For the multiple-leaf spring described in Problem 12.5 (Fig. 12.17), consider only one-half of the spring for modeling using five beam elements of equal length and derive the stiffness and mass matrices of each of the five beam elements. The Young's modulus is \(200 \mathrm{GPa}\) and the density is \(7800 \mathrm{~kg} / \mathrm{m}^{3}\) for the material.

Data From Problem 12.5:-

A multiple-leaf spring used in automobiles is shown in Fig. 12.17. It consists of five leaves, each of thickness \(t=0.65 \mathrm{~cm}\) and width \(w=3.8 \mathrm{~cm}\). For the multiple-leaf spring described in Fig. 12.17, derive the assembled stiffness and mass matrices. Consider only one-half of the spring for modeling using five beam elements of equal length.

Y U U3 U Joint 1 Joint 2 U6 Us UA FIGURE 12.16 A frame element in global system. X