Question 8.14: Solve Example 8.12 by the central difference method with h =...
Solve Example 8.12 by the central difference method with b=0.1 \mathrm{~s}.
On substituting for m, c, k, and h, Equation 8.144
\left(\frac{m}{h^2}+\frac{c}{2 h}\right) u_{n+1}=p_n+\left(-k+\frac{2 m}{h^2}\right) u_n+\left(\frac{c}{2 h}-\frac{m}{h^2}\right) u_{n-1} \qquad (8.144)
becomes
269.21 u_{n+1}=p_{n}+406.6 u_{n}-237.39 u_{n-1} \qquad (a)
Table E8. 14 Central difference method.
\begin{matrix} \\\hline & &&&u_{n+1} & \dot{u}_{n} & \ddot{u}_{n} & u_{n} \\ \text{Time }& {p_{n}} & {u_{n-1}} & {u_{n}} & (Eq. a) & { (Eq. b) } & { (Eq. c) } & { (\text{theoretical}) } \\\hline 0.0 & 0.0 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.000 & 0.0000 \\0.1 & 50.0 & 0.0000 & 0.0000 & 0.1857 & 0.9286 & 18.573 & 0.0323 \\0.2 & 86.6 & 0.0000 & 0.1857 & 0.6022 & 3.0110 & 23.074 & 0.2254 \\0.3 & 100.0 & 0.1857 & 0.6022 & 1.1172 & 4.6574 & 9.854 & 0.6204 \\0.4 & 86.6 & 0.6022 & 1.1172 & 1.4780 & 4.3792 & -15.418 & 1.0961 \\0.5 & 50.0 & 1.1172 & 1.4780 & 1.4329 & 1.5785 & -40.594 & 1.4251 \\0.6 & 0.0 & 1.4780 & 1.4329 & 0.8609 & -3.0859 & -52.693 & 1.3772 \\0.7 & 0.0 & 1.4329 & 0.8609 & 0.0366 & -6.9813 & -25.216 & 0.8683 \\0.8 & 0.0 & 0.8609 & 0.0366 & -0.7038 & -7.8231 & 8.381 & 0.1105 \\0.9 & 0.0 & 0.0366 & -0.7038 & -1.0953 & -5.6593 & 34.893 & -0.5974 \\1.0 & 0.0 & -0.7038 & -1.0953 & & & & -1.0073 \\ \hline \end{matrix}
Equations 8.139
\dot{u}_n=\frac{1}{2 h}\left(u_{n+1}-u_{n-1}\right)+R \qquad(8.139)
and 8.141
\ddot{u}_n=\frac{1}{h^2}\left(u_{n+1}-2 u_n+u_{n-1}\right)+R \qquad(8.141)
then give
\begin{aligned}&\dot{u}_{n}=5\left(u_{n+1}-u_{n-1}\right) \qquad \qquad (b) \\&\ddot{u}_{n}=100\left(u_{n+1}-2 u_{n}+u_{n-1}\right) \qquad (c) \end{aligned}
To start the integration, Equation 8.145
u_{-1}=u_0+\frac{h^2}{2} \ddot{u}_0-h \dot{u}_0 \qquad(8.145)
is used to obtain u_{-1}. In this case because u_{0}, \dot{u}_{0}, and \ddot{u}_{0} are all zero, u_{-1} is also zero. Equations a, b, and c are now used to carry out step-by-step integration. Response calculations are shown in Table E8.14 for the first 1 \mathrm{~s}. If the velocity and acceleration are not of interest, the corresponding columns may be omitted from the table.