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From Projectile to Equipment: the Ground-Shock Chain
The real job of a protective structure is to protect what is inside, not its walls. This article follows the chain from the point where the projectile stops to the shock seen by sensitive equipment inside: free-field ground shock, the equivalent pulse that carries it to the equipment, and isolator selection. In the end we find that a few metres of penetration decide whether isolation is possible at all.
The calculations and text of this article were prepared with Claude (Anthropic) and checked by the site's verification tests.
1. The links of the chain
In a buried protective structure, the shock reaching the equipment is computed in four steps: (1) where the projectile stops (PRT-01, PRT-02); (2) the distance between that point and the equipment; (3) the free-field shock at that distance; (4) the part of the shock that passes through the isolator to the equipment. Each link carries — and sometimes amplifies — the uncertainty of the previous one.
2. Free-field ground shock
Drake and Little (1983) compiled, from tests, the shock produced in soil by penetrating conventional weapons; their analysis underlies the ground-shock chapter of TM 5-855-1 (1986). With scaled distance Z = R / W1/3 (ft/lb1/3), peak particle velocity, acceleration and pressure are:
Here c is the seismic velocity of the soil, ρc its acoustic impedance and n the attenuation coefficient; f is the coupling factor of the charge to the soil (1 when fully buried). Soil types and coefficients:
| Soil | c (ft/s) | ρc (psi/(ft/s)) | n |
|---|---|---|---|
| computing… | |||
3. An independent check of the relations
The coefficient of the acceleration relation (50) can be derived from the velocity relation: with arrival time ta = R/c, rise time tr ≈ 0.1 · ta and a₀ ≈ V₀ / tr, the coefficient becomes 160 / (0.1 · g) = —. Agreement with the published 50 shows that the relations are internally consistent and that the unit conversions in the engine are right.
4. Distance and soil
Acceleration falls with Z−n−1 — in dense sand roughly with the 3.5th power of distance. For a burst of 2700 kg TNT equivalent:
| Distance R | Dense sand a₀ | Heavy clay/shale a₀ | Near field? |
|---|---|---|---|
| computing… | |||
Doubling the distance reduces acceleration roughly tenfold. The soil matters a great deal too: heavy clay and shale attenuate shock much less than dense sand (smaller n, larger c).
5. The pulse carried to the equipment
We idealise the motion entering the equipment's isolator as a half-sine pulse with peak acceleration a₀ and velocity change V₀; the pulse duration is τ = π · V₀ / (2 · a₀). This is an explicit idealisation that loses the detail of the real waveform but keeps the two most important quantities (peak and impulse).
6. Choosing an isolator
The equipment is treated as a single-degree-of-freedom system on a spring and damper, with a half-sine base acceleration. As the isolator becomes softer (lower natural frequency), the acceleration reaching the equipment falls, but the equipment's motion relative to the base — the required rattle space — grows. The design question is: which is the stiffest isolator that meets the equipment limit, and how much rattle space does it need? The engine solves this with the Newmark method; it is verified against the transmissibility formula under harmonic excitation and against the analytical solution for an undamped half-sine.
7. The whole chain: three cases
2700 kg TNT equivalent, dense sand, damping ratio ζ = 0.2 and an equipment limit of 40 g: the MIL-STD-810H functional-shock level for ground equipment (Method 516.8, Table 516.8-IV). Three distances: the worst case of the front-block arrangement from PRT-02 (R = 16.0 m), the reference distance in the reference calculation note (R = 13 m; the note used a parametric 50 g limit), and the worst non-perforating case of the five-thin-groups arrangement (R = 5.9 m).
| Case | R | a₀ | V₀ | τ | Stiffest isolator | Rattle space |
|---|---|---|---|---|---|---|
| computing… | ||||||
Try your own scenario in the Ground Shock Explorer; in the Penetration Explorer the stopping point is handed over to it directly.
8. Limits
9. Check yourself
- In dense sand, by roughly what factor does peak acceleration fall if the distance doubles? From which coefficient?
- Why does the same burst produce a larger acceleration in heavy clay/shale?
- Why is softening the isolator not always good? Which quantity limits it?
- Given a₀ and V₀, what is the duration of the equivalent half-sine? Is a short pulse good or bad for an isolator?
- Which design changes would you consider to meet the 40 g limit in the R = 5.9 m case?
References
- Drake, J. L., Little, C. D. (1983). Ground Shock from Penetrating Conventional Weapons. U.S. Army Waterways Experiment Station (DTIC ADP001706).
- U.S. Department of the Army (1986). TM 5-855-1: Fundamentals of Protective Design for Conventional Weapons.
- Harris, C. M., Piersol, A. G. (eds.). Harris' Shock and Vibration Handbook — transmissibility in base-excited isolation.
- Chopra, A. K. Dynamics of Structures — response of a single-degree-of-freedom system to base excitation.
- U.S. Department of Defense (2019). MIL-STD-810H: Environmental Engineering Considerations and Laboratory Tests, Method 516.8 Shock — Tables 516.8-III and 516.8-IV.
All figures in this article are produced by engine/ground-shock.js and engine/shock-isolation.js when the page loads. The ground-shock engine is within 0.3% of its source table (tests/sim-yer-soku.test.js); the isolation engine is verified against analytical solutions (tests/shock-isolation.test.js).
A few metres of penetration decide whether isolation is possible.