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.

Derivation

The equations of motion for this model are usually derived by summing the forces on the mass (including any applied external forces $F_{external})$ :

F = - k x - c \dot{x} + F_{external} = m \ddot{x}

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

\ddot{x} + 2 ζ ω_{n} \dot{x} + ω_{n}^{2} x = u

where

ω_{n} = \sqrt{\frac{k}{m}}; ζ = \frac{c}{2 m ω_{n}}; u = \frac{F_{external}}{m}

$ω_{n}$ is the undamped natural frequency and $ζ$ is the damping ratio. The homogeneous equation for the mass spring system is:

\ddot{x} + 2 ζ ω_{n} \dot{x} + ω_{n}^{2} x = 0

This has the solution:

x = A e^{- ω_{n} t (ζ + \sqrt{ζ^{2} - 1})} + B e^{- ω_{n} t (ζ - \sqrt{ζ^{2} - 1})}

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

References

↑ "Fast Simulation of Mass-Spring Systems" (PDF).
↑ "Introduction to Vibrations, Free Response Part 2: Spring-Mass Systems with Damping" (PDF). www.maplesoft.com. Retrieved 2024-09-22.

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[1] "Fast Simulation of Mass-Spring Systems" (PDF).

[2] "Introduction to Vibrations, Free Response Part 2: Spring-Mass Systems with Damping" (PDF). www.maplesoft.com. Retrieved 2024-09-22.

Mass-spring-damper model

Derivation

See also

References