Q:

9.1 (a) Using Taylor series expansions, derive a central finite difference approximation for a second total derivative at the jth time step where the error is of order ?t ⁴.

(b) Using Taylor series expansions, derive a central finite difference approximation for a fourth total derivative at the jth spatial step where the error is of order ?x ².

Q:

9.2 (a) Using spreadsheet software, repeat the step time integration of Example 9.1, but this time use the larger time step of 0.04 sec for 50 time steps. Compare this result with that obtained in Example 9.1.

(b) As in part (a), use a time step of 0.01 sec for 200 time steps.

Q:

9.3 Using spreadsheet software and (a) Heun s method; (b) the fourth order Runge-Kutta method; numerically integrate the linear, undamped, one-DOF equation of Example 9.1, but here let the base motion input be a step function. That is, numerically solve

Again, let the natural frequency have a unit value, and extend the integration from time zero to t = 10 sec. Choose your own time step. Compare your numerical result with the correct analytical solution.

Q:

9.4 (a) Using spreadsheet software and (a) Heun s method; (b) the fourth order Runge-Kutta method; find a numerical solution for the horizontal, one-DOF motion of the mass shown in Figure 9.12 for the time period (0. 6.28 sec). Note that the second set of springs...