Oscillations Test - 4

The free damping oscillations contains the forces as acceleration forces, damping and spring forces, But in the case of forced oscillations there is an external force that acts to maintain the vibrations that usually gets damped due to presence of the damping forces like friction, air drag etc. The presence of the force promotes to the Total Mechanical Energy relations.

When the mass and spring have no external force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, to efficiently pump energy into both mass and spring requires that the energy source feed the energy in at a rate equal to the natural frequency. Applying a force to the mass and spring is similar to pushing a child on swing, a push is needed at the correct moment to make the swing get higher and higher. As in the case of the swing, the force applied need not be high to get large motions, but must just add energy to the system.

The damper, instead of storing energy, dissipates energy. Since the damping force is proportional to the velocity, the more the motion, the more the damper dissipates the energy. Therefore, there is a point when the energy dissipated by the damper equals the energy added by the force. At this point, the system has reached its maximum amplitude and will continue to vibrate at this level as long as the force applied stays the same. If no damping exists, there is nothing to dissipate the energy and, theoretically, the motion will continue to grow into infinity.

v_max   = aw = 2m/s
a_max = aw²   =4m/s²
On dividing the two w = 2 radians / sec