The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems

The dynamics of mechanical structures are often described by linear algebraic systems of the form Ax=f. At high frequencies, A may represent the coupling loss factor matrix in a Statistical Energy Analysis (SEA) model, whereas at low frequencies, it may correspond to the dynamic stiffness matrix of...

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Detalles Bibliográficos
Autores: Guasch, Oriol, Deng, Jie
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Universitat Ramon Llull (URL)
Repositorio:DAU Arxiu Digital de la Universitat Ramon Llull
OAI Identifier:oai:dau.url.edu:20.500.14342/5991
Acceso en línea:http://hdl.handle.net/20.500.14342/5991
https://doi.org/10.1121/2.0002044
Access Level:acceso abierto
Palabra clave:Lippmann-Schwinger equation
Analysis
Mechanics
53
531/534
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spelling The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systemsGuasch, OriolDeng, JieLippmann-Schwinger equationAnalysisMechanics53531/534The dynamics of mechanical structures are often described by linear algebraic systems of the form Ax=f. At high frequencies, A may represent the coupling loss factor matrix in a Statistical Energy Analysis (SEA) model, whereas at low frequencies, it may correspond to the dynamic stiffness matrix of a system of oscillators. While such systems admit a Neumann series solution at high frequencies-where the terms can be interpreted as energy transmission paths of increasing order-this series typically fails to converge at low frequencies, rendering its physical interpretation unclear. In this work, we recast the system within the framework of the Lippmann-Schwinger equation and express the solution as a series in powers of a transmission matrix T, defined as the product of the system’s bare Green function and coupling matrix. To achieve convergence, we introduce a multi-parameter product renormalization scheme. We show that, with a suitable choice of parameters based on the eigenvalues of T, a finite expansion is obtained involving powers up to TN−1, where N is the system's dimension. That is, the expansion includes at most the longest open transmission paths between elements. In doing so, we recover-through purely algebraic methods-a result previously derived using considerations from graph theory.info:eu-repo/semantics/publishedVersionAcoustical Society of AmericaUniversitat Ramon Llull. La Salle2026202620252025info:eu-repo/semantics/article7 p.application/pdfhttp://hdl.handle.net/20.500.14342/5991https://doi.org/10.1121/2.0002044reponame:DAU Arxiu Digital de la Universitat Ramon Llullinstname:Universitat Ramon Llull (URL)InglésProceedings of Meetings on Acoustics, Vol. 57, 045001 (2025)© L'autor/aAttribution-NonCommercial 4.0 Internationalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccessoai:dau.url.edu:20.500.14342/59912026-06-21T06:40:37Z
dc.title.none.fl_str_mv The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
title The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
spellingShingle The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
Guasch, Oriol
Lippmann-Schwinger equation
Analysis
Mechanics
53
531/534
title_short The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
title_full The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
title_fullStr The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
title_full_unstemmed The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
title_sort The Lippmann–Schwinger equation and renormalization for transmission path analysis in discrete mechanical systems
dc.creator.none.fl_str_mv Guasch, Oriol
Deng, Jie
author Guasch, Oriol
author_facet Guasch, Oriol
Deng, Jie
author_role author
author2 Deng, Jie
author2_role author
dc.contributor.none.fl_str_mv Universitat Ramon Llull. La Salle
dc.subject.none.fl_str_mv Lippmann-Schwinger equation
Analysis
Mechanics
53
531/534
topic Lippmann-Schwinger equation
Analysis
Mechanics
53
531/534
description The dynamics of mechanical structures are often described by linear algebraic systems of the form Ax=f. At high frequencies, A may represent the coupling loss factor matrix in a Statistical Energy Analysis (SEA) model, whereas at low frequencies, it may correspond to the dynamic stiffness matrix of a system of oscillators. While such systems admit a Neumann series solution at high frequencies-where the terms can be interpreted as energy transmission paths of increasing order-this series typically fails to converge at low frequencies, rendering its physical interpretation unclear. In this work, we recast the system within the framework of the Lippmann-Schwinger equation and express the solution as a series in powers of a transmission matrix T, defined as the product of the system’s bare Green function and coupling matrix. To achieve convergence, we introduce a multi-parameter product renormalization scheme. We show that, with a suitable choice of parameters based on the eigenvalues of T, a finite expansion is obtained involving powers up to TN−1, where N is the system's dimension. That is, the expansion includes at most the longest open transmission paths between elements. In doing so, we recover-through purely algebraic methods-a result previously derived using considerations from graph theory.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.14342/5991
https://doi.org/10.1121/2.0002044
url http://hdl.handle.net/20.500.14342/5991
https://doi.org/10.1121/2.0002044
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Proceedings of Meetings on Acoustics, Vol. 57, 045001 (2025)
dc.rights.none.fl_str_mv © L'autor/a
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv © L'autor/a
Attribution-NonCommercial 4.0 International
http://creativecommons.org/licenses/by-nc/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 7 p.
application/pdf
dc.publisher.none.fl_str_mv Acoustical Society of America
publisher.none.fl_str_mv Acoustical Society of America
dc.source.none.fl_str_mv reponame:DAU Arxiu Digital de la Universitat Ramon Llull
instname:Universitat Ramon Llull (URL)
instname_str Universitat Ramon Llull (URL)
reponame_str DAU Arxiu Digital de la Universitat Ramon Llull
collection DAU Arxiu Digital de la Universitat Ramon Llull
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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