Mass change in a viscously damped spring oscillator (${\omega }_d^2=k/m-b^2/(4m^2)$)

For fixed spring constant $k$ and linear damping coefficient $b$, replacing mass $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma =b/(2m)$ to $\gamma /\lambda$. The damped angular frequency becomes ${\omega }_{d,\lambda }^2=({\omega }_d^2+{\gamma }^2)/\lambda -{\gamma }^2/{\lambda }^2$. Thus envelope decay times scale exactly as mass, while the familiar inverse-square-root frequency scaling is a weak-damping approximation. A constant gravitational force shifts equilibrium without changing either rate.