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Explicit Solution to Large Deformation of Cantilever Beam by Improved Homotopy Analysis Method II: Vertical and Horizontal Displacements
Article

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

School of Mechanical Engineering, Hebei University of Technology, Tianjin 300401, China
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Author to whom correspondence should be addressed.
Academic Editor: Valentino Paolo Berardi
Appl. Sci. 2022, 12(13), 6400; https://doi.org/10.3390/app12136400
Received: 12 December 2021 / Revised: 6 June 2022 / Accepted: 15 June 2022 / Published: 23 June 2022
An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures. View Full-Text
Keywords: improved homotopy analysis method; strong nonlinearity; large deformation of cantilever beam; convergence range; homotopy-Páde approximation improved homotopy analysis method; strong nonlinearity; large deformation of cantilever beam; convergence range; homotopy-Páde approximation
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MDPI and ACS Style

Li, Y.; Li, X.; Huo, S.; Xie, C. Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle. Appl. Sci. 2022, 12, 6400. https://doi.org/10.3390/app12136400

AMA Style

Li Y, Li X, Huo S, Xie C. Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle. Applied Sciences. 2022; 12(13):6400. https://doi.org/10.3390/app12136400

Chicago/Turabian Style

Li, Yinshan, Xinye Li, Shuhao Huo, and Chen Xie. 2022. "Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle" Applied Sciences 12, no. 13: 6400. https://doi.org/10.3390/app12136400

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