Misplaced Pages

Mass-spring-damper model

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Concept in physics
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

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. As well as engineering simulation, these systems have applications in computer graphics and computer animation.

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 ) {\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.

See also

References

  1. "Solving mass spring damper systems in MATLAB" (PDF).
  2. "Fast Simulation of Mass-Spring Systems" (PDF).
  3. "Introduction to Vibrations, Free Response Part 2: Spring-Mass Systems with Damping" (PDF). www.maplesoft.com. Retrieved 2024-09-22.


Stub icon

This engineering-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: