Calculating Logarithmic Decrement

The logarithmic decrement ($\delta$) quantifies the rate at which free vibrations decay in a damped system. It is directly related to the damping characteristics of the system, specifically the damping ratio ($\zeta$).

Determining the Damping Ratio

First, we find the damping ratio ($\zeta$) using the given damping coefficient ($c$) and critical damping coefficient ($c_c$).

Given values:

• Actual damping coefficient: $c = 35 \text{ N s/m}$
• Critical damping coefficient: $c_c = 350 \text{ N s/m}$

The damping ratio is calculated as:

$ \zeta = \frac{c}{c_c} $ $ \zeta = \frac{35 \text{ N s/m}}{350 \text{ N s/m}} $ $ \zeta = 0.1 $

Calculating Logarithmic Decrement

The logarithmic decrement ($\delta$) is related to the damping ratio ($\zeta$) by the formula:

$ \delta = \frac{2 \pi \zeta}{\sqrt{1 - \zeta^2}} $

Substitute the calculated damping ratio ($\zeta = 0.1$) into the formula:

$ \delta = \frac{2 \pi (0.1)}{\sqrt{1 - (0.1)^2}} $ $ \delta = \frac{0.2 \pi}{\sqrt{1 - 0.01}} $ $ \delta = \frac{0.2 \pi}{\sqrt{0.99}} $ $ \delta \approx \frac{2 \times 3.14159 \times 0.1}{0.994987} $ $ \delta \approx \frac{0.628318}{0.994987} $ $ \delta \approx 0.6315 $

Rounding to two decimal places, the logarithmic decrement is approximately 0.63.