International Journal of Engineering

International Journal of Engineering

Physics-Informed Neural Networks Techniques for Analyzing Forced Vibrations of Simply Supported Beams Featuring Variable Cross-Sections

Document Type : Original Article

Authors
S. M. Sahebdad 1
M. Eftekhari 1
M. Eftekhari 2
1 Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
2 Department of Computer Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
10.5829/ije.2026.39.05b.01
Abstract
This research explores the forced vibrations of isotropic beams with variable cross-sections, modeled by the Euler-Bernoulli beam theory. Using Hamilton's principle, the governing partial differential equations are derived, and the complex vibrational behaviors were analyzed. By introducing physics-informed neural networks (PINNs) as an innovative, mesh-free solution technique, the study highlights their ability to provide rapid and precise results by integrating physical laws directly into the machine learning framework. Compared to traditional methods like finite element (FEM) or finite difference schemes, PINNs significantly streamline the computational process by eliminating the need for mesh generation, which simplifies implementation and reduces computational effort, validation with the 6th-order Galerkin method and FEM confirms the high accuracy and efficiency of the proposed approach for analyzing vibrations in beams with varying cross-sections. Overall, this work enhances the application of PINNs in vibration assessment and offers valuable insights for optimizing design and performance across diverse engineering domains, including structural and mechanical systems.

Graphical Abstract

Physics-Informed Neural Networks Techniques for Analyzing Forced Vibrations of Simply Supported Beams Featuring Variable Cross-Sections
Keywords
Free and Forced Vibration
Isotropic Beam
Variable Cross Section
Physics-informed Neural Networks
Galerkin Method

Subjects

  1. Elshafei MA. FE modeling and analysis of isotropic and orthotropic beams using first order shear deformation theory. Materials Sciences and Applications. 2013;4(1):77-102. https://doi.org/10.4236/msa.2013.41010
  2. Thomsen DK, Søe-Knudsen R, Balling O, Zhang X. Vibration control of industrial robot arms by multi-mode time-varying input shaping. Mechanism and Machine Theory. 2021;155:104072. https://doi.org/10.1016/j.mechmachtheory.2020.104072
  3. Li S, Shen L. Seismic optimization design and application of civil engineering structures integrated with building robot system technology. High-tech And Innovation Journal. 2024;5(4):1118-34. https://doi.org/10.28991/HIJ-2024-05-04-017
  4. Makrup L, Pawirodikromo W, Muntafi Y. Comparison of structural response utilizing probabilistic seismic hazard analysis and design spectral ground motion. Civil Engineering Journal. 2024;10:235-51. https://doi.org/10.28991/CEJ-SP2024-010-012
  5. Canales F, Mantari J. Free vibration of thick isotropic and laminated beams with arbitrary boundary conditions via unified formulation and Ritz method. Applied Mathematical Modelling. 2018;61:693-708. https://doi.org/10.1016/j.apm.2018.05.005
  6. Moreno-García P, dos Santos JVA, Lopes H. A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Archives of Computational Methods in Engineering. 2018;25(3):785-815. https://doi.org/10.1007/s11831-017-9214-7
  7. Azzara R, Carrera E, Filippi M, Pagani A. Vibration analysis of thermally loaded isotropic and composite beam and plate structures. Journal of Thermal Stresses. 2023;46(5):369-86. https://doi.org/10.1080/01495739.2023.2188399
  8. Ghugal YM, Sharma R. A hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams. International Journal of Computational Methods. 2009;6(04):585-604. https://doi.org/10.1142/S0219876209002017
  9. Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics. 2019;378:686-707. https://doi.org/10.1016/j.jcp.2018.10.045
  10. Namdeo J, Dubey S, Dorji L. Dynamic analysis of isotropic homogeneous beams using the method of initial functions and comparison with classical beam theories and FEM. Complexity. 2023;2023(1):6636975. https://doi.org/10.1155/2023/6636975
  11. Zienkiewicz O, Taylor R, Ziaie-Moayed R, Kamalzare M, Safavian M, Zhou S, et al. Numerical Study of Floating Stone (static). Comput Geotech. 2013;2(2).
  12. Kumar R, Tiwari R, Prasad R. Numerical solution of partial differential equations: finite difference method. Mathematics and Computer Science Volume 1. 2023:353-72. https://doi.org/10.2307/2007849
  13. Lipnikov K, Manzini G, Shashkov M. Mimetic finite difference method. Journal of Computational Physics. 2014;257:1163-227. https://doi.org/10.1016/j.jcp.2013.07.031
  14. Wang J, Wu R. The extended Galerkin method for approximate solutions of nonlinear vibration equations. Applied sciences. 2022;12(6):2979. https://doi.org/10.3390/app12062979
  15. Polycarpou AC. Introduction to the finite element method in electromagnetics: Springer Nature; 2022.
  16. Chen Z, Lai S-K, Yang Z. AT-PINN: Advanced time-marching physics-informed neural network for structural vibration analysis. Thin-Walled Structures. 2024;196:111423. https://doi.org/10.1016/j.tws.2023.111423
  17. Bazmara M, Silani M, Mianroodi M, editors. Physics-informed neural networks for nonlinear bending of 3D functionally graded beam. Structures; 2023: Elsevier.
  18. Bazmara M, Mianroodi M, Silani M. Application of physics-informed neural networks for nonlinear buckling analysis of beams. Acta Mechanica Sinica. 2023;39(6):422438. https://doi.org/10.1007/s10409-023-22438-x
  19. Kapoor T, Wang H, Núñez A, Dollevoet R. Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulations. Engineering Applications of Artificial Intelligence. 2024;133:108085. https://doi.org/10.1016/j.engappai.2024.108085
  20. Kapoor T, Wang H, Núñez A, Dollevoet R. Physics-informed neural networks for solving forward and inverse problems in complex beam systems. IEEE Transactions on Neural Networks and Learning Systems. 2023;35(5):5981-95. https://doi.org/10.1109/TNNLS.2023.3310585
  21. Söyleyici C, Ünver HÖ. A Physics-Informed Deep Neural Network based beam vibration framework for simulation and parameter identification. Engineering Applications of Artificial Intelligence. 2025;141:109804. https://doi.org/10.1016/j.engappai.2024.109804
  22. Teloli RdO, Tittarelli R, Bigot M, Coelho L, Ramasso E, Le Moal P, et al. A physics-informed neural networks framework for model parameter identification of beam-like structures. Mechanical Systems and Signal Processing. 2025;224:112189. https://doi.org/10.1016/j.ymssp.2024.112189
  23. Barillaro L. Deep learning platforms: PyTorch. 2025. https://doi.org/10.1016/B978-0-323-95502-7.00093-2
  24. Vadyala SR, Betgeri SN, Betgeri NP. Physics-informed neural network method for solving one-dimensional advection equation using PyTorch. Array. 2022;13:100110. https://doi.org/10.1016/j.array.2021.100110
  25. Chaudhary A, Chouhan KS, Gajrani J, Sharma B. Deep learning with PyTorch. Machine learning and deep learning in real-time applications: IGI Global; 2020. p. 61-95.
  26. Hou Q, Li Y, Singh VP, Sun Z, Wei J. Physics-informed neural network for solution of forward and inverse kinematic wave problems. Journal of Hydrology. 2024;633:130934. https://doi.org/10.1016/j.jhydrol.2024.130934
  27. Mehmood F, Ahmad S, Whangbo TK. An efficient optimization technique for training deep neural networks. Mathematics. 2023;11(6):1360. https://doi.org/10.3390/math11061360
  28. Tong Q, Liang G, Cai X, Zhu C, Bi J. Asynchronous parallel stochastic Quasi-Newton methods. Parallel computing. 2021;101:102721. https://doi.org/10.1016/j.parco.2020.102721
  29. Jagtap AD, Kawaguchi K, Karniadakis GE. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. Journal of Computational Physics. 2020;404:109136. https://doi.org/10.1016/j.jcp.2019.109136
  30. Jagtap AD, Kawaguchi K, Em Karniadakis G. Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Proceedings of the Royal Society A. 2020;476(2239):20200334. https://doi.org/10.1098/rspa.2020.0334
International Journal of Engineering
Volume 39, Issue 5
TRANSACTIONS B: Applications
May 2026
Pages 1047-1062