The ideal pendulam can be modeled by the second-order, nonlinear differential eq
ID: 2981930 • Letter: T
Question
The ideal pendulam can be modeled by the second-order, nonlinear differential equation d2 theta/dt2 +sin(theta) = 0 where theta is the angle from the vertical. For small angles, sin ( theta ) theta , giving a linear approximation to the differential equation in (1), d2 theta/dt2 + theta = 0 using the first two terms of a power series expansion of sin ( theta ) about theta = 0 gives the cubic approximation The damped driven pendulum is modeled using For your report: Compare the results of numerical simulations of (1), (2), and (3) to see how closely the period of the periodic orbits relate. Perform a phase portrait ( theta '(t) vs. theta ) analysis for (1), (2). and (3). Compare and contrast from the point of view of how well (2) and (3) approximate (1). Study how the periods of the periodic orbits are related by considering the initial conditions theta (0) = theta 0 and theta '(0) = 0. For what intervals of theta 0 do the periodic orbits of (2) and (3) closely resemble the periodic orbits of (1)? Plot graphs of the period as a function of theta 0 using 10-15 initial values. (c) Consider theta (0) = 0 and theta '(0) = v0. What changes from part (b) above?Explanation / Answer
The comparisons show expected results.
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