Deriving the equations of motion for this model is usually done by summing the forces on the mass (including any applied external forces F_\text{external}): :\Sigma F = -kx - c \dot x +F_\text{external} = m \ddot x By rearranging this equation, one can obtain the standard form: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = u where \omega_n=\sqrt\frac{k}{m}; \quad \zeta = \frac{c}{2 m \omega_n}; \quad u=\frac{F_\text{external}}{m} \omega_n is the undamped
natural frequency and \zeta is the
damping ratio. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta then \zeta^2-1 is negative, meaning the square root will be
imaginary and therefore the solution will have an oscillatory component. ==See also==