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Optimization of Response of a Dynamic Vibration Absorber Forming Part of the Main System by the fixed-point theory

KSCE JOURNAL OF CIVIL ENGINEERING 2018年第7期22卷 2354-2361页

作者： Kalehsar, Houshyar Eimani Khodaie, Nahmat Univ Mohaghegh Ardabili Dept Civil Engn POB 56199-11367 Ardebil Iran

In the classic form of the Dynamic Vibration Absorber (DVA), a damped DVA is attached to an undamped primary mass, and the main tuning objective is to minimize the dynamic response of the primary mass. Recently proposed structural systems with the selfcontrol ability such as controlled mega-substructure and core tubes with suspension systems use part of the main structure as a dynamic vibration absorber. Because the DVA in the self-control structural systems is a part of the main structure, the control objective of these kinds of systems is dual not only to limit the motion of the primary system but also to reduce the vibration of the DVA. In the present study, the simplified classic model of the DVA with the primary mass under the harmonic excitation is studied and the fixed-point theory is used to derive the optimum parameters of the DVA to minimize the maximum displacement and acceleration responses of the DVA. The effects of different design parameters such as the mass and stiffness ratios of the DVA on the response of the system are investigated. It is shown that the coordinates of one of the fixed points in both displacement and acceleration responses of the DVA are independent of the natural frequency and damping ratios. This point plays an important role in the optimization procedure. For a specific range of the mass ratios of the DVA, the maximum response of the DVA is larger than that of the primary mass;therefore, optimization based on the response of the DVA yields a better result for the self-control structural system.

关键词： dynamic vibration absorber self-control systems fixed-point theory mega-substructure optimization

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Closed-form solutions to the optimization of dynamic vibration absorber attached to multi-degrees-of-freedom damped linear systems under torsional excitation using the fixed-point theory

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PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART K-JOURNAL OF MULTI-BODY DYNAMICS 2018年第2期232卷 237-252页

作者： Xuan-Truong Vu Duy-Chinh Nguyen Doan-Dien Khong Van-Canh Tong Hung Yen Univ Technol & Educ Dept Mech Engn Hung Yen Vietnam Vietnam Acad Sci & Technol Grad Univ Sci & Technol Hanoi Vietnam

Traditional design optimization of the Voigt-type dynamic vibration absorber often solved for the vertical or lateral vibration problems. However, for the damped primary system under torsional excitation, to the best of our knowledge, there is no study to solve this problem by algebraic approaches. This paper presents the analytical solutions to the optimization of dynamic vibration absorber, which is used to suppress torsional vibration of multi-degrees-of-freedom damped linear systems. The parameters considered in optimizing are dimensionless natural frequency of dynamic vibration absorber and viscous damping of absorber. First, the system equations of motion for shaft-dynamic vibration absorber system subjected to time-varying torsional moment were established. Then, closed-form formulae of optimized parameters were derived using the fixed-point theory. The obtained formulae provide exact solution for the proposed problem. To confirm the effectiveness of the obtained formulae, parametric studies on torsional vibration were performed for several sample multi-degrees-of-freedom systems with and without optimal dynamic vibration absorber. Numerical results showed that torsional vibrations of the primary system attached with optimal dynamic vibration absorber are effectively suppressed, even in the resonant conditions.

关键词： Closed-form solution damped linear system dynamic vibration absorber fixed-point theory torsional vibration suppression

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Non-stationary versions of fixed-point theory, with applications to fractals and subdivision

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JOURNAL OF fixed point theory AND APPLICATIONS 2019年第1期21卷 26-26页

作者： Levin, David Dyn, Nira Veedu, Viswanathan Puthan Tel Aviv Univ Sch Math Sci IL-6997801 Tel Aviv Israel Indian Inst Technol Delhi Dept Math New Delhi 110016 India

Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link fractals generated by IFSs to limits generated by subdivision schemes. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this paper investigates trajectories of maps defined by function systems which are considered as generalizations of the traditional IFS. The significance of 'forward' and 'backward' trajectories of general sequences of maps is studied. The convergence properties of these trajectories constitute a non-stationary version of the classical fixed-point theory. Unlike the ordinary fractals which are self-similar at different scales, the attractors of trajectories of maps defined by function systems may have different structures at different scales.

关键词： Fractals subdivision schemes fixed-point theory attractors

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Simulation of the fractional HIV/AIDS epidemic model using a novel predictor-corrector technique based on singular and non-singular kernels

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年 1-27页

作者： Garg, Rahul Prakash, Amit Natl Inst Technol Kurukshetra Dept Math Kurukshetra 136119 India

This study analyze the fractional HIV/AIDS epidemic model that incorporates an antiretroviral treatment compartment using Caputo and Caputo-Fabrizio fractional derivatives. Banach's fixed-point approach is used to construct the existence theory for the solution of the model. A novel predictor-corrector method is proposed for numerical simulation of the model. This predictor-corrector method is a modified form of the fractional Adams-Bashforth method, formulated in terms of Caputo and Caputo-Fabrizio derivatives. The error estimate of the proposed method is presented. This numerical approach can be easily computed using the programming language MATLAB, and simulation results for distinct values of the fractional order are presented both graphically and in tabular form.

关键词： HIV/AIDS model Caputo derivative Caputo Fabrizio (CF) derivative Predictor-corrector technique fixed-point theory

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Results concerning the optimal control for second-order Sobolev-type stochastic integrodifferential equations with impulses and nonlocal conditions

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年第2期71卷 2779-2802页

作者： Johnson, M. Vijayakumar, V. Kwon, Kiwoon Dongguk Univ Seoul Dept Math Seoul 04620 South Korea Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India

This work explores the optimal control outcomes for second-order Sobolev-type stochastic integrodifferential systems involving impulses and nonlocal conditions. We verify the existence of the mild solutions for the considered system through stochastic analysis and the fixed-point approach involving the cosine family. Additionally, we address the Lagrange problem to confirm the existence of optimal controls. A relevant case study illustrates the key findings of our research.

关键词： fixed-point theory Mild solution Cosine family of operators Stochastic differential systems Optimal controls

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Studies on Fractional Differential Equations With Functional Boundary Condition by Inverse Operators

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第0期

作者： Li, Chenkuan Brandon Univ Dept Math & Comp Sci Brandon MB Canada

Fractional differential equations (FDEs) generalize classical integer-order calculus to noninteger orders, enabling the modeling of complex phenomena that classical equations cannot fully capture. Their study has become essential across science, engineering, and mathematics due to their unique ability to describe systems with nonlocal interactions. In this paper, we study the uniqueness, existence, and stability for a new nonlinear FDE with functional boundary condition (which describes nonlocal properties) based on several well-known fixed-point theorems, the two-parameter Mittag-Leffler function, and an implicit integral equation obtained from inverse operators. Several examples are presented to demonstrate applications of our key theorems. Furthermore, we also indicate that the method used can deal with PDEs, with various initial or boundary conditions.

关键词： fixed-point theory fractional differential equation inverse operator Mittag-Leffler function

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Finite-time stability and numerical approximations of fractional neutral delay systems involving proportional Caputo derivative

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第4期48卷 4881-4892页

作者： Ben Makhlouf, Abdellatif Nagy, A. M. Univ Sfax Fac Sci Sfax Dept Math Sfax Tunisia Kuwait Univ Coll Sci Dept Math Safat Kuwait Benha Univ Fac Sci Dept Math Banha Egypt

This paper deals with a class of fractional neutral delay systems involving proportional Caputo derivative. Maintaining the finite-time stability of fractional-order systems is a major challenge, as their capacity to mimic complex dynamics draws more attention to them. Therefore, the paper presents a novel finite-time stability criterion based on the Banach fixed-point theorem. A decomposition formula for proportional Caputo derivative is provided. This formula allows us to derive a new numerical technique for efficiently solving the proposed problem. Finally, extensive numerical results are performed to illustrate and validate the proposed theoretical results.

关键词： finite-time stability fixed-point theory numerical methods time-delay systems

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Solving generalized nonlinear functional integral equations with applications to epidemic models

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第2期48卷 2318-2337页

作者： Halder, Sukanta Vandana Deepmala PDPM Indian Inst Informat Technol Design & Mfg Math Discipline Jabalpur 482005 Madhya Pradesh India Indian Inst Management Sirmaur Decis Sci Area Sirmaur Himachal Prades India

In this article, we investigate the existence and uniqueness of solutions to a generalized nonlinear functional integral equation (G-NLFIE) associated with certain epidemic models of infectious diseases, defined within the Banach space C[0,1]. Our existence results include several specific cases of nonlinear functional integral equations that commonly occur in nonlinear sciences. We then introduce an iterative algorithm that combines Adomian's decomposition method (ADM) with the modified homotopy perturbation method (mHPM) to approximate solutions to the G-NLFIE. The paper addresses the convergence properties and error analysis of this method. Finally, we present numerical examples to demonstrate the effectiveness and efficiency of our proposed approach.

关键词： Adomian's decomposition method convergence analysis error analysis fixed-point theory homotopy perturbation method nonlinear integral equations

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Results on the existence, uniqueness, and controllability of neutral fractional differential equations in the sense of Atangana-Baleanu derivative

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JOURNAL OF ANALYSIS 2025年第1期33卷 105-120页

作者： Kalamani, Palaniyappan Raj, A. Stephan Antony Kumar, Pushpendra Adithya Inst Technol Dept Math Coimbatore 641107 Tamil Nadu India Rathinam Grp Inst Dept Math Kovilpalayam Campus Coimbatore 641107 Tamil Nadu India Univ Johannesburg Inst Future Knowledge POB 524Auckland Pk ZA-2006 Johannesburg South Africa

In this manuscript, we study multi-derivative non-linear fractional neutral differential equations with non-local conditions using the Atangana-Baleanu (AB) fractional derivative. We derive some novel analyses of the existence, uniqueness and controllability of the solution. Fundamentals of the fixed-point theory are used to achieve the desired results. Finally, an illustrative example is given to verify the theoretical results.

关键词： Atangana-Baleanu derivative Fractional neutral differential equations Existence and uniqueness Controllability fixed-point theory

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Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 520卷

作者： Caucao, Sergio Gatica, Gabriel N. Medrado, Saulo R. Sobral, Yuri D. Univ Catolica Santisima Concepcion GIANuC 2 Casilla 297 Concepcion Chile Univ Catolica Santisima Concepcion Dept Matemat & Fis Aplicadas Casilla 297 Concepcion Chile Univ Concepcion CI2MA Casilla 160-C Concepcion Chile Univ Concepcion Dept Ingn Matemat Casilla 160-C Concepcion Chile Univ Brasilia Dept Matemat Campus Univ Darcy Ribeiro BR-70910900 Brasilia DF Brazil

We propose and analyze new mixed finite element methods for a regularized mu(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order l >= 0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree <= l + n (resp. <= l + 1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

关键词： Granular flows Nonlinear viscosity Twofold saddle point Mixed finite elements fixed-point theory A priori error analysis

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