A numerical approach for solving optimal control problem of fractional order vibration equation of large membranes

Aghchi, Sima; Fazli, Hossein; Sun, HongGunag

doi:10.1016/j.camwa.2024.03.037

Computers & Mathematics with Applications

2024

DOI: 10.1016/j.camwa.2024.03.037

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A numerical approach for solving optimal control problem of fractional order vibration equation of large membranes

Sima Aghchi,

Hossein Fazli,

HongGunag Sun

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Cited by 3 publications

References 33 publications

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New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

Fazli,

Sun

2025

Chaos, Solitons & Fractals

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New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

Fazli,

Sun

2025

Chaos, Solitons & Fractals

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Dealing with Stationary Sinusoidal Responses of Seven Types of Multi-Fractional Vibrators Using Multi-Fractional Phasor

2024

Symmetry

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The novelty and main contributions of this paper are reflected in four aspects. First, we introduce multi-fractional phasor in Theorem 1. Second, we propose the motion phasor equations of seven types of multi-fractional vibrators in Theorems 2, 12, 22, 32, 43, 54, and 65, respectively. Third, we present the analytical expressions of response phasors of seven types of multi-fractional vibrators in Theorems 10, 20, 30, 41, 52, 63, and 74, respectively. Fourth, we bring forward the analytical expressions of stationary sinusoidal responses of seven types of multi-fractional vibrators in Theorems 11, 21, 31, 42, 53, 64, and 75, respectively. In addition, by using multi-fractional phasor, we put forward the analytical expressions of vibration parameters (equivalent mass, equivalent damping, equivalent stiffness, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, equivalent frequency ratio) and frequency transfer functions of seven types of multi-fractional vibrators. Demonstrations exhibit that the effects of multi-fractional orders on stationary sinusoidal responses of those multi-fractional vibrators are considerable.

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Analytic Theory of Seven Classes of Fractional Vibrations Based on Elementary Functions: A Tutorial Review

2024

Symmetry

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This paper conducts a tutorial review of the analytic theory of seven classes of fractional vibrations based on elementary functions. We discuss the classification of seven classes of fractional vibrations and introduce the problem statements. Then, the analytic theory of class VI fractional vibrators is given. The analytic theories of fractional vibrators from class I to class V and class VII are, respectively, represented. Furthermore, seven analytic expressions of frequency bandwidth of seven classes of fractional vibrators are newly introduced in this paper. Four analytic expressions of sinusoidal responses to fractional vibrators from class IV to VII by using elementary functions are also newly reported in this paper. The analytical expressions of responses (free, impulse, step, and sinusoidal) are first reported in this research. We dissert three applications of the analytic theory of fractional vibrations: (1) analytical expression of the forced response to a damped multi-fractional Euler–Bernoulli beam; (2) analytical expressions of power spectrum density (PSD) and cross-PSD responses to seven classes of fractional vibrators under the excitation with the Pierson and Moskowitz spectrum, which are newly introduced in this paper; and (3) a mathematical explanation of the Rayleigh damping assumption.

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A numerical approach for solving optimal control problem of fractional order vibration equation of large membranes

Cited by 3 publications

References 33 publications

New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

Dealing with Stationary Sinusoidal Responses of Seven Types of Multi-Fractional Vibrators Using Multi-Fractional Phasor

Analytic Theory of Seven Classes of Fractional Vibrations Based on Elementary Functions: A Tutorial Review

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