Transient Dynamic Analysis
Time Integration, Stability & Solver Selection

In static loading you're implicitly assuming the structure has infinite time to reach equilibrium — so acceleration is zero and inertia vanishes. But apply a 5 kN load in 1 millisecond and the structure hasn't had time to redistribute it. The mass resists the sudden acceleration; the structure "lags" the load. This lag creates stress waves, vibration, and — critically — stresses that can be several times higher than the static peak. A bracket that survives 5 kN statically can fracture under an identical impulse that hits at the right frequency.

The Newmark-β method expresses the displacement and velocity at step n+1 as weighted combinations of both the current and the unknown acceleration at n+1:

Because \(\ddot{u}_{n+1}\) is unknown, you substitute these into the equation of motion at time n+1 and solve a modified stiffness system. It's more work per step, but you can take very large steps without going unstable.

The parameter γ controls numerical dissipation (artificial damping of high-frequency modes); β controls the accuracy order. The most common choice is γ = 0.5, β = 0.25 — the "constant average acceleration" or trapezoidal rule. This is second-order accurate and unconditionally stable — meaning you can choose any Δt and the solution won't blow up. If you push γ above 0.5 (say γ = 0.6, β = 0.3025, the HHT-α variant), you get controllable high-frequency dissipation that filters out noise from stiff modes without hurting low-frequency accuracy.

The Central Difference method computes the next state purely from known current quantities — no matrix solve needed. That makes each step extremely cheap. The catch: it's only conditionally stable. The critical time step is:

In a crash model with 2 mm steel elements, \(\Delta t_\text{crit}\) is roughly 0.3 microseconds. A 100 ms crash event needs ~330,000 steps. Still fast because each step is a vector multiply — no factorization. And because there's no convergence iteration, severe contact and material failure are handled naturally without divergence.

Here's my rule of thumb: use implicit when the event duration is longer than ~100 ms, nonlinearity is moderate, and you need accurate low-frequency response — think seismic analysis of a building, turbine blade vibration, engine mount fatigue. Use explicit when the event is very fast (crash <100 ms, impact, blast, bird strike), contact is severe, or material failure and erosion are involved. In the grey zone of 10–100 ms, run both and compare. Explicit is also often better for drop tests even at moderate durations, because contact changes too rapidly for Newton-Raphson to converge reliably.

Modal superposition is a completely different approach — and much faster for linear problems. You first extract the mode shapes \(\{\Phi_i\}\) and frequencies \(\omega_i\) from a modal analysis. Then you express the transient response as a sum of scaled modes: \(\{u(t)\} = \sum_i q_i(t)\{\Phi_i\}\). Each modal coordinate \(q_i\) satisfies a simple 1D damped oscillator equation — trivially cheap to integrate. For a model with 500,000 DOF, retaining just 50 modes reduces the problem to 50 scalar ODEs instead of a 500k-dimensional system. The big limitation: the moment you add contact or plasticity, the mode shapes change — you need direct integration.

The go-to method is Rayleigh damping, which builds [C] from the already-available mass and stiffness matrices:

The modal damping ratio for mode i becomes \(\xi_i = \frac{\alpha}{2\omega_i} + \frac{\beta\omega_i}{2}\). You pick two target frequencies \(\omega_1, \omega_2\) — say the first mode and the highest mode of interest — and the desired damping \(\xi\) at those points, then solve for \(\alpha\) and \(\beta\). For steel structures, \(\xi\) is typically 1–3%; for welded joints or bolted assemblies, 3–5%.