Chapter 8, Problem 19P

The displacement of a structure is defined by the following equation for a damped oscillation:

$y=8e^{-kt}\text{ cos }\omega t$

where $k=0.5\text{ and }\omega =3$.

(a) Use the graphical method to make an initial estimate of the time required for the displacement to decrease to 4.

(b) Use the Newton-Raphson method to determine the root to ${\epsilon }_s=0.01\%$.

(c) Use the secant method to determine the root to ${\epsilon }_s=0.01\%$.