Damped vibrations: Compute the logarithmic decrement from the damping ratio. Given: actual damping coefficient c = 40 N·s/m; critical damping coefficient c_c = 420 N·s/m. Calculate the logarithmic decrement (δ) for the underdamped system.

Given datac = 40 N·s/mc_c = 420 N·s/m

Concept/ApproachFor underdamped free vibration, damping ratio ζ = c / c_c.Logarithmic decrement: δ = (2πζ)/√(1 − ζ²).

Step-by-Step calculationζ = 40 / 420 = 2/21 ≈ 0.095238√(1 − ζ²) = √(1 − 0.00907) ≈ √(0.99093) ≈ 0.99546δ = (2π × 0.095238) / 0.99546 ≈ 0.598 ≈ 0.6

VerificationSmall ζ (≈0.095) should give a small δ (< 1), so ≈0.6 is reasonable.

Final Answerδ ≈ 0.6