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 model is well-suited for modelling object with complex material properties such as nonlinearity and viscoelasticity. Packages such as MATLAB may be used to run simulations of such models.^[1] As well as engineering simulation, these systems have applications in computer graphics and computer animation.^[2]

mass connected to the ground with a spring and damper in parallel — Classic model used for deriving the equations of a mass spring damper model

Derivation (Single Mass)

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, we can derive 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}{2m\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=Ae^{-\omega _{n}t\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)}+Be^{-\omega _{n}t\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)}

If $\zeta <1$ then $\zeta ^{2}-1$ is negative, meaning the square root will be imaginary and therefore the solution will have an oscillatory component.^[3]