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20 pages, 3672 KiB

Open AccessArticle

Vibration of a Flexible Follower in a Cam Mechanism with Time-Dependent Boundary Effect

by Jer-Rong Chang

Axioms 2023, 12(2), 177; https://doi.org/10.3390/axioms12020177 - 08 Feb 2023

Viewed by 1396

A vibration analysis of a flexible follower in an oscillating follower cam system undergoing a rise-dwell-fall-dwell (RDFD) motion is performed. Owing to the time-dependent boundary effect caused by considering simultaneously the axial and the lateral displacements of the follower, two geometric constraints are [...] Read more.

A vibration analysis of a flexible follower in an oscillating follower cam system undergoing a rise-dwell-fall-dwell (RDFD) motion is performed. Owing to the time-dependent boundary effect caused by considering simultaneously the axial and the lateral displacements of the follower, two geometric constraints are formulated and added to Hamilton’s principle to establish the vibration equation of the motion of the follower. The coupled axial and lateral vibration of the flexible follower has been studied for the first time. The fast Fourier transform (FFT) spectrum generated from the time history is used for parametric studies. The numerical results of the present study show some new findings. The major spectral peaks for the lateral follower response locate at the low frequencies of 1 Ω, 3 Ω, 5 Ω, and 7 Ω and the high frequency near the fundamental natural frequency where Ω is the cam speed. The largest peak locates mostly at the frequency of 3 Ω. For the ascending and descending motions of the follower RDFD motion, three types of cam profiles are designed. Important new results are found: although the three cam profiles nearly overlap, the vibration results of the follower are quite different. By using a modified sinusoidal acceleration motion, the magnitude of the main lateral peak at low frequencies is minimized. The lateral peak amplitude near the fundamental natural frequency of the follower is the smallest when the cycloid displacement motion is adopted. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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13 pages, 1655 KiB

Open AccessArticle

A Very Simple Procedure for Constructing the Solution of Nonlinear Elliptic PDEs (with Nonlinear Boundary Conditions) Arising from Heat Transfer Problems in Solids

by Rogério Martins Saldanha da Gama

Axioms 2022, 11(7), 311; https://doi.org/10.3390/axioms11070311 - 26 Jun 2022

Cited by 1 | Viewed by 1276

Abstract

This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, [...] Read more.

This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, the partial differential equations which govern the steady-state heat transfer process in bodies with temperature-dependent thermal conductivity and temperature-dependent internal heat source. In addition, it will be considered a large class of nonlinear boundary conditions, especially those arising from the description of the heat exchange process from/to bodies at high temperature levels. Proofs of the solution’s existence and uniqueness are presented. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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21 pages, 2066 KiB

Open AccessArticle

Dynamical Analysis of a Modified Epidemic Model with Saturated Incidence Rate and Incomplete Treatment

by Lazarus Kalvein Beay and Nursanti Anggriani

Axioms 2022, 11(6), 256; https://doi.org/10.3390/axioms11060256 - 27 May 2022

Cited by 1 | Viewed by 2219

Abstract

This paper addresses a modified epidemic model with saturated incidence and incomplete treatment. The existence of all equilibrium points is analyzed. A reproduction number

R_{0}

is determined. Next, it is found that the non-endemic point

P_{0}

is stable in case [...] Read more.

This paper addresses a modified epidemic model with saturated incidence and incomplete treatment. The existence of all equilibrium points is analyzed. A reproduction number

R_{0}

is determined. Next, it is found that the non-endemic point

P_{0}

is stable in case

R_{0} < 1

, but unstable in case

R_{0} > 1

. The special conditions to analyze the local and global stability of the non-endemic and endemic points are investigated. Globally, the sensitivity analysis of the system is studied by combining the Latin Hypercube Sampling and Partial Rating Correlation Coefficients methods. By using the Pontryagins maximum principle, the optimal control problem is studied. Various numerical results are given to support our analysis. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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16 pages, 2635 KiB

Open AccessArticle

Exact Solutions of Nonlinear Partial Differential Equations via the New Double Integral Transform Combined with Iterative Method

by Shams A. Ahmed, Ahmad Qazza and Rania Saadeh

Axioms 2022, 11(6), 247; https://doi.org/10.3390/axioms11060247 - 25 May 2022

Cited by 30 | Viewed by 3310

Abstract

This article demonstrates how the new Double Laplace–Sumudu transform (DLST) is successfully implemented in combination with the iterative method to obtain the exact solutions of nonlinear partial differential equations (NLPDEs) by considering specified conditions. The solutions of nonlinear terms of these equations were [...] Read more.

This article demonstrates how the new Double Laplace–Sumudu transform (DLST) is successfully implemented in combination with the iterative method to obtain the exact solutions of nonlinear partial differential equations (NLPDEs) by considering specified conditions. The solutions of nonlinear terms of these equations were determined by using the successive iterative procedure. The proposed technique has the advantage of generating exact solutions, and it is easy to apply analytically on the given problems. In addition, the theorems handling the mode properties of the DLST have been proved. To prove the usability and effectiveness of this method, examples have been given. The results show that the presented method holds promise for solving other types of NLPDEs. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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13 pages, 3634 KiB

Open AccessArticle

Investigation into the Explicit Solutions of the Integrable (2+1)—Dimensional Maccari System via the Variational Approach

by Kang-Jia Wang and Jing Si

Axioms 2022, 11(5), 234; https://doi.org/10.3390/axioms11050234 - 18 May 2022

Cited by 26 | Viewed by 1861

Abstract

In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct [...] Read more.

In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct the explicit solutions via the stationary conditions only taking two steps. Finally, the dynamic behaviors of the solutions are exhibited by choosing the appropriate parameters through the 3-D and density plots. It can be seen that the proposed method is concise and straightforward, and can be adopted to study the travelling wave theory in physics. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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8 pages, 235 KiB

Open AccessArticle

A New Equilibrium Version of Ekeland’s Variational Principle and Its Applications

by Yuqiang Feng, Juntao Xie and Bo Wu

Axioms 2022, 11(2), 68; https://doi.org/10.3390/axioms11020068 - 09 Feb 2022

Viewed by 1689

Abstract

In this note, a new equilibrium version of Ekeland’s variational principle is presented. It is a modification and promotion of previous results. Subsequently, the principle is applied to discuss the equilibrium points for binary functions and the fixed points for nonlinear mappings. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

16 pages, 277 KiB

Open AccessArticle

A Straightforward Sufficiency Proof for a Nonparametric Problem of Bolza in the Calculus of Variations

by Gerardo Sánchez Licea

Axioms 2022, 11(2), 55; https://doi.org/10.3390/axioms11020055 - 29 Jan 2022

Cited by 1 | Viewed by 1685

Abstract

We study a variable end-points calculus of variations problem of Bolza containing inequality and equality constraints. The proof of the principal theorem of the paper has a direct nature since it is independent of some classical sufficiency approaches invoking the Hamiltonian-Jacobi theory, Riccati [...] Read more.

We study a variable end-points calculus of variations problem of Bolza containing inequality and equality constraints. The proof of the principal theorem of the paper has a direct nature since it is independent of some classical sufficiency approaches invoking the Hamiltonian-Jacobi theory, Riccati equations, fields of extremals or the theory of conjugate points. In contrast, the algorithm employed to prove the principal theorem of the article is based on elementary tools of the real analysis. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

17 pages, 311 KiB

Open AccessArticle

On Constrained Set-Valued Semi-Infinite Programming Problems with ρ-Cone Arcwise Connectedness

by Koushik Das and Savin Treanţă

Axioms 2021, 10(4), 302; https://doi.org/10.3390/axioms10040302 - 12 Nov 2021

Cited by 3 | Viewed by 1092

Abstract

In this paper, we establish sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued semi-infinite programming problem (SP) via the notion of contingent epiderivative of set-valued maps. We also derive duality results of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types of the [...] Read more.

In this paper, we establish sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued semi-infinite programming problem (SP) via the notion of contingent epiderivative of set-valued maps. We also derive duality results of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types of the problem (SP) under

ρ

-cone arcwise connectedness assumptions. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

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