Amplitude falls to 25% after two cycles. Which expression gives the logarithmic decrement?

delta = (1/2) ln(1/0.25) = 0.693

How is the damping coefficient c obtained here?

c = zeta x c_c with c_c = 2 sqrt(km) = 1414 N s/m, so c is about 155 N s/m

c = 2 m wn with zeta ignored, about 1414 N s/m

Practice · 6 marks

Damping from logarithmic decrement

Short answer

Log decrement δ = (1/2) ln(1/0.25) = 0.6931. Damping ratio ζ = δ/√(4π²+δ²) = 0.1097. Natural frequency ω_n = 28.28 rad/s, damped f_d = ω_n√(1-ζ²)/(2π) = 4.474 Hz. Damping coefficient c = ζ·2√(km) = 155.1 N·s/m. Cycles to 10%: n = ln(10)/δ ≈ 3.32.

A vibrating system has a mass of 25 kg and a spring of stiffness 20 kN/m with a viscous damper. It is found that the amplitude of free vibration reduces to 25% of its initial value after two complete cycles. Find the logarithmic decrement, the damping ratio, the damping coefficient, and the damped natural frequency. How many cycles are needed for the amplitude to fall to 10% of the initial value?

Data: $m = 25$ kg, $k = 20000$ N/m, $x_2/x_0 = 0.25$ after $n = 2$ cycles.

Logarithmic decrement

\delta = \frac{1}{n}\ln\frac{x_0}{x_n} = \frac{1}{2}\ln\frac{1}{0.25} = \frac{\ln 4}{2} = 0.6931

Damping ratio

\delta = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}} \Rightarrow \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} = \frac{0.6931}{\sqrt{39.478 + 0.4805}} = 0.1097

Natural and damped frequency

\omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{20000}{25}} = 28.28\ \text{rad/s}

\omega_d = \omega_n\sqrt{1-\zeta^2} = 28.28\sqrt{1 - 0.1097^2} = 28.11\ \text{rad/s}, \qquad f_d = \frac{\omega_d}{2\pi} = 4.474\ \text{Hz}

Damping coefficient

c_c = 2\sqrt{km} = 2\sqrt{20000 \times 25} = 1414.2\ \text{N·s/m}

c = \zeta c_c = 0.1097 \times 1414.2 = 155.1\ \text{N·s/m}

Cycles to reach 10%

\frac{x_0}{x_n} = e^{n\delta} = 10 \Rightarrow n = \frac{\ln 10}{\delta} = \frac{2.3026}{0.6931} = 3.32\ \text{cycles}

Answer: $\delta = 0.6931$ ; $\zeta = 0.1097$ ; $c = 155.1$ N·s/m; $f_d = 4.474$ Hz; about 3.32 cycles to fall to 10%.

✓ Answer checked by Claude Opus 5.5 (high effort)

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