What you are trying to measure
Experimental modal analysis describes each mode by three things: its natural frequency, its damping, and its mode shape. They are normally found from a measured frequency response function (FRF) between a known input force and the structure's response.
For a sphere the picture has two features worth knowing. First, apart from the radial “breathing” mode, the modes of a perfect sphere come in groups of orientation-equivalent modes with the same frequency. Real spheres are not perfect: tests on thin spherical shells (marine floats) found that variations in thickness and radius split those groups into nearby peaks, while the axisymmetric modes matched theory to within about 1%. Second, torsional modes depend only on shear stiffness, while spheroidal modes depend on both shear and compression, which is why resonant ultrasound spectroscopy can recover elastic constants from sphere modes. Slight mode splitting of under 5% has been reported in measured spheres and attributed to material anisotropy.
Step 1: support the sphere so the support does not dominate
Most analysis assumes a free object. Approximate that with soft supports at or near nodal points, strings, or minimal-force contact. Resonant ultrasound work on spheres rests the sample on two transducers applied with minimal force to approximate free vibration. Record the support geometry, then remove and re-seat the sphere several times. The scatter between re-seatings is the first term in your uncertainty budget.
Step 2: excite the whole band
| Excitation | Good for | Watch out |
|---|---|---|
| Impact hammer | Fast, broadband survey with a measured input force | Operator-dependent hits, double strikes, contact time sets the usable bandwidth |
| Shaker with swept sine, chirp or multisine | Repeatable amplitude and phase; amplitude-dependence tests | The stinger and attachment change the boundary conditions |
| Piezoelectric actuator | Compact, wide bandwidth, usable in feedback loops | Bonding adds mass and local stiffness; voltage limits and hysteresis |
| Loudspeaker | Non-contact, easy to drive from a computer | Weak coupling at high frequency; the room and loudspeaker nonlinearity enter the result |
| Random or operational excitation | When input forces cannot be measured | Different assumptions and interpretation from controlled testing |
Repeat the measurement at two or more drive levels. If the peak frequency or shape changes with level, the response is not linear and the plain FRF analysis below needs care.
Step 3: choose a sensor that does not change the answer
| Sensor | What it measures | Strength | Watch out |
|---|---|---|---|
| Microphone | Air pressure at one point | Non-contact; shows radiated sound and decay | Not surface motion; depends on room and position |
| Microphone array | Pressure at several points | Maps radiation; supports near-field reconstruction | Needs calibration, dense sampling and inverse modelling |
| Accelerometer | Local acceleration, one or three axes | Standard in modal testing; synchronous multichannel | Adds mass and local stiffness; cable and mounting effects |
| Contact piezo | Local strain or acceleration | Compact, usable in feedback loops | Bonding alters the object; often narrowband |
| Single-point laser vibrometer | Surface velocity along the beam | No mass loading | One point, one direction; needs line of sight and a reflective surface |
| Scanning 3D laser vibrometer | Velocity maps, three components | Dense mode shapes on curved surfaces | Response must repeat during the scan; optical access and registration |
The sensor's mass should be small compared with the object's. For light objects the effect can be large: see the added-mass example on the sound and resonance page. A single microphone cannot give mode shapes; use it alongside a surface-motion sensor, not instead of one. Laser vibrometer calibration procedures are covered by ISO 16063-41.
Step 4: acquire synchronously and avoid aliasing
- Sample fast enough. The sampling rate must exceed twice the highest frequency present, and in practice with margin. A 20 kHz audio band is commonly recorded at 96 kHz; a 500 kHz band needs about 2.5 MS/s for a 5× margin.
- Filter before the converter. An analog anti-aliasing filter stops out-of-band energy folding into the band you care about.
- Share one clock. Force, response and any reference channels must be sampled synchronously so phase relationships survive.
- Mind the dynamic range. Set gain so the loudest resonance does not clip and the weakest is well above the noise floor.
- Record long enough. Frequency resolution of one FFT segment is 1/T, so resolving 0.1 Hz needs 10 s of data.
Step 5: compute the frequency response function
With force x in and response y out, the common H1 estimator divides the cross-spectrum by the input auto-spectrum, averaged over several records:
H1(f) = Sxy(f) / Sxx(f) coherence: gamma^2(f) = |Sxy(f)|^2 / (Sxx(f) * Syy(f))
Peaks in |H1| mark resonances. Coherence close to 1 at a peak says the output is linearly related to the measured input; low coherence flags noise, nonlinearity or an unmeasured input. A bare FFT peak without phase, coherence and repeats is not enough to claim a mode.
Step 6: extract frequency, damping and mode shape
- Frequency: read from the FRF peak, refined by interpolation or curve fitting.
- Damping: from the half-power bandwidth Δf,
Q ≈ f0/Δfandζ ≈ 1/(2Q), or from the decay of a windowed impulse response. - Mode shape: measure at many points and compare amplitude and phase across them; the imaginary part of the FRF is often used. Use the modal assurance criterion to check that modes from different runs are the same modes.
- Repeat after re-seating the sphere and on different days.
Common mistakes
- Calling the loudest peak “the” resonance without mapping its shape.
- Too few measurement points, so different high-order modes look alike (spatial aliasing).
- Ignoring temperature, which moves every frequency.
- Letting the mount, adhesive or cable change the object being measured.
- Reporting a single run with no repeatability.
In this project
The proposed S.P.H.E.R.E. measurement band is above 230 kHz on a thin hollow copper shell, so the project plans non-contact laser vibrometry and megasample-per-second acquisition rather than accelerometers and a sound card. The principles above are the same. See the hollow copper sphere page, the hardware guide and the error budget. These are proposals; no measurement has been made.
Frequently asked questions
How do you find the resonant frequency of a sphere?
Support it so the mount has little influence, excite it across the band with an impact, sweep or chirp, record the input and the response synchronously, compute the frequency response function, and read the resonances from its peaks. Confirm each one with phase, coherence and repeated runs.
What sampling rate do you need to measure resonances?
More than twice the highest frequency present, and in practice a margin of about 2.5 times or more, with an analog anti-aliasing filter ahead of the converter. A 500 kHz band therefore needs roughly 2.5 megasamples per second.
Why does a sphere show several close resonance peaks?
A perfect sphere has groups of modes with identical frequencies. Small variations in thickness, radius or material split each group into nearby peaks, so closely spaced peaks are expected in a real sphere.
Sources
- How to Determine the Modal Parameters of Simple Structures, Brüel & Kjær application note.
- The resonant ultrasound spectroscopy method for determining the Poisson's ratio of spheres over the full range, Li, Dong and Lakes, Materials Letters 143, 31-34 (2015).
- Vibrations of Complete Spherical Shells with Imperfections, Duffey, Pepin, Robertson and Steinzig, IMAC-XXIII (2005).
- ISO 16063-41:2011, Calibration of laser vibrometers.