Questions and discussion for this lecture live here. Fire away by hitting Reply below 
I think the reason the beam oscillates is the presence of three forces: the external force, the inertia force, and the internal stiffness force. The little red arrows you drew are not the inertia forces, but represent the restoring force caused by the enlarged beam deformations, which pulls the beam back toward its static equilibrium position. The inertia forces actually act in the opposite direction. In other words, the restoring forces drive the beam toward its static position, while the inertia forces cause it to deviate from it.
Hey @Kamil - this can be confusing, and I could have done a much better job explaining it. In fact, I will return to this lecture and improve it for clarity.
In the meantime, let me restate the force balance as follows for clarity:
Let’s assume downwards is the positive vertical direction. A downward applied force produces a positive downward displacement (x) at the beam tip. The beam’s stiffness provides a restoring force upward, proportional to the displacement (F_restore = -kx).
Because acceleration (a) is 180° out of phase with displacement (x), the mass accelerates upward when the tip is displaced downward (and vice versa). The inertial effect (ma) therefore acts upward as well, in the same direction as the stiffness force.
The faster the external load is applied, the larger the inertial reaction becomes (because it’s proportional to acceleration), adding to the stiffness reaction in opposing the downward motion.
Note that in the video I’m showing an inertial force in the opposite direction to the acceleration - this is particularly confusing so I will clarify this with an updated lecture.
Thanks for posting!
S