Transient (time-history) analysis#

When to use#

Transient analysis recovers the time-history response of a structure to time-varying loads. Use it when:

  • Loading is shock / impact / ramp / arbitrary time-history.

  • The structure’s response is governed by inertia (loading changes faster than natural-period scales).

  • You need displacement / stress at specific instants in time rather than steady-state amplitudes.

  • You’re doing modal-superposition response-spectrum analysis (shock-spectrum / earthquake / ASCE 7 design spectrum).

For periodic / sinusoidal forcing use Harmonic (frequency-response) analysis — the frequency-domain solve is much cheaper. For natural frequencies alone use Modal analysis (free vibration). For a steady-state- under-fixed-load snapshot use Linear static analysis.

Boundary conditions and loads#

  • Dirichlet constraints — same as Linear static analysis.

  • Time-varying nodal forces — supplied as a callable f(t) or a sampled (n_steps, n_dof) array.

  • Initial conditions — initial displacement and velocity passed to the solver. Default is zero / zero (the structure starts at rest at its undeformed configuration).

  • Damping — Rayleigh (\(\mathbf{C} = \alpha \mathbf{M} + \beta \mathbf{K}\)) via Model.rayleigh_damping is the supported path today; modal damping for mode-superposition is on the planned roadmap.

The math (one paragraph)#

Transient analysis solves the second-order ODE

\[\mathbf{M}\, \ddot{\mathbf{u}}(t) + \mathbf{C}\, \dot{\mathbf{u}}(t) + \mathbf{K}\, \mathbf{u}(t) = \mathbf{f}(t)\]

via the Newmark-β family of implicit time-integrators (Newmark 1959; Bathe §9.5). Default parameters are \(\beta = 1/4, \gamma = 1/2\) (the unconditionally-stable average-acceleration scheme). Each time step solves \((\mathbf{M} / (\beta \Delta t^{2}) + \mathbf{C} \gamma / (\beta \Delta t) + \mathbf{K})\, \mathbf{u}_{n+1} = \mathbf{f}_n\), re-using the same factorisation across all time steps.

Running the solve#

times = np.linspace(0.0, 0.1, 1001)  # 100 ms, 0.1 ms steps
forcing = np.zeros((len(times), n_dof))
forcing[:, tip_dof] = ramp_to_step(times)  # user-defined input

res = m.solve_transient(times=times, forcing=forcing)

Returns a TransientResult:

  • res.times — (n_steps,) time samples [s].

  • res.displacement — (n_dof, n_steps) displacement history.

  • res.velocity — (n_dof, n_steps) velocity history.

  • res.acceleration — (n_dof, n_steps) acceleration history (useful for inertial-stress checks).

For a single-DOF / shock-input lookup, see the focused recipe at SDOF transient — step-load response.

Time-step selection#

The Newmark-β scheme is unconditionally stable but accuracy depends on time-step:

  • For modal-superposition (when shipped), \(\Delta t \le 1 / (10\, f_\text{max-mode-included})\) resolves the mode-set.

  • For direct integration with the default Newmark-β, \(\Delta t \le 1 / (20\, f_\text{highest-frequency-of-interest})\) is the practical guidance.

  • For shock / impact, resolve the rise-time of the forcing input — typically \(\Delta t \le \tau_\text{rise} / 10\).

Post-processing recipes#

  • Time-history plots — index res.displacement by the DOF of interest, plot vs res.times.

  • Velocity / acceleration spectra — res.velocity and res.acceleration give the corresponding histories without numerical differentiation.

  • Snapshot stress at a chosen instant — pass the column res.displacement[:, k] to compute_nodal_stress.

Common gotchas#

  • No damping → infinite ringing. An undamped transient on a non-dissipative structure rings forever. Configure at least minimal Rayleigh damping (typically ζ ≈ 1-5 %) on the modes you care about.

  • Time-step too large smears high-frequency content into artificial damping. Verify by solving with halved time-step and checking if the response stabilises.

  • Initial conditions inconsistent with the loads. Starting from zero displacement under non-zero initial loading produces a transient ringing that has no physical meaning; pass u0 = static-equilibrium-solve(f(0)) to start from the consistent static state.

Solver mechanics + performance#

For Newmark-β derivation, generalised-α / HHT alternatives, mode-superposition mechanics, and time-step batching, see Transient analysis — that page is the solver-engineering deep-dive on the same Model.solve_transient() invocation.