In the previous examples, the oscillations just go on forever. In practice, this rarely happens. Oscillating systems are usually damped. That means the oscillations are reduced and die out in time due to energy being removed from the system. This is similar to the drag force, which dissipates kinetic energy of a moving object and makes it slow down. To implement damping in our springs, we need a damping force.
The damping force is usually modeled as being proportional to the velocity of the moving object. This means that, at any instant in time, it is given by this equation, where c is a constant known as the damping coefficient, and the minus sign indicates that the force is in the opposite direction to the ...