Mass-spring-damper model

Deriving the equations of motion for this model is usually done by examining the sum of forces on the mass:

${\displaystyle \Sigma F=-kx-c{\dot {x}}+F_{external}=m{\ddot {x}}}$

By rearranging this equation, we can derive the standard form:[3]

${\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=u}$ where ${\displaystyle \omega _{n}={\sqrt {\frac {k}{m}}};\quad \zeta ={\frac {c}{2m\omega _{n}}};\quad u={\frac {F_{\text{external}}}{m}}}$

${\displaystyle \omega _{n}}$ is the undamped natural frequency and ${\displaystyle \zeta }$ is the damping ratio.

• Numerical methods
• Soft body dynamics#Spring/mass models
• Finite element analysis