Step 2: Damping Analysis
Estimate damping ratios using logarithmic decrement method from free vibration decay
Methodology
The damping analysis uses the logarithmic decrement method on the free vibration decay portion:
- Decay Detection: Automatically identifies when forced vibration ends and free decay begins by detecting consistent negative slope in the amplitude envelope.
- Peak Extraction: Finds local maxima and minima in the decay portion using prominence and distance constraints based on the natural period.
- Logarithmic Decrement: Computes $\delta = \frac{1}{n}\ln\left(\frac{x_0}{x_n}\right)$ averaged over $n$ cycles for stability.
- Damping Ratio: Calculates $\zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} \approx \frac{\delta}{2\pi}$ for small damping.
- Multi-Floor Analysis: Estimates damping from each floor's modal acceleration to assess consistency and compute mean ± standard deviation.
Mode 1 Damping Analysis
Mode 2 Damping Analysis
Mode 3 Damping Analysis
Damping Summary
The visualizations above show four key analyses for each mode:
1. Full Time History
Complete acceleration response showing forced and free vibration portions.
Complete acceleration response showing forced and free vibration portions.
2. Decay with Peaks
Free vibration decay portion with detected peaks marked for logarithmic decrement calculation.
Free vibration decay portion with detected peaks marked for logarithmic decrement calculation.
3. Envelope Decay
Hilbert transform envelope showing exponential decay fit for damping estimation.
Hilbert transform envelope showing exponential decay fit for damping estimation.
4. Floor Comparison
Damping ratio estimates from each floor with mean value for consistency assessment.
Damping ratio estimates from each floor with mean value for consistency assessment.