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Irvine's 50 g, 11 ms half-sine shock: the Q = 10 shock response spectrum by Smallwood's ramp-invariant filter, its closed forms and the MIL-STD-810H test spectrum
| Tool | Shock Response Spectrum |
| Source | Irvine, T. (2012). An Introduction to the Shock Response Spectrum, Revision S. Vibrationdata, Madison, AL. Figure 5 (coordinates printed on the chart) and Table 1; Smallwood, D. O. (1981). "An Improved Recursive Formula for Calculating Shock Response Spectra." The Shock and Vibration Bulletin, Bulletin 51, Part 2, pp. 211–217, eqs. 7–9; U.S. Department of Defense (2019). MIL-STD-810H Method 516.8, Table 516.8-III and Figure 516.8-2. |
| Level | Professional |
Problem
A component is mounted on a structure that receives a 50 g, 11 ms half-sine shock. Compute the shock response spectrum for Q = 10 from 10 to 10,000 Hz at 1/6 octave with Smallwood's ramp-invariant recursive filters (absolute acceleration and relative displacement), separate the primary and residual parts, check the undamped closed forms of the half-sine against the digital result, find the velocity change, the peak of the spectrum, the crossover and knee frequencies and the pseudo-velocity, compare the spectrum with the MIL-STD-810H functional test SRS for flight equipment (20 g, F_co 45 Hz), and find the peak acceleration and the stress margin of a component with f_c = 80 Hz and 0.5 ksi per g against a 60 ksi allowable.
Inputs to Enter
- Half-sine, A = 50 g, τ = 11 ms; Q = 10 (ζ = 0.05); grid 10 to 10,000 Hz at 6 points per octave (61 frequencies); MIL-STD-810H category: functional test, flight equipment
- Component f_c = 80 Hz, stress per g 0.5 ksi, allowable 60 ksi, FS 1.0, limit level
Answers
- Velocity change ΔV = 2 A τ/π = 135.2 in/s (3.434 m/s); the pulse sampled every 5 µs (2,202 samples) so that the filter's straight lines follow it
- Maximax SRS 55.44 g at 30 Hz (residual governs; book 55 G), 82.41 g at 80 Hz (positive primary; book 82 G), 70.18 g at 140 Hz (book 70 G); negative primary zero below 100 Hz; peak of the spectrum 82.49 g at 71.3 Hz (f τ = 0.78, 1.650 A); convergence to 50.00 g at 10 kHz
- Undamped closed forms per unit A beside the filter at ζ = 10⁻⁶: residual 2β|cos(πβ/2)|/|β² − 1| = 0.9428 at f τ = 0.25 (digital 0.9427), 1.5708 at β = 1, zero at β = 3; primary 2/3 at f τ = 0.25, 1.768 at f τ = 0.80 (the undamped maximax peak 1.768 at f τ = 0.81)
- Crossover frequency (first reaches the convergence magnitude) 26.4 Hz, T_E = 2/F_co = 75.7 ms; knee A/(2π ΔV) = 22.7 Hz; pseudo-velocity 63.1 in/s and relative displacement 0.1256 in at 80 Hz, 124 in/s at 10 Hz approaching ΔV
- Against the 20 g / 45 Hz test SRS: +7.97 dB at 1,140 Hz to +13.2 dB at 10 Hz, outside the −1.5/+3 dB tolerance at every point (over-test); the 20 g, 11 ms sawtooth of Table 516.8-IV itself stays within −1.5/+3 dB of that line
- Component at 80 Hz: 82.41 g, σ = 0.5 × 82.41 = 41.21 ksi (284.1 MPa), M.S. = 60/41.21 − 1 = +0.46 (limit)
Notes
The tool's default inputs are this case; press Submit. Irvine's chart coordinates are printed to the nearest gram; the filter reproduces them within 1 % and agrees with a DOP853 integration of the oscillator equation to 4 × 10⁻⁶ (References/Shock-Response/checks/srs_check.py). MIL-STD-810H Method 516.8 has no 50 g half-sine functional shock: its classical pulse is the 20 g, 11 ms terminal-peak sawtooth, also checked in test_shock_response_spectrum.php (its Q = 10 spectrum first reaches 20 g at 37.6 Hz and is 22.2 g at the standard's 45 Hz crossover).