Technical Note · Shock Testing
Shock Response Spectrum (SRS)
What an SRS actually is, how to read one, why very different waveforms can share the same SRS, and why shaker-based SRS testing uses synthesized wavelets instead of classical pulses.
Key takeaways
- An SRS does not describe the shock itself. It describes what the shock would do to a set of simple resonant parts: the peak response of single-degree-of-freedom oscillators, plotted against their natural frequency.
- By convention SRS is computed with Q = 10 (5% damping) at 1/6- or 1/12-octave spacing. The Q used must always be stated.
- At high frequency the SRS approaches the peak acceleration of the input. At low frequency it is set by the shock's velocity change. In between it can exceed the input peak by 50–100% or more.
- An SRS discards duration and phase, so many different waveforms share the same SRS. That is the weakness of the method, and also what makes it practical to reproduce on a shaker.
- Classical pulses like a 50 g half-sine leave the table moving at several meters per second. SRS synthesis with wavelets produces waveforms with zero net velocity and displacement, which a shaker can actually reproduce.
Why a shock needs a different description
Random vibration is described by a PSD and sine vibration by an amplitude at each frequency. Neither works for a shock. A pyrotechnic separation, a hard landing, a dropped container or a gunfire burst lasts milliseconds, never repeats, and its damage depends on how the hardware responds to it, not on the waveform's exact shape.
Engineers needed a single description that answers the practical question: how hard will this shock drive the resonances in my equipment? The shock response spectrum, developed in the 1930s and 1940s and now standard in MIL-STD-810, NASA and ESA test standards, answers it directly.
How an SRS is computed
Imagine mounting a whole row of simple mass-spring-damper oscillators on the structure that receives the shock, each tuned to a different natural frequency but all with the same damping. Apply the shock, record the largest acceleration each oscillator reaches, and plot those peaks against the oscillators' frequencies. That plot is the SRS.
In practice nobody builds the oscillators. The controller or analysis software applies the measured shock to a digital model of each oscillator, typically using the Smallwood recursive filter algorithm, and records the peak. Figure 2 shows what three of those oscillators do when a 50 g, 11 ms half-sine is applied.
Repeat that for oscillators every 1/6 octave from 10 to 2000 Hz and plot the peaks, and you get the SRS of the half-sine.
Reading the three regions of an SRS
| Region | Oscillator behavior | What sets the SRS level |
|---|---|---|
| High frequency (oscillator period much shorter than the pulse) | Stiff oscillators simply follow the input | The peak acceleration of the input. Here 50 g. |
| Mid frequency (period comparable to the pulse duration) | Pulse and oscillator interact; amplification occurs | Pulse shape and duration. This is where the SRS can exceed the input peak. |
| Low frequency (period much longer than the pulse) | Soft oscillators barely move during the pulse, then ring afterward | The velocity change (area under the pulse). The SRS rises at about 6 dB/octave. |
The vocabulary: maximax, primary and residual
Each oscillator's response can be split in time and by sign, giving several related spectra. Specifications usually call for maximax, but the others are useful for diagnosis and pulse selection.
| Term | Meaning |
|---|---|
| Maximax SRS | The largest absolute response at any time, during or after the pulse. The default in most specifications. |
| Primary SRS | The largest response while the pulse is being applied. |
| Residual SRS | The largest response after the pulse has ended, while the oscillator rings freely. |
| Positive / negative SRS | Largest response in each direction. A big difference shows that the shock is directionally biased. |
| Q (damping) | Damping of the reference oscillators. Q = 10 (ζ = 5%) is the standard default. |
| Octave spacing | Frequency spacing of the oscillators: 1/6 octave is common; 1/12 octave gives finer detail. |
| Absolute acceleration vs. pseudo-velocity | Most test specifications plot absolute acceleration (g). Pseudo-velocity SRS, plotted on four-coordinate paper, is preferred by some for relating shock to stress. |
Classical pulses compared
Half-sine, terminal-peak sawtooth and trapezoid pulses have been specified for decades because they are easy to define and to produce on drop and pneumatic shock machines. With the same peak and duration they produce noticeably different SRS.
| Pulse | Velocity change | SRS peak | SRS at 20 Hz | SRS at 2000 Hz |
|---|---|---|---|---|
| Terminal-peak sawtooth | 106 in/s (2.7 m/s) | 59 g at 63 Hz | 31 g | 50 g |
| Half-sine | 135 in/s (3.4 m/s) | 82 g at 71 Hz | 39 g | 50 g |
| Trapezoid | 191 in/s (4.9 m/s) | 93 g at 50 Hz | 54 g | 52 g |
The terminal-peak sawtooth is often preferred when a classical pulse must be used. Its abrupt end sets every oscillator ringing, so its residual SRS stays within 2% of its maximax at every frequency, while the half-sine's residual has deep nulls where some resonances are barely excited after the pulse. In practice that makes the sawtooth's effect more uniform across frequency and less sensitive to exactly where the test article's resonances fall.
The role of Q
The SRS depends on the damping chosen for the reference oscillators, so an SRS without a stated Q is incomplete. Higher Q means less damping, larger oscillator responses and a higher SRS, especially in the mid-frequency region.
Why Q = 10? It is a convention, not a measurement of your hardware. Five percent damping is a reasonable middle value for many structures, and using one agreed value lets specifications and test results be compared. If the actual hardware is much more lightly damped, its real resonant responses will be higher than the Q = 10 SRS suggests.
Many waveforms, one SRS
Because an SRS keeps only the peak response of each oscillator, it discards the waveform's duration, phase and the number of times each response peak is repeated. Very different waveforms can therefore have the same SRS. Figure 7 shows two waveforms, built from different sets of wavelets, that both meet the same requirement.
This non-uniqueness cuts both ways. It is the main criticism of SRS-based testing: two "equivalent" shocks can stress real hardware differently, particularly in how many high-amplitude cycles they apply. But it is also what makes SRS testing on a shaker possible, as the next section explains.
SRS testing on a shaker
A classical pulse on a shaker has a fundamental problem: its velocity change. A 50 g, 11 ms half-sine leaves the table moving at 135 in/s (3.4 m/s), and if nothing stopped it, the table would travel 7.5 inches in the next 50 milliseconds, far beyond any electrodynamic shaker's stroke. Classical shock on a shaker is only possible by adding pre- and post-pulses that bring velocity and displacement back to zero, and those added pulses change the SRS.
SRS synthesis avoids the problem. The controller builds a waveform from a set of short wavelets, each a few cycles of a sine at one frequency shaped by a half-sine window. Each wavelet starts and ends at zero velocity and displacement, so their sum does too. The controller adjusts each wavelet's amplitude, iterating until the measured SRS matches the requirement.
| Waveform | Peak acceleration | Peak velocity | Net velocity at end | Peak displacement |
|---|---|---|---|---|
| 50 g, 11 ms half-sine (uncompensated) | 50 g | 135 in/s | 135 in/s | Unbounded (7.5 in after 50 ms) |
| Synthesized waveform A (Figure 7) | 16 g | 13 in/s | ≈ 0 | 0.19 in |
| Synthesized waveform B (Figure 7) | 17 g | 13 in/s | ≈ 0 | 0.16 in |
Wavelet parameters such as the number of half-cycles, delays and polarity give the controller freedom to reduce peak acceleration, velocity and displacement for a given SRS, so the same shaker can meet more demanding requirements. Some specifications also constrain the effective duration of the waveform, which limits that freedom and should be checked when a shaker test is substituted for a drop or pyroshock test.
Where SRS is used
| Application | Typical source of the requirement | Typical test method |
|---|---|---|
| Functional and crash-hazard shock, military equipment | MIL-STD-810 Method 516 | Shaker SRS synthesis or classical pulse; drop machine |
| Pyroshock from stage separation, bolt cutters | MIL-STD-810 Method 517, NASA and ESA standards | Resonant plate or bar, pyrotechnic simulators; shaker for lower-frequency portions |
| Shipboard shock | MIL-DTL-901 and related specifications | Shock machines and floating barge tests |
| Gunfire shock | MIL-STD-810 Method 519 | Shaker with SRS-based or time-waveform replication methods |
| Seismic qualification | IEEE 344 and building codes (response spectra at higher damping) | Shake tables, often multi-axis |
Common misconceptions
| Misconception | Reality |
|---|---|
| "The SRS is the spectrum of the shock." | It is the spectrum of oscillator responses, not a Fourier spectrum of the input. Its units are peak g, not g/Hz or g²/Hz. |
| "An SRS of 40 g means the shock reaches 40 g." | The input can peak well below the SRS level (Figure 8) or above it. Only at high frequency does the SRS approach the input peak. |
| "Same SRS means same damage." | The SRS ignores duration and the number of response cycles. Two shocks with the same SRS can produce different fatigue damage and different nonlinear behavior. |
| "Q = 10 reflects my hardware's damping." | It is a reference convention. Lightly damped hardware will respond more than the Q = 10 SRS suggests. |
| "A half-sine test can always be run on a shaker." | Only with compensating pre- and post-pulses, and only if the shaker's stroke and velocity limits allow. SRS synthesis is usually the practical route. |
Quick reference
| Quantity | Formula | Rule of thumb |
|---|---|---|
| Standard damping | Q = 10, ζ = 0.05 | Always state Q with the SRS |
| Oscillator spacing | fk+1 = fk · 21/6 | 1/6 octave ≈ 12% steps |
| Velocity change | ΔV = area under the acceleration pulse | Half-sine: ΔV = 2·A·T/π |
| Half-sine ΔV example | 2 × 50 g × 0.011 s / π × 386 in/s² per g | 135 in/s (3.4 m/s) |
| High-frequency SRS | → peak input acceleration | Read the input peak from the right-hand end |
| Low-frequency SRS | ≈ 2πf · ΔV / g (velocity-change line) | Rises ≈ 6 dB/octave |
| Half-sine SRS peak | ≈ 1.6–1.7 × A near 0.8/T | 50 g, 11 ms → ≈ 82 g near 71 Hz |
How Panther helps
- SRS Shock Synthesis: Panther Shock synthesizes waveforms to match a user-defined SRS, then controls the shaker in closed loop so the measured SRS falls within tolerance.
- Classic Shock Control: half-sine, sawtooth, trapezoidal and other classical pulses, for specifications that require them.
- Shock Response Analysis: computes the SRS of measured or recorded shocks, so field data can be compared with specifications.
- Accurate capture: sample rates up to 262,144 samples/s and 24-bit acquisition with more than 110 dB of dynamic range capture both the large low-frequency motion and the fast high-frequency content of a shock.
References and further reading
- D. O. Smallwood, "An Improved Recursive Formula for Calculating Shock Response Spectra," Shock and Vibration Bulletin 51, 1981.
- ISO 18431-4, Mechanical vibration and shock — Signal processing — Part 4: Shock-response spectrum analysis.
- MIL-STD-810H, Methods 516.8 (Shock), 517.3 (Pyroshock) and 519.8 (Gunfire Shock).
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- T. Irvine, "An Introduction to the Shock Response Spectrum," Vibrationdata.
- Spectral Dynamics Technical Library: Resonance; Power Spectral Density (PSD) and Grms.
