The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method
Source abstract
Abstract
For the optimal design and accurate prediction of structural behavior, the nonlinear analysis of large deformation of elastic beams has broad applications in various engineering fields. In this study, the nonlinear equation of flexure of an elastic beam, also known as an elastica, was solved by the Galerkin method for a highly accurate solution. The numerical results showed that the third-order solution of the rotation angle at the free end of the beam is more accurate and efficient in comparison with results of other approximate methods, and is perfectly consistent with the exact solution in elliptic functions. A general procedure with the Galerkin method is demonstrated for efficient solutions of nonlinear differential equations with the potential for adoption and implementation in more applications.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Composite Structure Analysis and Optimization · Vibration and Dynamic Analysis · Fractional Differential Equations Solutions
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Road, Ningbo 315211, China
Institute of Applied Mechanics, Ningbo Polytechnic, 1069 Xinda Road, Beilun District, Ningbo 315800, China
School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Ningbo 315211, China
Funding
National Natural Science Foundation of China · Technology Innovation 2025 Program
References (39)
Show source-provided references
1. Nayfeh, A.H., and Mook, D.T. (1979). Nonlinear Oscillations, Wiley.
2. Green function’s properties and existence theorems for nonlinear singular-delay-fractional differential equations
3. Hyers-Ulam stability and existence criteria for coupled fractional differential equations involving p-Laplacian operator10.1186/s13662-018-1899-x ↗
4. Stability analysis and a numerical scheme for fractional Klein-Gordon equations10.1002/mma.5375 ↗
5. Stability analysis of a dynamical model of tuberculosis with incomplete treatment10.1186/s13662-020-02950-0 ↗
6. Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler power law10.1016/j.aej.2020.07.014 ↗
7. Jin, G., Ye, T., and Su, Z. (2015). Structural Vibration, Science Press and Springer.10.1007/978-3-662-46364-2 ↗
8. Nayfeh, A.H., and Balachandran, B. (2008). Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley.
9. Bifurcation and stability of periodic solutions of Duffing equations10.1088/0951-7715/21/11/001 ↗
10. Hu, H.Y. (2000). Applied Nonlinear Mechanics, Aviation Industry Press. (In Chinese).
11. Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review10.1590/S1679-78252012000200003 ↗
12. Free vibration of rotating tapered beams using the dynamic stiffness method10.1016/j.jsv.2006.06.040 ↗
13. Liao, S.J. (2003). Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press.
14. Li, Y.S., Li, X.Y., Xie, C., and Huo, S.H. (2022). Explicit solution to large deformation of cantilever beam by improved homotopy analysis method II: Vertical and horizontal displacements. Appl. Sci., 12.10.3390/app12052513 ↗
15. An explicit solution of the large deformation of a cantilever beam under point load at the free tip10.1016/j.cam.2006.12.009 ↗
16. Homotopy perturbation technique10.1016/S0045-7825(99)00018-3 ↗
17. Improved Lindstedt-Poincaré method for the solution of nonlinear problem10.1016/j.jsv.2004.06.009 ↗
18. High-order approximate solutions of strongly nonlinear cubic-quantic Duffing oscillator based on the harmonic balance method10.1016/j.rinp.2017.10.008 ↗
19. Large deflections of an end supported beam subjected to a point load10.1016/S0020-7462(96)00017-0 ↗
20. Love, A.E.H. (1944). A Treatise on the Mathematical Theory of Elasticity, Dover Publications Inc.
21. Elliptic integral solution of the extensible elastica with a variable length under a concentrated force10.1007/s00707-011-0520-0 ↗
22. A Rayleigh-Ritz approach for postbuckling analysis of variable angle tow composite stiffened panels10.1016/j.compstruc.2017.10.009 ↗
23. Hybrid changeable basis Galerkin technique for nonlinear analysis of structures10.1007/BF02455251 ↗
24. Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds10.1016/j.jsv.2011.12.004 ↗
25. Two-dimensional cross-sectional analysis of composite beams using Rayleigh-Ritz-based dimensional reduction method10.1016/j.compstruct.2017.10.049 ↗
26. Momeni, M., Beni, M.R., Bedon, C., Najafgholipour, M.A., Dehghan, S.M., Sharifi, B.J., and Hadianfard, M.A. (2021). Dynamic response analysis of structures using Legendre-Galerkin matrix method. Appl. Sci., 11.10.3390/app11199307 ↗
27. Huang, K., Li, T.P., Xu, W., and Cao, L. (2022). Effects of nonlinear damping on vibrations of microbeam. Appl. Sci., 12.10.3390/app12063206 ↗
28. Wang, J. The extended Rayleigh-Ritz method for an analysis of nonlinear vibrations. Mech. Adv. Mater. Struct., 2021.10.1080/15376494.2021.1892888 ↗
29. Shi, B.Y., Yang, J., and Wang, J. Forced vibration analysis of multi-degree-of-freedom nonlinear systems with an extended Galerkin method. Mech. Adv. Mater. Struct., 2022.10.1080/15376494.2021.2023922 ↗
30. An analysis of nonlinear beam vibrations with the extended Rayleigh-Ritz method
31. Wang, J., and Wu, R.X. (2022). The extended Galerkin method for approximate solutions of nonlinear vibration equations. Appl. Sci., 12.10.3390/app12062979 ↗
32. Chen, S.H. (2009). Quantitative Analytical Methods of Strong Nonlinear Vibrations, Science Press. (In Chinese).
33. Gere, J.M., and Timoshenko, S.P. (1997). Mechanics of Materials, PWS Publishing Company.
34. Unified formulation of geometrically nonlinear refined beam theories10.1080/15376494.2016.1232458 ↗
35. Lee, J.K., and Lee, B.K. (2019). Elastica of non-prismatic and nonlinear elastic cantilever beams under combined loading. Appl. Sci., 9.10.3390/app9050877 ↗
36. Nonlinear phenomena in magnetostrictive elastic resonators10.1016/0020-7225(89)90155-9 ↗
37. The nonlinear thickness-shear vibrations of quartz crystal plates under an electric field10.1016/j.ijnonlinmec.2014.01.010 ↗
38. Analytical solution of the cantilevered elastica subjected to a normal uniformly distributed follower load10.1016/j.ijsolstr.2020.06.031 ↗
39. Extreme deformations of the cantilever Euler elastica under transverse aerodynamic load10.1016/j.eml.2020.101110 ↗