How Do You Derive the Equation for Damped Frequency in a Spring System?

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To derive the equation for damped frequency in a spring system, start with the differential equation for simple harmonic motion with damping. The damped frequency is expressed as ωd = ωn√(1-ζ²), where ωn is the natural frequency and ζ is the damping ratio. The solution involves standard techniques for solving second-order linear differential equations. No special tricks are necessary; a straightforward approach suffices. This foundational understanding is essential for analyzing damped oscillations in mechanical systems.
robmass
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Hiya can anyone show how to derive the euqation of damped frequency for a spring

ωd = ωnsqrt(1-ζ2)
 
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What is the differential equation describing the motion?
 
Google will find lots of web sites with the derivation.

As NascentOxygen said, you write down the differential equation for simple harmonic motion with damping, and solve it. There are no "clever tricks" required.
 

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