Mass-spring-damper model

Deriving the equations of motion for this model is usually done by summing the forces on the mass (including any applied external forces $F\_\backslash text\{external\})$:

:$\backslash Sigma\; F\; =\; -kx\; -\; c\; \backslash dot\; x\; +F\_\backslash text\{external\}\; =\; m\; \backslash ddot\; x$

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

:$\backslash ddot\; x\; +\; 2\; \backslash zeta\; \backslash omega\_n\; \backslash dot\; x\; +\; \backslash omega\_n^2\; x\; =\; u$ where $\backslash omega\_n=\backslash sqrt\backslash frac\{k\}\{m\};\; \backslash quad\; \backslash zeta\; =\; \backslash frac\{c\}\{2\; m\; \backslash omega\_n\};\; \backslash quad\; u=\backslash frac\{F\_\backslash text\{external\}\}\{m\}$

$\backslash omega\_n$ is the undamped natural frequency and $\backslash zeta$ is the damping ratio. The homogeneous equation for the mass spring system is:

:$\backslash ddot\; x\; +\; 2\; \backslash zeta\; \backslash omega\_n\; \backslash dot\; x\; +\; \backslash omega\_n^2\; x\; =\; 0$

This has the solution:

:$x\; =\; A\; e^\{-\backslash omega\_n\; t\; \backslash left(\backslash zeta\; +\; \backslash sqrt\{\backslash zeta^2-1\}\backslash right)\}\; +\; B\; e^\{-\backslash omega\_n\; t\; \backslash left(\backslash zeta\; -$

\sqrt{\zeta^2-1}\right)}

If then $\backslash zeta^2-1$ is negative, meaning the square root will be imaginary and therefore the solution will have an oscillatory component.{{Cite web |title=Introduction to Vibrations, Free Response Part 2: Spring-Mass Systems with Damping |url=https://www.maplesoft.com/content/EngineeringFundamentals/6/MapleDocument_32/Free%20Response%20Part%202.pdf |access-date=2024-09-22 |website=www.maplesoft.com}}

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

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Category:Classical mechanics

Category:Mechanical vibrations

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