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Open AccessArticle

Structural Damage Identification Using the First-Order Vibration-Mode-Based Frequency-Shift Flexibility Sensitivity Algorithm

by

Shanshan Cao

,

Qiuwei Yang

and

Xi Peng

Axioms 2023, 12(6), 551; https://doi.org/10.3390/axioms12060551 - 02 Jun 2023

Viewed by 289

Abstract

The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal [...] Read more.

The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal parameters. The core idea of the proposed method is to make the first-order vibration mode contribute the most to a structural flexibility matrix through the frequency-shift operation. A high-precision flexibility matrix can be obtained after the frequency-shift operation, which only needs the first mode of structural free vibration. Through this special advantage, structural damage coefficients can be accurately calculated by the frequency-shift flexibility sensitivity equation. Thus, a reliable identification result can be obtained according to the values of the calculated damage coefficients. In some engineering applications, another advantage of the proposed method is that it does not require a complete finite element modeling process, as long as a few lower-frequency vibration modes of the intact structure are measured. A truss structure and a beam structure are used as two numerical examples to demonstrate the proposed approach. The results show that the proposed method has higher calculation accuracy than the ordinary flexibility sensitivity method by using only the first-order vibration mode. The proposed method can overcome possible misdiagnosis of the ordinary flexibility sensitivity method. It also has been shown that the proposed method may have the potential to identify minor damage in a structure. Using the experimental data of a steel frame structure, the effectiveness and reliability of the proposed method have been further verified. The proposed method provides a simple way for structural damage identification with only a few vibration modal data. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

An Energy-Efficient Optimal Operation Control Strategy for High-Speed Trains via a Symmetric Alternating Direction Method of Multipliers

by

Shan Ma

,

Feng Ma

and

Chaoyu Tang

Axioms 2023, 12(5), 489; https://doi.org/10.3390/axioms12050489 - 18 May 2023

Viewed by 322

Abstract

Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control [...] Read more.

Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control problem for multiple train operation is formulated to reduce the energy consumption and improve the punctuality of trains. Then, we propose an efficient algorithm based on the framework of the symmetric alternating direction method of multipliers to solve this optimization problem. Finally, numerical simulations show that the method can obtain the optimal train control sequence in fewer iterative steps compared to the alternating direction multiplier method, thus illustrating the effectiveness of the algorithm. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

by

Heng-Pin Hsu

,

Te-Wen Tu

and

Jer-Rong Chang

Axioms 2023, 12(5), 416; https://doi.org/10.3390/axioms12050416 - 24 Apr 2023

Viewed by 473

Abstract

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case [...] Read more.

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models

by

,

,

and

Axioms 2023, 12(4), 370; https://doi.org/10.3390/axioms12040370 - 11 Apr 2023

Viewed by 752

Abstract

Three-dimensional printing is a layer-by-layer stacking process. It can realize complex models that cannot be manufactured by traditional manufacturing technology. The most common model currently used for 3D printing is the STL model. It uses planar triangles to simplify the CAD model. This [...] Read more.

Three-dimensional printing is a layer-by-layer stacking process. It can realize complex models that cannot be manufactured by traditional manufacturing technology. The most common model currently used for 3D printing is the STL model. It uses planar triangles to simplify the CAD model. This approach makes it difficult to fit complex surface shapes with high accuracy. The fitting result usually suffers from loss of local features of the model, poor fitting accuracy, or redundant data due to face piece subdivision, which will cause problems such as poor manufacturing accuracy or difficult data processing. To this end, this paper proposes a method for constructing Hermite surface models considering high-precision fitting of 3D printing models. The mapping relationship between different surface triangles and the same base triangle is established by analyzing the characteristics of Hermite surface triangles in AMF format files and using the radial variation property. By constructing a cubic surface model with general parameters and combining the vertex and tangent vector information, a cubic Hermite curve and surface triangle model are obtained. A sampling mapping point solution method is proposed, which transforms the volume integration problem between models into the summation problem of sampling point height difference. Considering the mean deviation and variance in multiple directions of the sampling points, a method for calculating and evaluating the model fitting error is constructed. Finally, the effectiveness of the proposed method is verified by rabbit and turbine. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

An Asymmetric Model Position Dependent Mass: Quantum Mechanical Study

by

,

,

,

,

and

Axioms 2023, 12(4), 318; https://doi.org/10.3390/axioms12040318 - 23 Mar 2023

Viewed by 485

Abstract

We propose an asymmetric model position dependent mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the [...] Read more.

We propose an asymmetric model position dependent mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the model asymmetric mass works well for weak coupling preserving the symmetric phase portrait. However, the dominance of asymmetric feature of the mass in the system clearly visible for higher values of the constant associated with the mass. Though, both position dependent mass and potential have significant role in controlling the spectral feature of the system, one may dominate over other for certain cases. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations

by

Leonid Plotnikov

Axioms 2023, 12(3), 244; https://doi.org/10.3390/axioms12030244 - 27 Feb 2023

Viewed by 502

Abstract

Conical diffusers of various configurations are used in many kinds of technical equipment and manufacturing processes. Therefore, it is a relevant objective to obtain reliable experimental and mathematical data on the aerodynamic characteristics of diffusers. This article presents experimental data on the aerodynamics [...] Read more.

Conical diffusers of various configurations are used in many kinds of technical equipment and manufacturing processes. Therefore, it is a relevant objective to obtain reliable experimental and mathematical data on the aerodynamic characteristics of diffusers. This article presents experimental data on the aerodynamics of stationary flows in a vertical conical diffuser when air is supplied through tubes with various cross sections (circle, square, and triangle). Instantaneous values of air flow velocity are measured with a constant-temperature hot-wire anemometer. Data are obtained on the velocity fields and turbulence intensity along the height and the diameter of the diffuser’s cylindrical part when air is supplied through tubes of various configurations. It is established that air supply through profiled tubes has a significant effect on the shape of the velocity field and turbulence intensity in a vertical conical diffuser. For example, higher values of turbulence intensity are typical of air supplied through profiled tubes (the differences reach 50%). A mathematical formulation (linear and exponential equations) of the change in the average speed and intensity of air flow turbulence along the height of the diffuser’s cylindrical part for various initial conditions and supply tube configurations is presented. The obtained findings will make it possible to refine mathematical models and update algorithms for engineering the design of diffusers for various engineering processes and pieces of technical equipment. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

LBM-MHD Data-Driven Approach to Predict Rayleigh–Bénard Convective Heat Transfer by Levenberg–Marquardt Algorithm

by

,

,

and

Axioms 2023, 12(2), 199; https://doi.org/10.3390/axioms12020199 - 13 Feb 2023

Viewed by 797

Abstract

This study aims to consider lattice Boltzmann method (LBM)–magnetohydrodynamics (MHD) data to develop equations to predict the average rate of heat transfer quantitatively. The present approach considers a 2D rectangular cavity with adiabatic side walls, and the bottom wall is heated while the [...] Read more.

This study aims to consider lattice Boltzmann method (LBM)–magnetohydrodynamics (MHD) data to develop equations to predict the average rate of heat transfer quantitatively. The present approach considers a 2D rectangular cavity with adiabatic side walls, and the bottom wall is heated while the top wall is kept cold. Rayleigh–Bénard (RB) convection was considered a heat-transfer phenomenon within the cavity. The Hartmann (

H a

) number, by varying the inclination angle (

θ

), was considered in developing the equations by considering the input parameters, namely, the Rayleigh (

R a

) numbers, Darcy (

D a

) numbers, and porosity (

ϵ

) of the cavity in different segments. Each segment considers a data-driven approach to calibrate the Levenberg–Marquardt (LM) algorithm, which is highly linked with the artificial neural network (ANN) machine learning method. Separate validations have been conducted in corresponding sections to showcase the accuracy of the equations. Overall, coefficients of determination (

R^{2}

) were found to be within 0.85 to 0.99. The significant findings of this study present mathematical equations to predict the average Nusselt number (

\bar{N u}

). The equations can be used to quantitatively predict the heat transfer without directly simulating LBM. In other words, the equations can be considered validations methods for any LBM-MHD model, which considers RB convection within the range of the parameters in each equation. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

Aggregative Game for Distributed Charging Strategy of PEVs in a Smart Charging Station

by

Ti Kang

,

Huaqing Li

and

Lifeng Zheng

Axioms 2023, 12(2), 186; https://doi.org/10.3390/axioms12020186 - 10 Feb 2023

Viewed by 577

Abstract

This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs [...] Read more.

This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs to be subject to overload limits, beyond which failures can occur. The time anxiety is presented to address some of the uncertainties that may arise while charging PEVs. Under an aggregative game framework, this paper constructs a price-driven charging model to minimize costs by choosing the optimal charging strategy. Meanwhile, since the driver information is an aggregated item in the PEV cost function, the drivers’ privacy can be protected. Then, a distributed reflected forward–backward (RFB) splitting method is developed to search for the generalized Nash equilibria (GNE) of the game. The convergence of the proposed algorithm and the effectiveness of the charging strategy are verified by the detailed simulation and results. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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Open AccessArticle

A Fast Calculation Method for Sensitivity Analysis Using Matrix Decomposition Technique

by

Qiuwei Yang

and

Xi Peng

Axioms 2023, 12(2), 179; https://doi.org/10.3390/axioms12020179 - 09 Feb 2023

Cited by 1 | Viewed by 582

Abstract

The sensitivity reanalysis technique is an important tool for selecting the search direction in structural optimization design. Based on the decomposition perturbation of the flexibility matrix, a fast and exact structural displacement sensitivity reanalysis method is proposed in this work. For this purpose, [...] Read more.

The sensitivity reanalysis technique is an important tool for selecting the search direction in structural optimization design. Based on the decomposition perturbation of the flexibility matrix, a fast and exact structural displacement sensitivity reanalysis method is proposed in this work. For this purpose, the direct formulas for computing the first-order and second-order sensitivities of structural displacements are derived. The algorithm can be applied to a variety of the modifications in optimal design, including the low-rank modifications, high-rank modifications, small modifications and large modifications. Two numerical examples are given to verify the effectiveness of the proposed approach. The results show that the presented algorithm is exact and effective. Compared with the existing two reanalysis methods, this method has obvious advantages in calculation accuracy and efficiency. This new algorithm is very useful for calculating displacement sensitivity in engineering problems such as structure optimization, model correction and defect detection. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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