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Chapter 3
Fundamentals of Structural Dynamics

## Excerpt

In the presence of damping but without external forces, the solution to the linear vibration problem takes the form (compare with free vibration in Chapter 2)

$y=a0eiΩte−(c∕2m)t$
where Ω, the natural frequency of the damped point-mass spring system, is given by:
$Ω=ω02−c24m2$

When the damping coefficient c > 2m ω0, energy dissipation is so large that vibration motion cannot be sustained. The motion becomes that of the exponentially decay type instead of vibration. The value of c at which this happens ( cc = 2mω0) is called the critical damping. The more commonly used damping ratio is the damping coefficient expressed as a fraction of its value at critical damping

$ζ=c∕cc,cc=2mω0$

In terms of the frequency (in Hz) and the damping ratio, the damped natural frequency is given by:

$f0′=f01−ζ2$

• Summary
• Nomenclature
• 3.1 The Equation of Motion with Damping but no External Force
• 3.2 Forced-Damped Vibration and Resonance
• 3.3 Transient Vibrations
• 3.4 Normal Modes
• 3.5 Structural Dynamics
• 3.7 Mode-Shape Function Normalization
• 3.8 Vibration Amplitudes, Bending Moments and Stresses
• Example 3.1
• 3.9 Equivalent Static Load Method
• Example 3.2
• 3.10 Power Dissipation in Vibrating Structures
• References
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