boundary between oscillations of the nonlinear pendulum rotations. Exercise i) The pendulum.c program was edited to apply the Runge-Kutta method for β = 1, θ = 3.14, ω = 0 it was demonstrated with the below illustration that we get equivalent results as the trapezoid method: 4
6 iii) For values of β = 1, k = 0.5 for the damped pendulum we compared the linear the nonlinear cases. We got the following plot: As we can see, after a sufficient amount of time the behaviour of the damped linear pendulum approximates very well to the non-linear case which is as expected. iv) The Runge-Kutta program was changed to analyse the damped-driven oscillator. The first 500 iterations were neglected as it took time to settle into the attractor when we plotted the phase portraits. This was achieved by adding a simple if statement into the for loop. The Phase Portraits are plotted below for certain values of A, θ, ω 5
7 Conclusions The seperatrix was successfully found in the case of the Non-Linear undamped pendulum. This is a very hy tool for it tells us at what value of θ we should expect to transform from oscillitory motion to rotary motion. As expected this should be in around a value of π which is where the separatrix was found to be. After a sufficient amount of time, roughly 0 seconds in our case, the linearly damped pendulum had approximately the same behaviour of the nonlinearly damped pendulum. We expect this as a result of the damping which results in θ decreasing when θ becomes small enough i.e after 0 seconds) we can apply the small angle approximation sinθ) θ resulting in equivalent behaviour. Analysing the behaviour of the damped-driven oscillator using phase portraits we found that at A = 1.50 the system exhibited chaotic behaviour. Increasing the number of iterations the program was run through it was found that the plot became quite a bit denser which indicate that the path in phase space was never repeated which is an indication of chaotic behaviour. Period doubling was witnessed at A = 1.07 which also would indicate that we should see chaotic behaviour for larger values of A. Finally, it was noted that that the initial values of θ, ω settled into the attractor as long as A was not in the chaotic regime. 6
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