Mass-spring-damper model explained

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]

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 _{external)} :\Sigma F = - kx - c • x + F _{external} = m \ddot x

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

\ddot x + 2 \zeta \omega _{n} • x + \omega ^{2} _{n} x = u where \omega _{n = \sqrt (k)/(m)} ; \zeta = (c)/(2 m \omega _{n}) ; u = (F _{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} • x + \omega ^{2} _{n} 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 < 1 then \zeta ^{2-1} is negative, meaning the square root will be negative and therefore the solution will have an oscillatory component.

See also

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

Notes and References

1. Web site: Solving mass spring damper systems in MATLAB.
2. Web site: Fast Simulation of Mass-Spring Systems.