Shock response spectrum

The formula

The core idea is a single-degree-of-freedom (SDOF) oscillator — a mass on a spring with a dashpot — driven by moving its base. The equation of motion in relative coordinates z = x - y (mass motion minus base motion):

z̈ + 2·ζ·ωₙ·ż + ωₙ²·z = -ÿ(t)

Reading it: the base acceleration ÿ(t) is the forcing; each oscillator answers with its own motion set by its natural frequency ωₙ and damping ratio ζ. That is the whole shock, filtered through one resonator.

The number you actually plot is the peak of the absolute acceleration the mass feels, over the entire response including ringdown:

SRS(fₙ) = max over t of | ẍ(t) |     for an oscillator tuned to fₙ = ωₙ/(2π)

Reading it: one oscillator gives one peak. Sweep fₙ across the band of interest (say 100 Hz to 10 kHz), keep damping fixed, and the collection of peaks versus frequency is the shock response spectrum.

Damping is carried by the quality factor Q:

Q = 1 / (2·ζ)

Reading it: Q = 10 means ζ = 0.05, five percent of critical — the near-universal default for SRS work. State it or the plot is meaningless.

Where you meet it

Pyroshock qualification on the bench. A separation nut, explosive bolt, or frangible joint fires and rings the structure at thousands of g above 1 kHz for a couple of milliseconds. Nobody specifies that jagged time history. They specify an SRS envelope, and your job is to reproduce it — mechanically, on a resonant plate rung by a projectile, or on a shaker synthesizing a decaying-sinusoid transient — until the measured SRS sits inside the required tolerance band.
Test-stand review board. You bring a shaker-synthesized shock. The customer's requirement is an SRS curve with a knee frequency, a low-frequency slope, and a plateau. The board compares your measured SRS, not your waveform, against the spec with its ±3 dB or ±6 dB tolerance lines. Arguments about waveform shape end the moment everyone agrees to talk in SRS.
Environments from a flight or field measurement. You instrument a real event — stage separation, landing gear touchdown, a hammer test on a bracket — and reduce the accelerometer record to an SRS to compare against the maximum predicted environment. The SRS is the common currency between what the hardware saw and what the spec allowed.
Component fragility. A relay, a crystal oscillator, or an optics mount has a shock rating expressed as an SRS. You check whether the ride it gets, expressed the same way, stays under the rating with margin.

How it works

The SRS trades the full, un-repeatable transient for a robust summary of what it does to structures. Two very different-looking shocks that stress hardware the same way land on nearly the same SRS — that is the point and the value.

The shape tells a story. At high frequency, where the oscillator is far stiffer than anything the pulse can bend, the mass simply follows the base and the SRS flattens to the peak input acceleration. Run the numbers on a 1 ms half-sine of 100 m/s² peak: the SRS climbs, peaks, then settles back toward 100 as fₙ runs past 20 kHz — dead-on the input peak. At low frequency, where the oscillator is soft and slow, the pulse is over before the mass gets moving, and the response rolls off with a slope tied to the pulse's velocity change. In between there's a resonant hump where the oscillator period is comparable to the pulse duration and the response amplifies above the input. For a lightly damped half-sine that amplification peaks near fₙ·T ≈ 0.8 at about 1.77× the input as damping goes to zero, dropping to roughly 1.65× at Q = 10 — both numbers you can reproduce with a filter and a peak-finder.

The honest computation is a digital recursive filter. The near-universal method is Smallwood's ramp-invariant algorithm: it models the oscillator as a digital filter whose impulse response matches the analog one when the input is interpolated with straight ramps between samples. Earlier impulse-invariant filters went badly wrong once natural frequencies climbed past about a sixth of the sample rate. Smallwood's version stays accurate even with natural frequencies near or above the sample rate, which is exactly the regime pyroshock lives in.

Now the gotchas, because this is where hardware passes a paper test and fails in flight.

The SRS is not invertible. Many time histories share one SRS. Passing the spectrum does not mean you reproduced the environment — you reproduced an environment with the same peak-oscillator-response signature. Velocity content and total energy can differ wildly between two waveforms with the same SRS, and overtest or undertest hides in that gap.
State the damping and the type. Q = 10 is convention, not law. And "SRS" alone is ambiguous: maximax takes the peak over the whole response; primary (initial) takes the peak while the pulse is still acting; residual takes the peak of the free ringdown after the pulse ends. Quote a maximax curve against a primary spec and you will lose an afternoon.
Positive and negative branches differ. A raw shock is not symmetric. Serious work carries both the positive and negative maximax, and often the max of the two absolute values.
Sample rate and filtering are load-bearing. Anti-alias filtering that clips real high-frequency content will quietly lower your SRS and let marginal hardware pass. For pyroshock, sample rates need to reach well past 100 kHz.
It's linear. The SDOF model is linear and elastic. It says nothing about yielding, joint slip, or plastic hinging in the real part. The SRS predicts stress in a structure that behaves like the model — a useful fiction that stops being useful once things go nonlinear.

History

The tool comes from earthquakes, not rockets. Maurice Anthony Biot, a Belgian-born engineer, worked out the response-spectrum idea as a young researcher at Caltech, in and around his doctoral work near 1932. His notion was a clean inversion of the usual approach: instead of chasing the exact, hopeless shape of a ground-motion record, characterize a shock by what it does — the peak response of a set of simple oscillators spanning a range of natural frequencies. He laid it out in papers in 1933 and 1934, framing the response spectrum as the practical way to reason about how buildings would answer a quake ^[1]^[2].

The idea sat mostly on paper until it met real data. In 1941, also at Caltech, George W. Housner took the newly recorded 1940 El Centro strong-motion accelerogram — one of the first good ones in the United States — and ground out response spectra by hand, using graphical integration, in his doctoral work. Through the late 1940s and 1950s Housner and colleagues at Caltech turned the response spectrum from a theorist's construct into standard earthquake-engineering practice, including analog machines built to compute spectra faster than pencil and paper allowed ^[3]^[4].

Aerospace inherited the method when the space age created a shock problem earthquakes never posed: pyrotechnics. Separation events, stage jettison, and bolt-cutter firings produce shocks that are too high in frequency and too brief for classic drop or half-sine testing to represent. The same SRS logic transferred cleanly, and by the 1960s the U.S. defense and space community had adopted it as the language for pyroshock. The 1981 arrival of Smallwood's ramp-invariant recursive filter made accurate digital computation routine ^[5]. Today the SRS is the specified quantity in the standards engineers actually cite — MIL-STD-810 Method 517 for pyroshock and NASA-STD-7003 for spacecraft pyroshock test criteria among them ^[6]^[7].

Related tools

/tools/vibration-natural-freq
/tools/spring-rate
/tools/db-converter

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →