Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force

Haruna Egbunu Abubakar; Ahmed Amoo Yinusa; Musbau Gbeminiyi Sobamowo; Obanishola Mufutau Sadiq

doi:doi:10.11648/j.es.20251002.12

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Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force

Haruna Egbunu Abubakar^*

, Ahmed Amoo Yinusa

, Musbau Gbeminiyi Sobamowo

, Obanishola Mufutau Sadiq

Published in Engineering Science (Volume 10, Issue 2)

Received: 1 February 2025 Accepted: 22 March 2025 Published: 29 April 2025

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Abstract

Nanotubes are known to have the strongest and stiffest fibers with exceptional tensile strength and higher modulus of elasticity, researchers have been very interested in studying their nonlinear vibrations. Nonetheless, using nanotubes to partially substitute reinforcing material has emerged as a revolutionary approach for applications of the future. This work examines the nonlinear vibration study of single-walled carbon (NTs) with geometric imperfection supported on linear and nonlinear Winkler and Pasternak Foundations in a thermal-magnetic-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model for this work is formulated using the Hamiliton principle, Euler-Bernoulli elasticity theory, and non-local elasticity beam theory. The Galerkin decomposition approach is used directly to break down the nonlinear partial differential equation of motions into two separate components, that is spatial and temporal parts of nonlinear duffing equations of motions. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). The method is used to solve the resulting nonlinear mathematical model and the following solutions were obtained that is the natural frequency of vibration, nonlinear frequency ratio and dynamic response analysis. Moreso, parametric study was conducted on the obtained semi-analytical solutions and the effects of the following parameters such as magnetic, thermal, electrostatic force, Van der Waals force, Casimir force, linear foundation, nonlinear foundation, Pasternak foundation and mass on stability and dynamic responses on four different boundary conditions considered. Afterwards, the outcomes are then compared with existing literature and found to be completely consistent.

Published in	Engineering Science (Volume 10, Issue 2)
DOI	10.11648/j.es.20251002.12
Page(s)	32-58
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Magnetic, Thermal, Iteration Perturbation, Casimir, Van Der Waal, Elastic Foundation, Electrostatic Force, Single-Walled Carbon Nanotubes

References

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[2]	Anh N. D. and Hieu D. V. (2021) Nonlinear Vibration of Nonlocal Strain Gradient Nanotubes Under Longitudinal Magnetic Field, Vietnam Journal of Mechanics, Vietnam Academy of Science and Technology, 1-23, https://doi.org/10.15625/0866-7136/15467
[3]	Belhadj A., Boukhalfa A., Belalia S. A., (2017) Free vibration modelling of single-walled carbon nanotubes using the differential quadrature method, Mathematical Modelling of Engineering Problems, 4(1), 33-37, https://doi.org/10.18280/mmep.040107
[4]	E H Abubakar, M G Sobamowo, M O Sadiq, A A Yinusa (2023) Vibration of Single-walled Carbon Nanotube Resting on Elastic Foundation with Magnetic and Thermal Effects under the Influence of Casimir Force, J Nanosci Technol 1(1) 103.
[5]	Egbunu H. Abubakar, Gbeminiyi M. Sobamowo, Obanishola M. Sadiq, Ahmed A. Yinusa (2024), Dynamic and Instability Analysis of Single Walled Carbon Nanotubes with Geometrical Imperfections Resting on Elastic Medium in a Magneto-Thermally-Electrostatic Environment with Impact of Casimir Force using Homotropy Perturbation Method, Engineering Today.
[6]	Hosseini M., Sadeghi-Goughari M., Atashipour S. A., Eftekhari M. (2014) Vibration analysis of single-walled carbon nanotubes conveying nanoflow embedded in a viscoelastic medium using modified nonlocal beam model, Arch. Mech., 66, 4, 217–244.
[7]	Matteo Strozzi, Isaac E. Elishakoff, Michele Bochicchio, Marco Cocconcelli, Riccardo Rubini 1 and Enrico Radi (2024) Nonlocal Strain Gradient Anisotropic Elastic Shell Model for Vibration Analysis of Single‒Walled Carbon Nanotubes.
[8]	Mustafa Oguz Nalbant, Süleyman Murat Bağdatli, Ayla Tekin (2023), Investigation of Free Vibrations of Stepped Nanobeam Embedded in Elastic Foundation, 5th International Conference on Applied Engineering and Natural Sciences July 10-12: Konya, Turkey.
[9]	Muzamal Hussain and Muhammad Nawaz Naeem (2017), Vibration analysis of single-walled carbon nanotubes using wave propagation approach, Mech. Sci., 8, 155–164.
[10]	Rafiu Olalekan Kuku, Nurudeen Adekunle Raji and Gbeminiyi Sobamowo (2024) Analytical Investigations of Thermal-Magnetic Effects on Nonlinear Vibration of Carbon Nanotube in a Multi-layer Elastic Media using Differential Transformation Method, World Scientific News 192, 1-21.
[11]	Ramgopal Reddy B., Ramji K. and Satyanarayana B. (2011) Free Vibration Analysis of Carbon Nanotube Reinforced Laminated Composite Panels, International Journal of Mechanical and Mechatronics Engineering 5(8) 1616-1620.
[12]	Ravi Kumar B., (2017) Nonlocal Vibration Analysis of Fluid Conveying Single-Walled Carbon Nanotube with Magnetic Effects, Rasayan J. Chem., 10(2), 643-651 http://dx.doi.org/10.7324/RJC.2017.1021669
[13]	Salvatore Brischetto, Francesco Tornabene, Nicholas Fantuzzi and Michele Bacciocchi (2015) Refined 2D and Exact 3D Shell Models for the Free Vibration Analysis of Single- and Double-Walled Carbon Nanotubes, Technologies, 3, 259-284, https://doi.org/10.3390/technologies304025
[14]	Shaba A. I., Jiya M., Aiyesimi Y. M. and Mohammed A. A. (2021), Vibration of Single – Walled Carbon Nanotube on a Winkler Foundation with Magnetic Field Effect, Science World Journal, 16(4), 402-406.
[15]	Shota Kuwahara, Yuki Kuwahara and Hisanori Shinohara (2014) Quantitative Analysis of Isolated Single-Wall Carbon Nanotubes with Their Molar Absorbance Coefficients, Journal of Nanomaterials Vol. 2014, 7, https://doi.org/10.1155/2014%2F262940
[16]	Subrat Kumar Jena and Chakraverty S., (2018) Free Vibration Analysis of Single Walled Carbon Nanotube with Exponentially Varying Stiffness, Curved and Layer. Struct., 5, 201–212, http://dx.doi.org/10.1515/cls-2018-0015
[17]	Subrat Kumar Jena, Chakraverty S., Mohammad Malikan, Francesco Tornabene (2019) Stability Analysis of Single-Walled Carbon Nanotubes Embedded in Winkler Foundation Placed in a Thermal Environment Considering the Surface Effect Using a New Refined Beam Theory, Mechanics Based Design of Structures and Machines, 1-27, http://dx.doi.org/10.1080/15397734.2019.1698437
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[19]	Yinusa A. A., Sobamowo M. G., Adelaja A. O., Ojolo S. J., Waheed M. A., Usman M. A., Antonio Marcos de Oliveira Siqueira, Júlio C´esar Costa Camposf, Ridwan Ola-Gbadamosi (2024) Numerical investigation on the thermal-nanofluidic flow induced transverse and longitudinal vibrations of single and multi-walled branched nanotubes resting on nonlinear elastic foundations in a magnetic environment, Partial Differential Equations in Applied Mathematics 9, 100602, https://doi.org/10.1016/j.padiff.2023.100602

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APA Style

Abubakar, H. E., Yinusa, A. A., Sobamowo, M. G., Sadiq, O. M. (2025). Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Engineering Science, 10(2), 32-58. https://doi.org/10.11648/j.es.20251002.12

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ACS Style

Abubakar, H. E.; Yinusa, A. A.; Sobamowo, M. G.; Sadiq, O. M. Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Eng. Sci. 2025, 10(2), 32-58. doi: 10.11648/j.es.20251002.12

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AMA Style

Abubakar HE, Yinusa AA, Sobamowo MG, Sadiq OM. Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Eng Sci. 2025;10(2):32-58. doi: 10.11648/j.es.20251002.12

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@article{10.11648/j.es.20251002.12,
  author = {Haruna Egbunu Abubakar and Ahmed Amoo Yinusa and Musbau Gbeminiyi Sobamowo and Obanishola Mufutau Sadiq},
  title = {Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force 
},
  journal = {Engineering Science},
  volume = {10},
  number = {2},
  pages = {32-58},
  doi = {10.11648/j.es.20251002.12},
  url = {https://doi.org/10.11648/j.es.20251002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.es.20251002.12},
  abstract = {Nanotubes are known to have the strongest and stiffest fibers with exceptional tensile strength and higher modulus of elasticity, researchers have been very interested in studying their nonlinear vibrations. Nonetheless, using nanotubes to partially substitute reinforcing material has emerged as a revolutionary approach for applications of the future. This work examines the nonlinear vibration study of single-walled carbon (NTs) with geometric imperfection supported on linear and nonlinear Winkler and Pasternak Foundations in a thermal-magnetic-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model for this work is formulated using the Hamiliton principle, Euler-Bernoulli elasticity theory, and non-local elasticity beam theory. The Galerkin decomposition approach is used directly to break down the nonlinear partial differential equation of motions into two separate components, that is spatial and temporal parts of nonlinear duffing equations of motions. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). The method is used to solve the resulting nonlinear mathematical model and the following solutions were obtained that is the natural frequency of vibration, nonlinear frequency ratio and dynamic response analysis. Moreso, parametric study was conducted on the obtained semi-analytical solutions and the effects of the following parameters such as magnetic, thermal, electrostatic force, Van der Waals force, Casimir force, linear foundation, nonlinear foundation, Pasternak foundation and mass on stability and dynamic responses on four different boundary conditions considered. Afterwards, the outcomes are then compared with existing literature and found to be completely consistent.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force 

AU  - Haruna Egbunu Abubakar
AU  - Ahmed Amoo Yinusa
AU  - Musbau Gbeminiyi Sobamowo
AU  - Obanishola Mufutau Sadiq
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.es.20251002.12
DO  - 10.11648/j.es.20251002.12
T2  - Engineering Science
JF  - Engineering Science
JO  - Engineering Science
SP  - 32
EP  - 58
PB  - Science Publishing Group
SN  - 2578-9279
UR  - https://doi.org/10.11648/j.es.20251002.12
AB  - Nanotubes are known to have the strongest and stiffest fibers with exceptional tensile strength and higher modulus of elasticity, researchers have been very interested in studying their nonlinear vibrations. Nonetheless, using nanotubes to partially substitute reinforcing material has emerged as a revolutionary approach for applications of the future. This work examines the nonlinear vibration study of single-walled carbon (NTs) with geometric imperfection supported on linear and nonlinear Winkler and Pasternak Foundations in a thermal-magnetic-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model for this work is formulated using the Hamiliton principle, Euler-Bernoulli elasticity theory, and non-local elasticity beam theory. The Galerkin decomposition approach is used directly to break down the nonlinear partial differential equation of motions into two separate components, that is spatial and temporal parts of nonlinear duffing equations of motions. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). The method is used to solve the resulting nonlinear mathematical model and the following solutions were obtained that is the natural frequency of vibration, nonlinear frequency ratio and dynamic response analysis. Moreso, parametric study was conducted on the obtained semi-analytical solutions and the effects of the following parameters such as magnetic, thermal, electrostatic force, Van der Waals force, Casimir force, linear foundation, nonlinear foundation, Pasternak foundation and mass on stability and dynamic responses on four different boundary conditions considered. Afterwards, the outcomes are then compared with existing literature and found to be completely consistent.

VL  - 10
IS  - 2
ER  -

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Author Information

Haruna Egbunu Abubakar

Civil and Environmental Engineering Department, University of Lagos, Lagos, Nigeria

Contact Email

http://orcid.org/0009-0005-7118-7022
Ahmed Amoo Yinusa

Mechanical Engineering Department, University of Lagos, Lagos, Nigeria

Contact Email

http://orcid.org/0000-0003-0874-0276
Musbau Gbeminiyi Sobamowo

Mechanical Engineering Department, University of Lagos, Lagos, Nigeria

Contact Email

http://orcid.org/0000-0003-2402-1423
Obanishola Mufutau Sadiq

Civil and Environmental Engineering Department, University of Lagos, Lagos, Nigeria

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Plain Text BibTeX RIS

APA Style

Abubakar, H. E., Yinusa, A. A., Sobamowo, M. G., Sadiq, O. M. (2025). Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Engineering Science, 10(2), 32-58. https://doi.org/10.11648/j.es.20251002.12

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ACS Style

Abubakar, H. E.; Yinusa, A. A.; Sobamowo, M. G.; Sadiq, O. M. Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Eng. Sci. 2025, 10(2), 32-58. doi: 10.11648/j.es.20251002.12

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AMA Style

Abubakar HE, Yinusa AA, Sobamowo MG, Sadiq OM. Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force. Eng Sci. 2025;10(2):32-58. doi: 10.11648/j.es.20251002.12

Copy | Download

@article{10.11648/j.es.20251002.12,
  author = {Haruna Egbunu Abubakar and Ahmed Amoo Yinusa and Musbau Gbeminiyi Sobamowo and Obanishola Mufutau Sadiq},
  title = {Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force 
},
  journal = {Engineering Science},
  volume = {10},
  number = {2},
  pages = {32-58},
  doi = {10.11648/j.es.20251002.12},
  url = {https://doi.org/10.11648/j.es.20251002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.es.20251002.12},
  abstract = {Nanotubes are known to have the strongest and stiffest fibers with exceptional tensile strength and higher modulus of elasticity, researchers have been very interested in studying their nonlinear vibrations. Nonetheless, using nanotubes to partially substitute reinforcing material has emerged as a revolutionary approach for applications of the future. This work examines the nonlinear vibration study of single-walled carbon (NTs) with geometric imperfection supported on linear and nonlinear Winkler and Pasternak Foundations in a thermal-magnetic-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model for this work is formulated using the Hamiliton principle, Euler-Bernoulli elasticity theory, and non-local elasticity beam theory. The Galerkin decomposition approach is used directly to break down the nonlinear partial differential equation of motions into two separate components, that is spatial and temporal parts of nonlinear duffing equations of motions. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). The method is used to solve the resulting nonlinear mathematical model and the following solutions were obtained that is the natural frequency of vibration, nonlinear frequency ratio and dynamic response analysis. Moreso, parametric study was conducted on the obtained semi-analytical solutions and the effects of the following parameters such as magnetic, thermal, electrostatic force, Van der Waals force, Casimir force, linear foundation, nonlinear foundation, Pasternak foundation and mass on stability and dynamic responses on four different boundary conditions considered. Afterwards, the outcomes are then compared with existing literature and found to be completely consistent.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Nonlinear Vibration of Single-Walled Carbon Nanotubes on Winkler and Pasternak Foundations in a Magneto-Thermal Environment Under the Influence of Casimir Force 

AU  - Haruna Egbunu Abubakar
AU  - Ahmed Amoo Yinusa
AU  - Musbau Gbeminiyi Sobamowo
AU  - Obanishola Mufutau Sadiq
Y1  - 2025/04/29
PY  - 2025
N1  - https://doi.org/10.11648/j.es.20251002.12
DO  - 10.11648/j.es.20251002.12
T2  - Engineering Science
JF  - Engineering Science
JO  - Engineering Science
SP  - 32
EP  - 58
PB  - Science Publishing Group
SN  - 2578-9279
UR  - https://doi.org/10.11648/j.es.20251002.12
AB  - Nanotubes are known to have the strongest and stiffest fibers with exceptional tensile strength and higher modulus of elasticity, researchers have been very interested in studying their nonlinear vibrations. Nonetheless, using nanotubes to partially substitute reinforcing material has emerged as a revolutionary approach for applications of the future. This work examines the nonlinear vibration study of single-walled carbon (NTs) with geometric imperfection supported on linear and nonlinear Winkler and Pasternak Foundations in a thermal-magnetic-electrostatic environment under the influence of Casimir Force. The nonlinear mathematical model for this work is formulated using the Hamiliton principle, Euler-Bernoulli elasticity theory, and non-local elasticity beam theory. The Galerkin decomposition approach is used directly to break down the nonlinear partial differential equation of motions into two separate components, that is spatial and temporal parts of nonlinear duffing equations of motions. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). The method is used to solve the resulting nonlinear mathematical model and the following solutions were obtained that is the natural frequency of vibration, nonlinear frequency ratio and dynamic response analysis. Moreso, parametric study was conducted on the obtained semi-analytical solutions and the effects of the following parameters such as magnetic, thermal, electrostatic force, Van der Waals force, Casimir force, linear foundation, nonlinear foundation, Pasternak foundation and mass on stability and dynamic responses on four different boundary conditions considered. Afterwards, the outcomes are then compared with existing literature and found to be completely consistent.

VL  - 10
IS  - 2
ER  -

Copy | Download