{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T05:15:44Z","timestamp":1788412544554,"version":"build-2803163510"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T00:00:00Z","timestamp":1787529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Mode veering is a modal-interaction phenomenon found in vibration systems. For inertially coupled structures, the combined influence of coupling delay, nonlinear restoring force, and stochastic coupling perturbation remain insufficiently understood. This work analyzes an inertially coupled two-coordinate prototype in which a discrete delay, a delayed cubic stiffness, and positive multiplicative stochastic modulation all enter through the same relative-coordinate coupling channel. We formulate the delayed linear spectrum through a quasi-polynomial characteristic equation. We also characterize the veering by the two positive-frequency characteristic-root branches descending from the mechanical modes. Coupling delay shifts the veering center, alters the minimum frequency gap, and moves the tracked rightmost roots toward the stability boundary. An analytical imaginary-axis-crossing criterion is derived to determine the delay-induced stability boundary of the deterministic linearized system, and the resulting boundary is independently validated by direct multi-start characteristic-root searches and Chebyshev-collocation approximation of the DDE generator. A fixed-reference modal-coordinate representation identifies the off-diagonal modal terms associated with branch exchange while retaining the full delayed characteristic equation. A first-harmonic treatment of the delayed cubic term can yield an amplitude-dependent nonlinear veering backbone. For the stochastic problem, frozen lognormal coupling samples and a time-dependent Ornstein\u2013Uhlenbeck-driven multiplier are constructed from the same unit-mean positive lognormal marginal law. The former is used to quantify realization-wise spectral broadening, whereas the latter retains temporal correlation and is used to evaluate finite-time branch residence and pathwise delayed-work statistics. The pathwise energy balance reveals that the delayed relative-coordinate work rate is sign-indefinite. This provides a common energy-transfer mechanism through which delay, nonlinearity, and stochastic modulation reshape mode veering in the inertially coupled system.<\/jats:p>","DOI":"10.3390\/e28090952","type":"journal-article","created":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:49:34Z","timestamp":1787582974000},"page":"952","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Delay-Modulated Nonlinear Stochastic Mode Veering in Inertially Coupled Vibration Systems"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8618-0898","authenticated-orcid":false,"given":"Lili","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-8796-6926","authenticated-orcid":false,"given":"Zikun","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8279-4128","authenticated-orcid":false,"given":"Qiubao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"},{"name":"Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1016\/0022-460X(88)90226-X","article-title":"Mode localization and eigenvalue loci veering phenomena in disordered structures","volume":"126","author":"Pierre","year":"1988","journal-title":"J. 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