Single-Frequency Decay Waveform Fitting

Strike a tuning fork and the sound fades; strike a structure and the vibration slowly stops. How fast it fades is described by physical parameters that can be measured precisely. The module turns the decay into a parameter table for a report.

The problem it solves

Engineers often need the natural frequency, damping, starting amplitude and half-decay time of a resonant part — a tuning fork, an oscillator, a beam or a sensor housing — after it is excited and vibrates freely. An oscilloscope gives only a visual estimate, not repeatable numbers. The module fits the complete decay waveform to a standard mathematical model by least squares, derives all parameters from the data and reports the goodness of fit so that the reliability of the result can be judged.

Functions

The main window shows acceleration, velocity and displacement from top to bottom with the measured curve and the fitted curve overlaid, channel and statistics information on the left and the fit parameter panel and single-degree-of-freedom fit report on the right.

The module applies wherever a part is struck and the way it comes to rest is evaluated.

Quality grading of resonant components

The key value of tuning forks, resonators and oscillators is the quality factor Q: the higher Q, the sharper the frequency selectivity and the longer the stored energy lasts. The module reads Q and the half-power bandwidth directly from a free decay, without a frequency response test rig. Example: an 80 Hz resonance with damping ratio 0.0344 and Q 14.53, all from a single strike.

Structural damping identification

A single-mode free decay in a modal test, or residual vibration after a machine stops, can be used to extract the damping ratio — a basic input for vibration control design and fatigue life assessment.

Comparing materials and processes

Samples with different materials, heat treatments or damping treatments are each struck once; their logarithmic decrement and decay coefficient rank them objectively.

Checking sensor and instrument resonance

Mounted accelerometers and displacement probes have resonances, and the damping at resonance limits the usable bandwidth. A light hammer tap is enough for the check.

Acceptance of vibration isolators

Measure the free-decay characteristics of isolation pads and dampers on delivery and compare them with contract values within minutes.

Teaching and training

Test mode generates a standard decay with user-defined parameters; the fit should reproduce the set values exactly, forming a generate-and-identify loop for teaching damped vibration.

Single-degree-of-freedom decay model

a(t) = A · e^(−αt) · cos(2π · fd · t + φ), where α = 2π · ζ · fn

The four unknowns are the amplitude A, the damped frequency fd, the damping ratio ζ and the initial phase φ. Once these are known, the acceleration, velocity and displacement curves are fully determined.

Two-stage parameter estimation

A direct search in four dimensions is slow and can converge to false solutions, so the estimation runs coarse first, then fine.

Why refine? The half-power bandwidth method is meant for steady-state frequency responses; applied to a transient decay it systematically overestimates the damping ratio (by almost a factor of two in a measured case). Least-squares refinement fits the original waveform in the time domain and is not affected by window leakage. A built-in check with a test signal of A = 7 g, fd = 80 Hz and ζ = 0.035 recovers the set values to four decimal places, and remains stable with 2 % added noise.

Three quantities and curve comparison

Measured velocity and displacement are integrated from acceleration with DC and low-frequency drift suppressed. The fitted velocity and displacement are integrated analytically from the model with initial-value compensation. Both use the same basis, so measured and fitted curves can be compared point by point; their agreement is a direct check of the fit.

Derived parameters

fn is the undamped natural frequency (calculated from fd and ζ) and m the equivalent mass entered by the user (1 kg by default). All derived values update with the fit result.

Recommended settings

1. Check that the fit is reliable: A = 6.898 g, fd = 80.00 Hz, ζ = 0.0344, φ = 84.402°, with a correlation coefficient of 0.99722. The measured and fitted curves agree throughout all three plots, so the single-frequency model describes the waveform and the values can be used.
2. Frequency and decay: undamped natural frequency fn = 80.05 Hz; damped period Td ≈ 12.5 ms; decay coefficient α = 17.3 /s. The amplitude halves every 0.04 s (T½) and falls to 10 % after 0.133 s (t90) — about three vibration periods per halving, typical of a lightly damped resonance.
3. Frequency selectivity: Q = 14.53 and Δf = 5.507 Hz, so the peak width is about 6.9 % of the frequency — moderate selectivity. For use as a filter or timing reference, the frequency stability for this Q can be estimated directly from Δf / fd.
4. Design parameters: with an equivalent mass of 1 kg, k = ωn² · m ≈ 2.53 × 10⁵ N/m and c ≈ 34.6 N·s/m, which can be entered in a dynamic simulation model. The maximum velocity of 0.1346 m/s and maximum displacement of 0.2676 mm are inputs for strength and clearance checks.