Pyroshock Analysis
Theory and Physics
What is Pyroshock?
Professor, what is pyroshock?
It is a high-frequency shock generated by the operation of pyrotechnic devices. It occurs during rocket stage separation, satellite separation, bolt cutters, etc.
Characteristics:
- Extremely high frequency components — 100 Hz to 100 kHz
- Short duration — Several ms or less
- Very high acceleration — Thousands to tens of thousands of G
- Little structural damage but destroys electronic equipment — Relays, crystal oscillators, HDDs, etc.
Tens of thousands of G acceleration! The structure doesn't break, but electronic components do?
Because pyroshock is dominated by high-frequency components, the structural response itself (low-frequency deflection) is small. However, electronic components are sensitive to high frequencies, leading to solder joint detachment or relay malfunctions.
SRS (Shock Response Spectrum)
SRS (Shock Response Spectrum) is used to evaluate pyroshock. It is a plot of the maximum response of single-degree-of-freedom systems at each natural frequency.
So it lists the maximum response at each frequency.
Using SRS, we determine "whether the response in this frequency band exceeds the allowable value." Allowable SRS values are specified in NASA-STD-7003.
Analysis with FEM
FEM for pyroshock requires accurately tracking high frequencies (on the order of kHz), necessitating very fine meshes and small $\Delta t$.
- Explicit FEM — $\Delta t$ is automatically small. Suitable for high frequencies
- SEA (Statistical Energy Analysis) — Statistical response for high frequencies
- FEM-SEA Hybrid — Low-frequency FEM + High-frequency SEA
Summary
Key Points:
- High-frequency shock (100 Hz to 100 kHz) — Generated by pyrotechnic device operation
- Evaluated with SRS (Shock Response Spectrum) — NASA-STD-7003
- Electronic components are the primary damage target — Structure usually remains intact
- FEM requires high-frequency meshing — Consider hybrid with SEA
Pyrotechnic Shock is the Greatest Challenge for Spacecraft
When a satellite's separation pyrotechnic device (bolt cutter/explosive bolt) operates, a shock acceleration of tens of thousands of G occurs within a few μs. This pyrotechnic shock is evaluated by SRS (Shock Response Spectrum), and the harsh environment exceeding 10,000 G at 2000 Hz is a major cause of electronic equipment destruction. It became a serious problem during the Apollo program in the 1960s-70s and was systematized as NASA HDBK-7005.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during a sudden brake? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration can be ignored." It absolutely cannot be omitted for shock loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), the force of the tires pushing the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"—sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extension
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | For mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N for mm system, N for m system |
Numerical Methods and Implementation
Pyroshock FEM
Mesh requirement to resolve high frequencies: $\lambda_{min} / 6$ or less. For 10 kHz elastic wave (steel: $c = 5000$ m/s):
If 80 mm elements are fine, then it's not that detailed.
That's true up to 10 kHz, but if 50 kHz is needed, then $h < 17$ mm. For 100 kHz, $h < 8$ mm. Covering the entire pyroshock frequency range with FEM alone incurs high computational cost.
SEA (Statistical Energy Analysis)
At high frequencies (above 1 kHz), modal density is high, making it less meaningful to track each discrete FEM mode individually. SEA is a method that statistically calculates the average energy flow between subsystems, optimal for high frequencies.
Solver
| Tool | Method |
|---|---|
| LS-DYNA | Explicit FEM. Up to mid-frequencies |
| VA One (ESI) | FEM-SEA Hybrid. Standard for pyroshock |
| Wave6 (Free Field Tech) | FEM-SEA Hybrid |
Summary
SRS was conceived by Shepard in 1932
The Shock Response Spectrum (SRS) concept was proposed by Charles Shepard in 1932 for earthquake motion evaluation. It plots the maximum response of single-degree-of-freedom systems with various natural frequencies to a shock input as a function of frequency. Test and analysis methods are specified in MIL-STD-810H Method 516.8 and ECSS-E-HB-32-25A. Standard settings for SRS calculation are 1/12 octave frequency steps with Q (quality factor) = 10.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (Locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Achieves quadratic convergence within convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once but in small increments. The arc-length method (Riks method) can trace beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Pyroshock in Practice
In spacecraft equipment design, evaluate the integrity of electronic components under pyroshock environments.
Practical Checklist
How JAXA Satellites are Protected by SRS Analysis
For JAXA's "Daichi 2" (ALOS-2, launched 2014), the satellite separation shock SRS environment from the H-IIA rocket was pre-evaluated using modal superposition response analysis with MSC Nastran. Simulation confirmed that the satellite's received shock environment (SRS) fell within component test specification values, and this was used to set vibration test input conditions. The industry acceptance criterion for FEM prediction of pyrotechnic shock is agreement within ±6 dB of actual measurements.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate (post-processing visualization). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If mesh quality is poor, the results will be a mess no matter how good the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver always returns "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize across at least three mesh densities—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraints is actually the most critical step in the entire analysis.
Software Comparison
Pyroshock Tools
Related Topics
なった
詳しく
報告