Dynamical system of a time-delayed of rigid rocking rod: analytical approximate solution

The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several i...

Full description

Saved in:
Bibliographic Details
Published inScientific reports Vol. 13; no. 1; pp. 5570 - 13
Main Authors Moatimid, Galal M., Amer, T. S.
Format Journal Article
LanguageEnglish
Published London Nature Publishing Group UK 05.04.2023
Nature Publishing Group
Nature Portfolio
Subjects
639/705
639/705/1041
Approximation
Humanities and Social Sciences
multidisciplinary
Science
Science (multidisciplinary)
Stability analysis
Online AccessGet full text

Cover

Abstract The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
AbstractList The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge-Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge-Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
Abstract The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.
ArticleNumber 5570
Author Moatimid, Galal M.
Amer, T. S.
Author_xml – sequence: 1
  givenname: Galal M.
  surname: Moatimid
  fullname: Moatimid, Galal M.
  organization: Department of Mathematics, Faculty of Education, Ain Shams University
– sequence: 2
  givenname: T. S.
  surname: Amer
  fullname: Amer, T. S.
  email: tarek.saleh@science.tanta.edu.eg
  organization: Department of Mathematics, Faculty of Science, Tanta University
BackLink https://www.ncbi.nlm.nih.gov/pubmed/37020045$$D View this record in MEDLINE/PubMed
BookMark eNp9Uk1v1TAQtFARLaV_gAOKxIVLwJ9xwgVVpUClSlxA4mb52ZvgR2I_7ISSf1_npYW2h1qW19qdGY-8-xwd-OABoZcEvyWY1e8SJ6KpS0xZyajkrLx6go4o5qKkjNKDO_dDdJLSFuclaMNJ8wwdMokpzvUj9OPj7PXgjO6LNKcRhiK0hS5GN0Bpodcz2CUTXedsEYP55XyXo31faK_7edwz9W4Xw1836BGKFPppdMG_QE9b3Sc4uYnH6Pun829nX8rLr58vzk4vSyM4GUvTSMEMGF2ZmrCWCcIFA1FxDo2xwlaUSQ21tRWpoaVG1NZYCZgA25i6ZewYXay6Nuit2sXsIs4qaKf2iRA7pWO22YMSG1nlTaqNbrgAq1srJQeMdWPqChatD6vWbtoMYA34Mer-nuj9inc_VRf-KIKxrBihWeHNjUIMvydIoxpcMtD32kOYkqKykYTnY3ns9QPoNkwxf-oeJRoia9pk1Ku7lv55ue1gBtQrwMSQUoRWGTfqpQPZoeuzNbXMi1rnReV5Uft5UVeZSh9Qb9UfJbGVlDLYdxD_236EdQ2TCtPc
CitedBy_id crossref_primary_10_59400_sv2358
crossref_primary_10_1038_s41598_023_38679_5
crossref_primary_10_1016_j_aej_2023_05_030
crossref_primary_10_1016_j_rinp_2024_107792
crossref_primary_10_1038_s41598_023_47519_5
Cites_doi 10.1016/0375-9601(92)90745-8
10.1007/s00419-021-02039-4
10.1016/j.matcom.2020.08.001
10.1016/S0045-7825(99)00018-3
10.3390/axioms10030191
10.1016/j.mcm.2010.04.010
10.1007/978-3-662-05030-9
10.1038/s41598-022-15121-w
10.1016/j.apm.2017.02.019
10.3390/axioms10040320
10.1002/eqe.4290060503
10.1007/s13369-018-03705-6
10.1177/1461348419844145
10.1023/B:NODY.0000009957.42817.4f
10.1177/14613484221130148
10.1177/14613484211026407
10.1006/jsvi.1996.0672
10.1002/eqe.4290120209
10.1007/s11071-006-9192-z
10.1007/s00419-023-02395-3
10.1007/s42417-022-00693-8
10.1007/s00419-022-02246-7
10.1109/ACCESS.2020.3044998
10.1016/S1007-5704(99)90026-6
10.1177/14613484221077474
10.1061/(ASCE)EM.1943-7889.0001764
ContentType Journal Article
Copyright The Author(s) 2023
2023. The Author(s).
The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Copyright_xml – notice: The Author(s) 2023
– notice: 2023. The Author(s).
– notice: The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
DBID C6C
AAYXX
CITATION
NPM
3V.
7X7
7XB
88A
88E
88I
8FE
8FH
8FI
8FJ
8FK
ABUWG
AEUYN
AFKRA
AZQEC
BBNVY
BENPR
BHPHI
CCPQU
DWQXO
FYUFA
GHDGH
GNUQQ
HCIFZ
K9.
LK8
M0S
M1P
M2P
M7P
PHGZM
PHGZT
PIMPY
PJZUB
PKEHL
PPXIY
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
Q9U
7X8
5PM
DOA
DOI 10.1038/s41598-023-32743-w
DatabaseName Springer Nature OA Free Journals
CrossRef
PubMed
ProQuest Central (Corporate)
Health & Medical Collection
ProQuest Central (purchase pre-March 2016)
Biology Database (Alumni Edition)
Medical Database (Alumni Edition)
Science Database (Alumni Edition)
ProQuest SciTech Collection
ProQuest Natural Science Journals
Hospital Premium Collection
Hospital Premium Collection (Alumni Edition)
ProQuest Central (Alumni) (purchase pre-March 2016)
ProQuest Central (Alumni)
ProQuest One Sustainability
ProQuest Central UK/Ireland
ProQuest Central Essentials
Biological Science Collection
ProQuest Central
Natural Science Collection
ProQuest One
ProQuest Central Korea
Health Research Premium Collection
Health Research Premium Collection (Alumni)
ProQuest Central Student
SciTech Premium Collection
ProQuest Health & Medical Complete (Alumni)
Biological Sciences
ProQuest Health & Medical Collection
Medical Database
Science Database
Biological Science Database
ProQuest Central Premium
ProQuest One Academic (New)
Publicly Available Content Database
ProQuest Health & Medical Research Collection
ProQuest One Academic Middle East (New)
ProQuest One Health & Nursing
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Applied & Life Sciences
ProQuest One Academic
ProQuest One Academic UKI Edition
ProQuest Central China
ProQuest Central Basic
MEDLINE - Academic
PubMed Central (Full Participant titles)
DOAJ Directory of Open Access Journals
DatabaseTitle CrossRef
PubMed
Publicly Available Content Database
ProQuest Central Student
ProQuest One Academic Middle East (New)
ProQuest Central Essentials
ProQuest Health & Medical Complete (Alumni)
ProQuest Central (Alumni Edition)
SciTech Premium Collection
ProQuest One Community College
ProQuest One Health & Nursing
ProQuest Natural Science Collection
ProQuest Central China
ProQuest Biology Journals (Alumni Edition)
ProQuest Central
ProQuest One Applied & Life Sciences
ProQuest One Sustainability
ProQuest Health & Medical Research Collection
Health Research Premium Collection
Health and Medicine Complete (Alumni Edition)
Natural Science Collection
ProQuest Central Korea
Health & Medical Research Collection
Biological Science Collection
ProQuest Central (New)
ProQuest Medical Library (Alumni)
ProQuest Science Journals (Alumni Edition)
ProQuest Biological Science Collection
ProQuest Central Basic
ProQuest Science Journals
ProQuest One Academic Eastern Edition
ProQuest Hospital Collection
Health Research Premium Collection (Alumni)
Biological Science Database
ProQuest SciTech Collection
ProQuest Hospital Collection (Alumni)
ProQuest Health & Medical Complete
ProQuest Medical Library
ProQuest One Academic UKI Edition
ProQuest One Academic
ProQuest One Academic (New)
ProQuest Central (Alumni)
MEDLINE - Academic
DatabaseTitleList
PubMed
CrossRef
MEDLINE - Academic


Publicly Available Content Database
Database_xml – sequence: 1
  dbid: C6C
  name: Springer Nature OA Free Journals
  url: http://www.springeropen.com/
  sourceTypes: Publisher
– sequence: 2
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 3
  dbid: NPM
  name: PubMed
  url: https://proxy.k.utb.cz/login?url=http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed
  sourceTypes: Index Database
– sequence: 4
  dbid: BENPR
  name: ProQuest Central
  url: https://www.proquest.com/central
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
Discipline Biology
EISSN 2045-2322
EndPage 13
ExternalDocumentID oai_doaj_org_article_5b76b7616ba945edafd774e00a9c86e3
PMC10076312
37020045
10_1038_s41598_023_32743_w
Genre Journal Article
GrantInformation_xml – fundername: Tanta University
– fundername: ;
GroupedDBID 0R~
3V.
4.4
53G
5VS
7X7
88A
88E
88I
8FE
8FH
8FI
8FJ
AAFWJ
AAJSJ
AAKDD
ABDBF
ABUWG
ACGFS
ACSMW
ACUHS
ADBBV
ADRAZ
AENEX
AEUYN
AFKRA
AJTQC
ALIPV
ALMA_UNASSIGNED_HOLDINGS
AOIJS
AZQEC
BAWUL
BBNVY
BCNDV
BENPR
BHPHI
BPHCQ
BVXVI
C6C
CCPQU
DIK
DWQXO
EBD
EBLON
EBS
ESX
FYUFA
GNUQQ
GROUPED_DOAJ
GX1
HCIFZ
HH5
HMCUK
HYE
KQ8
LK8
M0L
M1P
M2P
M48
M7P
M~E
NAO
OK1
PIMPY
PQQKQ
PROAC
PSQYO
RNT
RNTTT
RPM
SNYQT
UKHRP
AASML
AAYXX
AFPKN
CITATION
PHGZM
PHGZT
NPM
7XB
8FK
AARCD
K9.
PJZUB
PKEHL
PPXIY
PQEST
PQGLB
PQUKI
PRINS
Q9U
7X8
5PM
PUEGO
ID FETCH-LOGICAL-c541t-c9753ceca6c813f351453e5644e9cd5d6237ae8dd618ef2c58dcd7e01e3bc8f33
IEDL.DBID M48
ISSN 2045-2322
IngestDate Wed Aug 27 01:32:03 EDT 2025
Thu Aug 21 18:38:19 EDT 2025
Thu Jul 10 17:55:59 EDT 2025
Wed Aug 13 07:17:22 EDT 2025
Thu Apr 03 07:08:45 EDT 2025
Thu Apr 24 22:55:38 EDT 2025
Tue Jul 01 04:24:26 EDT 2025
Fri Feb 21 02:39:43 EST 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 1
Language English
License 2023. The Author(s).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c541t-c9753ceca6c813f351453e5644e9cd5d6237ae8dd618ef2c58dcd7e01e3bc8f33
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
content type line 23
OpenAccessLink http://journals.scholarsportal.info/openUrl.xqy?doi=10.1038/s41598-023-32743-w
PMID 37020045
PQID 2795917829
PQPubID 2041939
PageCount 13
ParticipantIDs doaj_primary_oai_doaj_org_article_5b76b7616ba945edafd774e00a9c86e3
pubmedcentral_primary_oai_pubmedcentral_nih_gov_10076312
proquest_miscellaneous_2797147973
proquest_journals_2795917829
pubmed_primary_37020045
crossref_citationtrail_10_1038_s41598_023_32743_w
crossref_primary_10_1038_s41598_023_32743_w
springer_journals_10_1038_s41598_023_32743_w
ProviderPackageCode CITATION
AAYXX
PublicationCentury 2000
PublicationDate 2023-04-05
PublicationDateYYYYMMDD 2023-04-05
PublicationDate_xml – month: 04
  year: 2023
  text: 2023-04-05
  day: 05
PublicationDecade 2020
PublicationPlace London
PublicationPlace_xml – name: London
– name: England
PublicationTitle Scientific reports
PublicationTitleAbbrev Sci Rep
PublicationTitleAlternate Sci Rep
PublicationYear 2023
Publisher Nature Publishing Group UK
Nature Publishing Group
Nature Portfolio
Publisher_xml – name: Nature Publishing Group UK
– name: Nature Publishing Group
– name: Nature Portfolio
References Moatimid (CR20) 2020; 146
Ganji, Barari, Karimpour, Bomairry (CR8) 2012; 50
Moatimid, Amer (CR21) 2022; 12
Lin (CR2) 1999; 4
Ren (CR32) 2018; 9
Khah, Ganji (CR9) 2010; 10
Hu, Wang (CR11) 2002
Moatimid (CR4) 2020; 146
Paragas (CR12) 1992; 78
Beléndez, Bernabeu, Francés, Méndez, Marini (CR5) 2010; 52
MacDonald (CR13) 1989
Tunç (CR15) 2008; 51
El-Dib (CR33) 2023; 42
Ghaleb, Abou-Dina, Moatimid, Zekry (CR3) 2021; 180
He (CR18) 1999; 178
Saeed, Moatimid, Elsabaa, Ellabban, El-Meligy, Sharaf (CR24) 2020; 9
He, Tian, Moatimid, Salman, Zekry (CR17) 2022; 41
Moatimid (CR19) 2020; 47
El-Dib (CR27) 2022; 92
Moatimid, Amer (CR16) 2022; 92
Moatimid, Amer (CR26) 2022
El-Dib, Moatimid (CR10) 2019; 44
Hamdan, Shabaneh (CR1) 1997; 199
Nayfeh, Mook (CR29) 1979
CR23
Moatimid, Amer, Zekry (CR28) 2023
He, Amer, Elnaggar, Galal (CR22) 2021; 10
He (CR30) 2019; 38
He, Yang, He, Khan (CR31) 2021; 10
Meek (CR6) 1978; 6
Wang, Hu (CR14) 2003; 33
Saeed, El-Ganaini (CR25) 2017; 44
Yim, Chopra (CR7) 1984; 12
YO El-Dib (32743_CR27) 2022; 92
CS Yim (32743_CR7) 1984; 12
NA Saeed (32743_CR25) 2017; 44
K Paragas (32743_CR12) 1992; 78
MN Hamdan (32743_CR1) 1997; 199
SS Ganji (32743_CR8) 2012; 50
HE Khah (32743_CR9) 2010; 10
J-H He (32743_CR22) 2021; 10
GM Moatimid (32743_CR16) 2022; 92
CH He (32743_CR17) 2022; 41
YO El-Dib (32743_CR10) 2019; 44
GM Moatimid (32743_CR4) 2020; 146
HY Hu (32743_CR11) 2002
YO El-Dib (32743_CR33) 2023; 42
GM Moatimid (32743_CR21) 2022; 12
A Beléndez (32743_CR5) 2010; 52
NA Saeed (32743_CR24) 2020; 9
JW Meek (32743_CR6) 1978; 6
AH Nayfeh (32743_CR29) 1979
JH He (32743_CR18) 1999; 178
GM Moatimid (32743_CR19) 2020; 47
GM Moatimid (32743_CR26) 2022
AF Ghaleb (32743_CR3) 2021; 180
Z Ren (32743_CR32) 2018; 9
J Lin (32743_CR2) 1999; 4
C Tunç (32743_CR15) 2008; 51
GM Moatimid (32743_CR20) 2020; 146
J-H He (32743_CR30) 2019; 38
H Wang (32743_CR14) 2003; 33
N MacDonald (32743_CR13) 1989
32743_CR23
GM Moatimid (32743_CR28) 2023
J-H He (32743_CR31) 2021; 10
References_xml – year: 1989
  ident: CR13
  publication-title: Biological Delay Systems: Linear Stability Theory
– volume: 78
  start-page: 421
  issue: 2
  year: 1992
  end-page: 428
  ident: CR12
  article-title: Continuous control of chaos by self-controlling feedback
  publication-title: Phys. Rev. Lett.
  doi: 10.1016/0375-9601(92)90745-8
– volume: 92
  start-page: 1
  year: 2022
  end-page: 19
  ident: CR27
  article-title: Criteria of vibration control in delayed third-order critically damped Duffing oscillation
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-021-02039-4
– volume: 180
  start-page: 129
  year: 2021
  end-page: 151
  ident: CR3
  article-title: Analytic approximate solutions of the cubic–quintic Duffing–van der Pol equation with two-external periodic forcing terms: Stability analysis
  publication-title: Math. Comput. Simul.
  doi: 10.1016/j.matcom.2020.08.001
– volume: 178
  start-page: 257
  year: 1999
  end-page: 262
  ident: CR18
  article-title: Homotopy perturbation technique
  publication-title: Comput. Methods Appl. Mech. Eng.
  doi: 10.1016/S0045-7825(99)00018-3
– volume: 10
  start-page: 191
  year: 2021
  ident: CR22
  article-title: Periodic property and instability of a rotating pendulum system
  publication-title: Axioms
  doi: 10.3390/axioms10030191
– volume: 52
  start-page: 637
  issue: 3–4
  year: 2010
  end-page: 641
  ident: CR5
  article-title: An accurate closed-form approximate solution for the quintic Duffing oscillator equation
  publication-title: Math. Comput. Model.
  doi: 10.1016/j.mcm.2010.04.010
– year: 2002
  ident: CR11
  publication-title: Dynamics of Controlled Mechanical Systems with Delayed Feedback
  doi: 10.1007/978-3-662-05030-9
– volume: 12
  start-page: 12628
  year: 2022
  ident: CR21
  article-title: Analytical solution for the motion of a pendulum with rolling wheel: Stability analysis
  publication-title: Sci. Rep.
  doi: 10.1038/s41598-022-15121-w
– volume: 44
  start-page: 523
  year: 2017
  end-page: 539
  ident: CR25
  article-title: Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system
  publication-title: Appl. Math. Model.
  doi: 10.1016/j.apm.2017.02.019
– volume: 10
  start-page: 320
  issue: 4
  year: 2021
  ident: CR31
  article-title: A simple frequency formulation for the tangent oscillator
  publication-title: Axioms
  doi: 10.3390/axioms10040320
– ident: CR23
– volume: 6
  start-page: 437
  issue: 5
  year: 1978
  end-page: 454
  ident: CR6
  article-title: Dynamic response of tipping core buildings
  publication-title: Earthq. Eng. Struct. Dyn.
  doi: 10.1002/eqe.4290060503
– volume: 44
  start-page: 6581
  year: 2019
  end-page: 6591
  ident: CR10
  article-title: Stability configuration of a rocking rigid rod over a circular surface using the homotopy perturbation method and Laplace transform
  publication-title: Arab. J. Sci. Eng.
  doi: 10.1007/s13369-018-03705-6
– volume: 38
  start-page: 1252
  year: 2019
  end-page: 1260
  ident: CR30
  article-title: The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/1461348419844145
– volume: 50
  start-page: 215
  issue: 1
  year: 2012
  end-page: 229
  ident: CR8
  article-title: Motion of a rigid rod rocking back and forth and cubic-quintic Duffing oscillators
  publication-title: J. Theor. Appl. Mech.
– volume: 33
  start-page: 379
  year: 2003
  end-page: 398
  ident: CR14
  article-title: Remarks on the perturbation methods in solving the second-order delay differential equations
  publication-title: Nonlinear Dyn.
  doi: 10.1023/B:NODY.0000009957.42817.4f
– volume: 146
  start-page: 05020001
  issue: 5
  year: 2020
  ident: CR20
  article-title: Stability analysis of a parametric duffing oscillator
  publication-title: J. Eng. Mech.
– volume: 9
  start-page: 86
  year: 2018
  end-page: 90
  ident: CR32
  article-title: Theoretical basis of He’s frequency-amplitude formulation for nonlinear oscillators
  publication-title: Nonlinear Sci. Lett. A
– volume: 42
  start-page: 110
  issue: 1
  year: 2023
  end-page: 125
  ident: CR33
  article-title: Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/14613484221130148
– volume: 10
  start-page: 447
  issue: 4
  year: 2010
  end-page: 451
  ident: CR9
  article-title: A study on the motion of a rigid rod rocking back and cubic-quintic Duffing oscillators by using He’s energy balance method
  publication-title: Int. J. Nonlinear Sci.
– volume: 41
  start-page: 244
  issue: 1
  year: 2022
  end-page: 268
  ident: CR17
  article-title: Hybrid Rayleigh–van der Pol-Duffing oscillator: Stability analysis and controller
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/14613484211026407
– volume: 199
  start-page: 711
  year: 1997
  end-page: 736
  ident: CR1
  article-title: On the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass
  publication-title: J. Sound Vib.
  doi: 10.1006/jsvi.1996.0672
– volume: 12
  start-page: 263
  issue: 2
  year: 1984
  end-page: 281
  ident: CR7
  article-title: Earthquake response of structures with partial uplift on Winkler foundation
  publication-title: Earthq. Eng. Struct. Dyn.
  doi: 10.1002/eqe.4290120209
– volume: 51
  start-page: 71
  year: 2008
  end-page: 81
  ident: CR15
  article-title: On the stability of solutions to a certain fourth-order delay differential equation
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-006-9192-z
– year: 2023
  ident: CR28
  article-title: Analytical and numerical study of a vibrating magnetic inverted pendulum
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-023-02395-3
– volume: 146
  start-page: 0502001
  issue: 5
  year: 2020
  ident: CR4
  article-title: Stability analysis of a parametric Duffing oscillator
  publication-title: J. Eng. Mech.
– year: 2022
  ident: CR26
  article-title: Analytical approximate solutions of a magnetic spherical pendulum: Stability analysis
  publication-title: J. Vib. Eng. Technol.
  doi: 10.1007/s42417-022-00693-8
– volume: 92
  start-page: 3515
  year: 2022
  end-page: 3531
  ident: CR16
  article-title: Nonlinear suppression using time-delayed controller to excited Van der Pol-Duffing oscillator: Analytical solution techniques
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-022-02246-7
– volume: 9
  start-page: 226151
  year: 2020
  end-page: 226166
  ident: CR24
  article-title: Time-delayed nonlinear feedback controllers to suppress the principal parameter excitation
  publication-title: IEEE Access
  doi: 10.1109/ACCESS.2020.3044998
– year: 1979
  ident: CR29
  publication-title: Nonlinear Oscillations
– volume: 4
  start-page: 132
  year: 1999
  end-page: 135
  ident: CR2
  article-title: A new approach to Duffing equation with strong and high order nonlinearity
  publication-title: Commun. Nonlinear Sci. Numer. Simul.
  doi: 10.1016/S1007-5704(99)90026-6
– volume: 47
  start-page: 18
  issue: 2
  year: 2020
  end-page: 33
  ident: CR19
  article-title: Sliding bead on a smooth vertical rotated parabola: Stability configuration
  publication-title: Kuwait J. Sci.
– volume: 44
  start-page: 6581
  year: 2019
  ident: 32743_CR10
  publication-title: Arab. J. Sci. Eng.
  doi: 10.1007/s13369-018-03705-6
– volume: 42
  start-page: 110
  issue: 1
  year: 2023
  ident: 32743_CR33
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/14613484221130148
– volume: 9
  start-page: 86
  year: 2018
  ident: 32743_CR32
  publication-title: Nonlinear Sci. Lett. A
– volume: 92
  start-page: 1
  year: 2022
  ident: 32743_CR27
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-021-02039-4
– volume: 52
  start-page: 637
  issue: 3–4
  year: 2010
  ident: 32743_CR5
  publication-title: Math. Comput. Model.
  doi: 10.1016/j.mcm.2010.04.010
– volume: 9
  start-page: 226151
  year: 2020
  ident: 32743_CR24
  publication-title: IEEE Access
  doi: 10.1109/ACCESS.2020.3044998
– volume-title: Nonlinear Oscillations
  year: 1979
  ident: 32743_CR29
– volume: 38
  start-page: 1252
  year: 2019
  ident: 32743_CR30
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/1461348419844145
– volume: 41
  start-page: 244
  issue: 1
  year: 2022
  ident: 32743_CR17
  publication-title: J. Low Freq. Noise Vib. Active Control
  doi: 10.1177/14613484211026407
– volume: 33
  start-page: 379
  year: 2003
  ident: 32743_CR14
  publication-title: Nonlinear Dyn.
  doi: 10.1023/B:NODY.0000009957.42817.4f
– volume: 12
  start-page: 263
  issue: 2
  year: 1984
  ident: 32743_CR7
  publication-title: Earthq. Eng. Struct. Dyn.
  doi: 10.1002/eqe.4290120209
– ident: 32743_CR23
  doi: 10.1177/14613484221077474
– volume: 10
  start-page: 320
  issue: 4
  year: 2021
  ident: 32743_CR31
  publication-title: Axioms
  doi: 10.3390/axioms10040320
– volume: 178
  start-page: 257
  year: 1999
  ident: 32743_CR18
  publication-title: Comput. Methods Appl. Mech. Eng.
  doi: 10.1016/S0045-7825(99)00018-3
– volume: 199
  start-page: 711
  year: 1997
  ident: 32743_CR1
  publication-title: J. Sound Vib.
  doi: 10.1006/jsvi.1996.0672
– volume: 50
  start-page: 215
  issue: 1
  year: 2012
  ident: 32743_CR8
  publication-title: J. Theor. Appl. Mech.
– volume-title: Biological Delay Systems: Linear Stability Theory
  year: 1989
  ident: 32743_CR13
– volume: 44
  start-page: 523
  year: 2017
  ident: 32743_CR25
  publication-title: Appl. Math. Model.
  doi: 10.1016/j.apm.2017.02.019
– volume: 10
  start-page: 447
  issue: 4
  year: 2010
  ident: 32743_CR9
  publication-title: Int. J. Nonlinear Sci.
– volume: 78
  start-page: 421
  issue: 2
  year: 1992
  ident: 32743_CR12
  publication-title: Phys. Rev. Lett.
  doi: 10.1016/0375-9601(92)90745-8
– volume: 6
  start-page: 437
  issue: 5
  year: 1978
  ident: 32743_CR6
  publication-title: Earthq. Eng. Struct. Dyn.
  doi: 10.1002/eqe.4290060503
– volume: 51
  start-page: 71
  year: 2008
  ident: 32743_CR15
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-006-9192-z
– volume: 180
  start-page: 129
  year: 2021
  ident: 32743_CR3
  publication-title: Math. Comput. Simul.
  doi: 10.1016/j.matcom.2020.08.001
– volume-title: Dynamics of Controlled Mechanical Systems with Delayed Feedback
  year: 2002
  ident: 32743_CR11
  doi: 10.1007/978-3-662-05030-9
– volume: 146
  start-page: 05020001
  issue: 5
  year: 2020
  ident: 32743_CR20
  publication-title: J. Eng. Mech.
  doi: 10.1061/(ASCE)EM.1943-7889.0001764
– volume: 4
  start-page: 132
  year: 1999
  ident: 32743_CR2
  publication-title: Commun. Nonlinear Sci. Numer. Simul.
  doi: 10.1016/S1007-5704(99)90026-6
– year: 2022
  ident: 32743_CR26
  publication-title: J. Vib. Eng. Technol.
  doi: 10.1007/s42417-022-00693-8
– volume: 47
  start-page: 18
  issue: 2
  year: 2020
  ident: 32743_CR19
  publication-title: Kuwait J. Sci.
– volume: 12
  start-page: 12628
  year: 2022
  ident: 32743_CR21
  publication-title: Sci. Rep.
  doi: 10.1038/s41598-022-15121-w
– year: 2023
  ident: 32743_CR28
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-023-02395-3
– volume: 92
  start-page: 3515
  year: 2022
  ident: 32743_CR16
  publication-title: Arch. Appl. Mech.
  doi: 10.1007/s00419-022-02246-7
– volume: 146
  start-page: 0502001
  issue: 5
  year: 2020
  ident: 32743_CR4
  publication-title: J. Eng. Mech.
  doi: 10.1061/(ASCE)EM.1943-7889.0001764
– volume: 10
  start-page: 191
  year: 2021
  ident: 32743_CR22
  publication-title: Axioms
  doi: 10.3390/axioms10030191
SSID ssj0000529419
Score 2.4289546
Snippet The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional...
The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional...
Abstract The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an...
SourceID doaj
pubmedcentral
proquest
pubmed
crossref
springer
SourceType Open Website
Open Access Repository
Aggregation Database
Index Database
Enrichment Source
Publisher
StartPage 5570
SubjectTerms 639/705
639/705/1041
Approximation
Humanities and Social Sciences
multidisciplinary
Science
Science (multidisciplinary)
Stability analysis
SummonAdditionalLinks – databaseName: DOAJ Directory of Open Access Journals
  dbid: DOA
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV3Na90wDDelMOil7LNL1xUPdttC4_gj9m5ru1IG22mFdzOOrdDCyBvrK23_-0l23lvftm6XQUjAUYKQpUix5J8Yey2V6-KgqVQnQq20SHUfrat7A8H0bQxDzuB_-mxOz9THmZ7dafVFNWEFHrgI7kD3ncFDmD44pSGFIWHEAk0TXLQGMs4n-rw7P1MF1bt1Srhpl0wj7cEleiraTdbKWraEy3m95okyYP-foszfiyV_yZhmR3TykG1PESR_Xzh_xDZgfMwelJ6St0_Y7Lj0mEeSgtLM5wMPnHrI14QIeQuJRqgfVuLovWipHK_pHQ-ET5KXtnlGGr-5wGgW-FI5n7Kzkw9fjk7rqX1CHbUSizrSntkIMZhohRyoZF9L0BgAgYtJJwx8ugA2JSMsDG3UNsXUQSNA4nQNUj5jm-N8hOeMd6l3Qg-mVwEUKBEa1aHxR2gC6oBpKiaWovRxwhanFhdffc5xS-uL-D2K32fx--uKvVk9860ga_yV-pBmaEVJqNh5AHXFT7ri_6UrFdtbzq-fTPXSt9RtXWCg5Cr2anUbjYwyJ2GE-VWm6YTCE75ip6jDihPZNfSl0RWza4qyxur6nfHiPAN5U4WKkaKt2NulTv3k635Z7P4PWbxgWy0ZAxUh6T22ufh-BS8xvlr0-9mUfgDGXyL1
  priority: 102
  providerName: Directory of Open Access Journals
– databaseName: Health & Medical Collection
  dbid: 7X7
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1bi9UwEA66Ivgi3q2uEsE3Dds01_oi3pZF0CcXzltIk6kuLO265yzr_nszadrleFkoLbRpSSeTzDAz-T5CXgrZmtArLNUJwKTikXXBtqzT4HXXBN_nDP6Xr_rgUH5eqVUJuK1LWeW8JuaFOo4BY-R7DZJi82TP2rcnPxmyRmF2tVBoXCc3ELoMS7rMyiwxFsxiSd6WvTK1sHvrZK9wT1kjmGgQnfN8yx5l2P5_-Zp_l0z-kTfN5mj_Drld_Ej6bhr4u-QaDPfIzYlZ8uI-WX2cmOZTkwmrmY499RSZ5BniQl5AxDvIihVpsmEYME_X-IZ6RCnJAW6a8cZ_HSWfFuisog_I4f6nbx8OWCFRYEFJvmEBd84GCF4Hy0WPhftKgEpuELQhqpjcH-PBxqi5hb4JysYQDdQcRBq0XoiHZGcYB3hMqIldy1WvO-lBguS-liYtAQFqnzRB1xXhsyhdKAjjSHRx7HKmW1g3id8l8bssfndekVfLOycTvsaVrd_jCC0tERs73xhPv7sy1ZzqjE4H151vpYLo-5h8XKhr3warQVRkdx5fVybs2l2qV0VeLI_TVMP8iR9gPMttDJfplD7xaFKHpSfC1LjeqIrYLUXZ6ur2k-HoR4bzxjoVLXhTkdezTl326_-yeHL1bzwltxpUcywyUrtkZ3N6Bs-S_7TpnudJ8huJEhoM
  priority: 102
  providerName: ProQuest
– databaseName: Springer Nature OA Free Journals
  dbid: C6C
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1baxUxEB5qRfBFrNetrUTwTRc3m8vu-qanliLok4XzFrLJrBbKHmlPqf33ncle5NgqCMsuJJMlzCUZMpNvAF4r3VShM5yqEzDXRsa8DXWTtxa9bcvguxTB__LVHh3rz0uz3IJyuguTkvYTpGVapqfssHfntNHwZbBS5apkWM3LO3CXodtZqxd2MZ-rcORKy2a8H1Oo-pahG3tQguq_zb-8mSb5R6w0bUGHD-HB6DuKD8Nsd2AL-0dwb6gmefUYlgdDdXkiGfCZxaoTXnD1-JyxIK8wcgtXwoqC9i0-JKdvfC88I5OkQ22RMMZ_nZAfi2JSyydwfPjp2-IoHwsn5MFouc4D35YNGLwNtVQdJ-sbhYZcH2xCNJFcnspjHaOVNXZlMHUMscJCoiJBdUo9he1-1eNzEFVsG2k622qPGrX0ha7I7AMWnqRviwzkxEoXRlRxLm5x6lJ0W9VuYL8j9rvEfneZwZt5zM8BU-Of1B9ZQjMl42GnhtXZdzfqhzNtZemRtvWNNhh9F8mvxaLwTagtqgz2Jvm60UjPXcl11iW5SE0Gr-ZuMi-OmfgeVxeJppKaXvSLZ4M6zDNRVcFrjMmg3lCUjalu9vQnPxKEN-emWCXLDN5OOvV7Xn_nxe7_kb-A-yWrPScamT3YXp9d4D75UOv2ZTKaa6J9GD4
  priority: 102
  providerName: Springer Nature
Title Dynamical system of a time-delayed of rigid rocking rod: analytical approximate solution
URI https://link.springer.com/article/10.1038/s41598-023-32743-w
https://www.ncbi.nlm.nih.gov/pubmed/37020045
https://www.proquest.com/docview/2795917829
https://www.proquest.com/docview/2797147973
https://pubmed.ncbi.nlm.nih.gov/PMC10076312
https://doaj.org/article/5b76b7616ba945edafd774e00a9c86e3
Volume 13
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV1ta9RAEB76guAX8d3UekTwm0az2bdEELmeLeWgRdSD-xY2u5taKDl7vdLev3dmk5ycnoIQEthMwjI7k3myO_sMwCsuCm1rSak61idCMpdUNi-SSnmjqsyaOqzgn5yq44kYT-V0C_pyR50Crzb-2lE9qcn84u3t5fIjOvyHdst4_u4KgxBtFMt4wjOi3LzZhl2MTJoc9aSD-y3Xd1YIVnR7ZzY_uhafAo3_Juz5Zwrlb-uoITwd3Yd7Ha6Mh60hPIAt3zyEO22lyeUjmH5qK8-jSMvdHM_q2MRUWT4hnsild9RCVbJcjDGNJtDx6t7HhlhLwoR3HPjHb88R4_q4N9nHMDk6_DY6TrqiComVgi0SSztprbdG2ZzxmhL5JfcSYZEvrJMO4ZA2PndOsdzXmZW5s077lHmOg1hz_gR2mlnjn0GsXVUwWatKGC-8YCYVGj8J1qcGLUOlEbBelaXtGMep8MVFGVa-eV626i9R_WVQf3kTwevVMz9avo1_Sh_QCK0kiSs7NMzmZ2XneqWstMKDqcoUQnpnaoeY16epKWyuPI9gvx_fsre_MqMa7AzhUxHBy9VtdD1aTzGNn10HGc0EnvAVT1tzWPWE65S-PzKCfM1Q1rq6fqc5_x7ovSlvRXGWRfCmt6lf_fq7Lvb-T_w53M3I7CkJSe7DzmJ-7V8gvlpUA9jWUz2A3eFw_HWM14PD089fsHWkRoMwZzEIbvUTDVYnuw
linkProvider Scholars Portal
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3ra9RAEF_KFdEv4tto1Qj6SUOz2UcSQcTalqttD5EW7tu62Z1oQXK1d-W8f8q_0ZlNcuV89FvhyEGyCZvZeWVmdn6MvRCyzF2tqFTHQSIV90nlijKpNFhdZc7WIYN_ONLDY_lxrMZr7Fe_F4bKKnudGBS1nziKkW9mBIrN0Z6V705_JIQaRdnVHkKjZYt9WMzxk236dm8b1_dllu3uHH0YJh2qQOKU5LPE0VZSB85qV3BRUyW7EqDQL4DSeeXRH8gtFN5rXkCdOVV453NIOQh8i5oCoKjy16XAT5kBW9_aGX36vIzqUN5M8rLbnZOKYnOKFpJ2sWUiERn1A52vWMAAFPAv7_bvIs0_MrXBAO7eYjc7zzV-37LabbYGzR12rcWyXNxl4-0W2x6HtN2h40kd25iw6xPqRLkAT2cIh8vHaDUpRI___k1sqS9KCKnHocP5zxP0oiHuheIeO74SAt9ng2bSwEMW574quap1JS1IkNymMkel4yC1yHs6jRjvSWlc19OcoDW-m5BbF4VpyW-Q_CaQ38wj9mp5z2nb0ePS0Vu0QsuR1I07nJicfTWdcBtV5Rp_XFe2lAq8rT161ZCmtnSFBhGxjX59TacipuaCoSP2fHkZhZsyNraByXkYk3OJB3zEg5YdljMReUoaTkWsWGGUlamuXmlOvoUG4lQZowXPIva656mLef2fFo8uf41n7Prw6PDAHOyN9h-zGxmxPJU4qQ02mJ2dwxP03mbV005kYvblqqX0Ny6UWVE
linkToPdf http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3da9RAEF9KRfFF_G5q1Qj6pOGy2a9EEFHPo7VafLBwb-tmd6IFuWt7V8771_zrnNkkV86PvhXCBZJN2MzO183Mzo-xp0JWxjeKSnU8ZFLxkNW-rLJag9N14V0TM_ifDvTuofwwVuMN9qvfC0Nllb1OjIo6TD3FyAcFgWJztGfVoOnKIj4PR6-PTzJCkKJMaw-n0bLIPiwX-Pdt9mpviGv9rChG77-82806hIHMK8nnmadtpR68077koqGqdiVAoY8AlQ8qoG9gHJQhaF5CU3hVBh8M5BwEflFDwVBU_1cMPkgyZsZmFd-hDJrkVbdPJxflYIa2kvazFSITBXUGXazZwggZ8C8_9-9yzT9yttEUjm6yG50Pm75pme4W24DJbXa1RbVc3mHjYYtyj0PaPtHptEldSij2GfWkXEKgK4TIFVK0nxSsx3N4mTrqkBKD62nsdf7zCP1pSHvxuMsOL4W899jmZDqBLZaaUFdcNbqWDiRI7nJpUP14yB1yoc4TxntSWt91NyeQjR82ZtlFaVvyWyS_jeS3i4Q9Xz1z3Pb2uHD0W1qh1Ujqyx0vTE-_2U7MraqNxoPr2lVSQXBNQP8a8txVvtQgErbTr6_tlMXMnrN2wp6sbqOYU-7GTWB6FscYLvEHX3G_ZYfVTITJSdephJVrjLI21fU7k6PvsZU41chowYuEveh56nxe_6fF9sWf8ZhdQ9m0H_cO9h-w6wVxPNU6qR22OT89g4foxs3rR1FeUvb1sgX0N1YXXCE
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Dynamical+system+of+a+time-delayed+of+rigid+rocking+rod%3A+analytical+approximate+solution&rft.jtitle=Scientific+reports&rft.au=Moatimid%2C+Galal+M.&rft.au=Amer%2C+T.+S.&rft.date=2023-04-05&rft.pub=Nature+Publishing+Group+UK&rft.eissn=2045-2322&rft.volume=13&rft.issue=1&rft_id=info:doi/10.1038%2Fs41598-023-32743-w&rft.externalDocID=10_1038_s41598_023_32743_w
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2045-2322&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2045-2322&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2045-2322&client=summon