Consider the mechanical system shown in Figure 5.90. A disk-shaft system is connected to a block of mass \(m\) through a translational spring of stiffness \(k\). The elasticity of the shaft and the fluid coupling are modeled as a torsional spring of stiffness \(K\) and a torsional viscous damper of damping coefficient \(B\), respectively. The radius of the disk is \(r\) and its mass moment of inertia about point \(\mathrm{O}\) is \(I_{\mathrm{O}}\). Assume that the friction between the block and the horizontal surface cannot be ignored and is modeled as a translational viscous damper of damping coefficient \(b\). The input to the system is the force \(f\). Draw the necessary free-body diagram and the kinematic diagram, and derive the equations of motion.

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FIGURE 5.90 Problem 3. K B m b