For the beam considered in Problem 12.8 (Fig. 12.20), derive the assembled stiffness and mass matrices of the system.

Data From Problem 12.8:-

For the beam shown in Fig. 12.20, one end (point \(A\) ) is fixed and a spring-mass system is attached to the other end (point \(B\) ). Assume the cross section of the beam to be circular with radius \(2 \mathrm{~cm}\) and the material of the beam to be steel with Young's modulus of \(207 \mathrm{GPa}\) and weight density of \(7650 \mathrm{~N} / \mathrm{m}^{3}\). Using two beam elements of equal length, derive the element stiffness and mass matrices of the two elements.

Figure 12.20:-

112 A 146 141 117 114 Beam -0.5 m- x m 183 m = 100 kg m B k - 1000 N/m us FIGURE 12.20 Cantilever beam carrying spring and mass at free end.