Problem

The mechanism shown is a pendulum consisting of a pendulum bob B with mass m and a T bar,...

The mechanism shown is a pendulum consisting of a pendulum bob B with mass m and a T bar, which is pinned at O and has negligible mass. The horizontal portion of the T bar is connected to two supports, each of which has an identical spring and dashpot system, each with spring constant k and damping coefficient c. The springs are unstretched when B is vertically aligned with the pin at O. Modeling B as a particle, derive the linearized equations of motion of the system. In addition, assuming that the system is underdamped, derive the expression for the damped natural frequency of vibration of the system.