Vibrations — the natural frequency of a viscously damped system is __________ the natural frequency of the corresponding undamped system.
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Amore than
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Bless than
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Csame as
Answer
Correct Answer: less than
Explanation
Introduction / Context: Damping modifies both the amplitude response and the oscillation rate of a vibrating system. For viscous damping, the damped natural frequency reflects the loss of “effective” stiffness–inertia energy exchange rate.
Given Data / Assumptions:
- Single degree-of-freedom mass–spring–damper model.
- Viscous damping ratio ζ between 0 and 1 (underdamped).
- Undamped natural frequency ωn = √(k/m).
Concept / Approach: In the underdamped case, the damped natural frequency is ωd = ωn√(1 − ζ²), which is strictly less than ωn for any ζ > 0.
Step-by-Step Solution:
1) Write characteristic equation for mẍ + cẋ + kx = 0.2) Define ζ = c/(2√(km)).3) Roots give ωd = ωn√(1 − ζ²) < ωn for 0 < ζ < 1.Verification / Alternative Check: As ζ → 0, ωd → ωn; as ζ increases, ωd decreases monotonically.
Why Other Options Are Wrong:more than — Contradicts ωd expression.same as — Only true for ζ = 0 (undamped), not with viscous damping.
Common Pitfalls: Confusing damped frequency with forced response peaks; the formula above is for free vibration.
Final Answer: less than.