Validation/Structural dynamics

Cantilever step response

It rings at 1634.5 Hz where the frequency-domain sweep on the same bar puts the resonance at 1637.6 — two different solvers agreeing to 0.19 % about a number neither was given.

VerificationLogarithmic decrement within 0.926 %

Why this case

The same steel cantilever again, with the same distributed load — but applied as a step at t = 0 and held, then integrated in time for twenty periods of the first mode.

Three quantities, and the third is the one that cannot be faked:

The settled value. Once the ringing has decayed the answer is the static one, exactly.
The ringing frequency, against the Euler–Bernoulli eigenvalue.
The logarithmic decrement — how fast the ringing dies away.

Physics	Transient dynamic, Newmark integration with Rayleigh damping
Material	Steel — E = 210 GPa, ν = 0.3, ρ = 7 850 kg/m³
Applied	10⁵ Pa stepped on at t = 0; damping ratio 0.02 at f₁
Time	20 periods in 800 steps — 40 steps per period
Closed form	Euler–Bernoulli beam theory, and δ = 2πζ/√(1−ζ²)
Mesh	Tetrahedral, from Netgen — 713 elements
Agreement	Logarithmic decrement within 0.926 %
Last run	2026-08-27

Result

The three checks against their closed forms
Quantity	Closed form	Measured	Difference
Settled value	8.928571 × 10⁻⁶ m	9.118779 × 10⁻⁶ m	+2.130 %
Ringing frequency	1 671.033 Hz	1 634.476 Hz	−2.188 %
Logarithmic decrement	0.122938	0.121799	−0.926 %
First overshoot	— (reported, not judged)	1.9527 × the settled value	—

Tip deflection against time, oscillating and decaying onto the static value — The measured ring-down over twenty periods, with the static closed form drawn through it. The overshoot on the first swing is 1.95 times the settled value.

Why the decrement is the measurement

A step response settles to the static answer whatever the damping is, and it rings at very nearly the undamped natural frequency for any small damping. So the first two quantities say nothing at all about the damping matrix.

The logarithmic decrement does, and like the frequency-domain case’s amplification it is a ratio of successive peaks on the same mesh, so the discretisation cancels. It lands at 0.926 % where the absolute deflection is 2.13 %.

Two failure modes, each invisible to the other’s check

The check guarding this case is tested against two runs, and between them they are the argument for all three quantities being watched:

Damping in the wrong keyword. The bar rings essentially forever — decrement −0.0015 against 0.1229 — while settling at +1.78 % and ringing at +2.96 %, both comfortably in tolerance. Its first overshoot is 2.007 times the settled value, which is the undamped step result exactly, and the clearest possible statement that no damping reached the model.
Six time steps per period instead of forty. The obvious guess is that a coarse step adds false damping. It does not — this integration scheme is non-dissipative, so what a coarse step costs is period elongation. Its decrement is only 3.4 % off and inside tolerance; its frequency is −8.9 %.

Each is caught by exactly the check the other passes.

Why the overshoot is reported and not judged

A step load excites every mode, and the familiar “twice the static deflection” is a single-degree-of-freedom result. The real first peak is twice the first mode’s share of the static deflection plus whatever the higher modes happen to be doing at that instant — which depends on the time step through the integration as much as on the physics.

It is worth printing, because a number near 2 is a good sign and a number near 1 would mean the load had been applied as a ramp rather than a step. It is not worth a tight tolerance, and this page does not give it one.

Repeat this yourself

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