The basic relationship
For a single mode, frequency rises with stiffness and falls with mass. Real structures are more subtle, because how much a change matters depends on where it happens relative to the mode shape.
f ~ (1/2pi) * sqrt(k / m) small added mass: df/f ~ -1/2 * (dm / m_eff) m_eff = effective mass of the mode at the attachment point
A small mass added where the mode moves most (an antinode) lowers the frequency the most. The same mass added at a node barely changes that mode. This is why one added mass can shift some modes and leave others untouched, and why mass is a useful calibration tool.
The levers
| Lever | Effect on frequency | Reversible | Notes and limits |
|---|---|---|---|
| Added mass | Lowers | Yes | In one published test, 144 g added to a 213 g aluminium plate lowered the first three resonances by about 12 to 16%. Adhesives also add damping. |
| Radius or overall size | Frequencies scale roughly as 1/radius | No | Permanent geometry change. For a thin spherical shell the radial breathing mode follows about (1/2πR)√(2E/ρ(1−ν)). |
| Wall thickness | Depends on the mode | No | To first order the membrane breathing mode of a thin shell does not depend on wall thickness, while bending modes do. Thickness also sets mass. |
| Temperature | Usually lowers as it rises | Yes | Moduli change with temperature. One steel benchmark structure shifted by about 2.43% over 30 °C; the value is material- and structure-specific. |
| Support and contact | Raises or lowers, mode-dependent | Yes | Clamps, preload and contact stiffness create a different system. |
| Fluid loading (immersion or fill) | Lowers, adds damping | Yes | The surrounding or enclosed fluid moves with the shell. |
| Prestress or pressure | Raises or lowers | Yes | Changes incremental stiffness. |
| Material state (anneal, work, damage) | Either way | No | Changes elastic constants and residual stress. Permanent, and cracks are damage, not tuning. |
| Large drive amplitude | Slightly lowers in some materials | Mostly | Nonlinear elasticity; check with an amplitude sweep. |
| Tuned absorber or shunted piezo | Local tuning and added damping | Yes | Mostly reduces amplitude; the attached mass also perturbs the host. |
| Active feedback control | Moves the closed-loop response | Yes | Changes the combined system. The passive object is unchanged when the loop is off. |
Passive eigenfrequency versus closed-loop response
A natural frequency of the passive object changes only when its mass, stiffness, geometry, material state or constraints change. A feedback controller leaves all of those alone. It adds forces computed from measurements, so the combined system has different dynamics. That can be useful, but it is a different claim. The test is simple: switch the controller off. If the original spectrum returns, the shift was active and reversible, not a change to the object.
Which levers suit a measurement experiment
For metrology the useful levers are those that are reversible, small and quantifiable. A known added mass calibrates how sensitive a mode is to a perturbation. A controlled temperature step measures the thermal coefficient that must be corrected in every run. Permanent changes such as annealing or cutting are valuable for characterisation but destroy the ability to repeat a measurement on the same object.
In this project
S.P.H.E.R.E. treats temperature, mass loading and mounting as nuisance variables to be measured and corrected, not as the effect being tested. The proposed tests of whether a replayed signal could retune a copper sphere are described in the record-and-replay article, and the design for confirming any claimed shift is on the verification protocol page. Nothing here reports a result.
Frequently asked questions
How do you change the resonant frequency of an object?
Change its mass, stiffness, geometry or how it is supported. Adding mass lowers frequencies, stiffening raises them, and heating usually lowers them. A feedback controller can change the response of the combined system without changing the passive object.
Does adding weight lower resonant frequency?
Yes, in general. The amount depends on how heavy the added mass is compared with the object and where it is attached. A mass placed at a vibration node barely affects that mode, and one at an antinode affects it most.
Sources
- Zero mass loading: effect and compensation in vibration analysis, Polytec application note.
- Effect of Temperature Changes on the Experimental Modal Analysis of a Galvanized Steel Benchmark Structure, Tuhta, Koç and Günday, Buildings 16(5), 1069 (2026).
- Vibrations of Complete Spherical Shells with Imperfections, Duffey, Pepin, Robertson and Steinzig, IMAC-XXIII (2005).