Vibrations — the natural frequency of a viscously damped system is __________ the natural frequency of the corresponding undamped system.

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 ω_n = √(k/m).

Concept / Approach: In the underdamped case, the damped natural frequency is ω_d = ω_n√(1 − ζ²), which is strictly less than ω_n for any ζ > 0.

Step-by-Step Solution:

Verification / Alternative Check: As ζ → 0, ω_d → ω_n; as ζ increases, ω_d decreases monotonically.

Why Other Options Are Wrong:
more than — Contradicts ω_d 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.