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] Objects may be described as volumetric meshes for simulation in this manner. As well as engineering simulation, these systems have applications in computer graphics and computer animation[2]

Derivation (Single Mass)

Classic model used for deriving the equations of a 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 ${\displaystyle \omega _{n}={\sqrt {\frac {k}{m}}};\quad \zeta ={\frac {c}{2m\omega _{n}}};\quad u={\frac {F_{external}}{m}}}$

${\displaystyle \omega _{n}}$ is the undamped natural frequency

${\displaystyle \zeta }$ is the damping ratio

See also

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

References

1. "Solving mass spring damper systems in MATLAB" (PDF).
2. "Fast Simulation of Mass-Spring Systems" (PDF).
3. Longoria, Prof. R.G. "Modeling and Experimentation: Mass-Spring-Damper System Dynamics" (PDF). Retrieved 2019-11-19.