2. Shock environments and loading specifications

Before any transient shock simulation can produce meaningful results, the analyst must translate a physical event — a drop, a pyrotechnic separation, a ballistic impact, a rail-car coupling — into a mathematical loading specification that the solver can consume. This chapter covers the four representations that dominate industrial practice: the half-sine pulse, the terminal peak sawtooth (TPS), the shock response spectrum (SRS), and the standards that codify them (MIL-STD-810, MIL-S-901, IEC 60068-2-27, and their industrial equivalents).

Time-domain pulses: half-sine and terminal peak sawtooth

A classical shock pulse is defined by an acceleration–time history applied to the equipment base. The two most common shapes are the half-sine and the terminal peak sawtooth.

The half-sine pulse is a single positive half-cycle of a sinusoid:

a(t) = A · sin(π·t / T)   for 0 ≤ t ≤ T
a(t) = 0                    otherwise

where A is the peak acceleration and T is the pulse duration. Its velocity change (the area under the curve) is Δv = 2·A·T/π. Because the pulse starts and ends at zero acceleration, it is relatively gentle on high-frequency content and is widely used to represent non-contact events such as package drops onto a padded surface.

The terminal peak sawtooth rises linearly to a peak and then drops abruptly to zero:

a(t) = A · t / T   for 0 ≤ t ≤ T
a(t) = 0           otherwise

Its velocity change is Δv = A·T/2. The instantaneous drop at t = T injects substantial high-frequency energy, which is why TPS pulses are the standard choice for representing pyrotechnic shock and impact-driven events where the structure experiences a sharp unloading.

Both pulses are fully described by the pair (A, T). A useful sanity check before running any simulation is to compute Δv and compare it against the physically expected velocity change of the event. If a 500 g, 2 ms half-sine is specified, Δv = 2 × 500 × 9.81 × 0.002 / π ≈ 6.24 m/s — a value the analyst should be able to justify from the drop height or impact velocity.

Always verify the sign convention and coordinate axis before applying a pulse. In most standards the acceleration is applied to the base of the equipment, and the equipment response is measured relative to that base. Mixing absolute and relative frames is a common source of sign errors in SRS post-processing.

From time domain to shock response spectrum

A single time-domain pulse is a poor predictor of how a complex piece of equipment will respond, because the response depends strongly on the natural frequencies of the mounted components. The shock response spectrum solves this by answering a different question: what is the peak response of a single-degree-of-freedom (SDOF) oscillator of a given natural frequency and damping, when subjected to this pulse?

The SRS is computed by sweeping a bank of SDOF oscillators across a frequency range (typically 10 Hz to 10 kHz for industrial equipment), each with a fixed damping ratio ζ (commonly 5%, or Q = 10). For each oscillator the absolute acceleration response is calculated, and its maximum is plotted against the oscillator's natural frequency. The result is a curve of peak acceleration versus frequency.

Two conventions matter and must be stated explicitly in any specification:

  • Absolute vs. relative acceleration: the absolute SRS plots the peak of the mass acceleration; the relative SRS plots the peak of the mass-to-base relative displacement or acceleration. MIL-STD-810 and most industrial specs use absolute acceleration.
  • Damping: a 5% damping ratio (Q = 10) is the near-universal default. Changing damping shifts the SRS peak by roughly a factor of Q, so a 10% damping SRS will be noticeably lower than a 5% one.

The SRS is not a unique description of a time history — many different pulses can produce the same SRS. This is both its strength (it captures the damaging potential across frequencies) and its weakness (it cannot be inverted to recover a unique time history). When a specification gives only an SRS, the analyst must synthesize a time history whose SRS envelopes the specified curve.

Standards and their loading specifications

Military and industrial standards provide the numerical values that populate A, T, and the SRS envelope. The table below summarizes the most frequently encountered ones.

Standard Typical loading Application
MIL-STD-810 Method 516 Half-sine, TPS, or SRS; e.g. 40 g / 11 ms half-sine General military equipment, functional and crash shock
MIL-S-901 (Grade A/B) Shock machine hammer blow; SRS envelope up to ~1000 g Shipboard equipment, underwater explosion
IEC 60068-2-27 Half-sine, TPS; e.g. 15 g / 11 ms Commercial and industrial electronics
MIL-STD-810 Method 517 Pyrotechnic SRS, 100 Hz–10 kHz Ordnance and separation shock

When a standard specifies an SRS, it is usually given as a piecewise log-log envelope with breakpoints. For example, a pyrotechnic SRS might rise at +6 dB/octave from 100 Hz to 1 kHz, plateau at 3000 g from 1 kHz to 3 kHz, then roll off at −6 dB/octave to 10 kHz. The analyst must digitize these breakpoints and either synthesize a time history or apply the SRS directly in a transient solver.

Applying a specification in practice

Consider a representative industrial example: a 12 kg control cabinet mounted on a skid, specified to survive a 30 g / 11 ms half-sine pulse per IEC 60068-2-27. The workflow is:

  1. Define the input: A = 30 g = 294.3 m/s², T = 0.011 s. Compute Δv = 2 × 294.3 × 0.011 / π ≈ 2.06 m/s.
  2. Build the model: mesh the cabinet and skid, assign material properties, and define the base as a rigid boundary condition with the prescribed acceleration.
  3. Run the transient solve: use a time step small enough to resolve the pulse — at least 20 points per period of the highest frequency of interest. For a 2 kHz cutoff, Δt ≤ 25 µs.
  4. Post-process: extract the acceleration at critical component locations (e.g. PCB mounts, relay brackets) and compute their SRS. Compare against the component's own qualification SRS.
  5. Judge: if the computed SRS at a component's natural frequency exceeds its rated limit, the design fails and requires isolation or stiffening.
A common failure symptom is a simulation that runs to completion but shows unrealistically low response. This usually means the pulse was applied as a displacement rather than an acceleration, or the time step was too coarse to capture the pulse. Check that the applied base acceleration integrates to the expected Δv before trusting any downstream result.

Visualizing the relationship

The SVG below shows how a time-domain pulse maps to an SRS, and how the two standard pulse shapes differ in their high-frequency content. The left panel shows the half-sine and TPS pulses in the time domain; the right panel shows their corresponding SRS curves, illustrating why the TPS produces higher response at high frequencies.

Time-Domain Pulses and Their Shock Response Spectra Time Domain (acceleration vs. time) time (ms) accel (g) half-sine TPS A A Shock Response Spectrum (5% damping) frequency (Hz, log) peak accel (g) 10 100 1k 10k half-sine SRS TPS SRS TPS retains more high-frequency energy → higher SRS above ~1 kHz

The key takeaway from the figure is that the two pulses, despite having the same peak acceleration A, produce different SRS curves. The half-sine rolls off more steeply above its corner frequency, while the TPS maintains higher response at high frequencies. This is why the choice of pulse shape is not arbitrary — it must match the physical event being represented. A padded drop is well modeled by a half-sine; a pyrotechnic separation is better modeled by a TPS or a synthesized SRS-compatible transient.

Practical guidance for specification handling

  • Always record the damping ratio when an SRS is specified. An SRS without a stated damping is ambiguous.
  • Check the velocity change of any time-domain pulse against the physical event. A pulse that implies an unrealistic Δv will produce unrealistic response.
  • Use the SRS as the acceptance criterion for component-level qualification, and the time-domain pulse as the input for system-level simulation.
  • When only an SRS is given, synthesize a time history (e.g. via a sum of decaying sinusoids or a wavelet-based method) and verify that its SRS envelopes the specification across the full frequency range.

With the loading specification properly defined and understood, the next step is to build a finite element model that can faithfully propagate this input through the equipment structure — the subject of the following chapter.

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