Solve example 8.5 using Newmark's method with two different time steps \(\Delta t=42 \mu\) sec and \(\Delta t=75 \mu \mathrm{sec}\). Compare the tip displacement using the average acceleration method (unconditionally stable) and the linear acceleration method (conditionally stable).

Data From Exercise 8.5:

Consider a clamped uniaxial bar subjected to a tip load as shown in figure 8.9. Use three elements of equal length to determine the tip displacement as a function of time. Use the central difference method for time integration and the consistent mass matrix. The properties of the bar are: \(E=100 \mathrm{GPa}, L=1 \mathrm{~m}, A=10^{-5} \mathrm{~m}^{2}, ho=4000 \mathrm{~kg} / \mathrm{m}^{3}\). The tip force is given by: \(F(t)=2 F_{\text {max }} t / t_{\max }\) for \(0t_{\max } / 2\), where \(F_{\text {max }}=0.001 E A\) and \(t_{\max }=3 \mathrm{~ms}\). Calculate the tip displacement \(u(t)\) for \(0

u1 E 2 3 E2 E3 L= 1 m F. tmax 2 tmax Figure 8.9 A clampedfree uniaxial bar subjected to a tip force F(t) F(t) U 4