Frequency response curve · H(r) for current damping · operating point marked
— "where you're sitting on the curve"
Forcing freq f
1.00 Hz
Forcing amp F₀
1.00 kip
Damping ζ
2.0% (steel)
Span L
40 ft
EI stiffness
1.0e7 k·in²
Mass μ (k/ft)
0.50 k/ft
Forcing preset
Crowd visual
Natural freq fₙ—first bending mode
Frequency ratio r—f / fₙ
Amplification H—vs static
Phase φ—beam lags forcing
Static deflection—if load just sat
Peak dynamic deflection—amplification × static
Regime—r < 0.5
About this lab
The Live Beam — v1.0 (TRUE dynamic)
A browser dynamics lab. The live in this lab's name means live loads — loads that move. A crowd jumping in sync. A column of soldiers marching. A 42-mph crosswind tickling a bridge deck just right. The beam isn't a passive piece of steel; it has a natural frequency, and when the forcing frequency lines up, the response amplifies. Sometimes by 50×. Sometimes catastrophically.
How to read the lab
The top canvas shows the beam in real-time, oscillating in its first bending mode. The forcing is a midspan arrow that pulses at frequency f. The beam's natural frequency fₙ depends on span L, stiffness EI, and mass-per-unit-length μ. When f matches fₙ, the amplitude blows up — held in check only by damping ζ.
What that arrow really is: it's the moving load made visible — the marching feet, the crowd swaying in sync, the weight of people shifting back and forth. Push the forcing toward fₙ and watch the arrow and the beam turn jittery: that growing jitter is resonance building, the structure starting to answer the load harder than the load itself pushes. It's the same runaway that hit Broughton's marching soldiers and the Millennium Bridge crowd. (The Hyatt walkways are the other lesson on this page — there the crowd's weight alone overloaded a bad connection; no resonance required.)
The bottom canvas is the frequency-response curve. It plots the amplification factor H(r) for the current damping. Your operating point is the bright dot. Slide forcing frequency, watch the dot ride the curve.
What to try
Set the Walking preset (1 Hz). Note H ≈ 1; the beam barely cares. Quasi-static.
Set the Jumping preset (2 Hz). Drag span L until fₙ lands on 2 Hz. Watch H climb past 25. That's the amplification a crowd synced at ~2 Hz can drive on a floor — the regime engineers design against (and the mechanism behind Broughton's marching soldiers).
Drop damping ζ to 0.005. Set forcing at resonance. Watch H spike toward 100. That's a structure with no damping — the kind we deliberately don't build.
Set the Soldier-march preset (2.3 Hz). Find a span where fₙ matches. That's Broughton 1831.
What isn't here
Single-degree-of-freedom (SDOF) first-mode approximation. The beam is rendered as a half-sine, amplified by the frequency response of the underlying SDOF oscillator. The lab doesn't model: higher modes (2nd, 3rd bending), modal coupling, geometric nonlinearity, aeroelastic flutter (Tacoma's actual physics is more subtle than vortex-shedding-driven resonance — v0.3 will add a torsion-mode panel), or human-structure feedback (the Millennium-Bridge synchronous lateral excitation needs a coupled model the v0.1 doesn't have).
For real design work see AISC Design Guide 11 (Floor Vibrations Due to Human Activity) or AASHTO LRFD Bridge Design Spec for vehicular dynamic load allowance.
The static counterpart
The dual of this lab — the case where the load just sits — lives at The Static Beam. Same chassis (simply supported / continuous / cantilever), different physics regime.
Suite context
Lab of the OPA Engineering Suite · dual of The Static Beam · sibling to The Horseshoe Vortex and The Wing. Filed under College X (Engineering). Course anchor: CIVL 360 (Dynamics of Structures), STRC 360 (Dynamic Loads).
v0.3: Tacoma Narrows torsion-flutter mode — separate panel for the torsion-bending coupling that actually killed the bridge (it wasn't simple resonance).
v0.4: human-structure feedback — Millennium Bridge synchronous lateral excitation modeled as a coupled crowd-bridge SDOF system.
Math under the hood
SDOF first-mode forced response
Natural frequency — simply supported, first bending mode
fₙ = (π/2) · √(EI / (μ · L⁴)) Hz
EI is flexural stiffness (k·in²), μ is mass per unit length (converted from k/ft to slug/in), L is span in inches. A steel W14×30 spanning 40 ft with typical floor loading lands fₙ around 3–5 Hz — right in the range walkers and joggers can excite.
Frequency response (amplification factor)
H(r) = 1 / √( (1 − r²)² + (2ζr)² )
where r = f / fₙ
At r = 0 (static): H = 1. At r = 1 (resonance, undamped): H → ∞. At r = 1 with damping ζ: Hₘₐ₧ ≈ 1/(2ζ). Steel buildings ζ ≈ 0.02 → Hₘₐ₧ ≈ 25. Reinforced concrete ζ ≈ 0.05 → Hₘₐ₧ ≈ 10. Tacoma-Narrows steel deck with effectively no torsional damping, ζ ≈ 0.005, Hₘₐ₧ ≈ 100.
Phase
φ(r) = − atan2( 2ζr , 1 − r² )
Below resonance the beam moves in phase with the forcing (φ ≈ 0°). At resonance, beam lags forcing by 90°. Above resonance, beam lags by 180° (moves opposite the forcing).
Damping ratios in real structures (working values)
Welded steel building, bare: 0.005–0.01
Steel building with cladding + partitions: 0.02–0.03
2.0–2.5 Hz — running, jumping in unison, soldier march cadence
3.0–5.0 Hz — first bending mode of typical office floor (the danger zone for vibration complaints)
5.0–10.0 Hz — first mode of stiff floor systems, second mode of typical floors
Sources
Chopra, Dynamics of Structures, 5th ed., Ch. 3 (forced harmonic response, SDOF). AISC Design Guide 11: Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2nd ed. (2016) — the 1st ed. (1997) was titled Floor Vibrations Due to Human Activity. Petroski, To Engineer Is Human, Ch. 8 (Hyatt). Levy & Salvadori, Why Buildings Fall Down (general). Dallard et al., "The London Millennium Footbridge," Structural Engineer 79(22), 2001.
The stories in full
Four bridges, four lessons
Hyatt Regency, Kansas City · July 17, 1981
Three skywalks spanned the atrium. The original design called for single 24-ft hanger rods to support the 4th-floor walkway, which would also support the 2nd-floor walkway via separate rods to the same upper connection. The fabricator changed the detail at shop drawings: two shorter rods instead of one long one, with the 4th-floor walkway's bolt now also carrying the 2nd-floor walkway load. The change doubled the load on the 4th-floor nut. Late-evening tea dance: ~1,500 in the atrium, ~40–65 on the walkways, swaying in sync to the music at roughly 1 Hz. The nut failed. Both walkways collapsed onto the dancers in the atrium. 114 dead, 216 injured. The structural-engineering profession's worst day. Changed peer-review requirements forever.
Tacoma Narrows Bridge · November 7, 1940
Opened July 1, 1940. Unusually light and shallow for its span (2,800 ft suspended). Nicknamed "Galloping Gertie" from day one for its modest-wind roll. November 7, a 42-mph wind set up a torsional oscillation. Aeroelastic flutter: vortices shedding off the deck synchronized with the deck's own twisting motion, feeding energy in. Amplitude grew. The deck twisted through ~45 degrees at midspan. After about an hour, the center span tore free and fell. No human deaths — one dog, Tubby, trapped in a car, went down with the deck. The film footage is the most-watched engineering-disaster footage ever made.
v0.3 will add a torsion-mode panel: Tacoma's failure wasn't simple SDOF resonance; it was self-excited aeroelastic flutter, a coupled bending-torsion physics that needs its own model.
Millennium Bridge, London · June 10, 2000
Pedestrian footbridge across the Thames between St. Paul's and the Tate Modern. ~90,000 people crossed the first day; peak ~2,000 simultaneously. Within an hour the bridge began to sway laterally at ~1 Hz — the lateral frequency of human walking, side-to-side (the central span's own lateral mode was nearer ~0.5 Hz). Pedestrians felt the sway and adjusted their stride to match it (helps you stay balanced), feeding more energy in. Sway amplitude reached ~70 mm. Closed two days later. Reopened February 2002 with 37 fluid viscous dampers and 52 tuned mass dampers, providing ~20% additional damping. The phenomenon — synchronous lateral excitation — was understood theoretically before 2000, but the Millennium Bridge made it canonical.
Broughton Suspension Bridge, England · April 12, 1831
A 74-man detachment of the 60th Rifle Corps marched in step, four abreast, across the Broughton suspension bridge on their way back to barracks. The bridge began to oscillate noticeably; the soldiers found it amusing and whistled a marching tune in cadence. The bridge collapsed. Forty fell into the river; twenty were injured but none died. The British Army afterward ordered troops to "break step" when crossing any bridge — a practice armies have kept ever since. (The order is usually told as the Army's, not any one commander's; the popular "Duke of Wellington" version isn't supported by the record.)
What links them
All four are excitation frequency matching a structure's natural frequency. Hyatt and Broughton: vertical bending mode of a horizontal span, excited by walking-cadence-locked crowd. Millennium: lateral mode of a footbridge, excited by walkers' lateral sway. Tacoma: torsion mode of a suspended deck, excited by aeroelastic feedback with crosswind. Same equation, four geometries. This lab models the first one cleanly. The others get their own modes in v0.3 and v0.4.
College X · Structural Engineering · Faculty
Oskar Simpson
Instructor · Structural Dynamics · CIVL 360 + 420
Six-three, two-twenty, buttoned into a shirt and tie — the structural engineer who takes the handoff the moment a building stops holding still. Ray Ray teaches you statics, the beam that isn’t going anywhere; Oskar takes over the instant it starts to move. Forced vibration, a crowd finding a floor’s resonance, the footfall that couples to the span and amplifies until the amplification factor blows up — this is his turf. He cross-references Belvedere Shaketon’s seismology one college over: Belvedere studies what the ground does, Oskar studies what your building does about it.
A note from the builder
Single developer. A 49‑year‑old father and solo builder.
If you see something wrong, please let me know — I’ll change it as soon as I can. I’ll credit you if you want the credit.
Some labs are better than others; a few have obvious broken bits I’m still working through. Some were built early, some late — they don’t all look the same, but they share a feel.