Shock Response Spectrum Simulator

Numerically integrate the response of a single-degree-of-freedom (SDOF) system to a half-sine shock pulse. Adjust pulse amplitude, pulse duration, natural frequency and damping ratio to see the peak response acceleration, response amplification, full Shock Response Spectrum and time-history update in real time — for sizing electronics, PCBs and packaging against drop and qualification shocks.

Parameters

Pulse amplitude G_p

Peak input acceleration of the shock pulse, in g

Pulse duration τ

Half-sine pulse width. MIL-STD-810 standard values are 6 ms and 11 ms

System natural frequency f_n

First natural frequency of the structure you are protecting (PCB, chassis, etc.)

Damping ratio ζ

Viscous damping ratio of the structure. 0.02-0.05 is typical for electronics

Allowable peak acceleration

Shock limit of the structure / mounted component you must protect

Results

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Freq. ratio ν = f_n·τ

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Response amplification

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Peak response accel. (g)

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Input acceleration (g)

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Safety factor allow/peak

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Shock verdict

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Shock pulse and SDOF response — animated

Left: half-sine input pulse (G_p, τ). Right: absolute response acceleration of an SDOF system at f_n. Colour reflects the peak-to-allowable ratio.

Shock Response Spectrum — peak response vs natural frequency

Time history — input pulse and SDOF response acceleration

Theory & Key Formulas

$$\ddot u+2\zeta\omega\dot u+\omega^{2}u=G_p\,g\,\sin\!\left(\frac{\pi t}{\tau}\right)\ \ (0\le t\le\tau)$$

Equation of motion of a viscously-damped SDOF system. u is the displacement relative to the base, ω = 2π·f_n, G_p·g is the input acceleration amplitude and τ the pulse width. For t>τ the right-hand side vanishes and the system rings down as residual motion.

$$\text{SRS}(f_n) = \max_t|\ddot u_{\text{abs}}(t)|, \qquad \ddot u_{\text{abs}} = -2\zeta\omega\dot u-\omega^{2}u$$

The SRS plots the peak absolute response acceleration of SDOF oscillators of varying f_n. For an undamped half-sine pulse the SRS peaks near ν = f_n·τ ≈ 0.5 with an amplification of about 1.77; damping flattens both the peak and the residual response. This tool solves the equation by explicit time integration.

What is the Shock Response Spectrum (SRS)?

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"Shock response spectrum" — that's a heavy-sounding name. How is it different from a normal shock waveform (time vs. acceleration)?

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Confusing at first, isn't it? The SRS is not the waveform — it is a map of how a structure responds to that waveform. The horizontal axis is natural frequency f_n; the vertical axis is the peak response acceleration. A 50 g shock shakes a soft structure with f_n = 10 Hz very differently from a stiff PCB with f_n = 500 Hz. The SRS lets you look up your structure's natural frequency on the x-axis and read off what acceleration it actually sees. That makes it the everyday tool of every shock-qualification engineer.

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Got it. So what does the SRS curve generally look like — flat?

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A pretty interesting shape. At low frequencies (natural period much longer than the pulse) the amplification is small, because the shock is already over before the structure has time to respond. At high frequencies (natural period much shorter than the pulse) the amplification approaches 1 — the structure follows the input quasi-statically, so response equals input. The fun part is in between: around ν = f_n·τ ≈ 0.5, where the natural period is about twice the pulse width, an undamped half-sine response shoots up to about 1.77×. You can see that hump clearly on the SRS chart at the top right.

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1.77×! So when I design electronics I should keep the natural frequency away from that "hump", right?

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Exactly. For example MIL-STD-810 classical shock tests commonly use τ = 6 ms or 11 ms. With τ = 11 ms, the danger band sits around 45 Hz. If a PCB's first natural frequency lands there, the resonance amplification can kill it. The classic countermeasures are (1) add ribs or more mount points to push f_n above the band, (2) put a cushion under the assembly so the pulse gets longer (τ ↑) and the danger frequency shifts down, or (3) add a viscoelastic damper so ζ goes up and the peak comes down. Even going from ζ = 0 to 0.05 already trims a lot off the peak — try the ζ slider on the left.

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Damping also kills the residual ringing — I can see in the time-history chart that after the pulse ends the oscillation dies out fast. Does that "afterswing" really matter in real drop testing?

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It matters a lot. In some cases the residual response after the pulse ends is actually larger than the peak during the pulse — that's why test labs sometimes split the SRS into the "primary" (during the pulse) and "residual" (after the pulse) parts. Solder-joint fatigue, for example, doesn't accumulate during the pulse itself; it accumulates over the tens of cycles of ringdown that follow. That's why a real shock spec looks at the whole SRS shape and at the residual duration, not just the peak g. The time-history chart on the right shows exactly the free vibration after the pulse, and you'll see it collapse much faster as you raise ζ.

Frequently Asked Questions

The Shock Response Spectrum (SRS) is a plot of the peak response acceleration of an array of imaginary single-degree-of-freedom (SDOF) oscillators, each with a different natural frequency, when subjected to a given shock pulse. It does NOT show the shock waveform; instead it answers the question 'how hard does a structure with natural frequency f_n get shaken when hit by this shock?'. Two shocks with the same peak g-level can have very different SRS curves and therefore very different risks for an electronic assembly. Avionics, automotive ECUs and defence electronics are all qualified by demonstrating that their internal stress responses to a specified test SRS stay within limits.

The half-sine pulse is the classical shape adopted by international shock-test standards such as MIL-STD-810 Method 516, IEC 60068-2-27 and JIS C 60068-2-27. It is a reasonable approximation of a free-fall impact on a floor, a packaged-goods drop through a cushion or a handling bump, and it can be fully specified with just two parameters — pulse width τ and peak acceleration G_p. Drop towers and electrodynamic shakers reproduce it precisely. This tool simulates exactly that test condition with an SDOF model.

For an undamped SDOF system, the SRS of a half-sine pulse peaks near a frequency ratio ν = f_n·τ ≈ 0.5 (natural period ≈ 2τ), where the maximum response amplification is about 1.77. At lower frequencies the shock is over before the structure can respond — the response is impulse-governed and amplification is small. At higher frequencies (ν » 1) the structure follows the input quasi-statically and the amplification approaches 1. Adding 5-10% of damping suppresses the residual response and lowers the peak. The SRS curve in this tool shows that characteristic asymmetric hump.

Take the natural frequency f_n of your PCB or chassis (from a modal analysis or hammer-impact test) and read off the response ratio at that f_n on the SRS for the worst expected shock. PCBs with f_n in the 100-500 Hz range hit by millisecond-scale pulses (τ = 5-20 ms) typically land at ν = 0.5-10 with amplification 1.5-1.7×, so a 50 g external shock becomes 75-85 g for the resonant component. The three classical fixes are: (1) move f_n away from the test pulse band (add ribs, increase mount points), (2) lengthen the input pulse with foam or elastomer cushions so the dangerous frequency drops, and (3) raise the damping ratio ζ with viscoelastic dampers to flatten the peak.

Real-World Applications

Avionics and defence electronics: Ejection seats, missile pyrotechnic events and naval gun-firing shocks hit on-board electronics with peaks of hundreds of g over a few milliseconds. Equipment is qualified by showing its internal stress response stays inside the allowable when subjected to the specified test SRS. MIL-STD-810 Method 516.6 codifies a family of standard SRS shapes (terminal-peak sawtooth, trapezoidal and half-sine); during development engineers use an SDOF tool like this to estimate the response and to keep resonant bands clear of the test pulse spectrum.

Automotive ECUs and cameras: Curb hits, door slams and dropped units during tyre swaps subject vehicle electronics to tens of g routinely. Combining the ECU enclosure's natural frequency (typically 200-800 Hz) with an assumed pulse width (5-15 ms) and running the SRS in this tool gives the peak g seen by BGA and QFP packages on the PCB, which feeds directly into a solder-joint fatigue life estimate. The AEC-Q100/Q200 shock spec (500 g / 1 ms half-sine, 3 hits in each of 6 directions) is a natural fit for the inputs here.

Packaging design and logistics: Drop testing through corrugated boxes and EPE/EPS foam, the cushion's dynamic-compression curve fixes the equivalent shock pulse width τ. Sweeping G_p and τ in this tool shows how much amplification the contents see for each candidate cushion thickness and material — giving an engineering basis for the choice before any ISTA or ASTM D5276 drop test is run, and for picking a "fragility" g-level that the product survives.

HDD, SSD and mobile-device drop survival: Smartphones and external storage units that fall 1-1.5 m onto concrete see thousands of g for pulses under 0.5 ms. HDD actuator arms with natural frequencies of 1-3 kHz sit right in the ν = 0.5-1.5 amplification band for that pulse, which is the main reason head-slap damage occurs. Putting τ = 0.5 ms, G_p = 1500 g and f_n = 2000 Hz into this tool reproduces the kind of SRS peak that HDD designers fight every day.

Common Misconceptions and Pitfalls

The biggest misconception is treating the "peak input acceleration as the acceleration the component sees". The 50 g coming out of the shock machine is the input at the mounting base. SDOF theory shows that a component sitting in the resonance band (ν ≈ 0.5) is amplified by 1.5-1.7×, so it actually sees 75-85 g. A rigid block with a very high f_n, on the other hand, sees almost the same 50 g as the input. A datasheet line that reads "Shock: 50 g" is a test condition, not a component allowable — always keep them separate. The SRS curve in this tool visualises that amplification at every frequency.

Next, "a half-sine pulse equals a real shock" is not true either. Real-world shocks are irregular waveforms with noise and high-frequency content; the half-sine is chosen by test standards because it is repeatable and easy to specify, not because it matches reality. A proper ruggedisation programme compares the field SRS (obtained by SRS-transforming measured field data) against the test SRS to make sure the test envelopes the field. This tool is excellent for estimating the test side, but for field qualification you still need to cross-check it against measured SRS.

Finally, "just pick 5% damping and you're done" is dangerously sloppy. The typical value for electronics is ζ = 0.02-0.05, but it depends strongly on mounting stiffness, joint slip, solder viscosity and the presence of conformal coating or potting. Without a hammer-impact test on the real assembly to measure ζ (half-power method or logarithmic-decrement method), the computed SRS can differ from the actual response by a factor of two. At very high acceleration levels non-linear effects (bolt slip, solder yielding) increase ζ further. Sweep ζ from 0 to 0.05 to 0.10 in this tool to see how sensitive the peak is, and only commit to a design number once you understand that sensitivity.

Shock Response Spectrum Simulator

What is the Shock Response Spectrum (SRS)?

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

How to Use

Worked Example

Practical Notes