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Mass-spring-damper model - Wikipedia

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Concept in physics

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 F external ) {\displaystyle F_{\text{external}})} :

Σ F = − k x − c x ˙ + F external = m x ¨ {\displaystyle \Sigma F=-kx-c{\dot {x}}+F_{\text{external}}=m{\ddot {x}}}

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

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

ω n {\displaystyle \omega _{n}} is the undamped natural frequency and ζ {\displaystyle \zeta } is the damping ratio. The homogeneous equation for the mass spring system is:

x ¨ + 2 ζ ω n x ˙ + ω n 2 x = 0 {\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=0}

This has the solution:

x = A e − ω n t ( ζ + ζ 2 − 1 ) + B e − ω n t ( ζ − ζ 2 − 1 ) {\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 ζ < 1 {\displaystyle \zeta <1} then ζ 2 − 1 {\displaystyle \zeta ^{2}-1} is negative, meaning the square root will be imaginary and therefore the solution will have an oscillatory component.^[2]

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

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