Impacts
From whether to how much — measuring thickness with the same tap
Tap testing, in all its instrumented forms, answers one question: is this point intact or damaged. It says nothing about the thickness of what is vibrating, and in conservation that number matters, because it is part of what determines the state of a detachment and the risk it carries. The most recent study in this research line sets out to extract it from the very same measurement the instrument already performs.
The starting point is classical impact theory. Hertz described perfectly elastic contact between curved bodies, but assumed bodies of infinite dimensions and neglected the energy carried away by elastic waves. Zener extended it to large thin plates, where energy is dissipated mainly through flexural waves propagating radially from the contact area, and derived a nonlinear differential equation governing the process. That equation has no closed-form solution — but a recent approximation proposed by Tsai does, valid for values of the inelasticity parameter λ up to 1.5 and reproducing the numerical solution of Zener's equation to within about 7% on the normalised force–time history.
From this the authors work backwards. The contact time TC — the duration of the physical contact between striker and surface — fixes the inelasticity parameter, and the inelasticity parameter fixes the thickness. Knowing the impact velocity, the mass, radius and elastic constants of the striker, and the density, elastic modulus and Poisson's ratio of the plate, the thickness follows from the measured contact time alone. Thickness varies as the inverse square root of λ, which is also the reason the method becomes delicate on very thin plates: there, small variations in λ move the estimate a long way.
The approach was checked twice over. A finite element model of the sphere–plate system, with local mesh refinement in the impact zone, simulated a steel sphere dropped from 150 mm onto steel and aluminium plates; the end of contact was identified by the reversal of the plate's velocity along the impact axis, and the simulated contact times agreed closely with the theory — slightly longer for the more compliant aluminium, as the flexural-wave mechanism predicts.
The experimental campaign is disarmingly simple, and deliberately so. Plates of steel, brass and aluminium measuring 15 × 30 cm were clamped at the edges between bolted rectangular frames, putting them under tension like the skin of a drum, and mounted in a baffle. A hollow steel ball 1 cm in diameter, threaded inside so that a thin conductor could be attached and weighed separately on a precision scale, was released by an electromagnet so that it fell freely without rotation or initial velocity. The ball and plate were wired into a circuit: during contact the ball closes the switch, and an oscilloscope reads the duration directly, with microsecond resolution and no high-speed camera or laser system involved. Each series was repeated more than ten times and the mean reported.
- Within the validity range of the model (λ < 1.5), estimated thicknesses fell within about 3% of values measured with a thickness gauge: 2.58 mm against a real 2.56 mm for steel (+0.78%), 3.02 against 3.00 mm for aluminium (+0.66%), 4.02 against 4.00 mm for brass (+0.5%), 2.90 against 2.96 mm for Al 1050 (−2.02%).
- Applied to a measurement reported in the literature, the method returned 1.56 mm against an experimental 1.59 mm, a relative error of −1.88%.
- Outside that range the method degrades as predicted: on a 1.50 mm aluminium plate (λ = 2.71) and a 1.08 mm brass plate (λ = 2.85) the errors reached 18.6% and 16.7%, where the increased flexibility of thin plates amplifies edge effects and localised plastic deformation violates the purely elastic assumption.
- A sensitivity analysis explains why the results are better than Tsai's 7% approximation might suggest: the inversion depends on the zero-crossing of the contact-force waveform rather than on its amplitude, and is therefore comparatively insensitive to that error. The predicted thickness uncertainty falls from roughly 30% at λ ≈ 0.1 to below 5% for λ above about 1.2.
The algorithm has been added to the software already running on the PICUS tester, which measures contact time as part of its normal operation; the operator supplies the mechanical properties of the material through the instrument's menus, and the thickness at the measured point becomes available alongside the other readings. The authors state explicitly that the measurements obtained this way with PICUS are the subject of a future publication, together with a full discussion of the errors involved.
The motivation is stated just as plainly at the end of the paper: the study was carried out in response to the need to measure the thickness of detached surfaces in architectural coatings and ancient frescoes. It also points outward, to tanks, pipelines and structural elements that can only be reached from one side, and where speed, low cost and safety are the binding constraints. Future work will extend the model to viscoelastic and plastic effects, and examine the influence of temperature and surface roughness.
Publications
- G. Caliano, F. Mariani, M. Lo Giudice, A. Salvini, "The Measurement of the Impact Time to Evaluate the Plate Thickness", Applied Sciences, 16(1), 89, 2026. DOI: 10.3390/app16010089