Logarithmic Decrement (δ): Quantifying Damped Oscillations
Logarithmic decrement, symbolised δ, measures the rate at which successive peaks of a damped oscillation diminish. It equals the natural logarithm of the ratio between two consecutive peak amplitudes, providing a direct estimate of viscous damping when a system behaves as a lightly damped single-degree-of-freedom oscillator.
Combine this reference with the vibration exposure points calculator to translate measured decay envelopes into actionable maintenance decisions.
Definition and Formulae
For a damped system with peak amplitudes x₁ and x₂ measured one period apart, logarithmic decrement is δ = ln(x₁ / x₂). When more than two peaks are available, averaging improves accuracy: δ = (1/n) ln(x₁ / xₙ₊₁). This formulation assumes the damping ratio ζ is much less than one so that successive peaks are well defined and the exponential decay approximation holds.
The unit is dimensionless, yet its interpretation relies on consistent measurement of displacement, velocity, or acceleration. Signal conditioning, filtering, and sampling rate all influence the precision of peak detection, making well-calibrated sensors and data acquisition systems essential.
Historical Perspective
The metric traces back to 19th-century studies of pendulum clocks and bridge vibrations, where engineers observed exponential decay under air resistance and joint friction. With the advent of structural dynamics in the mid-20th century, logarithmic decrement became standard in modal testing, enabling rapid estimation of damping ratios without solving differential equations for each mode.
Today, vibration analysts still rely on the technique when commissioning buildings, aircraft, and rotating machinery. Digital signal processing has expanded the toolset, but logarithmic decrement remains a foundational concept in standards such as ISO 7626 for modal analysis.
Conceptual Links
Relationship to Damping Ratio
For lightly damped systems, ζ = δ / √(4π² + δ²). When δ is small, this simplifies to ζ ≈ δ / (2π). Engineers often determine δ from free-decay measurements and then compute ζ to compare against design specifications or control requirements.
Connection with Quality Factor
The mechanical quality factor Q equals π/δ for lightly damped oscillators. This relationship bridges energy-based interpretations of damping with amplitude-based ones, providing consistency between modal testing and frequency-response measurements obtained from swept-sine experiments.
Multiple-Degree-of-Freedom Systems
In complex structures, each mode possesses its own logarithmic decrement. Experimental modal analysis isolates modes using impact hammers or shaker excitation, then applies curve fitting to the decay envelope of each mode separately. Documentation should clearly state which mode and measurement location each reported δ corresponds to.
Practical Measurement Considerations
Signal processing. Apply band-pass filtering around the mode of interest before peak picking to minimise contamination from other modes or noise. Use logarithmic decrement in tandem with the vector magnitude calculator when combining triaxial accelerometer readings.
Data sufficiency. Collect enough cycles to average out measurement noise but stop before structural non-linearities appear. The resonant frequency calculator illustrates the importance of staying near the natural frequency when exciting modes.
Documentation. Record test temperatures, boundary conditions, and sensor mounting methods. These factors alter damping through changes in material properties or joint preload, and they help analysts compare future decay tests with the same baseline.
Importance for Engineering Decisions
Logarithmic decrement informs maintenance schedules, tuned mass damper design, and certification of aerospace and civil structures. Excessive decay rates may signal energy dissipation through friction or material damping that will generate heat, while low decay may indicate insufficient damping and risk of resonance.
Integrating δ data into digital twins helps teams predict vibration response under new loads or configurations. Aligning reports with standards and linking to calculators such as the vibration exposure tool ensures stakeholders can translate measurements into occupational safety and structural reliability insights.