Suppose a cantilever beam stands uprigt, such as a tube attached to the floor in one end and free at the other end. Suppose then that a force is applied normal to the beam at some point along the beam causing a deflection. Then suppose that at t=0 the force disappears, and the intertia of the beam as its deflection diminishes causes an underdamped motion. The question I cannot figure out is how to accurately determine a second order differential equation, which will describe deflection as a function of time. All I have is the idea of assuming no damping, but then I end up with harmonic vibrations, which never change. That which I'm missing from my equation is a way to determine the damping constant or function in my second order differential equation, such that the damping force is proportional to the velocity of the beam at a given time, and is in opposite direction to the motion of the beam. The beam is assumed to have uniform E and I. How do I determine the damping effect theoretically?