Technical Note · Vibration Fundamentals
Resonance
Why a 1 g input can become a 20 g response, how to read it on a transmissibility plot, and what it means for sine, random and shock testing.
Key takeaways
- Every structure has natural frequencies. When the shaker drives a test article at or near one of them, the article's response can be many times larger than the input. That amplification is resonance.
- Damping is the only thing that limits the peak. It is summarized by the quality factor Q: at resonance, response ≈ Q × input.
- Resonance has three signatures you can measure on a controller: a peak in transmissibility, a 90° phase lag, and a bandwidth that tells you Q (Δf = fn/Q).
- Resonances take time to build up (roughly Q cycles), which is why sweep rate, dwell strategy and control loop design all matter.
- Finding resonances is often the point of the test. Controlling them safely, without over- or under-testing, is the job of the vibration controller.
What resonance is
Push a child on a swing in time with its natural rhythm and each small push adds a little energy; the arc grows until friction and air drag remove energy as fast as you add it. Push at the wrong rhythm and your pushes fight the motion instead. That is resonance in a sentence: a system driven at its natural frequency accumulates energy, cycle after cycle, until damping balances the input.
Mechanically, a resonating structure is trading energy back and forth between two stores: kinetic energy in its moving mass and potential energy in its flexing stiffness. At the natural frequency, that exchange happens with almost no help from outside, so even a small, well-timed force keeps topping it up. Damping (material hysteresis, friction in joints, air) leaks a little energy away every cycle and sets the ceiling.
On a shaker, this is exactly what happens to a bracket, a circuit card, a fixture, or the entire test article. The controller holds the table at, say, 1 g, but an accelerometer on a resonant component can read 10, 20 or 50 g. Whether that amplification is the failure mechanism you are trying to find, or an artifact you need to keep under control, is a central question in almost every vibration test.
The single-degree-of-freedom model
The simplest useful model is a mass on a spring and damper, mounted on a moving base (the shaker table). Real test articles have many natural frequencies, but near each one the structure behaves much like this simple system, so it is the right place to build intuition.
Three numbers describe this system completely:
| Parameter | Symbol | What it means on the test floor |
|---|---|---|
| Natural frequency | fn = (1/2π)·√(k/m) | Where the peak will be. Stiffer parts resonate higher; heavier parts resonate lower. Doubling the stiffness raises fn by only about 41%. |
| Damping ratio | ζ = c / (2√(km)) | How much energy is lost each cycle, as a fraction of "critical" damping. Most test articles and fixtures are lightly damped: ζ = 0.5% to 5%. |
| Quality factor | Q = 1 / (2ζ) | The amplification at resonance. A part with Q = 20 sees roughly 20 g when the table is held at 1 g at its natural frequency. |
Natural frequency vs. resonant frequency. Strictly, the frequency of peak response depends slightly on damping and on what you measure (displacement, velocity, acceleration, or transmissibility). For the light damping typical of test articles the differences are well under 1% and are usually ignored. With heavy damping (ζ above about 0.2) they become visible, and the peak also becomes broad and low.
Transmissibility: the curve every test engineer should know
Drive the base with a slow sine sweep and plot response divided by input against frequency, and you get the transmissibility curve. Plotting it against the frequency ratio r = f/fn makes it universal: the same shape applies to a 30 Hz pallet load and a 1,500 Hz circuit card.
The curve has three distinct regions, and each one shows up in real tests:
| Region | Frequency | Behavior | Test-floor consequence |
|---|---|---|---|
| Rigid (stiffness-controlled) | f well below fn | T ≈ 1. The part moves with the table. | The control accelerometer is a good proxy for what the part experiences. This is why fixtures are designed to resonate above the test band. |
| Resonant (damping-controlled) | f near fn | T rises to ≈ Q. Only damping limits the peak. | Highest stress, fastest fatigue damage. Where most vibration failures originate, and where response limiting or notching may be applied. |
| Isolation (mass-controlled) | f above √2·fn | T < 1 and falling. The mass can't keep up with the table. | The principle behind isolators. Also why a resonating fixture can "hide" high-frequency input from the part bolted to it. |
Damping and Q
Because Q sets the height of the peak, it is the single most important number for estimating how a part will respond. The table below shows how damping ratio translates to amplification and bandwidth, along with representative values for hardware commonly seen on a shaker. Treat the typical values as orders of magnitude, not design data; measured Q varies with amplitude, assembly torque and temperature.
| Damping ratio ζ | Q = 1/(2ζ) | Peak transmissibility | Half-power bandwidth at fn = 100 Hz | Typical of |
|---|---|---|---|---|
| 0.005 (0.5%) | 100 | 100 | 1 Hz | Monolithic cast or welded fixtures, bare metal plates |
| 0.01 (1%) | 50 | 50 | 2 Hz | Well-made bolted fixtures, head expanders |
| 0.02 (2%) | 25 | 25 | 4 Hz | Circuit card assemblies, brackets, sheet-metal chassis |
| 0.05 (5%) | 10 | 10.1 | 10 Hz | Bolted assemblies with many joints, potted or conformal-coated electronics |
| 0.10 (10%) | 5 | 5.1 | 20 Hz | Elastomeric isolators, structures with added damping treatment |
| 0.25 (25%) | 2 | 2.3 | 50 Hz | Heavily damped isolators, constrained-layer treatments |
Measuring Q: the half-power bandwidth
Q can be read directly from a measured transmissibility curve. Find the peak, draw a line at the peak divided by √2 (−3 dB, the "half-power" level), and note the two frequencies where the curve crosses it. The narrower the band, the higher the Q.
Two practical cautions follow from Figure 3. First, the resonance is only 12.5 Hz wide at 250 Hz; a coarse frequency resolution or a fast sweep can step straight over the true peak and under-report Q. Second, in a high-Q part, a few percent shift in natural frequency (from temperature, preload, or accumulating damage) moves the peak completely off a fixed dwell frequency.
Phase: the fingerprint of a resonance
A peak in amplitude alone does not prove a resonance; noise, a transducer problem or an input spectrum feature can also produce peaks. Phase is the confirmation. Below resonance the part moves in step with the table. Through resonance the response lags progressively, passing through about 90° at the natural frequency. Just above resonance, a lightly damped part moves nearly opposite to the table (close to 180° lag).
| Where you are | Transmissibility | Phase (response vs. control) | Physical picture |
|---|---|---|---|
| Well below fn | ≈ 1 | ≈ 0° | Part and table move together |
| At fn | ≈ Q | ≈ −90° | Peak velocity of the part coincides with peak table acceleration: energy input is maximized |
| Well above fn | < 1, falling | Toward −180° (lightly damped parts) | Part moves opposite to the table; the mass is effectively isolated |
This 90° signature is what makes phase-tracked resonance dwells possible: instead of dwelling at a fixed frequency, the controller adjusts the drive frequency to hold the response-to-control phase at the value measured during the resonance search, so the dwell follows the resonance as the part heats up or fatigues. A drifting natural frequency during a dwell is itself useful data, since a downward shift is a common early indicator of structural damage such as a cracking solder joint or loosening fastener.
Resonance takes time: build-up and ring-down
Steady-state transmissibility describes what happens after the response has settled. Getting there takes time, because energy is added a little each cycle. The same is true in reverse: when the drive stops, a lightly damped part keeps ringing at its natural frequency.
A convenient rule of thumb falls out of the math: a resonance needs roughly Q cycles to reach about 95% of its steady-state amplitude. At 250 Hz with Q = 20, that is only 80 ms. At 20 Hz with Q = 50, it is 2.5 seconds. This has direct consequences:
- Sweep rate. If a sine sweep passes through the half-power bandwidth faster than the resonance can build up, the measured peak is too low and appears at a slightly shifted frequency. Low-frequency, high-Q resonances are the most vulnerable. Our technical note Understanding Sine Test Methodologies covers when to use stepped sine or a resonance search and dwell instead.
- Control loop response. As the drive passes through a resonance of the shaker/fixture/article system, the plant gain the controller sees changes rapidly. A controller must measure and correct fast enough to keep the control point on profile without overshooting, which is a key differentiator between controllers on lightly damped loads.
- Shock testing. A short pulse delivers energy faster than a resonance can fully build, then leaves the part ringing. That ringing is what the shock response spectrum (SRS) captures: the SRS is literally the peak response of a bank of SDOF oscillators, usually computed at Q = 10.
Resonance under random vibration
A random test excites every frequency in the band at once, so every resonance in the band responds simultaneously. Each resonance acts as a narrow band-pass filter: the response spectrum is the input spectrum multiplied by the square of the transmissibility, because ASD is a power quantity.
For an SDOF resonance on a reasonably flat input spectrum, Miles' equation estimates the overall rms response:
| Quantity | Value | Note |
|---|---|---|
| Natural frequency, fn | 250 Hz | |
| Quality factor, Q | 20 | From a sine sweep, half-power method |
| Input ASD at fn | 0.04 g²/Hz | Input overall: 8.9 g rms over 20–2000 Hz |
| Miles' estimate | 17.7 g rms | √(1.571 × 250 × 20 × 0.04) |
| Exact integration of response ASD | 17.7 g rms | Agrees within 0.01% for this case |
| Approximate 3σ peak | ≈ 53 g | Assumes a Gaussian response; peaks can exceed this |
Note the result: the part's rms response is about twice the overall table level, even though the input band runs to 2 kHz. Miles' equation is an estimate, not a substitute for measurement. It assumes a single, lightly damped mode and an input that is roughly flat across the resonance; it does not account for multiple modes, notched or sloped profiles, or non-Gaussian (high-kurtosis) inputs.
Real structures: multiple modes, fixtures and response limiting
A real test article has a natural frequency for each way it can flex, called its modes. A circuit card has a first bending mode, a twisting mode, and higher modes; the chassis it mounts in has its own set; and the fixture and head expander add theirs. Each mode looks like its own SDOF peak in the transmissibility, and where modes are close together they interact.
Fixture resonances
A fixture is supposed to transmit the shaker's motion to the test article unchanged, which means operating in the rigid region of Figure 2. Ideally, its first resonance sits above the highest test frequency. When that is impractical (large fixtures, slip tables or tests to 2 kHz), a fixture resonance in the band means the control accelerometer no longer represents the input at every mounting point: one corner may be at the specified level while another is amplified or attenuated. Common remedies are adding damping, stiffening the fixture, and using multiple control accelerometers with an averaging or extremal control strategy. Panther's MISO control is designed for exactly this: several input channels contribute to a single drive, so the controlled level reflects the whole interface rather than a single point.
Response limiting (notching)
The opposite problem occurs when a test article's own resonance would drive a critical component beyond what it will see in service, typically because a rigid shaker table is a far stiffer support than the real mounting structure. Where the test specification allows, the controller can add a limit channel on the response point and automatically reduce the drive in the narrow band where the response would exceed its allowable level.
Notching needs justification. A response limit changes the test. It should be based on an engineering rationale (measured field response, interface force limits, or component allowables) and agreed with whoever owns the specification. Programs such as MIL-STD-810 and the NASA force-limited vibration approach allow it with approval; unrestricted notching can turn a qualification test into an under-test.
Common misconceptions
| Misconception | Reality |
|---|---|
| "Resonance is always bad and should be avoided in the test." | Revealing resonances is often the purpose of a vibration test. Field environments excite them too. The goal is to test them at the correct level, not to hide them. |
| "More damping is always better." | Damping lowers the resonant peak but increases transmission above √2·fn (Figure 2). Isolator design always balances the two. |
| "The peak in the response spectrum is the part's resonance." | Only if the input is flat and the phase confirms it. Peaks can come from the input profile, a fixture or shaker armature mode, cable whip or a loose accelerometer. Check transmissibility and phase, not response alone. |
| "Natural frequency doesn't change." | It shifts with temperature, preload, amplitude (nonlinear joints) and damage. A frequency drop during a dwell is an important warning sign. |
| "The Tacoma Narrows Bridge collapsed because of simple resonance." | It is the famous example, but the 1940 failure is generally attributed to aeroelastic flutter, a self-excited instability in which wind and structural motion feed each other. Forced resonance in the classic sense is better illustrated by the 1831 Broughton Suspension Bridge collapse under marching soldiers. |
Quick reference
| Quantity | Formula | Rule of thumb |
|---|---|---|
| Natural frequency | fn = (1/2π)·√(k/m) | Stiffness ×4 → frequency ×2 |
| Quality factor | Q = 1/(2ζ) | ζ = 2% → Q = 25 |
| Peak transmissibility | Tmax ≈ Q | 1 g in → ~Q g out at resonance |
| Half-power bandwidth | Δf = fn/Q | Q = 20 at 500 Hz → 25 Hz wide |
| Phase at resonance | ≈ −90° (response vs. input) | Confirms a true resonance |
| Isolation begins | f > √2·fn | Independent of damping |
| Build-up time | ≈ Q cycles to 95% | Slow sweeps for low-frequency, high-Q parts |
| Random response peak | ASDout ≈ Q² · ASDin at fn | Q = 20 → 400× the input level |
| Random rms response (Miles) | Grms ≈ √((π/2)·fn·Q·ASDin) | Single mode, flat input only |
How Panther helps
Every stage of working with resonance, from finding it, through measuring it, to controlling the test around it, depends on clean measurements and responsive control:
- Finding resonances: Panther Sine provides swept and stepped sine for low-level resonance searches, with transmissibility and phase between any response channel and the control.
- Measuring them accurately: 24-bit acquisition with more than 110 dB of dynamic range and channel-to-channel phase match on the order of 0.01° keeps both the peak and the 90° phase signature trustworthy, even when the response is 40 dB above the input.
- Controlling the test: MISO control with multiple control and limit channels lets the controlled level represent the whole fixture interface and keeps critical responses within their allowables.
- Analyzing results: Panther Analyzer adds FFT, transfer function and modal data tools for a deeper look at mode shapes and damping.
References and further reading
- "Resonance." Wikipedia. en.wikipedia.org/wiki/Resonance (accessed October 2026).
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- D. S. Steinberg, Vibration Analysis for Electronic Equipment, 3rd ed., Wiley.
- J. W. Miles, "On Structural Fatigue Under Random Loading," Journal of the Aeronautical Sciences, 1954.
- K. Y. Billah and R. H. Scanlan, "Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks," American Journal of Physics 59(2), 1991.
- MIL-STD-810H, Method 514.8, Vibration; NASA-HDBK-7004, Force Limited Vibration Testing.
- Spectral Dynamics, Understanding Sine Test Methodologies: Swept Sine, Stepped Sine, and Resonance Search and Dwell.
