The logarithmic decrement dd is a measure of the damping in a system. If the amplitude of the n th peak in a free vibration response is A n and the amplitude of the (n + 1) th peak is A n+1 , what is the formula for the logarithmic decrement dd?
- Aδ = (1/n) ln(An/An+1)
- Bδ = (1/n) ln(An+1/An)
- Cδ = ln(An/An+1)Correct
- Dδ = ln(An+1/An)
Solution
Concept
Logarithmic decrement in structural dynamics
Formula
δ = ln(An / An+1)
Step-by-step solution
1. Logarithmic decrement is defined as the natural logarithm of the ratio of any two consecutive successive amplitudes in a free damped vibration response. 2. For amplitudes An and An+1 of successive peaks, the formula for logarithmic decrement is δ = ln(An / An+1). Answer: C
Explanation
The logarithmic decrement is defined as the natural logarithm of the ratio of two consecutive peak amplitudes in a free damped vibration response. When comparing adjacent peaks, the formula simplifies directly to the natural logarithm of An divided by An+1. Therefore, option C correctly states the fundamental equation for logarithmic decrement between successive cycles.
Why the other options are wrong
A) The factor (1/n) is used when comparing amplitudes separated by n cycles, not adjacent peaks. B) The logarithmic decrement is based on the ratio of successive amplitudes An/An+1, not An+1/An, and does not include 1/n for adjacent peaks. D) The formula uses the ratio of current amplitude to next amplitude An/An+1, not An+1/An.
Shortcut
Log decrement is the natural log of earlier peak over next peak: δ = ln(An / An+1).
Common mistake
Selecting option A by incorrectly including the factor (1/n) when considering two consecutive peaks.
Correct answer: C) δ = ln(An/An+1)