Mass-spring-damper model: Difference between revisions
No edit summary |
No edit summary |
||
Line 19: | Line 19: | ||
This has the solution: |
This has the solution: |
||
:<math> x = A e^{- |
:<math> 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)} </math> |
|||
If <math>\zeta < 1</math> then <math>\zeta^2-1</math> is negative, meaning the square root will be negative the solution will have an oscillatory component. |
If <math>\zeta < 1</math> then <math>\zeta^2-1</math> is negative, meaning the square root will be negative the solution will have an oscillatory component. |
Revision as of 15:45, 6 February 2023
![mass connected to the ground with a spring and damper in parallel](http://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Mass_spring_damper.svg/220px-Mass_spring_damper.svg.png)
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 examining the sum of forces on the mass:
By rearranging this equation, we can derive the standard form:[3]
- where
is the undamped natural frequency and is the damping ratio. The homogeneous equation for the mass spring system is:
This has the solution:
If then is negative, meaning the square root will be negative the solution will have an oscillatory component.
See also
References
- ^ "Solving mass spring damper systems in MATLAB" (PDF).
- ^ "Fast Simulation of Mass-Spring Systems" (PDF).
- ^ Longoria, Prof. R.G. "Modeling and Experimentation: Mass-Spring-Damper System Dynamics" (PDF). Retrieved 2019-11-19.
{{cite web}}
: CS1 maint: url-status (link)