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Applied Mathematics to Mechanisms and Machines
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Applied Mathematics to Mechanisms and Machines

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 40351

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Higinio Rubio Alonso

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Guest Editor
Department of Mechanical Engineering, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
Interests: kinematical and dynamical systems; vibration analysis and machine learning
Special Issues, Collections and Topics in MDPI journals
Dr. Alejandro Bustos Caballero

E-Mail Website
Guest Editor
Department of Mechanics, National University of Distance Education, Calle Juan del Rosal 12, 28040 Madrid, Spain
Interests: railways; condition monitoring; vibration analysis; multibody dynamics systems; FEM
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jesus Meneses Alonso

E-Mail Website
Guest Editor
Department of Mechanical Engineering, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
Interests: Kinematics and Dynamics of mechanisms and machines; Multibody Dynamics; Contact Models; Biped robots; Biomedical mechanical devices.
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Enrique Soriano-Heras

E-Mail Website
Guest Editor
Department of Mechanical Engineering, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
Interests: mechanisms; machines; machine-tools; grippers; collet-chuck
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In mechanism and machine science, theoretical and computational analysis leads to problems where mathematical modelling and processing are transcendent aspects for solving these problems.

The different ways of approaching problems in the area of mechanical engineering, or related areas, include the use of integral and differential equations; interpolation; polynomial computation; least squares and regularization; vector algebra; matrix equations; control theory; synthesis process; discrete and continuous  transforms; image and signal processing; Galerkin, finite-element, finite-difference and finite-volume methods; etc.

This Special Issue compiles articles with the aim of showing and explaining/describing the different mathematical processes used to solve mechanisms and machines problems. Special attention is devoted to: analysis and synthesis of mechanisms and machines; linear and nonlinear mechanical systems; transportation engineering (cars, trains, ships, …); vibrations data analysis and machine learning; and kinematics, dynamics and control in biomechanics, mechatronics and micro-machines. 

Prof. Dr. Higinio Rubio Alonso
Dr. Alejandro Bustos Caballero
Prof. Dr. Jesus Meneses Alonso
Prof. Dr. Enrique Soriano-Heras
Guest Editors

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Analytic synthesis
  • Mechanisms and machines
  • Kinematics and dynamics
  • Vibrations analysis techniques
  • Linear/nonlinear systems
  • Machine learning
  • Control algorithms
  • Transportation engineering

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Related Special Issue

Published Papers (16 papers)

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Research

25 pages, 9589 KiB  
Article
Lateral Dynamic Simulation of a Bus under Variable Conditions of Camber and Curvature Radius
by Ester Olmeda, Enrique Roberto Carrillo Li, Jorge Rodríguez Hernández and Vicente Díaz
Mathematics 2022, 10(17), 3081; https://doi.org/10.3390/math10173081 - 26 Aug 2022
Cited by 2 | Viewed by 1694
Abstract
The objective of this paper is to describe a model for the simulation of the lateral dynamics of a vehicle, specifically buses, under variable trajectory conditions, such as camber and radius of curvature; in addition, a variable speed is added as a simulation [...] Read more.
The objective of this paper is to describe a model for the simulation of the lateral dynamics of a vehicle, specifically buses, under variable trajectory conditions, such as camber and radius of curvature; in addition, a variable speed is added as a simulation parameter. The objective of this study is the prevention of vehicle rollover and sideslip. An 8 degrees of freedom model was developed, considering a front and a rear section of the bus with its respective suspension system, and both sections have been connected by a torsion spring that emulates the torsional stiffness of the vehicle chassis. A Panhard bar is also added at the rear as an additional element to the suspension and the behavior of the bus when it is added is analyzed. This model also allows the evaluation of the force on each suspension component, which allows for future controllability of the active suspension components. The results show the dynamic behavior of the vehicle, and some indicators are introduced to show the possible sideslip or rollover. As a conclusion, the influence of the road parameters on the dynamic behavior of the bus and the effect of the Panhard bar on the dynamic behavior of the bus can be pointed out. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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29 pages, 12253 KiB  
Article
Fault Classification in a Reciprocating Compressor and a Centrifugal Pump Using Non-Linear Entropy Features
by Ruben Medina, Mariela Cerrada, Shuai Yang, Diego Cabrera, Edgar Estupiñan and René-Vinicio Sánchez
Mathematics 2022, 10(17), 3033; https://doi.org/10.3390/math10173033 - 23 Aug 2022
Cited by 5 | Viewed by 2161
Abstract
This paper describes a comparison of three types of feature sets. The feature sets were intended to classify 13 faults in a centrifugal pump (CP) and 17 valve faults in a reciprocating compressor (RC). The first set comprised 14 non-linear entropy-based features, the [...] Read more.
This paper describes a comparison of three types of feature sets. The feature sets were intended to classify 13 faults in a centrifugal pump (CP) and 17 valve faults in a reciprocating compressor (RC). The first set comprised 14 non-linear entropy-based features, the second comprised 15 information-based entropy features, and the third comprised 12 statistical features. The classification was performed using random forest (RF) models and support vector machines (SVM). The experimental work showed that the combination of information-based features with non-linear entropy-based features provides a statistically significant accuracy higher than the accuracy provided by the Statistical Features set. Results for classifying the 13 conditions in the CP using non-linear entropy features showed accuracies of up to 99.50%. The same feature set provided a classification accuracy of 97.50% for the classification of the 17 conditions in the RC. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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18 pages, 2732 KiB  
Article
The General Dispersion Relation for the Vibration Modes of Helical Springs
by Leopoldo Prieto, Alejandro Quesada, Ana María Gómez Amador and Vicente Díaz
Mathematics 2022, 10(15), 2698; https://doi.org/10.3390/math10152698 - 30 Jul 2022
Viewed by 1449
Abstract
A system of mathematical equations was developed for the calculation of the natural frequencies of helical springs, its predictions being compared with finite element simulation with ANSYS®. Authors derive the general equations governing the helical spring vibration relative to the Frenet [...] Read more.
A system of mathematical equations was developed for the calculation of the natural frequencies of helical springs, its predictions being compared with finite element simulation with ANSYS®. Authors derive the general equations governing the helical spring vibration relative to the Frenet trihedral representing the normal, binormal and tangent unit vectors to the spring medium line. The dispersion relation ω=f(k) has been obtained to model a wave traveling along the axis of the wire. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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17 pages, 4435 KiB  
Article
Jump and Initial-Sensitive Excessive Motion of a Class of Relative Rotation Systems and Their Control via Delayed Feedback
by Ziyin Cui and Huilin Shang
Mathematics 2022, 10(15), 2676; https://doi.org/10.3390/math10152676 - 29 Jul 2022
Viewed by 1074
Abstract
Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback [...] Read more.
Jump and excessive motion are undesirable phenomena in relative rotation systems, causing a loss of global integrity and reliability of the systems. In this work, a typical relative rotation system is considered in which jump, excessive motion, and their suppression via delayed feedback are investigated. The Method of Multiple Scales and the Melnikov method are applied to analyze critical conditions for bi-stability and initial-sensitive excessive motion, respectively. By introducing the fractal of basins of attraction and the erosion of the safe basin to depict jump and initial-sensitive excessive motion, respectively, the point mapping approach is used to present numerical simulations which are in agreement with the theoretical prediction, showing the validity of the analysis. It is found that jump between bistable attractors can be due to saddle–node bifurcation, while initial-sensitive excessive motion can be due to heteroclinic bifurcation. Under a positive coefficient of the gain, the types of delayed feedback can both be effective in reducing jump and initial-sensitive excessive motion. The results may provide some reference for the performance improvement of rotors and main bearings. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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15 pages, 19970 KiB  
Article
Miura-Ori Inspired Smooth Sheet Attachments for Zipper-Coupled Tubes
by Dylan C. Webb, Elissa Reynolds, Denise M. Halverson and Larry L. Howell
Mathematics 2022, 10(15), 2643; https://doi.org/10.3390/math10152643 - 28 Jul 2022
Cited by 3 | Viewed by 2825
Abstract
Zipper-coupled tubes are a broadly applicable, deployable mechanism with an angular surface that can be smoothed by attaching an additional smooth sheet pattern. The existing design for the smooth sheet attachment, however, leaves small gaps that can only be covered by adding flaps [...] Read more.
Zipper-coupled tubes are a broadly applicable, deployable mechanism with an angular surface that can be smoothed by attaching an additional smooth sheet pattern. The existing design for the smooth sheet attachment, however, leaves small gaps that can only be covered by adding flaps that unfold separately, limiting applicability in situations requiring a seamless surface and simultaneous deployment. We provide a novel construction of the smooth sheet attachment that unfolds simultaneously with zipper-coupled tubes to cover the entire surface without requiring additional actuation and without inhibiting the tubes’ motion up to an ideal, unfolded state of stability. Furthermore, we highlight the mathematics underlying the design and motion of the new smooth sheet pattern, thereby demonstrating its rigid-foldability and compatibility with asymmetric zipper-coupled tubes. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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34 pages, 16213 KiB  
Article
Mechanical Model and FEM Simulations for Efforts on Biceps and Triceps Muscles under Vertical Load: Mathematical Formulation of Results
by Emilio Lechosa Urquijo, Fernando Blaya Haro, Juan David Cano-Moreno, Roberto D’Amato and Juan Antonio Juanes Méndez
Mathematics 2022, 10(14), 2441; https://doi.org/10.3390/math10142441 - 13 Jul 2022
Viewed by 4857
Abstract
Although isometric contractions in human muscles have been analyzed several times, there are no FEA models that allow us to use the same modeled joint (the elbow under our case) in different conditions. Most elbow joints use 3D elements for meshing. Representing the [...] Read more.
Although isometric contractions in human muscles have been analyzed several times, there are no FEA models that allow us to use the same modeled joint (the elbow under our case) in different conditions. Most elbow joints use 3D elements for meshing. Representing the muscles in the joint is quite useful when the study is focused on the muscle itself, knowing stress distribution on muscle, and checking damage in muscle in a detailed manner (tendon–muscle insertion, for example). However, this technique is not useful for studying muscle behavior at different positions of the joint. This study, based on the mechanical model of the elbow joint, proposes a methodology for modelling muscles that will be studied in different positions by meshing them with 1D elements. Furthermore, the methodology allows us to calculate biceps and triceps efforts under load for different angles of elbow joint aperture. The simulation results have been mathematically modelled to obtain general formulations for these efforts, depending on the load and the aperture angle. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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32 pages, 12076 KiB  
Article
Mathematical Analysis of a Low Cost Mechanical Ventilator Respiratory Dynamics Enhanced by a Sensor Transducer (ST) Based in Nanostructures of Anodic Aluminium Oxide (AAO)
by Jesús Alan Calderón Chavarri, Carlos Gianpaul Rincón Ruiz, Ana María Gómez Amador, Bray Jesús Martin Agreda Cardenas, Sebastián Calero Anaya, John Hugo Lozano Jauregui, Alexandr Toribio Hinostroza and Juan José Jiménez de Cisneros y Fonfría
Mathematics 2022, 10(14), 2403; https://doi.org/10.3390/math10142403 - 8 Jul 2022
Cited by 3 | Viewed by 2812
Abstract
Mechanical ventilation systems require a device for measuring the air flow provided to a patient in order to monitor and ensure the correct quantity of air proportionated to the patient, this device is the air flow sensor. At the beginning of the COVID-19 [...] Read more.
Mechanical ventilation systems require a device for measuring the air flow provided to a patient in order to monitor and ensure the correct quantity of air proportionated to the patient, this device is the air flow sensor. At the beginning of the COVID-19 pandemic, flow sensors were not available in Peru because of the international supply shortage. In this context, a novel air flow sensor based on an orifice plate and an intelligent transducer was developed to form an integrated device. The proposed design was focused on simple manufacturing requirements for mass production in a developing country. CAD and CAE techniques were used in the design stage, and a mathematical model of the device was proposed and calibrated experimentally for the measured data transduction. The device was tested in its real working conditions and was therefore implemented in a breathing circuit connected to a low-cost mechanical ventilation system. Results indicate that the designed air flow sensor/transducer is a low-cost complete medical device for mechanical ventilators that is able to provide all the ventilation parameters by an equivalent electrical signal to directly display the following factors: air flow, pressure and volume over time. The evaluation of the designed sensor transducer was performed according to sundry transducer parameters such as geometrical parameters, material parameters and adaptive coefficients in the main transduction algorithm; in effect, the variety of the described results were achieved by the faster response time and robustness proportionated by transducers of nanostructures based on Anodic Aluminum Oxide (AAO), which enhanced the designed sensor/transducer (ST) during operation in intricate geographic places, such as the Andes mountains of Peru. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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27 pages, 558 KiB  
Article
Singularities of Serial Robots: Identification and Distance Computation Using Geometric Algebra
by Isiah Zaplana, Hugo Hadfield and Joan Lasenby
Mathematics 2022, 10(12), 2068; https://doi.org/10.3390/math10122068 - 15 Jun 2022
Cited by 3 | Viewed by 2105
Abstract
The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail [...] Read more.
The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6×n or n×n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C, such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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24 pages, 1634 KiB  
Article
Complete Balancing of the Six-Bar Mechanism Using Fully Cartesian Coordinates and Multiobjective Differential Evolution Optimization
by María T. Orvañanos-Guerrero, Mario Acevedo, Claudia N. Sánchez, Daniel U. Campos-Delgado, Amir Aminzadeh Ghavifekr, Paolo Visconti and Ramiro Velázquez
Mathematics 2022, 10(11), 1830; https://doi.org/10.3390/math10111830 - 26 May 2022
Cited by 1 | Viewed by 2484
Abstract
The high-speed operation of unbalanced machines may cause vibrations that lead to noise, wear, and fatigue that will eventually limit their efficiency and operating life. To restrain such vibrations, a complete balancing must be performed. This paper presents the complete balancing optimization of [...] Read more.
The high-speed operation of unbalanced machines may cause vibrations that lead to noise, wear, and fatigue that will eventually limit their efficiency and operating life. To restrain such vibrations, a complete balancing must be performed. This paper presents the complete balancing optimization of a six-bar mechanism with the use of counterweights. A novel method based on fully Cartesian coordinates (FCC) is proposed to represent such a balanced mechanism. A multiobjective optimization problem was solved using the Differential Evolution (DE) algorithm to minimize the shaking force (ShF) and the shaking moment (ShM) and thus balance the system. The Pareto front is used to determine the best solutions according to three optimization criteria: only the ShF, only the ShM, and both the ShF and ShM. The dimensions of the counterweights are further fine-tuned with an analysis of their partial derivatives, volumes, and area–thickness relations. Numerical results show that the ShF and ShM can be reduced by 76.82% and 77.21%, respectively, when importance is given to either of them and by 45.69% and 46.81%, respectively, when equal importance is given to both. A comparison of these results with others previously reported in the literature shows that the use of FCC in conjunction with DE is a suitable methodology for the complete balancing of mechanisms. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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26 pages, 6877 KiB  
Article
Adaptive Rejection of a Sinusoidal Disturbance with Unknown Frequency in a Flexible Rotor with Lubricated Journal Bearings
by Gerardo Amato, Roberto D’Amato and Alessandro Ruggiero
Mathematics 2022, 10(10), 1703; https://doi.org/10.3390/math10101703 - 16 May 2022
Cited by 2 | Viewed by 1749
Abstract
Frequency-dependent adaptive noise cancellation-based tracking controller (ANC-TC) is a known technique for the stabilization of several nonlinear dynamical systems. In recent years, this control strategy has been introduced and applied for the stabilization of a flexible rotor supported on full-lubricated journal bearings. This [...] Read more.
Frequency-dependent adaptive noise cancellation-based tracking controller (ANC-TC) is a known technique for the stabilization of several nonlinear dynamical systems. In recent years, this control strategy has been introduced and applied for the stabilization of a flexible rotor supported on full-lubricated journal bearings. This paper proposes a theoretical investigation, based on robust immersion and invariance (I&I) theory, of a novel ANC-frequency estimation (FE) technique designed to stabilize a flexible rotor shaft affected by self-generated sinusoidal disturbances, generalized to the case of unknown frequency. A structural proof, under assumptions on closed-loop output signals, shows that the sinusoidal disturbance rejection is exponential. Numerical simulations are presented to validate the mathematical results in silico. The iterative Inexact Newton method is applied to the disturbance frequency and phase estimation error point series. The data fitting confirms that the phase estimation succession has an exponential convergence behavior and that the asymptotical frequency estimation is a warm-up phase of the overall close-loop disturbance estimation process. In two different operating conditions, the orders of convergence obtained by phase and frequency estimate timeseries are pφ=1, pω,unc=0.9983 and pω,cav=1.005. Rejection of the rotor dynamic disturbance occurs approximately 76% before in the cavitated than in the uncavitated condition, 2 (s) and 8.5 (s), respectively. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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21 pages, 3543 KiB  
Article
A Novel Denoising Method for Retaining Data Characteristics Brought from Washing Aeroengines
by Zhiqi Yan, Ming Zu, Zhiquan Cui and Shisheng Zhong
Mathematics 2022, 10(9), 1485; https://doi.org/10.3390/math10091485 - 29 Apr 2022
Viewed by 1486
Abstract
Airlines evaluate the energy-saving and emission reduction effect of washing aeroengines by analyzing the exhaust gas temperature margin (EGTM) data of aeroengines so as to formulate a reasonable washing schedule. The noise in EGTM data must be reduced because they interfere with the [...] Read more.
Airlines evaluate the energy-saving and emission reduction effect of washing aeroengines by analyzing the exhaust gas temperature margin (EGTM) data of aeroengines so as to formulate a reasonable washing schedule. The noise in EGTM data must be reduced because they interfere with the analysis. EGTM data will show several step changes after cleaning the aeroengine. These step changes increase the difficulty of denoising because they will be smoothed in the denoising. A denoising method for aeroengine data based on a hybrid model is proposed to meet the needs of accurately evaluating the washing effect. Specifically, the aeroengine data is first decomposed into several components by time and frequency. The amplitude of the component containing the most noise is amplified, and Gaussian noise is added to generate noise-amplified data. Second, a Gated Recurrent Unit Autoencoder (GAE) model is proposed to capture engine data features. The GAE is trained to reconstruct the original data from the amplified noise data to develop its noise reduction ability. The experimental results show that, compared with the current popular algorithms, the proposed denoising method can achieve a better denoising effect, retaining the key characteristics of the aeroengine data. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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20 pages, 6261 KiB  
Article
Predictive Suspension Algorithm for Land Vehicles over Deterministic Topography
by Alejandro Bustos, Jesus Meneses, Higinio Rubio and Enrique Soriano-Heras
Mathematics 2022, 10(9), 1467; https://doi.org/10.3390/math10091467 - 27 Apr 2022
Viewed by 1607
Abstract
A good suspension system is mandatory for ensuring stability, comfort and safety in land vehicles; therefore, advanced semi and fully active suspension systems have been developed along with their associated management strategies to overcome the limitations of passive suspensions. This paper presents a [...] Read more.
A good suspension system is mandatory for ensuring stability, comfort and safety in land vehicles; therefore, advanced semi and fully active suspension systems have been developed along with their associated management strategies to overcome the limitations of passive suspensions. This paper presents a suspension algorithm for land vehicles traveling through a deterministic topography. The kinematics of a half-vehicle model and the algorithm are implemented in Simulink. The algorithm’s inputs are the measurements provided by a position scanner located on the front wheel of the vehicle. Based on this input, the algorithm reconstructs the topography in real-time and sends the corresponding command to an actuator located on the rear wheel to compensate for the irregularities of the terrain. The actuation is governed by the parameter “D”, which represents the distance over which the algorithm averages the height of the terrain. Two ground profiles were tested and sensitivity analysis of the parameter “D” was performed. Results show that larger values of “D” usually yield less vibration on the actuated mass, but this value also depends on the irregularities of the terrain. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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30 pages, 4834 KiB  
Article
Three-Legged Compliant Parallel Mechanisms: Fundamental Design Criteria to Achieve Fully Decoupled Motion Characteristics and a State-of-the-Art Review
by Minh Tuan Pham, Song Huat Yeo and Tat Joo Teo
Mathematics 2022, 10(9), 1414; https://doi.org/10.3390/math10091414 - 22 Apr 2022
Cited by 2 | Viewed by 2427
Abstract
A three-legged compliant parallel mechanism (3L-CPM) achieves fully decoupled motions when its theoretical 6 × 6 stiffness/compliance matrix is a diagonal matrix, which only contains diagonal components, while all non-diagonal components are zeros. Because the motion decoupling capability of 3L-CPMs is essential in [...] Read more.
A three-legged compliant parallel mechanism (3L-CPM) achieves fully decoupled motions when its theoretical 6 × 6 stiffness/compliance matrix is a diagonal matrix, which only contains diagonal components, while all non-diagonal components are zeros. Because the motion decoupling capability of 3L-CPMs is essential in the precision engineering field, this paper presents the fundamental criteria for designing 3L-CPMs with fully decoupled motions, regardless of degrees-of-freedom and the types of flexure element. The 6 × 6 stiffness matrix of a general 3L-CPM is derived based on the orientation of each flexure element, e.g., thin/slender beam and notch hinge, etc., and its relative position to the moving platform. Based on an analytical solution, several requirements for the flexure elements were identified and needed to be satisfied in order to design a 3L-CPM with a diagonal stiffness/compliance matrix. In addition, the developed design criteria were used to analyze the decoupled-motion capability of some existing 3L-CPM designs and shown to provide insight into the motion characteristics of any 3L-CPM. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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17 pages, 11882 KiB  
Article
Adaptive Levenberg–Marquardt Algorithm: A New Optimization Strategy for Levenberg–Marquardt Neural Networks
by Zhiqi Yan, Shisheng Zhong, Lin Lin and Zhiquan Cui
Mathematics 2021, 9(17), 2176; https://doi.org/10.3390/math9172176 - 6 Sep 2021
Cited by 24 | Viewed by 4365
Abstract
Engineering data are often highly nonlinear and contain high-frequency noise, so the Levenberg–Marquardt (LM) algorithm may not converge when a neural network optimized by the algorithm is trained with engineering data. In this work, we analyzed the reasons for the LM neural network’s [...] Read more.
Engineering data are often highly nonlinear and contain high-frequency noise, so the Levenberg–Marquardt (LM) algorithm may not converge when a neural network optimized by the algorithm is trained with engineering data. In this work, we analyzed the reasons for the LM neural network’s poor convergence commonly associated with the LM algorithm. Specifically, the effects of different activation functions such as Sigmoid, Tanh, Rectified Linear Unit (RELU) and Parametric Rectified Linear Unit (PRLU) were evaluated on the general performance of LM neural networks, and special values of LM neural network parameters were found that could make the LM algorithm converge poorly. We proposed an adaptive LM (AdaLM) algorithm to solve the problem of the LM algorithm. The algorithm coordinates the descent direction and the descent step by the iteration number, which can prevent falling into the local minimum value and avoid the influence of the parameter state of LM neural networks. We compared the AdaLM algorithm with the traditional LM algorithm and its variants in terms of accuracy and speed in the context of testing common datasets and aero-engine data, and the results verified the effectiveness of the AdaLM algorithm. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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26 pages, 2364 KiB  
Article
Path Analysis for Hybrid Rigid–Flexible Mechanisms
by Oscar Altuzarra, David Manuel Solanillas, Enrique Amezua and Victor Petuya
Mathematics 2021, 9(16), 1869; https://doi.org/10.3390/math9161869 - 6 Aug 2021
Cited by 4 | Viewed by 2243
Abstract
Hybrid rigid–flexible mechanisms are a type of compliant mechanism that combines rigid and flexible elements, being that their mobility is due to rigid-body joints and the relative flexibility of bendable rods. Two of the modeling methods of flexible rods are the Cosserat rod [...] Read more.
Hybrid rigid–flexible mechanisms are a type of compliant mechanism that combines rigid and flexible elements, being that their mobility is due to rigid-body joints and the relative flexibility of bendable rods. Two of the modeling methods of flexible rods are the Cosserat rod model and its simplification, the Kirchhoff rod model. Both of them present a system of differential equations that must be solved in conjunction with the boundary constraints of the rod, leading to a boundary value problem (BVP). In this work, two methods to solve this BVP are applied to analyze the influence of external loads in the movement of hybrid compliant mechanisms. First, a shooting method (SM) is used to integrate directly the shape of the flexible rod and the forces that appear in it. Then, an integration with elliptic integrals (EI) is carried out to solve the workspace of the compliant element, considering its buckling mode. Applying both methods, an algorithm that obtains the locus of all possible trajectories of the mechanism’s coupler point, and detects the buckling mode change, is developed. This algorithm also allows calculating all possible circuits of the mechanism. Thus, the performance of this method within the path analysis of mechanisms is demonstrated. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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17 pages, 4110 KiB  
Article
Hybrid Optimization Based Mathematical Procedure for Dimensional Synthesis of Slider-Crank Linkage
by Alfonso Hernández, Aitor Muñoyerro, Mónica Urízar and Enrique Amezua
Mathematics 2021, 9(13), 1581; https://doi.org/10.3390/math9131581 - 5 Jul 2021
Cited by 6 | Viewed by 2635
Abstract
In this paper, an optimization procedure for path generation synthesis of the slider-crank mechanism will be presented. The proposed approach is based on a hybrid strategy, mixing local and global optimization techniques. Regarding the local optimization scheme, based on the null gradient condition, [...] Read more.
In this paper, an optimization procedure for path generation synthesis of the slider-crank mechanism will be presented. The proposed approach is based on a hybrid strategy, mixing local and global optimization techniques. Regarding the local optimization scheme, based on the null gradient condition, a novel methodology to solve the resulting non-linear equations is developed. The solving procedure consists of decoupling two subsystems of equations which can be solved separately and following an iterative process. In relation to the global technique, a multi-start method based on a genetic algorithm is implemented. The fitness function incorporated in the genetic algorithm will take as arguments the set of dimensional parameters of the slider-crank mechanism. Several illustrative examples will prove the validity of the proposed optimization methodology, in some cases achieving an even better result compared to mechanisms with a higher number of dimensional parameters, such as the four-bar mechanism or the Watt’s mechanism. Full article
(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines)
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