Shock Response Spectrum: Method

Shock Response Spectrum: the SDOF Shock Model, Smallwood's Ramp-Invariant Filter, Primary and Residual Spectra, Pseudo-Velocity and the MIL-STD-810H Test Spectra

The problem

A piece of equipment is dropped, a stage separates with a pyrotechnic charge, a store is ejected, a vehicle crashes, a ship takes a wave slam: the mounting of the equipment sees a short, strong transient acceleration. The transient lasts a few milliseconds and never repeats exactly, so neither a sine level nor a power spectral density describes it. The shock response spectrum, introduced by Biot in 1932 for earthquake engineering, describes a transient by what it does to a simple structure: a mass on a spring and damper fixed to the moving base, the single-degree-of-freedom (SDOF) oscillator of MIL-STD-810H Figure 516.8-1 and NASA-HDBK-7005 Figure 2.4. The transient is applied to a whole family of such oscillators, one at each natural frequency, and the largest response of each one is plotted against its natural frequency. That plot is the shock response spectrum (SRS). A component whose first mounted natural frequency is f_c and whose damping is near the assumed value will see about the spectrum value at f_c as its peak acceleration, whatever the detailed shape of the transient; two transients with the same spectrum are, for design and test purposes, the same shock.

The oscillator and its two responses

With y the base motion, x the motion of the mass and z = x − y their relative displacement (the stretch of the spring), Newton's law for the mass gives z̈ + 2ζω_nż + ω_n²z = −ÿ(t), where ω_n = 2πf_n = √(k/m) is the undamped natural frequency and ζ = c/(2√(km)) the damping ratio, written in shock work as the amplification Q = 1/(2ζ): Q = 10 is 5 % of critical damping, the standard assumption of MIL-STD-810H Method 516.8 and of most specifications; Q = 50 is 1 %, Q = 5 is 10 %. The absolute acceleration of the mass follows from the same equation, ẍ = −(2ζω_nż + ω_n²z). Two response quantities are in use:

Absolute acceleration ẍ(t): the acceleration the equipment feels, which gives the inertia load on its parts and the stress through a stress per g. Its largest magnitude over all time is the maximax absolute acceleration SRS, the standard measure of shock severity (NASA-HDBK-7005 section 2.2.10, MIL-STD-810H Method 516.8 Annex A paragraph 1.4). The transfer function from base acceleration to absolute acceleration is (2ζω_ns + ω_n²)/(s² + 2ζω_ns + ω_n²), which is 1 at zero frequency: a very stiff oscillator follows the base exactly, so every acceleration spectrum converges to the peak input acceleration at high natural frequency.

Relative displacement z(t): the deflection of the mounting, which strains the spring (the bracket, the isolator, the circuit-board bend). It is reported in three equivalent forms: z_max itself (the stroke a mounting must allow), the pseudo-velocity PV = ω_nz_max, and the equivalent static acceleration ω_n²z_max, which for an undamped oscillator equals the absolute acceleration exactly and for Q = 10 differs from it by a fraction of a percent. Smallwood's relative displacement model has the transfer function ω_n²/(s² + 2ζω_ns + ω_n²), also 1 at zero frequency. Gaberson and Chalmers (1969) showed that the peak modal stress of a structure is proportional to its modal velocity, which is why MIL-STD-810H asks for the pseudo-velocity spectrum beside the acceleration spectrum and why the pseudo-velocity is the better indicator of structural damage.

Primary, residual and maximax

While the pulse acts (0 ≤ t ≤ τ) the oscillator's response is the primary response; its largest positive and largest negative values are the positive and negative primary spectra. After the pulse the oscillator is in free vibration, e^−ζω_nt(P cos ω_dt + R sin ω_dt) with ω_d = ω_n√(1 − ζ²), decaying from the state the pulse left it in; its first positive and first negative extremes (the later ones are smaller) are the positive and negative residual spectra. The maximax spectrum is the largest of the four magnitudes. For a one-sided pulse such as the half-sine the negative primary is zero at low frequency (the mass is only pushed one way while the pulse lasts), the residual governs at low frequency, and the positive primary governs above about f_nτ = 0.7. A laboratory test must satisfy the spectrum in both polarities, so MIL-STD-810H asks for the positive and negative maximax (the "maximum and minimum") spectra and, for a classical pulse, a shock in each direction of each axis.

The three regions of a pulse spectrum

Low frequency. An oscillator much softer than the pulse is long (f_nτ ≪ 1) barely moves while the pulse acts: it sees an impulse, the base simply acquires the velocity change ΔV = ∫a dt, and the mass then swings with relative velocity ΔV. Its pseudo-velocity is therefore ΔV, its relative displacement ΔV/ω_n, and its peak acceleration ω_nΔV = 2πf_nΔV: the acceleration spectrum rises at 6 dB per octave and the pseudo-velocity spectrum is flat. Irvine's example of a 1 g, 11 ms half-sine (ΔV = 2 A τ/π = 2.70 in/s) gives 0.04, 0.2 and 0.4 g at 1, 5 and 10 Hz with Q = 10, which is the 6 dB per octave line of a constant 2.5 in/s: the damped oscillator reaches e^−ζπ/2ΔV = 0.924 ΔV, because its first residual peak comes a quarter period after the pulse and the response has decayed by then. The velocity change is the single number that governs this region, which is why MIL-STD-810H lets a half-sine replace a sawtooth of the same velocity change (Table 516.8-IV Note 1: A_half-sine = (π/4) A_sawtooth for the same duration) and why Procedure III specifies fragility shocks by ΔV.

The peak. When the natural period is about twice the pulse length the pulse pushes the mass through half a cycle in step with it and the response overshoots the input. For the undamped half-sine the maximax is 1.768 A at f_nτ = 0.81; with Q = 10 it is 1.650 A at f_nτ = 0.82. A rectangular pulse reaches 2 A (the step overshoot) once f_nτ ≥ 1/2; the sawtooth and haversine are gentler. Irvine's 50 g, 11 ms half-sine peaks at 82 g near 80 Hz.

High frequency. An oscillator much stiffer than the pulse is short follows the base: the spectrum converges to the peak input acceleration A from above (the terminal-peak sawtooth has an almost flat spectrum from its crossover frequency on, which is why MIL-STD-810H uses it as the reference classical pulse; the half-sine approaches A through decaying ripples). A pulse that begins with a jump is the exception: the rectangle and the initial-peak sawtooth hit the oscillator with a step, whose overshoot (1 + e^{−ζπ/√(1 − ζ²)}) A, 1.85 A at Q = 10, persists at every high frequency, so their spectra converge to that value and not to A (the terminal-peak sawtooth ends with a jump but starts smoothly, and converges to A). MIL-STD-810H calls the lowest frequency at which the spectrum reaches this convergence magnitude the crossover frequency F_co, and defines the effective duration of a synthesized test transient as T_E = 2/F_co; for the 11 ms sawtooth the standard's F_co = 45 Hz and for the 6 ms sawtooth 80 Hz, about 1/(2τ). The tool also reports a knee frequency A/(2πΔV), where the velocity asymptote of the low-frequency region meets the plateau.

Smallwood's ramp-invariant digital filter

The response of each oscillator is computed from the sampled input by a second-order recursion, Smallwood (1981) eq. 9:

y_m = b₀a_m + b₁a_m−1 + b₂a_m−2 − a₁y_m−1 − a₂y_m−2, with a₁ = −2C, a₂ = E², E = e^−ζω_nT, K = ω_dT, C = E cos K, S = E sin K, S′ = S/K, T the sample interval.

Absolute acceleration (eq. 7): b₀ = 1 − S′, b₁ = 2(S′ − C), b₂ = E² − S′.

Relative displacement as ω_n²z (eq. 8): b₀ = [2ζ(C − 1) + (2ζ² − 1)S/√(1 − ζ²) + Tω_n]/(Tω_n), b₁ = [−2CTω_n + 2ζ(1 − E²) − 2(2ζ² − 1)S/√(1 − ζ²)]/(Tω_n), b₂ = [E²(Tω_n + 2ζ) − 2ζC + (2ζ² − 1)S/√(1 − ζ²)]/(Tω_n).

The earlier recursion of Kelly and Richman (1969) is impulse invariant: it is exact when the input is a train of impulses at the sample instants, which makes it in error whenever the natural frequency exceeds about a sixth of the sample rate (its spectra dip at half the sample rate and rise at the sample rate, as Smallwood's Figures 4 and 5 show). Smallwood derived the filter from the ramp-invariant transformation H̃(z) = [(z − 1)²/(Tz)] Z{L⁻¹[H(s)/s²]}: it is the exact response to the straight lines joining the samples, so by superposition it is exact for any piecewise-linear input at any natural frequency, above the sample rate included, at the cost of one more multiply-add per sample. The only approximation left is the input itself, which is why the tool samples a classical pulse at least 400 times over its length and 20 times per cycle of the highest natural frequency, and refines each sampled extreme by the parabola through its three samples. Both filters have unit gain at zero frequency; the paper's small-ω_nT power series (eqs 14–21) are reproduced by the exact weights to better than 1 % at f_nT = 10⁻³ and are checked in the test script. Smallwood warns that with ζ exactly zero and a small natural frequency round-off can make the filter unstable; the tool's undamped comparison uses ζ = 10⁻⁶. MIL-STD-810H Method 516.8 names Smallwood's ramp-invariant algorithm (its 2005 generalization) as the way the SRS is to be computed.

The residual spectrum needs the response for at least one full cycle after the pulse, which at a low natural frequency is a long tail of zero-input samples. The tool follows Rubin's remark in the discussion printed with the paper: after the last input sample the response is the free vibration above, fixed exactly by two consecutive filter outputs (P from the first, R from the second), and its extremes are at tan ω_dt = (ω_dR − ζω_nP)/(ω_dP + ζω_nR) and half a period later. For a pasted record sampled too coarsely for that construction (more than about a tenth of a cycle per sample at the natural frequency in question) the tail is filtered instead.

Closed forms for the classical pulses

For an undamped oscillator the state at the end of any pulse is Duhamel's integral, and the residual amplitude is ω_n|A(ω_n)| where A(ω) = ∫a(t)e^−iωtdt is the Fourier transform of the pulse: for the half-sine, with β = 2f_nτ, A·2β|cos(πβ/2)|/|β² − 1| (A π/2 in the resonant case β = 1), which vanishes at β = 3, 5, 7, …; for the rectangle 2A|sin(πf_nτ)|; for the sawtooths, haversine and trapezoid the sums of the elementary transforms of their straight segments, which the tool evaluates. During the half-sine the undamped response is ẍ/A = β²/(β² − 1)[sin(πt/τ) − sin(βπt/τ)/β], whose extremes lie at πt/τ = 2πk/(β ± 1) and at the end of the pulse; the rectangle gives A(1 − cos ω_nt). The tool runs its filter with ζ = 10⁻⁶ and prints these closed forms beside the digital result at every grid frequency of a classical pulse, which checks the filter, the residual construction and the sampling together. The decaying sinusoid a = A e^{−ζ_d2πf_dt} sin(2πf_dt), the building block of pyroshock synthesis (Smallwood and Nord 1974), has no end; the tool follows it until its envelope has fallen to 0.5 % of A.

MIL-STD-810H Method 516.8 test spectra and classical pulses

When no measured data exist, Method 516.8 defines the functional and crash-hazard test shocks by a spectrum (Figure 516.8-2, Table 516.8-III): a line proportional to frequency from 10 Hz to the crossover frequency F_co, then a flat level to 2000 Hz, at Q = 10, with the effective durations T_e = 2.5/f_min and T_E = 2/F_co. The categories are functional test for flight equipment 20 g (F_co 45 Hz), functional test for ground equipment 40 g (20 g for materiel mounted only in trucks and semi-trailers), launch/eject during captive carry 30 g, crash hazard for flight equipment 40 g, and crash hazard for ground equipment 75 g with F_co = 80 Hz. The spectrum "approximates that of the perfect terminal-peak sawtooth pulse", and the standard gives the sawtooth itself as the last-resort classical alternative: Table 516.8-IV, 20 g / 30 g / 40 g peak at 11 ms for the functional test, and Table 516.8-XIII, 40 g at 11 ms (flight) and 75 g at 6 ms (ground) for the crash hazard, each to be applied in both directions of each axis. The half-sine appears in the standard for the high-speed-craft functional shock (Table 516.8-V: 20 g and 5 g at 23 ms) and as the velocity-equivalent substitute of Table 516.8-IV Note 1; the trapezoid is the fragility pulse of Procedure III, 10 to 50 g with the duration set by the drop height (Table 516.8-VIII). A 50 g, 11 ms half-sine is therefore not a MIL-STD-810H functional shock; the tool's default case is Irvine's published example of that pulse, chosen because its spectrum values are printed.

The standard's tolerance on a test shock specified by its spectrum (paragraph 4.2.2.2) is −1.5 dB to +3 dB over at least 90 % of the one-twelfth-octave points from 10 to 2000 Hz and −3 dB to +6 dB over the rest, with the maximax spectrum computed at Q = 10. The tool evaluates 20 log₁₀(computed/specification) at every grid point inside the band and reports the extremes and the fraction of points inside each tolerance; a classical pulse seldom meets a spectrum tolerance over the whole band, which is why the standard asks for a synthesized complex transient on a shaker and allows the sawtooth only when nothing else is available.

Units and conventions

Accelerations are in g (9.80665 m/s², 386.09 in/s²) and durations in milliseconds in both unit systems; frequencies in Hz. A pasted record is two columns, time and acceleration, with its own time unit (s or ms) and acceleration unit (g, m/s², in/s², ft/s²); the sample interval is the median spacing of the time column, a record whose spacing varies by more than 1 % is resampled by linear interpolation, and a record that does not end at zero gets one zero sample appended so that the residual is that of the record as pasted. The pseudo-velocity and velocity change are in in/s (USCS) or m/s (SI), the relative displacement in inches or millimetres. The stress per g and the allowable stress are the only inputs that go through the unit system (ksi or MPa).

Limits

The oscillator is linear and has one degree of freedom, and its mass does not load the base (the zero-impedance assumption of NASA-HDBK-7005 section 2.2.10). A component with several modes in the band needs the spectrum applied mode by mode with the effective modal masses (the generalized modal shock spectrum procedure of NASA-HDBK-7005 section 5.1.4), or a transient finite-element analysis; isolators with snubbers, friction or rate-dependent stiffness are not linear oscillators; the spectrum of a pasted record is only as good as the record (zero shift, clipping and noise floor are discussed in Method 516.8 Annex B). The tool does not synthesize a test transient from a required spectrum, nor check a pulse against the time-domain tolerance bands of Figures 516.8-3 to 516.8-5.

Reading the results

The input section states the pulse, its peak, duration and velocity change and shows it in time together with the response of the component at its natural frequency. The filter section prints Smallwood's weights at three frequencies of the grid. The spectrum section gives the peak of the maximax spectrum and its frequency, the ratio to the peak input, the convergence magnitude at the top of the grid (raise the highest frequency until it equals the peak input), the crossover and knee frequencies and T_E, the log-log chart with the chosen parts and any specification lines, and a table of every spectrum at every grid frequency. For a classical pulse the next section compares the filter at ζ = 10⁻⁶ with the undamped closed forms. The pseudo-velocity chart shows the ΔV plateau and the A g/(2πf) asymptote. The specification section gives the dB extremes and the tolerance fractions, and the component section the peak acceleration, pseudo-velocity and stroke at f_c, the peak stress and its margins of safety.

Component check and the load-level convention

The stress at the critical point follows the peak acceleration of the component through a stress per g, σ_1g, from a static 1 g analysis or a unit-acceleration load case in a finite-element model. That peak stress is a limit stress: a shock test level or a flight shock environment is the limit load of MIL-STD-810H Method 516.8 and NASA-HDBK-7005 practice, so the tool offers no load-level radio and writes the margins in the site's convention. Strength margins against F_tu and F_cy are written at ultimate load, M.S. = F / (FS · σ_1g SRS(f_c)) − 1 with FS the ultimate factor (default 1.5), and labelled ULT; yield margins against F_ty and F_cy are written at limit load times the yield factor, M.S. = F / (FS_yld · σ_1g SRS(f_c)) − 1 (default 1.15, the factor several agencies require on limit load for no detrimental permanent set), and labelled LIM. The response oscillates and reaches both signs, so the compressive allowable is checked on the same peak magnitude, F_cy standing for the compression allowable of a stable element; a part that can buckle or cripple needs that allowable instead. Each box names the level of its allowable. A shock is a few cycles and rarely a fatigue question; the random vibration of the same mission usually is (Random Vibration Fatigue tool).