Topic 2: The pendulum

More documents

Recommendations

Info

PHY321F — cp 2005 29 You will need to compute the solution as an array of values and take the FFT of the result. 3. Include the drag force term in your solution. Assume the pendulum consists of a 1.0 cm diameter steel ball bearing suspended 1.0 m from the pivot, and is started from an angle of 40.0 ◦ . How long will it take for the amplitude to decrease to 5.0 ◦ ? (See e.g. Halliday and Resnick for drag coefficients and air density). Interestingly, the Euler solution might be better than Euler-Cromer for this case. 4. Program the case of linear damping. Compare the decrease in energy of the pendulum using the Euler and Euler-Cromer methods. How do they agree with the analytical solution? 2.6 Literature 1. Stable solutions using the Euler approximation. A. Cromer, Am. J. Phys. 49 455-459 (1981). 2. The pendulum — rich physics from a simple system. R.A. Nelson and M.G. Olsson, Am. J. Phys. 54 112-121 (1986). 3. Numerical integration of Newton’s equations including velocity-dependent forces. I.R. Gatland, Am. J. Phys. 62 259-265 (1994).
PHY321F — cp 2005 30 D A digression: Ordinary differential equations D.1 Explicit and implicit methods Consider an equation of the form dy(x) dx The explicit Euler-method solution is = −λ y(x) y(x + h) = (1 − h λ)y(x) (where we use now a generic variable x and stepsize h). This solution is unstable if h > 2/λ. We can see this by considering the error e(x), i.e. the difference between the true solution ŷ(x) and the computed solution y(x), Then e(x + h) ≈ e(x)(1 − h λ) This error term grows if |1 − h λ| > 1. This can be compared to an implicit scheme y(x + h) = y(x) − h λ y(x + h) = y(x) 1 + h λ Note that we have to solve an equation in order to obtain y(x + h). In this case, the solution is trivial, but in more complicated cases, the solution could involve substantial numerical effort. However, the stability is now assured as e(x + h) ≈ e(x) 1 + h λ so that the error term decreases for all h > 0. Implicit equations are important in the solution of stiff systems, where a rapidly decaying part of the solution dominates the stability.
Page 1 and 2: PHY321F — cp 2005 24 Topic 2: The
Page 3 and 4: PHY321F — cp 2005 26 of two coupl
Page 5: PHY321F — cp 2005 28 2.4 Higer-or
Page 9 and 10: PHY321F — cp 2005 32 More control
Page 11 and 12: PHY321F — cp 2005 34 E Fourier tr
Page 13 and 14: PHY321F — cp 2005 36 ... F=fft(f)

Topic 2: The pendulum

Create successful ePaper yourself

Delete template?

Save as template?