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Applied Mathematical Methods in Mechanical Engineering

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

Deadline for manuscript submissions: closed (31 May 2020) | Viewed by 30929

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Special Issue Editors

Prof. Dr. Natalie Baddour

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Guest Editor

Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, ON K1N 6N5, Canada
Interests: wearable sensors; mHealth; signal processing; mobility
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Victor Eremeyev

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Co-Guest Editor

Department of Mechanics of Materials and Structures, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, Gdansk, Poland
Interests: mechanics; continuum mechanics; applied mathematics; nonlinear elasticity; solid mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The importance of modelling and simulations in mechanical engineering cannot be understated. Mathematical models and the corresponding simulations can be used to predict outcomes, develop an understanding of complex systems, conduct parametrical experiments quickly, and cost-effectively understand the implications of design decisions. Mathematical approaches can also be used for inverse problems that seek to find answers to the opposite question: if we want a particular behavior, how can we produce it?

The purpose of this Special Issue is to gather a collection of articles reflecting the latest developments in the application of mathematical methods to problems in mechanical engineering.

Prof. Dr. Natalie Baddour
Prof. Dr. Victor Eremeyev
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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

Mechanical engineering;
Mathematical models;
Simulations;
Mathematical approaches;
Inverse problems

Published Papers (11 papers)

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Research

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15 pages, 1243 KiB

Open AccessArticle

Coordination in Coupled Arrays of Stiff Filaments—Modelling and Simulation

by Davide Spinello

Mathematics 2020, 8(8), 1282; https://doi.org/10.3390/math8081282 - 03 Aug 2020

Viewed by 1670

Abstract

We present the mechanical model of an array of elastic filaments and simulate the response with different mechanical couplings. This class of systems is inspired by robust and elegant solutions for locomotion mechanics that have emerged in several small-scale biological entities in the form of beating protrusions, such as cellular cilia and eukaryotic flagella. The collective dynamics of cilia arrays reveals important features such as array alignments, two-phase asymmetric beating of individual filaments, and the emergence of metachronal coordination, which make them suitable for bio-inspired terrestrial and aquatic locomotion. The model presented here is the basis for further developments towards the design of terrestrial and aquatic locomotion systems for general purpose robotic devices. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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14 pages, 5492 KiB

Open AccessArticle

Bending Analysis of Functionally Graded Nanoscale Plates by Using Nonlocal Mixed Variational Formula

by Ashraf M. Zenkour, Zahra S. Hafed and Ahmed F. Radwan

Mathematics 2020, 8(7), 1162; https://doi.org/10.3390/math8071162 - 15 Jul 2020

Cited by 16 | Viewed by 1829

Abstract

This work is devoted to the bending analysis of functionally graded (FG) nano-scale plate by using the nonlocal mixed variational formula under simply supported edge conditions. According to Eringen’s nonlocal elasticity theory, the mixed formula is utilized in order to obtain the governing equations. The system of equations is derived by using the principle of virtual work. The governing equations include both the small and the mechanical effects. The impact of the small-scale parameter, aspect and thickness nano-scale plate ratios, and gradient index on the displacement and stresses are explored, numerically presented, and discussed in detail. Different comparisons are made to check the precision and validity of the bending outcomes obtained from the present analysis of FG nano-scale plates. Parametric examinations are then performed to inspect the impacts of the thickness of the plate on the by and large mechanical reaction of the practically evaluated plates. The displayed outcomes are valuable for the configuration procedures of keen structures and examination from materials. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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16 pages, 5300 KiB

Open AccessArticle

Parameter Identification in Nonlinear Mechanical Systems with Noisy Partial State Measurement Using PID-Controller Penalty Functions

by R. Manikantan, Sayan Chakraborty, Thomas K. Uchida and C. P. Vyasarayani

Mathematics 2020, 8(7), 1084; https://doi.org/10.3390/math8071084 - 03 Jul 2020

Cited by 9 | Viewed by 3426

Abstract

Dynamic models of physical systems often contain parameters that must be estimated from experimental data. In this work, we consider the identification of parameters in nonlinear mechanical systems given noisy measurements of only some states. The resulting nonlinear optimization problem can be solved efficiently with a gradient-based optimizer, but convergence to a local optimum rather than the global optimum is common. We augment the dynamic equations with a morphing parameter and a proportional–integral–derivative (PID) controller to transform the objective function into a convex function; the global optimum can then be found using a gradient-based optimizer. The morphing parameter is used to gradually remove the PID controller in a sequence of steps, ultimately returning the model to its original form. An optimization problem is solved at each step, using the solution from the previous step as the initial guess. This strategy enables use of a gradient-based optimizer while avoiding convergence to a local optimum. The efficacy of the proposed approach is demonstrated by identifying parameters in the van der Pol–Duffing oscillator, a hydraulic engine mount system, and a magnetorheological damper system. Our method outperforms genetic algorithm and particle swarm optimization strategies, and demonstrates robustness to measurement noise. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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14 pages, 6871 KiB

Open AccessArticle

Simplified Analytical Solution of the Contact Problem on Indentation of a Coated Half-Space by a Conical Punch

by Andrey S. Vasiliev, Sergey S. Volkov, Evgeniy V. Sadyrin and Sergei M. Aizikovich

Mathematics 2020, 8(6), 983; https://doi.org/10.3390/math8060983 - 16 Jun 2020

Cited by 14 | Viewed by 2425

Abstract

The contact problem on indentation of an elastic coated half-space by a conical punch is considered. To obtain an explicit analytical solution suitable for applications, the bilateral asymptotic method is used in a simplified form. For that purpose, kernel transform of the integral equation is approximated by a ratio of two quadratic functions containing only one parameter. Such an approach allows us to obtain explicit analytical expressions for the distribution of contact stresses and relations between the indentation force, depth, stiffness and contact radius. The obtained solution is suitable both for homogeneous and functionally graded coatings. The dependence of the characteristics of contact interaction on a relative Young’s modulus of the coating and relative coating thickness is analyzed and illustrated by the numerical examples. Ranges of values of elastic and geometrical parameters are obtained, for which the presence of a coating sufficiently changes the contact characteristics. The accuracy of the obtained simplified expressions is studied in detail. Results of the paper sufficiently simplify engineering calculations and are suitable for inverse analysis, e.g., analysis of indentation experiments of coated materials using either a conical or a pyramidal (Berkovich) indenter. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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22 pages, 6651 KiB

Open AccessArticle

Application of the Combined ANN and GA for Multi-Response Optimization of Cutting Parameters for the Turning of Glass Fiber-Reinforced Polymer Composites

by Azhar Equbal, Mohammad Shamim, Irfan Anjum Badruddin, Md. Israr Equbal, Anoop Kumar Sood, Nik Nazri Nik Ghazali and Zahid A. Khan

Mathematics 2020, 8(6), 947; https://doi.org/10.3390/math8060947 - 09 Jun 2020

Cited by 13 | Viewed by 2782

Abstract

Glass fiber-reinforced polymer (GFRP) composites find wide applications in automobile, aerospace, aircraft and marine industries due to their attractive properties such as lightness of weight, high strength-to-weight ratio, high stiffness, good dimensional stability and corrosion resistance. Although these materials are required in a wide range of applications, their non-homogeneous and anisotropic properties make their machining troublesome and consequently restrict their use. It is thus important to study not only the machinability of these materials but also to determine optimum cutting parameters to achieve optimum machining performance. The present work focuses on turning of the GFRP composites with an aim to determine the optimal cutting parameters that yield the optimum output responses. The effect of three cutting parameters, i.e., spindle rotational speed (N), feed rate (f) and depth of cut (d) in conjunction with their interactions on three output responses, viz., Material Removal Rate (MRR), Tool Wear Rate (TWR), and Surface roughness (R_a), is studied using full factorial design of experiments (FFDE). The statistical significance of the cutting parameters and their interactions is determined using analysis of variance (ANOVA). To relate the output response and cutting parameters, empirical models are also developed. Artificial Neural Network (ANN) combined with Genetic Algorithm (GA) is employed for multi-response optimization to simultaneously optimize the MRR, TWR and R_a. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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21 pages, 904 KiB

Open AccessArticle

General Homogenised Formulation for Thick Viscoelastic Layered Structures for Finite Element Applications

by Ondiz Zarraga, Imanol Sarría, Jon García-Barruetabeña, María Jesús Elejabarrieta and Fernando Cortés

Mathematics 2020, 8(5), 714; https://doi.org/10.3390/math8050714 - 03 May 2020

Cited by 8 | Viewed by 2471

Abstract

Viscoelastic layered surface treatments are widely used for passive control of vibration and noise, especially in passenger vehicles and buildings. When the viscoelastic layer is thick, the structural models must account for shear effects. In this work, a homogenised formulation for thick N-layered viscoelastic structures for finite element applications is presented, which allows for avoiding computationally expensive models based on solids. This is achieved by substituting the flexural stiffness in the governing thin beam or plate equation by a frequency dependent equivalent flexural stiffness that takes shear and the properties of the different layers into account. The formulation is applied to Free Layer Damping (FLD) and Constrained Layer Damping (CLD) beams and plates and its ability to accurately compute the eigenpairs and dynamic response is tested by implementing it in a finite element model and comparing the obtained results to those given by the standard for the application—Oberst for the FLD case and RKU for the CLD one—and to a solid model, which is used as reference. For the cases studied, the homogenised formulation is nearly as precise as the model based on solids, but requires less computational effort, and provides better results than the standard model. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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19 pages, 695 KiB

Open AccessArticle

Characterization of the Functionally Graded Shear Modulus of a Half-Space

by Vladimir B. Zelentsov, Polina A. Lapina, Boris I. Mitrin and Victor A. Eremeyev

Mathematics 2020, 8(4), 640; https://doi.org/10.3390/math8040640 - 21 Apr 2020

Cited by 11 | Viewed by 2527

Abstract

In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of “approximate-homogeneous” state in the functionally graded half-space are developed. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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11 pages, 1208 KiB

Open AccessArticle

Photo-Thermal Interactions in a Semiconducting Media with a Spherical Cavity under Hyperbolic Two-Temperature Model

by Faris S. Alzahrani and Ibrahim A. Abbas

Mathematics 2020, 8(4), 585; https://doi.org/10.3390/math8040585 - 15 Apr 2020

Cited by 29 | Viewed by 2405

Abstract

This article highlights the study of photo-thermoelastic interaction in an unbounded semiconductor medium containing a spherical cavity. This problem is solved using the new hyperbolic two-temperature model. The bounding surface of the cavity is traction free and loaded thermally by exponentially decaying pulse boundary heat flux. In addition, the carrier density is prescribed on the inner surface of the cavity in terms of the recombination speed. The techniques of Laplace transforms are used to get the analytical solutions of the problem in the transformed domain by the eigenvalues method. The inversions of Laplace transformations have been carried out numerically. The outcomes also display that the analytical schemes can overcome the mathematical problem to analyze this problem. Numerical outcomes for a semiconductor material are performed and demonstrated graphically. According to the numerical results, this new hyperbolic two-temperature model of thermoelasticity offers finite speed of the thermal wave and mechanical wave propagation. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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9 pages, 799 KiB

Open AccessArticle

A Parameter Identification Method for Stewart Manipulator Based on Wavelet Transform

by Chenyang Zhang

Mathematics 2020, 8(2), 257; https://doi.org/10.3390/math8020257 - 15 Feb 2020

Cited by 1 | Viewed by 2707

Abstract

Aiming at inertial and viscous parameter identification for the Stewart manipulator regardless of the influence of Coulomb friction, a simple and effective dynamical parameter identification method based on wavelet transform and joint velocity analysis is proposed in this paper. Compared with previously known identification methods, the advantages of the new approach are that (1) the excitation trajectory is easy to design, and (2) it can not only identify the inertial matrix, but also the viscous matrix accurately regardless of the influence of Coulomb friction. Comparison is made among the identification method proposed in this paper, another identification method proposed previously, and the true value calculated with a formula. The errors from results of different identification methods demonstrate that the method proposed in this paper shows great adaptability and accuracy. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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17 pages, 6213 KiB

Open AccessArticle

Optimising Parameters for Expanded Polystyrene Based Pod Production Using Taguchi Method

by Sheikh Imran Ishrat, Zahid Akhtar Khan, Arshad Noor Siddiquee, Irfan Anjum Badruddin, Ali Algahtani, Shakeel Javaid and Rajan Gupta

Mathematics 2019, 7(9), 847; https://doi.org/10.3390/math7090847 - 14 Sep 2019

Cited by 13 | Viewed by 4348

Abstract

Expanded polystyrene (EPS) is used in the building and construction industry for insulation and under flooring purposes. The objective of the study is to investigate the impact of the application of the total quality management (TQM) technique on the significant parameters of the pod production process in a New Zealand based EPS manufacturing facility. In this work, Taguchi’s L₂₇ orthogonal array (OA) is considered for conducting experiments through three input parameters i.e., weight of untreated beads, batch duration, and temperature is investigated. Based on the results, the analyses are carried out while using statistical approaches, such as analysis of the means (ANOM) and analysis of variance (ANOVA). The results from confirmatory experiment indicate that, at optimal parameters setting (17 kg of untreated bead, 130 s of batch duration and 155 °F of temperature), a reasonably streamlined pod manufacturing process can be achieved for sustainable operations. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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Review

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15 pages, 390 KiB

Open AccessReview

Mathematical Models with Buckling and Contact Phenomena for Elastic Plates: A Review

by Aliki D. Muradova and Georgios E. Stavroulakis

Mathematics 2020, 8(4), 566; https://doi.org/10.3390/math8040566 - 11 Apr 2020

Cited by 3 | Viewed by 3153

Abstract

A review of mathematical models for elastic plates with buckling and contact phenomena is provided. The state of the art in this domain is presented. Buckling effects are discussed on an example of a system of nonlinear partial differential equations, describing large deflections of the plate. Unilateral contact problems with buckling, including models for plates, resting on elastic foundations, and contact models for delaminated composite plates, are formulated. Dynamic nonlinear equations for elastic plates, which possess buckling and contact effects are also presented. Most commonly used boundary and initial conditions are set up. The advantages and disadvantages of analytical, semi-analytical, and numerical techniques for the buckling and contact problems are discussed. The corresponding references are given. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)

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