Vibration

The Ritz method is applied to an in-plane vibration analysis to obtain accurate frequencies of isotropic annular plates. The method is formulated in a manner that allows all combinations of free boundary conditions, two types of supported (constraining only either radial or circumferential displacement) boundary conditions, and clamped boundary conditions. Admissible functions for the two displacement components are chosen as products of trigonometric functions in the circumferential coordinate and special algebraic polynomials in the radial coordinate, enabling all possible boundary-condition combinations to be satisfied. In the numerical study, after the solution’s accuracy is verified through convergence and comparison tests, extensive and accurate frequency parameters are presented to cover all combinations of the four in-plane boundary conditions along the outer and inner edges of the annular plates.

Variational Mode Decomposition (VMD) is a powerful formalism for the time-scale analysis of vibration signals from rotating machinery. However, its performance is often compromised by complex parameter configuration, where subjective manual tuning leads to mode mixing or information loss. In this study, we present a physics-guided framework that generalizes VMD optimization across diverse operating conditions. We utilized a meta-dataset combining three distinct sources (CWRU, HUST, UO) to validate the approach. Through a shaft-normalized segmentation strategy and K-Means++ clustering, we identified six distinct signal archetypes based on spectral morphology. Central to this framework is the Energy-Dispersion Index (EDI), a novel physically interpretable metric designed to differentiate between structured fault transients and stochastic noise. Extensive validation via a full-factorial Design of Experiments (8640 trials) confirmed the statistical superiority of EDI over benchmarks like kurtosis and envelope entropy, yielding an 8.3% improvement in modal fidelity. Furthermore, a rigorous ablation study demonstrated that the proposed archetype-based parameterization is highly efficient. This strategy achieved a $392\times$ speedup over online optimization while maintaining statistically equivalent diagnostic accuracy. Additionally, by generalizing parameters from high-quality archetype representatives, the framework reduced spectral leakage (Orthogonality Index) by 51.4% compared to instance-wise optimization. The resulting framework provides a mathematically rigorous, real-time solution for automated vibration signal decomposition.

Impact-echo/impact response testing is widely used to detect cracks, voids, and delamination, but transient signals and crowded spectra can complicate diagnosis. This study presents a nonlinear, harmonic-based framework that characterizes delamination using higher-order harmonics in the impact-free response, instead of the amplitude-dependent resonance–frequency shift. The delaminated region is formulated as a locally vibrating nonlinear plate/oscillator with polynomial material and geometric nonlinearities, predicting harmonic components whose levels depend on impact intensity and nonlinearity parameters. The approach is validated on a concrete slab containing an artificial delamination, excited by repeatable impacts, and measured with an accelerometer. Frequency-domain analysis shows that intact regions exhibit a distinct spectral pattern, whereas the delaminated region produces a clear fundamental component and, with modestly increased impacts, a strong second harmonic that serves as a defect signature; time series metrics corroborate nonlinearity. The results demonstrate a nondestructive technique that can localize and characterize delamination without driving the specimen into damaging strain. Looking ahead, the same harmonic signature principle can be extended to vibroacoustic/impact monitoring of lithium-ion batteries to flag mechanically induced internal defects (e.g., separator/electrode delamination) that can precipitate internal short circuits and elevate thermal runaway risk, improving quality control and in-service safety.