3. Structural dynamics fundamentals for shock simulation

Transient shock simulation rests on three structural-dynamics pillars: modal superposition, which lets us express a complex response as a weighted sum of simple modes; transient excitation, which defines how energy enters the structure over time; and damping mechanisms, which govern how that energy dissipates. Getting these three right is what separates a simulation that merely runs from one whose peak accelerations, stresses, and relative displacements can be trusted for design decisions.

From Continuous Structure to Modal Coordinates

A physical structure has effectively infinite degrees of freedom (DOF). The equation of motion for a linear, viscously damped system is:

[M]{ẍ} + [C]{ẋ} + [K]{x} = {F(t)}

where [M], [C], and [K] are the mass, damping, and stiffness matrices, and {F(t)} is the time-varying load vector. Solving this directly for thousands of DOF at every time step is expensive. Modal superposition transforms the problem into a small set of independent single-DOF equations. First, solve the undamped eigenvalue problem:

([K] − ωₙ²[M]){φ} = 0

This yields natural frequencies ωₙ (rad/s) and mode shapes {φ}. The response is then written as {x} = [Φ]{q}, where [Φ] contains the mass-normalized mode shapes and {q} are modal coordinates. Substituting and pre-multiplying by [Φ]ᵀ diagonalizes the system into:

q̈ᵢ + 2ζᵢωₙᵢq̇ᵢ + ωₙᵢ²qᵢ = Γᵢ F(t)

Here ζᵢ is the modal damping ratio and Γᵢ is the modal participation factor. Each mode can now be integrated independently, and the physical response is recovered by summing the contributions of all retained modes.

Transient Excitation: How Shock Enters the Model

Shock is not a steady-state vibration; it is a short, high-amplitude event. The excitation can be applied in several forms, and the choice affects both accuracy and cost:

• Direct time-history force — a measured or synthesized force {F(t)} applied at specific nodes. Most faithful, but requires a known input.
• Base acceleration — the structure is mounted on a moving base, and the input is a prescribed acceleration a(t). This is the standard form for equipment mounted on a vehicle, floor, or test shaker.
• Response spectrum — a peak-response envelope (SRS) used when only the maximum response matters, not the full time history.

For base excitation, the equation of motion in modal coordinates becomes:

q̈ᵢ + 2ζᵢωₙᵢq̇ᵢ + ωₙᵢ²qᵢ = −Γᵢ a(t)

The negative sign reflects that the base acceleration drives the relative motion. A half-sine pulse is a common idealized shock:

a(t) = A·sin(πt/T),  0 ≤ t ≤ T
a(t) = 0,            t > T

With A = 50 g and T = 11 ms, this approximates a classic drop or pyrotechnic shock. The response depends strongly on the ratio of pulse duration T to the structure's natural period Tₙ. When T ≪ Tₙ, the structure barely responds (impulse-like); when T ≈ Tₙ, amplification peaks; when T ≫ Tₙ, the structure follows the base quasi-statically.

Damping Mechanisms

Damping is the hardest parameter to know and the easiest to get wrong. In shock simulation, several mechanisms coexist:

Rayleigh damping is the most common engineering choice because it produces a diagonal [C] in modal coordinates. The coefficients α and β are fitted from two target damping ratios at two frequencies:

ζᵢ = α/(2ωᵢ) + βωᵢ/2

Given ζ₁ at ω₁ and ζ₂ at ω₂, solve the 2×2 system for α and β. Note that Rayleigh damping is only accurate between the two anchor frequencies; outside that band, the effective ζ drifts. For broadband shock, this drift can under- or over-damp high modes significantly.

Putting It Together: A Worked Modal Superposition Example

Consider a simplified equipment bracket modeled with 5 retained modes. Natural frequencies are 45, 120, 310, 680, and 1250 Hz. Modal damping is assigned as ζ = 0.03, 0.025, 0.02, 0.015, and 0.01 respectively. The base input is a half-sine of 50 g, 11 ms.

The workflow is:

1. Extract modes and mass-normalize the mode shapes.
2. Compute participation factors Γᵢ for the excitation direction.
3. Integrate each modal equation with a time step small enough to resolve the highest retained frequency (typically Δt ≤ 1/(20·f_max)).
4. Sum modal contributions to recover physical response at points of interest.
5. Compare peak acceleration and relative displacement against the SRS or test specification.

With f_max = 1250 Hz, Δt ≤ 1/(20×1250) = 40 µs. The simulation runs for 50 ms, giving 1250 time steps — trivial for a 5-mode model but instructive for understanding the cost scaling when hundreds of modes are retained.

The figure below summarizes how the three pillars interact in a shock simulation workflow.

The diagram shows the dependency chain: the physical model feeds the eigenvalue solve, which produces the modal basis. Damping and excitation enter the modal equations, which are integrated in time and summed back to physical coordinates. Mode selection and damping assignment are the two decisions most likely to change the predicted peak response, so they deserve explicit justification in any shock simulation report.

Practical Checks Before Trusting a Shock Result

• Convergence in modes: rerun with 1.5× the retained modes. If peak acceleration changes by more than 5%, add more modes.
• Time-step sensitivity: halve Δt. A well-resolved integration should show less than 2% change in peak values.
• Damping sensitivity: vary ζ by ±50% on the highest modes. If the peak is highly sensitive, the damping model needs experimental validation.
• Energy balance: check that input energy equals dissipated plus kinetic plus strain energy at the end of the event.

These checks are cheap compared to a physical shock test and catch the most common modeling errors: too few modes, too coarse a time step, and unrealistic damping.