Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method

Masjedi, Pedram Khaneh

doi:10.1016/j.apm.2021.10.037

Cited by 20 publications

(20 citation statements)

References 34 publications

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“…The HAM has been broadly used and its above-mentioned advantages have been verified and confirmed in thousands of articles by scientists and engineers all over the world [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].…”

Section: Basic Ideas Of the Hammentioning

confidence: 87%

“…So, even for given auxiliary linear operator L and initial guess u 0 , the convergence-control parameter c 0 provides us an additional way to guarantee the convergence of the solution series, which can overcome the limitations of perturbation methods mentioned above, as illustrated below in this paper and other publications [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. Substituting the power series (17) into the zeroth-order deformation equation ( 14) and equating the like-power of q, we have the high-order deformation equation…”

Section: Basic Ideas Of the Hammentioning

confidence: 97%

“…(A) solution series (17) is expanded in the homotopy parameter q ∈ [0, 1], which has no physical meanings at all. So, the HAM has nothing to do with any small/large physical parameters: it works no matter whether small/large physical parameters exist or not; (B) the HAM provides us great freedom to choose its auxiliary linear operator; (C) the HAM provides us great freedom to choose its initial approximation; (D) the so-called convergence-control parameter c 0 has no physical meanings but can guarantee the convergence of solution series even for high nonlinearity, as illustrated below and verified in many related publications [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].…”

Section: Basic Ideas Of the Hammentioning

confidence: 99%

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Avoiding small denominator problems by completely abandoning perturbation methods

Liao¹

2022

Preprint

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The so-called "small denominator problem" was the fundamental problem of dynamics, as pointed out by Poincaré. Small denominators are found most commonly in the perturbative theory, and the Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to the small denominators. In this paper, using the Duffing equation as an example, we illustrate that the famous "small denominator problems" never appear for arbitrary physical parameters if a non-perturbative technique, namely the homotopy analysis method (HAM), is used. A new HAM-based approach is proposed, which provides us great freedom to directly define the inverse operator of an auxiliary linear operator so that all small denominators can be completely avoided. Convergent series of multiple limit-cycles of the Duffing equation are successfully obtained, although such kind of directly defined inverse operators might be beyond the traditional mathematical theories. Thus, from the viewpoint of the HAM, the famous "small denominator problems" are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods, one would be never troubled by small denominators. This HAM-based approach has general meanings and can be used to attack many open problems related to "small denominators".

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Section: Basic Ideas Of the Hammentioning

confidence: 87%

Section: Basic Ideas Of the Hammentioning

confidence: 97%

Section: Basic Ideas Of the Hammentioning

confidence: 99%

See 1 more Smart Citation

Avoiding small denominator problems by completely abandoning perturbation methods

Liao¹

2022

Preprint

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“…Closed-form series solutions for the static deflection of anisotropic composite beams resting on elastic foundations were obtained by both the homotopy analysis method (HAM) and iterative HAM (iHAM) [46]. The iterative homotopy analysis method (iHAM) was used to obtain analytical solutions for the arbitrary large deflection of geometrically exact beams subjected to distributed and tip loads based on follower and conservative loading scenarios [47]. Wang et al [48] derived an explicit solution to the large deformation of a cantilever beam under point load at the free end with HAM.…”

Section: Introductionmentioning

confidence: 99%

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

Yin-shan¹,

Li²,

Huo³

et al. 2022

Applied Sciences

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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.

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“…This hypothesis, although not best representing surface forces and partial cross-sectional loads, is widely accepted in beam modeling approaches (Masjedi et al. 2019 ; Masjedi and Weaver 2020a , b , 2022 ; Masjedi et al. 2021 ; Doeva et al.…”

Section: Introductionmentioning

confidence: 99%

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

Vilar

Hadjiloizi

Masjedi

2022

Int J Mech Mater Des

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Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, offer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to interactions between stiffness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efficient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress field recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of effects where stiffness and external load distributions differ. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to $$15^{\circ }$$ 15 ∘ . The model is validated by comparison with finite element analysis and relevant analytical formulations.

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Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method

Cited by 20 publications

References 34 publications

Avoiding small denominator problems by completely abandoning perturbation methods

Avoiding small denominator problems by completely abandoning perturbation methods

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

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

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