Shock Response Spectrum (SRS)
Theory and Physics
What is SRS?
Professor, is SRS (Shock Response Spectrum) the same concept as the seismic response spectrum?
The concept is the same, but the input is different. The seismic response spectrum is the maximum response to "seismic waveforms", while SRS is the maximum response to shock waveforms (half-sine wave, pyroshock, etc.).
Shock test standards are often specified using SRS, right?
MIL-STD-810 Shock tests and NASA-STD-7003 pyroshock environments are specified using SRS. It's used to verify if the test input waveform satisfies the SRS specification.
Types of SRS
SRS Calculation in FEM
1. Perform time history analysis (modal method or direct method or explicit method) to calculate the response
2. Generate SRS in post-processing — Plot the maximum response of a single-degree-of-freedom system for each natural frequency
Summary
Key Points:
- SRS = Maximum response for each natural frequency to shock — The shock version of the seismic spectrum
- Specified in MIL-STD-810, NASA-STD-7003 — Standards for shock testing
- Primary / Residual / Maximax — During shock vs. after shock
- FEM time history → SRS generation in post-processing — Nastran PARAM,SRS
SRS originated from predicting shock damage in nuclear tests
The Shock Response Spectrum (SRS) was developed by NASA/the military in the 1960s for evaluating the shock resistance of equipment/structures during nuclear tests. The formal mathematical formulation was published by C.V. Nagel and D.S. Bernstein in 1969. It was later incorporated into IEC 60068-2-27 (shock testing) and MIL-STD-810G (transportation vibration/shock), and is now a design standard for space equipment and military electronics.
Physical meaning of each term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced your body being thrown forward during sudden braking? 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 enough that acceleration can be ignored". It absolutely cannot be omitted in shock loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Here's 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" is incorrect. 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), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but modeling "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated 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 away. 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 continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and application 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 the stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed
Dimensional analysis and unit systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the 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 | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
SRS Calculation
Numerical calculation of SRS is "simultaneous time integration of many single-degree-of-freedom systems":
For each natural frequency $f_n$ (10 Hz to 10 kHz, e.g., 1/3 octave intervals):
1. Time-integrate the equation of motion for a single-degree-of-freedom system (Newmark method, etc.)
2. Record the maximum response (acceleration or displacement or pseudo-velocity)
3. Plot $f_n$ vs. maximum response → SRS
Solver SRS Output
Summary
Minimum damping for SRS calculation is 2%, an industry standard
For SRS calculation, a damping ratio ζ = 5% is the MIL-STD-810 standard, but for ultra-sensitive equipment (gyros, accelerometers), actual damping is often below 2%, so calculating SRS with ζ = 2% is also required per NASA-STD-7003A (2011). Changing the damping ratio from 5% to 2% can increase SRS peak values by up to 1.5 times, significantly impacting design margin evaluation.
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 mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (ZZ estimator, etc.). 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 every iteration. Provides 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. Lower cost per iteration but linear convergence speed.
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
Instead of applying full load at once, apply 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: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching sequentially from the first page (direct method).
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 with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
SRS in Practice
Essential for pyroshock environment evaluation of space equipment and shock testing of military electronics.
Practical Checklist
Satellite separation shock reaches 1000–10000G
During the instant of satellite separation from a rocket via pyrobolt (explosive device) detonation, the structure experiences instantaneous peak accelerations of 1000–10000G. JAXA's H-IIA-mounted satellites specify separation shock SRS (ζ=5%, 10–10000Hz) in their specifications, with pass/fail determined by pre-launch shock tests. For ASTROS-H (launched 2016), detailed SRS prediction using ESI Crash/PAM-SHOCK was performed for separation shock analysis.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). 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 excellent the solver is.
Pitfalls Beginners Often Fall Into
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 will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Verify that results stabilize across at least three mesh density levels—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 a test. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most critical step in the entire analysis.
Software Comparison
SRS Tools
Selection Guide
Dewesoft and Data Physics Compete for SRS Analysis Market Share
The SRS analysis software market is divided between Dewesoft (Slovenia, founded 2000) and Data Physics (USA, founded 1984). Dewesoft's DS-NET PRO integrates data acquisition, SRS calculation, and MIL-STD-810 compliance reporting, with industry-leading processing speed (1 million points/sec). For CAE integration, a plugin connecting Ansys Motion's SRS input function directly with Dewesoft measurement values has been available since 2022, enabling rapid correlation verification between experiment and analysis.
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