Mathematical Modeling and Study of Nonlinear Dynamic Processes

A special issue of Computation (ISSN 2079-3197). This special issue belongs to the section "Computational Engineering".

Deadline for manuscript submissions: 31 October 2024 | Viewed by 22434

Special Issue Editor


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Guest Editor
Department of Higher Mathematics, Tambov State Technical University, Sovetskaja Str. 106, 392000 Tambov, Russia
Interests: differential equations; numerical methods; chaos

Special Issue Information

Dear Colleagues,

The development of computer technology has opened up new opportunities for the study of nonlinear dynamic processes. For example, using high-precision calculations, one can construct good approximations to unstable cycles contained in attractors of systems with quadratic nonlinearities. Of particular interest now are mathematical models whose equations have a non-smooth or discontinuous right-hand side.

The development of qualitative and numerical methods brings about new ideas about the structure of attractors of dynamical systems. In recent years, the recurrent motions of dynamical systems have been studied in many papers. The classical results of general systems theory were generalized to the non-autonomous case, and the Poincaré recurrences statistics were also studied.

These problems are the focus of this Special Issue. Particular attention is paid to modeling nonlinear dynamic systems with regular and chaotic behavior using modern numerical methods.

Prof. Dr. Alexander Pchelintsev
Guest Editor

Manuscript Submission Information

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Keywords

  • nonlinear dynamics
  • regular and chaotic dynamics
  • mathematical modeling
  • poisson stability
  • recurrent motions
  • lyapunov exponents
  • numerical methods

Published Papers (14 papers)

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Research

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21 pages, 3957 KiB  
Article
Manufacture of Microstructured Optical Fibers: Problem of Optimal Control of Silica Capillary Drawing Process
by Daria Vladimirova, Vladimir Pervadchuk and Yuri Konstantinov
Computation 2024, 12(5), 86; https://doi.org/10.3390/computation12050086 - 23 Apr 2024
Viewed by 596
Abstract
The effective control of any technological process is essential in ensuring high-quality finished products. This is particularly true in manufacturing knowledge-intensive and high-tech products, including microstructured photonic crystal fibers (PCF). This paper addresses the issues of stabilizing the optimal control of the silica [...] Read more.
The effective control of any technological process is essential in ensuring high-quality finished products. This is particularly true in manufacturing knowledge-intensive and high-tech products, including microstructured photonic crystal fibers (PCF). This paper addresses the issues of stabilizing the optimal control of the silica capillary drawing process. The silica capillaries are the main components of PCF. A modified mathematical model proposed by the authors is used as the basic model of capillary drawing. The uniqueness of this model is that it takes into account the main forces acting during drawing (gravity, inertia, viscosity, surface tension, pressure inside the drawn capillary), as well as all types of heat transfer (heat conduction, convection, radiation). In the first stage, the system of partial differential equations describing heat and mass transfer was linearized. Then, the problem of the optimal control of the drawing process was formulated, and optimization systems for the isothermal and non-isothermal cases were obtained. In the isothermal case, optimal adjustments of the drawing speed were obtained for different objective functionals. Thus, the proposed approach allows for the constant monitoring and adjustment of the observed state parameters (for example, the outer radius of the capillary). This is possible due to the optimal control of the drawing speed to obtain high-quality preforms. The ability to control and promptly eliminate geometric defects in the capillary was confirmed by the analysis of the numerical calculations, according to which even 15% deviations in the outer radius of the capillary during the drawing process can be reduced to 4–5% by controlling only the capillary drawing speed. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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36 pages, 970 KiB  
Article
Global Properties of Cytokine-Enhanced HIV-1 Dynamics Model with Adaptive Immunity and Distributed Delays
by Elsayed Dahy, Ahmed M. Elaiw, Aeshah A. Raezah, Hamdy Z. Zidan and Abd Elsattar A. Abdellatif
Computation 2023, 11(11), 217; https://doi.org/10.3390/computation11110217 - 2 Nov 2023
Viewed by 1301
Abstract
In this paper, we study a model that enhances our understanding of cytokine-influenced HIV-1 infection. The impact of adaptive immune response (cytotoxic T lymphocytes (CTLs) and antibodies) and time delay on HIV-1 infection is included. The model takes into account two types of [...] Read more.
In this paper, we study a model that enhances our understanding of cytokine-influenced HIV-1 infection. The impact of adaptive immune response (cytotoxic T lymphocytes (CTLs) and antibodies) and time delay on HIV-1 infection is included. The model takes into account two types of distributional delays, (i) the delay in the HIV-1 infection of CD4+T cells and (ii) the maturation delay of new virions. We first investigated the fundamental characteristics of the system, then found the system’s equilibria. We derived five threshold parameters, i, i = 0, 1,…, 4, which completely determine the existence and stability of the equilibria. The Lyapunov method was used to prove the global asymptotic stability for all equilibria. We illustrate the theoretical results by performing numerical simulations. We also performed a sensitivity analysis on the basic reproduction number 0 and identified the most-sensitive parameters. We found that pyroptosis contributes to the number 0, and then, neglecting it will make 0 underevaluated. Necrosulfonamide and highly active antiretroviral drug therapy (HAART) can be effective in preventing pyroptosis and at reducing viral replication. Further, it was also found that increasing time delays can effectively decrease 0 and, then, inhibit HIV-1 replication. Furthermore, it is shown that both CTLs and antibody immune responses have no effect on 0, while this can result in less HIV-1 infection. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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22 pages, 2979 KiB  
Article
Time-Dependent Two-Dimensional Model of Overlimiting Mass Transfer in Electromembrane Systems Based on the Nernst–Planck, Displacement Current and Navier–Stokes Equations
by Aminat Uzdenova
Computation 2023, 11(10), 205; https://doi.org/10.3390/computation11100205 - 16 Oct 2023
Viewed by 1405
Abstract
Electromembrane processes underlie the functioning of electrodialysis devices and nano- and microfluidic devices, the scope of which is steadily expanding. One of the main aspects that determine the effectiveness of membrane systems is the choice of the optimal electrical mode. The solution of [...] Read more.
Electromembrane processes underlie the functioning of electrodialysis devices and nano- and microfluidic devices, the scope of which is steadily expanding. One of the main aspects that determine the effectiveness of membrane systems is the choice of the optimal electrical mode. The solution of this problem, along with experimental studies, requires tools for the theoretical analysis of ion-transport processes in various electrical modes. The system of Nernst–Planck–Poisson and Navier–Stokes (NPP–NS) equations is widely used to describe the overlimiting mass transfer associated with the development of electroconvection. This paper proposes a new approach to describe the electrical mode in a membrane system using the displacement current equation. The equation for the displacement current makes it possible to simulate the galvanodynamic mode, in which the electric field is determined by the given current density. On the basis of the system of Nernst–Planck, displacement current and Navier–Stokes (NPD–NS) equations, a model of the electroconvective overlimiting mass transfer in the diffusion layer at the surface of the ion-exchange membrane in the DC current mode was constructed. Mathematical models based on the NPP–NS and NPD–NS equations, formulated to describe the same physical situation of mass transfer in the membrane system, differ in the peculiarities of numerical solution. At overlimiting currents, the required accuracy of the numerical solution is achieved in the approach based on the NPP–NS equations with a smaller time step than the NPD–NS equation approach. The accuracy of calculating the current density at the boundaries parallel to the membrane surface is higher for the model based on the NPD–NS equations compared to the model based on the NPP–NS equations. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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12 pages, 1361 KiB  
Article
Diffusion Kinetics Theory of Removal of Assemblies’ Surface Deposits with Flushing Oil
by Michael Vigdorowitsch, Valery V. Ostrikov, Alexander N. Pchelintsev and Irina Yu. Pchelintseva
Computation 2023, 11(8), 164; https://doi.org/10.3390/computation11080164 - 20 Aug 2023
Viewed by 880
Abstract
The diffusion kinetics theory of cleaning assemblies such as combustion engines with flushing oil has been introduced. Evolution of tar deposits on the engine surfaces and in the lube system has been described through the erosion dynamics. The time-dependent concentration pattern related to [...] Read more.
The diffusion kinetics theory of cleaning assemblies such as combustion engines with flushing oil has been introduced. Evolution of tar deposits on the engine surfaces and in the lube system has been described through the erosion dynamics. The time-dependent concentration pattern related to hydrodynamic (sub)layers around the tar deposit has been uncovered. Nonlinear equations explaining the experimentally observed dependences for scouring the contaminants off with the oil have been derived and indicate the power law in time. For reference purposes, a similar analysis based on formal chemical kinetics has been accomplished. Factors and scouring parameters for the favor of either mechanism have been discussed. Any preference for either diffusion or chemical kinetics should be based on a careful selection of washing agents in the flushing oil. Future directions of studies are proposed. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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18 pages, 1128 KiB  
Article
Analysis of the Dynamics of Tuberculosis in Algeria Using a Compartmental VSEIT Model with Evaluation of the Vaccination and Treatment Effects
by Bouchra Chennaf, Mohammed Salah Abdelouahab and René Lozi
Computation 2023, 11(7), 146; https://doi.org/10.3390/computation11070146 - 21 Jul 2023
Cited by 1 | Viewed by 1048
Abstract
Despite low tuberculosis (TB) mortality rates in China, Europe, and the United States, many countries are still struggling to control the epidemic, including India, South Africa, and Algeria. This study aims to contribute to the body of knowledge on this topic and provide [...] Read more.
Despite low tuberculosis (TB) mortality rates in China, Europe, and the United States, many countries are still struggling to control the epidemic, including India, South Africa, and Algeria. This study aims to contribute to the body of knowledge on this topic and provide a valuable tool and evidence-based guidance for the Algerian healthcare managers in understanding the spread of TB and implementing control strategies. For this purpose, a compartmental mathematical model is proposed to analyze TB dynamics in Algeria and investigate the vaccination and treatment effects on disease breaks. A qualitative study is conducted to discuss the stability property of both disease-free equilibrium and endemic equilibrium. In order to adopt the proposed model for the Algerian case, we estimate the model parameters using Algerian TB-reported data from 1990 to 2020. The obtained results using the proposed mathematical compartmental model show that the reproduction number (R0) of TB in Algeria is less than one, suggesting that the disease can be eradicated or effectively controlled through a combination of interventions, including vaccination, high-quality treatment, and isolation measures. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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18 pages, 3337 KiB  
Article
Extended Online DMD and Weighted Modifications for Streaming Data Analysis
by Gyurhan Nedzhibov
Computation 2023, 11(6), 114; https://doi.org/10.3390/computation11060114 - 9 Jun 2023
Cited by 1 | Viewed by 1243
Abstract
We present novel methods for computing the online dynamic mode decomposition (online DMD) for streaming datasets. We propose a framework that allows incremental updates to the DMD operator as data become available. Due to its ability to work on datasets with lower ranks, [...] Read more.
We present novel methods for computing the online dynamic mode decomposition (online DMD) for streaming datasets. We propose a framework that allows incremental updates to the DMD operator as data become available. Due to its ability to work on datasets with lower ranks, the proposed method is more advantageous than existing ones. A noteworthy feature of the method is that it is entirely data-driven and does not require knowledge of any underlying governing equations. Additionally, we present a modified version of our proposed approach that utilizes a weighted alternative to online DMD. The suggested techniques are demonstrated using several numerical examples. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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23 pages, 6080 KiB  
Article
Modeling of Nonlinear Dynamic Processes of Human Movement in Virtual Reality Based on Digital Shadows
by Artem Obukhov, Denis Dedov, Andrey Volkov and Daniil Teselkin
Computation 2023, 11(5), 85; https://doi.org/10.3390/computation11050085 - 23 Apr 2023
Cited by 3 | Viewed by 1686
Abstract
In virtual reality (VR) systems, a problem is the accurate reproduction of the user’s body in a virtual environment using inverse kinematics because existing motion capture systems have a number of drawbacks, and minimizing the number of key tracking points (KTPs) leads to [...] Read more.
In virtual reality (VR) systems, a problem is the accurate reproduction of the user’s body in a virtual environment using inverse kinematics because existing motion capture systems have a number of drawbacks, and minimizing the number of key tracking points (KTPs) leads to a large error. To solve this problem, it is proposed to use the concept of a digital shadow and machine learning technologies to optimize the number of KTPs. A technique for movement process data collecting from a virtual avatar is implemented, modeling of nonlinear dynamic processes of human movement based on a digital shadow is carried out, the problem of optimizing the number of KTP is formulated, and an overview of the applied machine learning algorithms and metrics for their evaluation is given. An experiment on a dataset formed from virtual avatar movements shows the following results: three KTPs do not provide sufficient reconstruction accuracy, the choice of five or seven KTPs is optimal; among the algorithms, the most efficient in descending order are AdaBoostRegressor, LinearRegression, and SGDRegressor. During the reconstruction using AdaBoostRegressor, the maximum deviation is not more than 0.25 m, and the average is not more than 0.10 m. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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16 pages, 480 KiB  
Article
A Design Concept of an Intelligent Onboard Computer Network
by Alexander N. Pchelintsev, Andrey M. Solovyov, Mikhail E. Semenov, Nikolay I. Selvesyuk, Vladislav V. Kosyanchuck and Evgeniy Yu. Zybin
Computation 2023, 11(3), 55; https://doi.org/10.3390/computation11030055 - 8 Mar 2023
Viewed by 1410
Abstract
The article suggests design principles of an advanced onboard computer network with an intelligent control system. It describes the main advantages of designing an onboard computer network based on fibre optics, which allows the implementation of an integrated intellectual system performing intelligent inference [...] Read more.
The article suggests design principles of an advanced onboard computer network with an intelligent control system. It describes the main advantages of designing an onboard computer network based on fibre optics, which allows the implementation of an integrated intellectual system performing intelligent inference in emergency situations. The suggested principles significantly increase the reliability and fault tolerance of avionics suits, which, in turn, enhances flight safety. The suggested concept aims to solve a number of important problems including the design of a switchless computing environment, the development of the methods for dynamic reconfiguration of avionics suits with such an environment, and the implementation of a specialised multilevel intelligent avionics system within this environment. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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34 pages, 3237 KiB  
Article
Discretization and Analysis of HIV-1 and HTLV-I Coinfection Model with Latent Reservoirs
by Ahmed M. Elaiw, Abdualaziz K. Aljahdali and Aatef D. Hobiny
Computation 2023, 11(3), 54; https://doi.org/10.3390/computation11030054 - 7 Mar 2023
Cited by 2 | Viewed by 2285
Abstract
This article formulates and analyzes a discrete-time Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) coinfection model with latent reservoirs. We consider that the HTLV-I infect the CD4+T cells, while HIV-1 has two classes of [...] Read more.
This article formulates and analyzes a discrete-time Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) coinfection model with latent reservoirs. We consider that the HTLV-I infect the CD4+T cells, while HIV-1 has two classes of target cells—CD4+T cells and macrophages. The discrete-time model is obtained by discretizing the original continuous-time by the non-standard finite difference (NSFD) approach. We establish that NSFD maintains the positivity and boundedness of the model’s solutions. We derived four threshold parameters that determine the existence and stability of the four equilibria of the model. The Lyapunov method is used to examine the global stability of all equilibria. The analytical findings are supported via numerical simulation. The impact of latent reservoirs on the HIV-1 and HTLV-I co-dynamics is discussed. We show that incorporating the latent reservoirs into the HIV-1 and HTLV-I coinfection model will reduce the basic HIV-1 single-infection and HTLV-I single-infection reproductive numbers. We establish that neglecting the latent reservoirs will lead to overestimation of the required HIV-1 antiviral drugs. Moreover, we show that lengthening of the latent phase can suppress the progression of viral coinfection. This may draw the attention of scientists and pharmaceutical companies to create new treatments that prolong the latency period. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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21 pages, 7894 KiB  
Article
Application of Orthogonal Functions to Equivalent Linearization Method for MDOF Duffing–Van der Pol Systems under Nonstationary Random Excitations
by Amir Younespour, Hosein Ghaffarzadeh and Shaohong Cheng
Computation 2023, 11(1), 8; https://doi.org/10.3390/computation11010008 - 5 Jan 2023
Viewed by 1442
Abstract
Many mechanical systems manifest nonlinear behavior under nonstationary random excitations. Neglecting this nonlinearity in the modeling of a dynamic system would result in unacceptable results. However, it is challenging to find exact solutions to nonlinear problems. Therefore, equivalent linearization methods are often used [...] Read more.
Many mechanical systems manifest nonlinear behavior under nonstationary random excitations. Neglecting this nonlinearity in the modeling of a dynamic system would result in unacceptable results. However, it is challenging to find exact solutions to nonlinear problems. Therefore, equivalent linearization methods are often used to seek approximate solutions for this kind of problem. To overcome the limitations of the existing equivalent linearization methods, an orthogonal-function-based equivalent linearization method in the time domain is proposed for nonlinear systems subjected to nonstationary random excitations. The proposed method is first applied to a single-degree-of-freedom (SDOF) Duffing–Van der Pol oscillator subjected to stationary and nonstationary excitations to validate its accuracy. Then, its applicability to nonlinear MDOF systems is depicted by a 5DOF Duffing–Van der Pol system subjected to nonstationary excitation, with different levels of system nonlinearity strength considered in the analysis. Results show that the proposed method has the merit of predicting the nonlinear system response with high accuracy and computation efficiency. In addition, it is applicable to any general type of nonstationary random excitation. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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15 pages, 4444 KiB  
Article
Parallelization of Runge–Kutta Methods for Hardware Implementation
by Petr Fedoseev, Konstantin Zhukov, Dmitry Kaplun, Nikita Vybornov and Valery Andreev
Computation 2022, 10(12), 215; https://doi.org/10.3390/computation10120215 - 7 Dec 2022
Viewed by 1869
Abstract
Parallel numerical integration is a valuable tool used in many applications requiring high-performance numerical solvers, which is of great interest nowadays due to the increasing difficulty and complexity in differential problems. One of the possible approaches to increase the efficiency of ODE solvers [...] Read more.
Parallel numerical integration is a valuable tool used in many applications requiring high-performance numerical solvers, which is of great interest nowadays due to the increasing difficulty and complexity in differential problems. One of the possible approaches to increase the efficiency of ODE solvers is to parallelize recurrent numerical methods, making them more suitable for execution in hardware with natural parallelism, e.g., field-programmable gate arrays (FPGAs) or graphical processing units (GPUs). Some of the simplest and most popular ODE solvers are explicit Runge–Kutta methods. Despite the high implementability and overall simplicity of the Runge–Kutta schemes, recurrent algorithms remain weakly suitable for execution in parallel computers. In this paper, we propose an approach for parallelizing classical explicit Runge–Kutta methods to construct efficient ODE solvers with pipeline architecture. A novel technique to obtain parallel finite-difference models based on Runge–Kutta integration is described. Three test initial value problems are considered to evaluate the properties of the obtained solvers. It is shown that the truncation error of the parallelized Runge–Kutta method does not significantly change after its known recurrent version. A possible speed up in calculations is estimated using Amdahl’s law and is approximately 2.5–3-times. Block diagrams of fixed-point parallel ODE solvers suitable for hardware implementation on FPGA are given. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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16 pages, 1068 KiB  
Article
On Alternative Algorithms for Computing Dynamic Mode Decomposition
by Gyurhan Nedzhibov
Computation 2022, 10(12), 210; https://doi.org/10.3390/computation10120210 - 1 Dec 2022
Cited by 1 | Viewed by 2268
Abstract
Dynamic mode decomposition (DMD) is a data-driven, modal decomposition technique that describes spatiotemporal features of high-dimensional dynamic data. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations. The main purpose of this article is [...] Read more.
Dynamic mode decomposition (DMD) is a data-driven, modal decomposition technique that describes spatiotemporal features of high-dimensional dynamic data. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations. The main purpose of this article is to introduce new alternatives to the currently accepted algorithm for calculating the dynamic mode decomposition. We present two new algorithms which are more economical from a computational point of view, which is an advantage when working with large data. With a few illustrative examples, we demonstrate the applicability of the introduced algorithms. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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12 pages, 829 KiB  
Article
Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems
by Călin-Ioan Gheorghiu
Computation 2022, 10(7), 116; https://doi.org/10.3390/computation10070116 - 8 Jul 2022
Cited by 2 | Viewed by 1658
Abstract
The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite [...] Read more.
The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness). Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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Review

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28 pages, 2214 KiB  
Review
A Circuit Theory Perspective on the Modeling and Analysis of Vibration Energy Harvesting Systems: A Review
by Michele Bonnin, Kailing Song, Fabio L. Traversa and Fabrizio Bonani
Computation 2023, 11(3), 45; https://doi.org/10.3390/computation11030045 - 25 Feb 2023
Viewed by 1582
Abstract
This paper reviews advanced modeling and analysis techniques useful in the description, design, and optimization of mechanical energy harvesting systems based on the collection of energy from vibration sources. The added value of the present contribution is to demonstrate the benefits of the [...] Read more.
This paper reviews advanced modeling and analysis techniques useful in the description, design, and optimization of mechanical energy harvesting systems based on the collection of energy from vibration sources. The added value of the present contribution is to demonstrate the benefits of the exploitation of advanced techniques, most often inherited from other fields of physics and engineering, to improve the performance of such systems. The review is focused on the modeling techniques that apply to the entire energy source/mechanical oscillator/transducer/electrical load chain, describing mechanical–electrical analogies to represent the collective behavior as the cascade of equivalent electrical two-ports, introducing matching networks enhancing the energy transfer to the load, and discussing the main numerical techniques in the frequency and time domains that can be used to analyze linear and nonlinear harvesters, both in the case of deterministic and stochastic excitations. Full article
(This article belongs to the Special Issue Mathematical Modeling and Study of Nonlinear Dynamic Processes)
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