Mathematical and numerical modeling of resonant frequencies of fluid-structure systems for digital twins

Bauyrzhan  Amirkhanov; Gulshat  Amirkhanova; Murat  Kunelbayev; Alina  Raeva

doi:10.59400/sv3981

Mathematical and numerical modeling of resonant frequencies of fluid-structure systems for digital twins

Bauyrzhan Amirkhanov
Department of Artificial Intelligence and Big Data, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan
Gulshat Amirkhanova
Department of Artificial Intelligence and Big Data, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan
Murat Kunelbayev
Department of Artificial Intelligence and Big Data, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan; Institute of Information and Computational Technologies CS MSHE RK, Almaty 050010, Kazakhstan
Alina Raeva
Department of Artificial Intelligence and Big Data, Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan

Article ID: 3981

Keywords: fluid–structure interaction (FSI); resonant frequency; added mass effect; thin-walled structures; vibroacoustics; finite element method (FEM); ANSYS finite element environment; digital twins

Abstract

This research establishes a comprehensive mathematical and numerical framework for accurately predicting the resonant frequencies of fluid-filled thin-walled structures, a critical factor in the safety and performance of engineering systems such as storage tanks, pipelines, and aerospace components. The study addresses the inherent limitations of classical analytical solutions, which are typically restricted to idealized geometries and cannot account for the complexities of real-world applications. By coupling the Navier-Lamé equations for elastic shell motion with the Laplace equation for the fluid domain, the model effectively captures the "added mass effect"—the phenomenon where internal fluid interaction significantly reduces a structure's natural frequencies. This effect is particularly pronounced in systems with denser fluids, larger radii, and thinner walls. The proposed framework was rigorously validated against a diverse dataset of 54 experimental cases from peer-reviewed literature, covering various geometries (spheres, cylinders, and square plates) and materials including glass, steel, and aluminum. The results demonstrated exceptional reliability, with an average relative error of less than 0.5% across all tested configurations. Statistical analysis, including Boxplots and Empirical CDFs, further confirmed that the model maintains sub-percent-level accuracy, with 100% of cases showing less than 1.1% error. Implemented within the ANSYS finite element environment, the model's computational efficiency and high precision make it an ideal tool for integration into digital twin systems. Such integration enables real-time dynamic monitoring and predictive maintenance of complex industrial infrastructure. Future developments aim to enhance the model by incorporating nonlinear fluid effects, such as sloshing, and integrating these simulations into Industrial Internet of Things (IIoT) frameworks.

Published

2026-06-02

How to Cite

Amirkhanov, B., Amirkhanova, G., Kunelbayev, M., & Raeva, A. (2026). Mathematical and numerical modeling of resonant frequencies of fluid-structure systems for digital twins. Sound & Vibration, 60(3). https://doi.org/10.59400/sv3981

Download Citation

Issue

Vol. 60 No. 3 (2026): In Progress

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Ghaheri A, Ahmadian MT, Fallah F. Free vibration analysis of a fluid-filled functionally graded spherical shell subjected to internal pressure. Acta Mechanica. 2022; 233(8): 3095–3112. doi: 10.1007/s00707-022-03262-y

[2]Sekine K. Free Vibration Characteristics of Thin Spherical Shells. EPI International Journal of Engineering. 2021; 4(2): 196–203. doi: 10.25042/10.25042/epi-ije.082021.12

[3]Khosravi AE, Shahabian F, Aftabi Sani A. Dynamic examination of closed cylindrical shells utilizing the differential transform method. Scientific Reports. 2024; 14(1): 15290. doi: 10.1038/s41598-024-66095-w

[4]D’Alessio SJD. Forced free vibrations of a square plate. SN Applied Sciences. 2021; 3(1): 60. doi: 10.1007/s42452-020-04062-6

[5]Krishna BV, Ganesan N. Polynomial approach for calculating added mass for fluid-filled cylindrical shells. Journal of Sound and Vibration. 2006; 291(3–5): 1221–1228. doi: 10.1016/j.jsv.2005.06.031

[6]Zhang XM, Liu GR, Lam KY. Coupled vibration analysis of fluid-filled cylindrical shells using the wave propagation approach. Applied Acoustics. 2001; 62(3): 229–243. doi: 10.1016/S0003-682X(00)00045-1

[7]Junger MC, Feit D. Sound, Structures, and Their Interaction. MIT Press; 1986.

[8]Vu VH, Thomas M, Lakis AA, et al. Effect of added mass on submerged vibrated plates. In: Proceedings of the 25th Seminar on machinery vibration, Canadian Machinery Vibration Association; 24–26 October 2007; Saint John, NB, Canada.

[9]Fahy FJ. Sound and Structural Vibration: Radiation, Transmission, and Response. Academic Press; 2007.

[10]Shah AG, Mahmood T, Naeem MN, et al. Vibrational Study of Fluid-Filled Functionally Graded Cylindrical Shells Resting on Elastic Foundations. ISRN Mechanical Engineering. 2011; 2011: 1–13. doi: 10.5402/2011/892460

[11]Lakis AA, Païdoussis MP. Free vibration of cylindrical shells partially filled with liquid. Journal of Sound and Vibration. 1971; 19(1): 1–15. doi: 10.1016/0022-460X(71)90417-2

[12]Blackstock DT, Atchley AA. Fundamentals of Physical Acoustics. The Journal of the Acoustical Society of America. 2001; 109(4): 1274–1276. doi: 10.1121/1.1354982

[13]Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Courier Corporation; 2003.

[14]Landau LD, Lifshitz EM. Theory of Elasticity. Elsevier; 2012.

[15]Gonçalves PB, Ramos NRSS. Free Vibration Analysis of Cylindrical Tanks Partially Filled with Liquid. Journal of Sound and Vibration. 1996; 195(3): 429–444. doi: 10.1006/jsvi.1996.0436

[16]Fackrell SA. Study of the Added Mass of Cylinders and Spheres [Master's Thesis]. University of Windsor; 2011.

[17]Durán RG, Hervella-Nieto L, Liberman E, et al. Finite element analysis of the vibration problem of a plate coupled with a fluid: Numerische Mathematik. 2000; 86(4): 591–616. doi: 10.1007/PL00005411

[18]Cho DS, Kim BH, Vladimir N, et al. Natural vibration analysis of rectangular bottom plate structures in contact with fluid. Ocean Engineering. 2015; 103: 171–179. doi: 10.1016/j.oceaneng.2015.04.078

[19]Thai HT, Kim SE. A review of theories for the modeling and analysis of functionally graded plates and shells. Composite Structures. 2015; 128: 70–86. doi: 10.1016/j.compstruct.2015.03.010

[20]Ismail G, Moatimid G, Yamani M. Periodic Solutions of Strongly Nonlinear Oscillators Using He’s Frequency Formulation. European Journal of Pure and Applied Mathematics. 2024; 17(3): 2155–2172. doi: 10.29020/nybg.ejpam.v17i3.5339

[21]Ismail GM, Moatimid GM, Alraddadi I, et al. Scrutinizing highly nonlinear oscillators using He’s frequency formula. Sound & Vibration. 2025; 59(2): 2358. doi: 10.59400/sv2358

[22]Ismail GM, El-Moshneb MM, Zayed M. A modified global error minimization method for solving nonlinear Duffing-harmonic oscillators. AIMS Mathematics. 2023; 8(1): 484–500. doi: 10.3934/math.2023023

[23]Ismail GM, Hosen MA, Mohammadian M, et al. Nonlinear Vibration of Electrostatically Actuated Microbeam. Mathematics. 2022; 10(24): 4762. doi: 10.3390/math10244762

[24]Tao F, Zhang H, Liu A, et al. Digital Twin in Industry: State-of-the-Art. IEEE Transactions on Industrial Informatics. 2019; 15(4): 2405–2415. doi: 10.1109/TII.2018.2873186

[25]Liu M, Fang S, Dong H, et al. Review of digital twin about concepts, technologies, and industrial applications. Journal of Manufacturing Systems. 2021; 58: 346–361. doi: 10.1016/j.jmsy.2020.06.017

[26]Li Q, Xin L, Li R. Application of digital twin technology in monitoring system of pump turbine. Discover Mechanical Engineering. 2024; 3(1): 30. doi: 10.1007/s44245-024-00068-1

[27]Mironova TB, Prokofiev AB, Sverbilov VY. The Finite Element Technique for Modelling of Pipe System Vibroacoustical Characteristics. Procedia Engineering. 2017; 176: 681–688. doi: 10.1016/j.proeng.2017.02.313

[28]Hossain MI, Sakib MSR, Begum M. Digital Twin-Driven Structural Health Monitoring: Emerging Paradigms, Simulation Strategies and Predictive Intelligence. In: Proceedings of the 8th International Conference on Civil Engineering for Sustainable Development (ICCESD 2026); 5–7 February 2026; Khulna, Bangladesh.

Editor-in-Chief

Prof. Jun Yang

Institute of Acoustics, Chinese Academy of Sciences, China

ISSN

1541-0161 (Print)

2693-1443 (Online)

Publication Frequency

Bi-monthly

Indexing

Web of Science Coverage

Emerging Sources Citation Index (2024 Impact Factor 1.0)

Elsevier Solutions

Scopus (2025 CiteScore 2.0; 2024 SNIP 0.5);

Portico, etc.

About the Publisher

Academic Publishing insists on taking academic exchange and publication as the main line, carrying out comprehensive management based on science and technology, and fully exploring excellent international publishing resources. Within 5 years, it will form a strategic framework and scale with science (S), technology (T), medicine (M), education (E), and humanities and arts (H) as the main publishing fields. Academic Publishing is headquartered in Singapore and based in Malaysia, with the United States and China providing the main scientific and academic resources. At the same time, it has established long-term good cooperative relations with other publishing companies, scientific research communities, and academic organizations in more than a dozen countries and regions. Academic Publishing uses English and Chinese as its main publishing languages, mainly publishing books, journals, and conference papers in print and online. The vast majority of publications follow the international open access policy, providing stable and long-term quality and professional publications. With the joint efforts of the expert team and our professional editorial team, our publications will gradually be indexed by international databases in stages to provide convenient and professional retrieval for various scholars. At the same time, manuscripts we accept will be subject to the peer review principle, and cutting-edge and innovative research articles will be preferentially accepted for peer reference and discussion. All kinds of our publications are welcome for peer to contribute, access, and download.

more

Member of ASC

Volume Arrangement

Featured Articles

New scaling of critical damping and reduced frequency for mechanically excited systems

This paper introduces a universal framework for understanding the vibration responses of systems subjected to harmonic excitation. By examining a simplified cylinder-spring-damper model, the study refurbishes traditional scaling methods for the excitation frequency ratio and critical damping ratio. The findings indicate that in damped systems, the maximum amplitude of vibration does not align with the natural frequency. This observation leads to the introduction of a new scaling method for reduced frequency. This new approach aligns resonance peaks at the new reduced velocity of 1.0 across different damping ratios, providing a consistent characterization of vibration behavior. A new critical damping ratio of 0.707 is identified for an excited system as opposed to the traditional damping ratio of 1.0 for an unexcited system. Key properties such as maximum amplitude, phase lag, bandwidth, and quality factor are analyzed, demonstrating that the proposed reduced frequency and critical damping ratio effectively capture the dynamics of both damped and undamped excited systems. The findings offer significant insights for practical applications in engineering and various scientific fields.

Ultrasonic wave velocity as a universal metric for defect detection in timber structures: A case study on Japanese cedar wood (Cryptomeria japonica)

This study makes significant contributions to the field of ultrasonic testing (UT) by offering a novel approach to the identification of artificially introduced defects within Japanese cedar wood (Cryptomeria japonica). The findings are of particular relevance for the heritage conservation and construction sectors, where non-invasive defect detection is paramount. The study establishes a robust framework for assessing the structural integrity of timber by correlating ultrasonic wave velocity reductions with defect size and distribution. Big-sized defects led to more substantial decreases in wave velocity. The study establishes a robust framework for assessing the structural integrity of historical timber by correlating ultrasonic wave velocity reductions with defect size and distribution. This framework has the potential to be applicable to diverse wood species and defect types.

Vehicle structural road noise prediction based on an improved Long Short-Term Memory method

The control of vehicle interior noise has become a critical metric for assessing noise, vibration, and harshness (NVH) in vehicles. During the initial phases of vehicle development, accurately predicting the impact of road noise on interior noise is essential for reducing noise levels and expediting the product development cycle. In recent years, data-driven methods based on machine learning have gained significant attention due to their robust capability in navigating complex data mapping relationships. Notably, surrogate models have demonstrated exceptional performance in this domain. Numerous researchers have integrated diverse intelligent algorithms into the study of vehicle noise, leveraging advantages such as the elimination of precise modeling requirements, extensive solution space exploration, continuous learning from data, and robust algorithmic versatility. However, in NVH engineering applications, data-driven models face inherent limitations, particularly in interpretability and stability. To address these issues, this paper introduces an improved Long Short-Term Memory (LSTM) network that combines knowledge and data. Inspired by the physical information neural network concept, this approach incorporates values calculated through empirical formulas into the neural network as constraints. Comparative assessments with traditional LSTM networks highlight the advantages of this deep learning model. By integrating empirical formulas constraints, the model not only enhances interpretability but also achieves robust generalization with fewer data samples. The proposed method is validated on a specific vehicle model, showing significant improvements in prediction accuracy and efficiency.