Mathematics

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24 pages, 3438 KiB

Open AccessArticle

AE-XGBoost: A Novel Approach for Machine Tool Machining Size Prediction Combining XGBoost, AE and SHAP

by Mu Gu, Shuimiao Kang, Zishuo Xu, Lin Lin and Zhihui Zhang

Mathematics 2025, 13(5), 835; https://doi.org/10.3390/math13050835 - 2 Mar 2025

Viewed by 736

To achieve intelligent manufacturing and improve the machining quality of machine tools, this paper proposes an interpretable machining size prediction model combining eXtreme Gradient Boosting (XGBoost), autoencoder (AE), and Shapley additive explanation (SHAP) analysis. In this study, XGBoost was used to establish an [...] Read more.

To achieve intelligent manufacturing and improve the machining quality of machine tools, this paper proposes an interpretable machining size prediction model combining eXtreme Gradient Boosting (XGBoost), autoencoder (AE), and Shapley additive explanation (SHAP) analysis. In this study, XGBoost was used to establish an evaluation system for the actual machining size of computer numerical control (CNC) machine tools. The XGBoost model was combined with SHAP approximation to effectively capture local and global features in the data using autoencoders and transform the preprocessed data into more representative feature vectors. Grey correlation analysis (GRA) and principal component analysis (PCA) were used to reduce the dimensions of the original data features, and the synthetic minority overstimulation technique of the Gaussian noise regression (SMOGN) method was used to deal with the problem of data imbalance. Taking the actual size of the machine tool as the response parameter, based on the size parameters in the milling process of the CNC machine tool, the effectiveness of the model is verified. The experimental results show that the proposed AE-XGBoost model is superior to the traditional XGBoost method, and the prediction accuracy of the model is 7.11% higher than that of the traditional method. The subsequent SHAP analysis reveals the importance and interrelationship of features and provides a reliable decision support system for machine tool processing personnel, helping to improve processing quality and achieve intelligent manufacturing. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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24 pages, 5390 KiB

Open AccessFeature PaperArticle

Mathematical Dimensional Synthesis of Four-Bar Linkages Based on Cognate Mechanisms

by Enrique Soriano-Heras, Carlos Pérez-Carrera and Higinio Rubio

Mathematics 2025, 13(1), 11; https://doi.org/10.3390/math13010011 - 24 Dec 2024

Viewed by 1096

Abstract

In the field of mechanical engineering, understanding mechanisms is essential for designing and developing devices and systems. Mechanisms, composed of interconnected elements, transform the energy applied to the input link into motion or force in the output link. Mechanisms are found in a [...] Read more.

In the field of mechanical engineering, understanding mechanisms is essential for designing and developing devices and systems. Mechanisms, composed of interconnected elements, transform the energy applied to the input link into motion or force in the output link. Mechanisms are found in a wide variety of machines, from industrial machines to household machines. In this paper, a mechanism synthesis method is developed that can model four-bar linkages and build their cognate mechanisms to be able to select the mechanism that best suits the required work. Studying four-bar mechanisms offers a strong foundation for grasping more complex mechanical systems. The concepts and principles learned from four-bar mechanisms are widely applicable to advanced mechanical systems, making them a crucial starting point in mechanical engineering education and research. The mechanism synthesis method proposed in this article is organized into three main sections. The first section provides a comprehensive overview of the theoretical and mathematical foundations required for modeling mechanisms, laying the groundwork for understanding the subsequent calculations. The second section delves into the process of obtaining and analyzing the initial mechanism and constructing cognate mechanisms, detailing the procedures and algorithms used for modeling and calculating the coupling curve. Finally, the third section discusses the practical implementation of the method, including the graphical representation of mechanisms and a comparative analysis of the solutions obtained, assessing dimensional differences, design and manufacturing efficiency, and their suitability for various practical applications. The proposed four-bar mechanism synthesis method serves as a valuable tool for mechanism design, offering versatile and adaptable solutions that can optimize both technical performance and economic viability across a wide range of engineering applications. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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25 pages, 2072 KiB

Open AccessFeature PaperArticle

Full Forward Kinematics of Lower-Mobility Planar Parallel Continuum Robots

by Oscar Altuzarra, Mónica Urizar, Kerman Bilbao and Alfonso Hernández

Mathematics 2024, 12(22), 3562; https://doi.org/10.3390/math12223562 - 14 Nov 2024

Viewed by 782

Abstract

In rigid lower-mobility parallel manipulators the motion of the end-effector is partially constrained due to a combination of passive kinematic pairs and rigid components. Translational mechanisms, such as the Delta manipulator, are the most common ones among this type of mechanisms. When flexible [...] Read more.

In rigid lower-mobility parallel manipulators the motion of the end-effector is partially constrained due to a combination of passive kinematic pairs and rigid components. Translational mechanisms, such as the Delta manipulator, are the most common ones among this type of mechanisms. When flexible elements are introduced, as in Parallel Continuum Manipulators, the constraint is no longer rigid, and new challenges arise in performing certain motions depending on the degree of compliance. Mobility analysis shifts from being purely a geometric issue to one that heavily relies on force distribution within the mechanism. Simply converting classical lower-mobility rigid parallel mechanisms into Parallel Continuum Mechanisms may yield unexpected outcomes. This work, making use of a planar parallel continuum Delta manipulator, on the one hand, presents two different approaches to solve the Forward Kinematics of planar continuum manipulators, and, on the other hand, explores some challenges and issues in assessing the resultant workspace for different design alternatives of this kind of flexible manipulators. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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15 pages, 2753 KiB

Open AccessArticle

An Analysis of Power Friction Losses in Gear Engagement with Intermediate Rolling Elements and a Free Cage

by Egor A. Efremenkov, Nikita V. Martyushev, Svetlana K. Efremenkova and Egor S. Chavrov

Mathematics 2024, 12(6), 873; https://doi.org/10.3390/math12060873 - 16 Mar 2024

Cited by 1 | Viewed by 1188

Abstract

Currently, mechanical gears with cycloid engagement are increasingly used in mechanisms along with involute ones. In modern drive mechanisms, using pin gears and gears with intermediate rolling elements (IRE) is widespread, which simultaneously use cycloid gears. To a greater extent, pin gears are [...] Read more.

Currently, mechanical gears with cycloid engagement are increasingly used in mechanisms along with involute ones. In modern drive mechanisms, using pin gears and gears with intermediate rolling elements (IRE) is widespread, which simultaneously use cycloid gears. To a greater extent, pin gears are now being investigated, but IRE gears have their undeniable advantages. Many works are devoted to the study of cycloid toothing for certain gears, but the efficiency, especially that of IRE gears, has practically not been investigated. Therefore, the analysis of power losses in the engagement of a gear with IRE and a free cage (IREFC) is relevant. In this analysis, the authors of the work have used laws of mechanics, methods of energy flows and a secant normal. Mathematical expressions have been obtained to estimate slip speeds and power friction losses in the engagement of a gear with IREFC, and a formula has been derived to determine the efficiency of a mentioned mechanical transmission. The calculation of slip speeds and power losses at the points of contact of a rolling element with cycloid profiles of wheels for selected initial parameters of a gear with IREFC has been presented. The friction power and the overall efficiency of the entire gear engagement have also been calculated. This work shows that power friction losses at the points of contact of a rolling element with cycloid profiles of tooth wheels of a gear with IREFC are not the same. The friction power in the contact of a rolling element with a cycloid profile of a cam is an order of magnitude higher than the friction power in the contact of a rolling element with a cycloid profile of a crown. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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19 pages, 2440 KiB

Open AccessArticle

A Comprehensive Evaluation Method of Machining Center Components’ Importance Based on Combined Variable Weight

by Lan Luan, Guixiang Shen and Yingzhi Zhang

Mathematics 2024, 12(2), 330; https://doi.org/10.3390/math12020330 - 19 Jan 2024

Cited by 1 | Viewed by 1018

Abstract

The fault transitivity of machining center components causes their fault propagation indexes to demonstrate dynamic time variability, which affects their importance. The method proposed in this study overcomes the biases of existing methods of evaluating the importance of system components, as they are [...] Read more.

The fault transitivity of machining center components causes their fault propagation indexes to demonstrate dynamic time variability, which affects their importance. The method proposed in this study overcomes the biases of existing methods of evaluating the importance of system components, as they are mostly based on single indexes; the fault propagation probability and fault propagation risk are selected to perform a comprehensive evaluation. This study first establishes a network hierarchical structure model for machining center components, and the degree of influence of fault propagation among the components is calculated. On this basis, the improved adjacent spreading paths (ASP) algorithm is used to calculate the fault propagation index of each component; from the two perspectives of fault propagation probability and fault propagation risk, an evaluation mechanism involving the combined variable weight is used to comprehensively evaluate components’ importance. Taking a certain type of machining center as an example, through a comparison with ranking results from other node importance methods, it is verified that the proposed method can more effectively distinguish the differences in the importance of each component, thus illustrating the effectiveness and practical value of this method. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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21 pages, 6281 KiB

Open AccessArticle

A New RP1PR Type Coupling for Shafts with Crossed Axes

by Stelian Alaci, Ioan Doroftei, Florina-Carmen Ciornei, Ionut-Cristian Romanu, Ioan-Alexandru Doroftei and Mariana-Catalina Ciornei

Mathematics 2023, 11(9), 2025; https://doi.org/10.3390/math11092025 - 24 Apr 2023

Cited by 3 | Viewed by 1420

Abstract

There are few examples of mechanical coupling solutions for the transmission of high torques between two rotating shafts that have non-coplanar, non-parallel axes. Based on the structural analysis, the paper proposes a solution for an RP1PR-type symmetrical coupling. The Hartenberg–Denavit methodology is not [...] Read more.

There are few examples of mechanical coupling solutions for the transmission of high torques between two rotating shafts that have non-coplanar, non-parallel axes. Based on the structural analysis, the paper proposes a solution for an RP1PR-type symmetrical coupling. The Hartenberg–Denavit methodology is not applicable for performing the kinematical analysis, hence the solution starts from the geometrical condition of the creation of planar pairs of the mechanism, expressed in vector form. The absolute motion of all elements of the mechanism’s structure can be expressed after developing the kinematical analysis. The theoretical results are validated via numerical analysis. By comparing the analytical results with the CATIA-modeled results, excellent compatibility is obtained. We also propose a constructive solution for the newly designed coupling mechanism. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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12 pages, 904 KiB

Open AccessArticle

Analytical Modelling of an Active Vibration Absorber for a Beam

by Carlos Rincon, Jorge Alencastre and Richard Rivera

Mathematics 2023, 11(9), 2009; https://doi.org/10.3390/math11092009 - 24 Apr 2023

Cited by 3 | Viewed by 2547

Abstract

Attenuation of mechanical vibrations is an ongoing field of research in engineering aiming at reducing damage and improving performance in the presence of dynamical forces. Different alternatives have been proposed over time; the active vibration absorber can be highlighted as an alternative which [...] Read more.

Attenuation of mechanical vibrations is an ongoing field of research in engineering aiming at reducing damage and improving performance in the presence of dynamical forces. Different alternatives have been proposed over time; the active vibration absorber can be highlighted as an alternative which can absorb the vibration from system in real time. In this study, an active vibration absorber was modelled as an electromechanical device. It was applied to a cantilever beam, mathematically modelled as a continuous beam. A set of differential equations representing the dynamical behaviour of the cantilever beam and active vibration absorber was obtained and it was simulated in Matlab Simulink^®. Results indicated that the active vibration absorber is able to significantly reduce the vibration amplitudes of a system, especially in resonance conditions. The analytical model and procedure developed here can easily spread to any more complex system. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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15 pages, 3406 KiB

Open AccessArticle

Development of an Algorithm for Computing the Force and Stress Parameters of a Cycloid Reducer

by Egor A. Efremenkov, Sergey A. Shanin and Nikita V. Martyushev

Mathematics 2023, 11(4), 993; https://doi.org/10.3390/math11040993 - 15 Feb 2023

Cited by 20 | Viewed by 2001

Abstract

The paper is devoted to the development of an algorithm for the automated calculation of force characteristics of cycloid toothing when the initial parameters vary widely. The algorithm forms a structured data array that accelerates finding and outputting the necessary parameters and reduces [...] Read more.

The paper is devoted to the development of an algorithm for the automated calculation of force characteristics of cycloid toothing when the initial parameters vary widely. The algorithm forms a structured data array that accelerates finding and outputting the necessary parameters and reduces the probability of error in determining these parameters. The algorithm serves the basis for a program that allows for the examination of the change in force and geometrical parameters in various combinations. The study includes the analysis of the dependence of forces and contact stresses in transmission toothing with intermediate rolling elements and a free cage on the initial parameters of this transmission. The obtained results will make it possible to select optimal combinations of initial parameters in order to minimize the force impact on the mechanism parts when designing modern compact mechanisms based on the cycloid with intermediate rolling elements and a free cage. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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