Shock Response Spectrum

Shock Response Spectrum (structures/loads/shock-response-spectrum)

Use when the task is computing the shock response spectrum (SRS) of a transient base acceleration pulse for equipment shock qualification: each SDOF oscillator on the frequency grid responds to the pulse at a fixed damping, the peak pseudo acceleration of every oscillator forms the spectrum, and the frequency of the maximum response shows which structural or equipment mode the shock amplifies most. This leaf implements the standard SRS model in pure Python, stdlib only, with RK4 time integration of the oscillator equation. It pairs with structures/loads/random-vibration-analysis, the steady-state counterpart that covers the random PSD regime of the same SDOF equipment model, and sits alongside structures/loads/gust-maneuver-loads in the loads pack. The transient shock regime is this leaf's claim; the DO-160 test condition planner for equipment lives in avionics/do160/ environmental-qualification and consumes these response levels.

Domain quick reference

• Quality factor and damping: zeta = 1/(2Q), Q_DEFAULT = 10.0.
• Base-excited SDOF oscillator (relative displacement x, natural frequency wn = 2pifn in rad/s): x_ddot + 2zetawnx_dot + wn^2x = -a_base(t).
• Peak pseudo acceleration: wn^2 * max|x(t)|, the SRS ordinate. For a lightly damped oscillator this equals the peak absolute acceleration of the oscillator mass within the Q-squared correction.
• Half-sine pulse: a(t) = Asin(pit/T) for 0 <= t < T, else 0.
• Decaying-sine pulse: a(t) = Asin(2pifdt)exp(-t/tau) for t >= 0 with fd = 1/T and tau default 3T.
• Integration: fixed-step RK4 from rest with dt = min(1/(fn50), T/200) per oscillator. Each oscillator is integrated over its excitation support: the pulse end T for the half-sine, and the time the envelope Aexp(-t/tau) first falls below 1% of A (ENVELOPE_FLOOR) for the decaying sine. The SRS ordinate is the peak of the forced response over that support. The ideal half-sine leaves a residual base velocity; the low-frequency free ring after the pulse is an artifact of that idealization and is excluded, which keeps the classical half-sine SRS shape (see the worked example anchors).
• Output: one dict per grid frequency {freq_hz, peak_ms2, peak_g}; the curve maximum carries the amplifying frequency.
• Units: amplitude m/s2 (G = 9.80665), durations s, frequencies Hz, ordinates m/s2 and g. SI throughout.
• FAR 25 frames the equipment and structure dynamic loads context; the relations above are standard engineering methodology, summary-only.

Workflow

1. State the shock input: pulse type (half-sine or decaying-sine), amplitude A in m/s2 (or g times 9.80665), pulse duration T in s, and the quality factor Q (default 10). For the decaying sine set the decay time tau (default 3*T).
2. Choose the SRS frequency grid: the natural frequencies of the equipment modes under evaluation, typically a log-spaced band around the pulse content, for example [5..1000] Hz for a 10 ms pulse.
3. Call srs_curve with the pulse and grid. The module computes zeta = 1/(2Q), integrates each oscillator with RK4 at dt = min(1/(fn*50), T/200) over the excitation support, and returns the curve list of {freq_hz, peak_ms2, peak_g}.
4. Call max_response on the curve for the peak pseudo acceleration and its amplifying frequency, the shock qualification driver.
5. Read individual ordinates off the curve: the low-frequency branch (oscillators slow relative to the pulse) rises from near zero, the amplification peak sits where the pulse spectrum meets the resonance, and the high-frequency branch approaches the input amplitude as the oscillator follows the base.
6. Confirm the deterministic checks with the contract test scripts/test_shock_response_spectrum.py.

Worked example

Half-sine pulse, amplitude 10 g (98.0665 m/s2), duration 10 ms, Q 10, grid [5, 10, 20, 30, 40, 50, 60, 80, 100, 150, 200, 300, 500, 1000] Hz:

• At 5 Hz the oscillator is slow relative to the pulse: 0.309 g, consistent with the anchor 0.31 g.
• The curve maximum sits at 80 Hz with 16.462 g, consistent with the anchor 16.5 g near 80 Hz.
• At 1000 Hz the oscillator follows the base: 10.111 g, consistent with the anchor 10.1 g high-frequency asymptote.
• Shape: 20 Hz gives 4.367 g and 500 Hz gives 10.427 g, both below the 80 Hz peak, so the amplification band is monotone up to and down from the maximum.

Intermediate half-sine ordinates (g): 10 Hz 1.202, 30 Hz 8.459, 40 Hz 12.222, 50 Hz 14.557, 60 Hz 15.678, 100 Hz 16.201, 150 Hz 14.192, 200 Hz 12.110, 300 Hz 11.009.

Decaying-sine case, amplitude 10 g, T 10 ms, tau 30 ms, Q 10: the pulse content sits at fd = 1/T = 100 Hz and the curve maximum is 35.780 g at exactly 100 Hz, the resonance of the decaying transient.

Pitfalls

• Reading the low-frequency branch as zero response: oscillators slow relative to the pulse still respond (0.309 g at 5 Hz for the worked 10 g pulse); the SRS rises from near zero only at very low frequency.
• Expecting the maximum at the pulse's own frequency: for the half-sine the amplification peak sits where the pulse spectrum meets the resonance (80 Hz for a 10 ms pulse, not 1/T = 100 Hz); the decaying-sine maximum, in contrast, sits at 1/T exactly.
• Trusting the free ring after a half-sine: the ideal pulse leaves a residual base velocity whose low-frequency ring is an artifact of the idealization and is excluded from the support, keeping the classical SRS shape.
• Choosing a grid that skips the peak: the maximum and its amplifying frequency only appear if a grid point lands in the amplification band, so the grid must cover the equipment modes and the pulse content.
• Confusing pseudo acceleration with peak absolute acceleration: the SRS ordinate is wn^2 * max|x(t)|, which equals the peak absolute acceleration only within the Q-squared correction for light damping.
• Feeding non-physical pulse inputs: amplitude or duration at or below zero, q <= 0.5, empty grids, frequencies at or below zero, unknown pulse types and non-positive decay tau all raise ValueError.

Verification

• Confirm the half-sine worked example values above: 0.309 g at 5 Hz (within 10% and within 0.05 g of the 0.31 g anchor), maximum at 80 Hz at 16.462 g (frequency in [60, 100] Hz, peak within 10% of the 16.5 g anchor), 10.111 g at 1000 Hz (within 10% of the 10.1 g anchor), and the monotonic checks peak(20 Hz) < peak(80 Hz) and peak(500 Hz) < peak(80 Hz).
• Confirm the decaying-sine maximum is finite and sits at 100 Hz, inside [70, 130] Hz.
• Confirm srs_curve output is deterministic: repeated calls return identical curves (no RNG anywhere).
• Confirm ValueError rejection: amplitude <= 0, pulse duration <= 0, quality factor q <= 0.5, empty frequency grid, any natural frequency <= 0, unknown pulse type, non-positive decay tau for the decaying sine, and non-positive wn, out-of-range zeta, non-positive total time or time step in sdof_peak.
• Run the contract test offline: python3 scripts/test_shock_response_spectrum.py (32 tests, deterministic).

Related leaves

• structures/loads/random-vibration-analysis: the steady-state counterpart for the random regime of the same SDOF equipment response model; together the two leaves cover the shock and vibration qualification regimes.
• structures/loads/gust-maneuver-loads: the loads pack context for airframe limit loads from discrete gust and maneuver conditions.
• avionics/do160/environmental-qualification: test condition planning for equipment environmental qualification, the consumer of these response levels.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_shock_response_spectrum.py

The test covers the analytic base acceleration histories (half-sine support and decaying-sine envelope), the half-sine worked example anchors (5 Hz 0.31 g, maximum near 80 Hz at 16.5 g, 1000 Hz 10.1 g, monotonic shape), the exact deterministic module values, the decaying- sine maximum near 1/T with its finite resonant peak, determinism of repeated runs, the g to m/s2 unit consistency of the curve dicts, the direct sdof_peak round trip, the effect of damping on the resonant peak, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: FAR 25 frames the dynamic loads and equipment context; the SDOF response relations above are standard engineering methodology, summary-only per standards-map.yaml (far-25 reference-only).
• compliance: STANDARDS-REF, gated: false.