Damping in vibrations: The ratio of the actual damping coefficient c to the critical damping coefficient cc is termed what, and is the statement ‘‘This ratio is called the damping factor’’ correct?
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AYes
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BNo
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COnly in electrical analogies
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DOnly for logarithmic decrement
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ETrue only for Coulomb friction
Answer
Correct Answer: Yes
Explanation
Introduction / Context
In single-degree-of-freedom vibration systems, characterizing how strongly the system is damped is essential for predicting overshoot, settling time, and resonance behavior. A common nondimensional measure is the ratio of actual damping to critical damping.
Given Data / Assumptions
- Linear viscous damping model with damping coefficient c.
- Critical damping coefficient cc = 2mωn (where ωn = √(k/m)).
- Standard mechanical vibration terminology.
Concept / Approach
The nondimensional ratio ζ = c / cc is called the damping factor or damping ratio. It partitions regimes: ζ < 1 (underdamped), ζ = 1 (critically damped), ζ > 1 (overdamped). This parameter directly enters transmissibility, phase angle, and time-response formulas.
Step-by-Step Solution
1) Define ωn = √(k/m); critical damping cc = 2mωn.2) Form ζ = c / cc.3) Recognize ζ as the damping factor used in standard solutions for free and forced vibrations.Verification / Alternative check
Time response x(t) for underdamped systems includes e−ζωn t; transmissibility and phase formulas use ζ explicitly, confirming its canonical status.
Why Other Options Are Wrong
- No: contradicts standard terminology.
- Only in electrical analogies: ζ is universal across analogous domains (mechanical, electrical RLC).
- Only for logarithmic decrement: log decrement δ relates to ζ but is not the same quantity.
- True only for Coulomb friction: ζ pertains to viscous damping models, not dry friction.
Common Pitfalls
- Interchanging ζ with the percentage of critical damping (which is simply 100ζ%).
- Confusing damping ratio with damping factor vs. damping coefficient c (dimensional).
Final Answer
Yes