Screening bolt loosening in a four-bolt plate with global FRF correlation and local FRAC maps from full-field laser Doppler vibrometry

• Joint friction, preload-dependent nonlinearity, and assembly conditions can alter resonance frequencies, damping, higher-order FRF behavior, effective stiffness, and measured vibration response.Source: §1 · p. 1
• Full-field LDV provides non-contact vibration measurements with high spatial sampling density, supporting comparison of response changes across an inspected surface.Source: §1 · p. 2
• A raw frequency label is specimen-specific, while assigning an FRF peak a mode number requires modal-parameter identification rather than peak picking alone.Source: §1 · p. 2
• The workflow uses ordered experimental resonance-group labels and reports a dimensionless plate-frequency coefficient alongside each measured frequency to separate response identity from scaled frequency information.Source: §1 · p. 2
• The contribution links resonance-group selection with amplitude-only and complex global FRF comparison and local FRAC-based spatial interpretation.Source: §1 · p. 2

Resonance-group selection and tracking

• Seven resonance groups are selected from the all-tight spatial RMS spectrum and retained only when corresponding peaks remain trackable in all four loose-bolt cases.Candidate selection uses smoothed-spectrum maxima, prominence and width criteria, de-duplication, and group-specific search windows.Source: §2 · p. 2, §2.2 · p. 3

Specimen and measurements

• The experiment uses a 100 × 150 × 7 mm aluminum plate, four M6 bolts, an all-tight 10 N m reference, and four single-bolt 0 N m cases.The plate was measured under soft-sponged, approximately free-free conditions using scanned H1 displacement-to-force FRFs up to 20 kHz.Source: §2.1 · p. 2
• The analysis compares 51 LDV coordinates common to amplitude and phase exports across all analyzed states.The complete scan contained 79 measured positions, of which 51 were retained for comparisons.Source: §2.1 · p. 2

Resonance-group selection and tracking

• RG1–RG7 are ordered by increasing all-tight frequency, with each group paired with a geometry- and material-normalized plate-frequency coefficient.The coefficient uses representative material properties because the alloy grade was not documented; the measured FRF metrics do not depend on these assumptions.Source: §2.2 · p. 3

Global and local screening metrics

• Global screening uses amplitude-only 1 − MAC_a and phase-retaining 1 − CMAC dissimilarities, averaged across the four single-bolt cases with exhaustive-bootstrap intervals.The complex metric retains measured phase information, whereas the amplitude-only metric compares spatial amplitude vectors.Source: §2.3 · pp. 3–4
• Local screening compares equal-width matched frequency windows with FRAC-deficit maps and a normalized high-deficit area fraction.The hotspot-area calculation uses the common LDV-coordinate hull and a fixed descriptive threshold τ = 0.75 rather than a dataset-optimized threshold.Source: §2.3 · p. 4, Table 2 · p. 5

Tracked resonance groups

• RG1−RG4 shift upward in all four loose-bolt cases, whereas RG5−RG7 show bidirectional changes from approximately Δf = −180 to +180 Hz.The lower-frequency shifts are 74−85 Hz for RG1, 134−169 Hz for RG2, 71−96 Hz for RG3, and 124−177 Hz for RG4.Source: §3 · p. 5

Global screening

• RG1−RG3 form a low-distortion set, with mean 1−MAC_a values of 0.049−0.052 and 1−CMAC values of 0.066−0.073.Source: §3 · p. 5
• RG4 is intermediate, with 1−MAC_a = 0.146 and 1−CMAC = 0.185, while RG5−RG7 have amplitude dissimilarities of 0.276, 0.335, and 0.288.Source: §3 · p. 5
• Phase-aware dissimilarities reach 0.624, 0.599, and 0.608 for RG5−RG7, indicating complex spatial FRF redistribution beyond resonance relocation.The uncertainty intervals also broaden for the high-frequency groups, particularly for 1−CMAC.Source: §3 · p. 5

Local screening

• RG3 has the clearest compact joint-centered field, with an L_g ≥ 0.75 core occupying 1.49% of the scanned area, while RG1 and RG2 occupy 2.87% and 2.01%.Despite similarly low global dissimilarities, RG1−RG3 do not exhibit equivalent local behavior.Source: §3 · p. 5
• RG4−RG6 have normalized high-deficit areas of 9.63%, 9.91%, and 7.98%, whereas RG7 has a 0.81% thresholded core despite 1−CMAC = 0.608.The full local map remains necessary because thresholded area does not describe absolute magnitude, position, or the remainder of the response field.Source: §3 · p. 5

Global distortion

• RG1−RG3 show low global distortion, while RG5−RG7 show much larger phase-aware changes; phase retention increases mean dissimilarity by 0.0183 versus 0.3107 across these regimes.The corresponding mean 1−CMAC values are 0.0690 and 0.6103, respectively.Source: §4 · p. 6

Limitations

• High-frequency uncertainty is strongly case dependent: the mean 95% 1−CMAC interval width is 0.032 for RG1−RG3 and 0.642 for RG5−RG7.With four bolt locations resampled, these intervals represent bolt-location variability rather than population-level uncertainty.Source: §4 · p. 6

Local spatial extent

• RG1−RG3 have a mean hotspot area of 2.12%, whereas RG4−RG6 average 9.17%, indicating broader multi-region redistribution in the latter group.RG4−RG6 therefore average 4.32 times the RG1−RG3 hotspot area.Source: §4 · p. 6

Complementary descriptors

• RG7 demonstrates that global dissimilarity and hotspot extent are not monotonic: its 1−CMAC is 0.608, but its hotspot core is only 0.81%.RG4 and RG5 have nearly identical hotspot areas despite markedly different global dissimilarities.Source: §4 · p. 6
• The pair (1−CMAC, η0.75) should be interpreted with the complete local map because the hotspot fraction measures relative core area, not absolute FRAC-deficit magnitude.The map additionally identifies hotspot location and fragmentation.Source: §4 · p. 6

Limitations

• The study is constrained by one plate, four 0 N m cases, approximate free-free support, overlapping high-frequency windows, asymmetric excitation during symmetry averaging, and an analyst-selected 0.75 threshold.Future work should test additional geometries, preload levels, bolt layouts, boundary conditions, and tracking or threshold sensitivities.Source: §4 · pp. 6–7

Workflow

• Seven trackable resonance groups were selected from the spatial RMS spectrum and reported with one plate-normalized frequency coefficient.The analysis used an all-tight reference and four single-bolt 0 N m cases.Source: §5 · p. 7

Findings

• RG1−RG3 form a low-distortion set, RG4 an intermediate response, and RG5−RG7 a strongly phase-aware distorted set.Local FRAC maps identify RG3 as the clearest compact joint-centered hotspot, while RG4−RG6 show broader high-deficit regions.Source: §5 · p. 7

Implication

• The normalized hotspot-area fraction complements, but does not replace, global dissimilarity measures.Together, the workflow links resonance selection, global correlation, and local FRF mapping without a trained classifier or specimen-specific node labels.Source: §5 · p. 7

Data availability

• The repository contains measurement data, metadata, supporting documentation, and analysis scripts associated with the experiments.A companion data-descriptor manuscript has been submitted for publication.Source: Data availability · p. 7

Contributions

• Kullukcu led conceptualization, methodology, software, formal analysis, and the original draft, with Pianowski contributing to methodology, analysis, and writing.Özer contributed to formal analysis and writing.Source: CRediT authorship contribution statement · p. 7
• Altinsoy and Hannebauer provided supervision and resources, with Hannebauer also contributing to review and editing.Source: CRediT authorship contribution statement · p. 7