Calculate logarithmic decrement, damping ratio, natural frequency, and damped frequency from oscillation amplitude data. Analyze underdamped vibration systems.
The Logarithmic Decrement Calculator determines damping characteristics from free vibration data. The logarithmic decrement (δ) is the natural log of the ratio of two successive peak amplitudes in an underdamped oscillating system — a fundamental measurement in structural dynamics, mechanical vibration analysis, and control systems. It gives a fast way to quantify how quickly a vibration dies out after an impulse or release.
From the logarithmic decrement, the calculator derives the damping ratio (ζ), quality factor (Q), half-life of oscillation, and the relationship between natural and damped frequencies. Engineers use these values to assess structural integrity, tune vibration isolators, and design stable control systems.
Enter peak amplitudes from successive oscillation cycles or directly input the logarithmic decrement value. The tool supports multi-cycle averaging for better accuracy, visual amplitude decay curves, and comparisons against common damping ratio ranges for different materials and structures. It gives a quick damping estimate without fitting the whole response by hand.
Use this calculator when you have peak decay data and need damping ratio, quality factor, or decay rate without fitting the whole response by hand. It is useful for vibration testing, structural checks, and control-system tuning when you want a quick damping estimate from measured peaks. That makes it easier to turn oscilloscope data into a usable damping number.
Logarithmic Decrement: δ = (1/n) × ln(x₁/x₂), where n = number of cycles between peaks. Damping Ratio: ζ = δ / √(4π² + δ²). Quality Factor: Q = π/δ. Damped Frequency: ωd = ωn × √(1 - ζ²). Half-life: t₁/₂ = ln(2) / (ζ × ωn).
Result: δ = 0.2877, ζ = 0.0458, Q = 10.92
With amplitudes of 10 and 7.5 over 1 cycle: δ = ln(10/7.5) = 0.2877. Damping ratio ζ = 0.2877 / √(4π² + 0.2877²) = 0.0458 (4.58%). Quality factor Q = π/0.2877 = 10.92. The system completes about 11 oscillations before amplitude drops to 1/e.
In an underdamped single-degree-of-freedom system, free vibration follows the equation x(t) = A × e^(-ζωnt) × cos(ωdt - φ). The peak amplitudes form a geometric series with ratio e^(-δ), where δ = 2πζ/√(1-ζ²) is the logarithmic decrement.
This elegant relationship means that measuring just two successive peaks gives you the damping ratio — one of the most important parameters in vibration engineering. The method is popular because it only requires free vibration data, not forced response testing.
In real testing, logarithmic decrement is measured using accelerometers or displacement sensors. The structure is excited (impact hammer, step release, or ambient vibration) and allowed to vibrate freely. Peak amplitudes are then extracted from the time history.
For better accuracy, measure amplitude ratios over n cycles and compute δ = (1/n)ln(x₁/xₙ₊₁). This averages out individual measurement errors. Some advanced methods use the entire envelope of the decay curve for least-squares fitting.
Damping assessment using logarithmic decrement is used across civil, mechanical, and aerospace engineering. Building designers verify that structural damping meets code requirements. Machine designers check that rotating equipment doesn't resonate excessively. Aerospace engineers ensure aircraft wings and fuselages have adequate flutter margins.
The logarithmic decrement (δ) is the natural logarithm of the ratio of successive peak amplitudes in free vibration. It quantifies how quickly oscillations decay due to damping. A δ of 0 means no damping; larger values mean faster decay.
The damping ratio ζ = δ / √(4π² + δ²). For small damping (δ << 2π), this simplifies to ζ ≈ δ / (2π). The damping ratio ranges from 0 (undamped) to 1 (critically damped), with values above 1 being overdamped.
Bare steel structures: ζ ≈ 0.5-1%. Reinforced concrete: 2-5%. Bolted steel: 2-3%. Welded steel with concrete fill: 5-7%. More damping means vibrations die out faster.
The quality factor Q = π/δ ≈ 1/(2ζ) for small damping. It represents the number of oscillation cycles for amplitude to decay to 1/e (≈37%) of its original value. Higher Q means less damping and longer ring-down.
Measuring over multiple cycles and averaging reduces measurement error. Instead of comparing two adjacent peaks, measure peaks n cycles apart and divide δ by n. This is especially useful when decay per cycle is small.
It only applies to underdamped systems (ζ < 1) where oscillations are visible. For critically damped or overdamped systems, use time-domain curve fitting or step response methods instead.