Mass-spring-damper model

The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers.

This form of model is also well-suited for modelling objects with complex material behavior such as those with nonlinearity or viscoelasticity.

As well as engineering simulation, these systems have applications in computer graphics and computer animation.¹

Deriving the equations of motion for this model is usually done by summing the forces on the mass (including any applied external forces ${\displaystyle F_{\text{external}})}$:

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

By rearranging this equation, one can obtain the standard form:

${\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=u}$ 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. The homogeneous equation for the mass spring system is:

${\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=0}$

This has the solution:

${\displaystyle x=Ae^{-\omega _{n}t\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)}+Be^{-\omega _{n}t\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)}}$

If ${\displaystyle \zeta <1}$ then ${\displaystyle \zeta ^{2}-1}$ is negative, meaning the square root will be imaginary and therefore the solution will have an oscillatory component.²