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Search Results (4)

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Keywords = Schwarz-Christoffel transformation

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17 pages, 5378 KB  
Article
An Innovative Method for Wind Load Estimation in High-Rise Buildings Based on Green’s Function
by Lin Song, Yang Yu, Jianxing Yu, Shibo Wu, Jiandong Ma and Zihang Jin
Mathematics 2024, 12(11), 1747; https://doi.org/10.3390/math12111747 - 4 Jun 2024
Viewed by 2203
Abstract
High-rise buildings are inherently vulnerable to substantial wind-induced forces. The increasing complexity of building designs has posed challenges in calculating wind loads, while traditional methods involving physical models have proven to be intricate and time-consuming. In order to overcome these obstacles, this paper [...] Read more.
High-rise buildings are inherently vulnerable to substantial wind-induced forces. The increasing complexity of building designs has posed challenges in calculating wind loads, while traditional methods involving physical models have proven to be intricate and time-consuming. In order to overcome these obstacles, this paper investigates a theoretical methodology aimed at streamlining the computation of wind loads. In the initial theoretical exploration, a simplified mathematical model based on Green’s function is introduced to take into account the interaction between wind loads and building geometry, while the model is not user-friendly and difficult to solve for complex polygonal buildings. To overcome this challenge, the study incorporates numerical simulations to extend the ideas and refine the methodology. To simplify the problem from a three-dimensional to a two-dimensional context, a bold tangential field assumption is made, assuming the wind pressure distribution remains similar across horizontal sections at different heights. The Schwarz–Christoffel formulation is then employed to facilitate the transformation. By integrating Green’s functions and conformal mapping to solve potential flow problems beyond the boundary layer, a comprehensive mathematical derivation is established. The above broadens the applicability of the mathematical theory and provides a new direction for estimations of high-speed wind load on buildings. Full article
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16 pages, 1963 KB  
Article
A Two-Dimensional Liquid Sloshing Analysis in a Partially Filled Complicated-Shape Tank by the Schwarz–Christoffel Transformation
by Jing Lü, Xiaolong Zhu and Yang Yu
Acoustics 2024, 6(2), 346-361; https://doi.org/10.3390/acoustics6020018 - 19 Apr 2024
Viewed by 2491
Abstract
The nonlinear sloshing of an incompressible fluid with irrotational flow in a complicated-shape tank due to horizontal excitation is studied with a semi-analytical method proposed in this study. In this method, the velocity potential function of a liquid in a complicated-shape tank is [...] Read more.
The nonlinear sloshing of an incompressible fluid with irrotational flow in a complicated-shape tank due to horizontal excitation is studied with a semi-analytical method proposed in this study. In this method, the velocity potential function of a liquid in a complicated-shape tank is estimated by using an approximate analytical transformation function from a complicated-shape region to a rectangular region. This function is obtained through Schwarz–Christoffel mapping and polynomial fitting. Nonlinear dynamic equations for the fluid–structure coupled system are developed based on the Hamilton–Ostrogradskiy principle. Nonlinear kinematic equations for the fluid–structure coupled system are derived based on the relationship between the liquid velocity and the free-surface equation. The Galerkin method is used to convert partial differential equations into ordinary differential equations. When tank movement is given, nonlinear models for the coupled system can be reduced to simple ones for liquid sloshing. Natural frequencies for the coupled system and liquid sloshing are analyzed, and the semi-analytical results agree with the numerical ones calculated with the software DampSlosh. Hydrodynamic forces and moments are also analyzed, and the semi-analytical results agree well with the numerical ones calculated with the Flow3D v10.1.1. Full article
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17 pages, 4945 KB  
Article
Modeling and Analysis of New Power Devices Based on Linear Phase-Shifting Transformer
by Jie Xue, Jinghong Zhao, Sinian Yan, Hanming Wang, Changduo Zhou, Dongao Yan and Hansi Chen
Processes 2022, 10(8), 1596; https://doi.org/10.3390/pr10081596 - 12 Aug 2022
Cited by 3 | Viewed by 2026
Abstract
With the rapid development of new power systems, various new power devices have also been developed. It is very important to establish analytical models of new power devices to ensure or even improve the reliability and stability of the power system. A linear [...] Read more.
With the rapid development of new power systems, various new power devices have also been developed. It is very important to establish analytical models of new power devices to ensure or even improve the reliability and stability of the power system. A linear phase-shifting transformer (LPST) is a new type of power device that mainly relies on air gaps to transfer energy, so establishing an accurate air-gap magnetic field model is very important for improving the efficiency of this system. In this paper, an analytical model of an unequal-pitch linear phase-shifting transformer (UP-LPST) was established by combining the distributed magnetic circuit method (DMCM) and Schwartz–Christopher transformation (SCT). Taking the magnetic field strength as a variable, an accurate magnetic field analysis model for a UP-LPST considering saturation, cogging, and edge was established. Taking a 1 kw UP-LPST as a prototype, the accuracy of the model was verified by the finite element method and experiments. This modeling method could also be used to establish magnetic field models of other similar structures in new energy power systems, especially those with cogging structures. Full article
(This article belongs to the Special Issue Modeling, Analysis and Control Processes of New Energy Power Systems)
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19 pages, 235 KB  
Article
On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions
by Thomas Ernst
Axioms 2015, 4(3), 235-253; https://doi.org/10.3390/axioms4030235 - 8 Jul 2015
Cited by 1 | Viewed by 4528
Abstract
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations [...] Read more.
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15. Full article
(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)
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