What Resonance Is and Why It Finds Your Machine
Why this matters
Techs treat resonance as bad luck, a freak coincidence that happened to this one machine. It is not a coincidence and it is not rare. An ordinary belt-driven unit emits six or seven distinct forcing frequencies at once, spread over a range of more than ten to one, and the assembly it sits on has several natural frequencies of its own scattered through the same range. Somewhere in that grid, something lines up on most installations. Knowing why coincidence is the normal case changes what you do on the roof: instead of hunting for a defective part that explains an unexplainable shake, you spend twenty minutes taking a census of what the machine emits and what the structure rings at, and the answer is usually sitting in the comparison.
Before the hammer comes out
The tests below are done on a machine that is locked out and stopped, or coasting down under observation from a safe distance, and much of this work happens on a roof. Open the disconnect, apply your own lock and tag, and confirm the rotor has stopped rather than slowed before striking any part of the assembly, under 29 CFR 1910.147 and its verification-of-isolation requirement. If the unit's edge-to-parapet distance puts you near an unprotected side or a skylight, fall protection is a duty before it is a preference: for service work on an existing building this is general industry under 29 CFR 1910.28(b)(1), with the construction counterpart at 29 CFR 1926.501 if the work is part of a construction project, and the two Parts set different trigger heights, so know which one your job falls under before you step out of the hatch. Watch a coast-down from outside an intact guard, never with a panel off, since 29 CFR 1910.219 applies to a decelerating drive exactly as it does to a running one.
A natural frequency is stiffness and mass, and nothing else
Any structure that can deflect and spring back has at least one frequency it prefers to vibrate at. That frequency rises with stiffness and falls with mass:
natural frequency = (1 / 2 pi) x square root of (stiffness / mass)
Two consequences follow directly, and they are the whole toolkit for moving a resonance. Adding stiffness raises it. Adding mass lowers it. Because both sit under a square root, you need a big change to move the frequency much: quadrupling stiffness only doubles the natural frequency. This is why bolting on one more angle-iron brace often does nothing measurable, and why an effective fix usually means a gusset that shortens the unsupported span rather than a member added alongside it.
In the field you rarely know stiffness in engineering units, but you can get a usable estimate from deflection. For a mass sitting on a spring, the natural frequency in cycles per minute is about 187.8 divided by the square root of the static deflection in inches under that mass. That relationship is derived for a single mass on a linear spring, which is exactly what a machine on isolators is; applied to a beam, a deck panel or a bracket it is a lumped approximation that gets the order of magnitude and the direction of change right, and should not be quoted as a design value.
Half an inch of deflection puts a mount system near 266 cpm. A tenth of an inch puts it near 594 cpm. A bracket that deflects five thousandths under load rings near 2,660 cpm. That spread is why brackets and sheet-metal panels resonate at running speeds while mounts resonate far below them.
Resonance multiplies a force, it does not create one
This is the sentence to keep. At resonance, nothing new is generating vibration. The same unbalance force, the same blade pulse, the same coupling reaction that was always there is now being fed into a structure that stores and returns energy every cycle instead of absorbing it, so the motion builds until damping bleeds off as much energy per cycle as the force puts in.
How much it builds is set by damping, and only by damping. At the peak, the amplification is roughly 1 divided by twice the damping ratio. A bolted steel structure with something on the order of 5 percent of critical damping, a common assumption for this kind of assembly and one that is almost never actually measured, amplifies about tenfold at the peak. A welded, grouted, well-damped assembly might reach four or five. A light sheet-metal panel with almost no damping can reach twenty or more, which is why panels crack and brackets break while the machine itself looks fine.
Two practical consequences. First, a resonance never explains a new force appearing, so if the excitation itself has grown, you still have a mechanical fault underneath the amplification. Second, because the multiplier is set by damping rather than by the force, the fix at a resonance is to move the frequency or add damping, not to make the machine quieter. The library's diagnostics side has a separate article on telling a resonance from a genuine mechanical fault when you are reading the symptom; this one is about why the condition exists.
Every machine emits a spread of forcing frequencies
Here is the census that makes coincidence normal. A single belt-driven fan puts out all of these simultaneously:
| Source | Frequency | Why it exists |
|---|---|---|
| Driver running speed | 1x motor rpm | Residual unbalance in the rotor and coupling half |
| Driven running speed | 1x fan rpm | Residual unbalance in the wheel |
| Twice running speed | 2x either shaft | Coupling reaction, looseness, a per-revolution geometry that repeats twice |
| Belt pass | pi x sheave pitch diameter x rpm, divided by belt length | A joint, a set, or thickness variation passing the sheaves |
| Blade or vane pass | blades x fan rpm | Each blade passing a cutoff, strut or housing tongue |
| Twice line frequency | 7,200 cpm on a 60 Hz supply | Magnetic pull in the motor, present regardless of speed |
| Bearing defect frequencies | non-integer multiples of shaft speed | Element and race geometry; the vibration articles cover reading these |
Seven bands on one ordinary machine, spanning roughly 400 cpm to over 10,000 cpm. Against that, the assembly typically has a mount or base mode down low, a pedestal or frame mode in the middle, and panel or duct modes higher up. The grid is dense enough that near-misses are the default outcome, not the exception.
A frequency census from one rooftop unit
A belt-driven rooftop exhaust fan: motor at 1,750 rpm, fan wheel at 890 rpm, twelve blades, 5.0 in pitch diameter motor sheave, 63 in belt, 60 Hz supply. Natural frequencies were taken with a soft-face hammer on the locked-out, stopped assembly at three points, plus one coast-down watched from outside the intact belt guard, standing beside the curb rather than in the discharge path.
What the machine emits
| Source | Frequency (cpm) |
|---|---|
| Belt pass (pi x 5.0 x 1750 / 63) | 436 |
| Fan running speed | 890 |
| Motor running speed | 1,750 |
| Twice fan speed | 1,780 |
| Twice motor speed | 3,500 |
| Twice line frequency | 7,200 |
| Blade pass (12 x 890) | 10,680 |
What the structure rings at
| Location struck | Natural frequency (cpm) |
|---|---|
| Curb and base rail assembly | 940 |
| Discharge duct side panel | 1,790 |
| Motor pedestal, across the rails | 3,300 |
The comparison
| Structure mode | Nearest excitation | Ratio | Amplification at 5 percent damping |
|---|---|---|---|
| Curb and base, 940 | Fan speed, 890 | 0.95 | about 7.4x |
| Duct panel, 1,790 | Twice fan speed, 1,780 | 0.99 | about 9.9x |
| Motor pedestal, 3,300 | Twice motor speed, 3,500 | 1.06 | about 6.1x |
Three near-coincidences on one unremarkable machine, from a census of seven excitation bands against three modes. Nobody was unlucky. The grid was dense and the odds were always poor.
Reading a census like that
The ratios do the work, and the further from 1.00 the better. The duct panel at a ratio of 0.99 is the one to fix first, not because its amplification number is highest but because a panel that close to a strong excitation is the classic source of cracked seams and fastener fatigue, and it is the cheapest to move: a stiffener that halves the unsupported span raises that panel's frequency substantially, far more than adding a second bracket alongside the first.
The curb mode at 0.95 is the one that will be blamed on the fan wheel. At that ratio, a residual unbalance well inside its tolerance produces a motion roughly seven times what the same rotor would produce on a stiff base, so the machine reads as needing balancing when the rotor is fine. Before anyone pulls the wheel, check whether the excitation itself has grown.
The motor pedestal at 1.06 is a watch item rather than an action item. Twice-line-frequency and twice-running-speed excitations are usually modest unless something else is wrong, so the amplification only matters if a fault develops that feeds them.
What a census like this cannot tell you
Three limits, each of which has burned somebody. A hammer test finds the modes you happen to excite from the direction you happen to strike, so a mode that lives in the axial direction can hide from a radial strike. The measurement is taken on a cold, stopped machine, and stiffness changes when the assembly is running, loaded and hot, so a ratio near the edge can move. And nothing in the census identifies which excitation is actually strong; it only tells you where amplification is waiting. Pair it with a running measurement before you commit to a fix, and use the companion article on working out whether you are near a critical speed to separate a rotor's own critical speed from the support modes this method finds.
References
- 29 CFR 1910.147 for isolation, lock and tag, and stopped-rotor verification before impact testing an assembly
- 29 CFR 1910.28(b)(1) for general-industry fall protection on a rooftop service call, and 29 CFR 1926.501 where the work falls under construction instead
- 29 CFR 1910.219 for guarding during a coast-down observation
- See related: Why a Machine Is Worse at One Speed Than at a Higher One; How to Work Out Whether You Are Near a Critical Speed; The Difference Between a Resonance and a Genuine Mechanical Fault