I am working with the equation of a conveyor belt system with buckets, where $\beta$ is the damper and $k$ the spring constant:
$$m \frac{ \mathrm{d}^2 x}{ \mathrm{d} t^2} + \beta \frac{ \mathrm{d} x }{ \mathrm{d} t} + kx = 0 $$
What I would like to find is a model where we can work out how much mass is lost every time we drop mass into the bucket.
As the conveyor belt moves along at a steady speed, what would the loss be from dropping mass in the bucket?