Study and control of thermoelastic damping of in-plane vibration of the functionally graded nano-plate

Rahimi, Zaher; Sumelka, Wojciech; Ahmadi, Samrand Rash; Băleanu, Dumitru

doi:10.1177/1077546319861009

Cited by 25 publications

(17 citation statements)

References 36 publications

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“…First, we will present the data simulation u(1, t) and u(2, t) with t > 0 by considering the fractional diffusion equation We recall the formulation (7) u t (x, ) = Ae x(i ) ∕2 + Be −x(i ) ∕2 .…”

Section: Data Simulationmentioning

confidence: 99%

“…Then, we have ||g − g n || 2 = ||h − h n || 2 → 0 as n → +∞. From Equation (7) and choosing x > 2, we obtain…”

Section: Ill-posedness Of the Problemmentioning

confidence: 99%

“…conduction problem arises from many branches of engineering sciences. 6,7 Then the fractional calculus has encountered much success in many fields of science: mathematics, physics, [3][4][5][8][9][10] chemistry, 11 biological systems 12 and so on. 2,13,14 In the present paper, we will consider the fractional diffusion equation D t u(x, t) = u xx (x, t), (x, t) ∈ Ω.…”

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confidence: 99%

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A fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Trong

Nhung

Minh

et al. 2020

Math Methods in App Sciences

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In a lot of engineering applications, one has to recover the temperature of a heat body having a surface that cannot be measured directly. This raises an inverse problem of determining the temperature on the surface of a body using interior measurements, which is called the sideways problem. Many papers investigated the problem (with or without fractional derivatives) by using theor using an interior data u(x 0 , t) and the assumption lim x→∞ u(x, t) = 0. However, the flux u x (x 0 , t) is not easy to measure, and the temperature assumption at infinity is inappropriate for a bounded body. Hence, in the present paper, we consider a fractional sideways problem, in which the interior measurements at two interior point, namely, x = 1 and x = 2, are given by continuous data with deterministic noises and by discrete data contaminated with random noises. We show that the problem is severely ill-posed and further construct an a posteriori optimal truncation regularization method for the deterministic data, and we also construct a nonparametric regularization for the discrete random data. Numerical examples show that the proposed method works well. KEYWORDS fractional diffusion equation, ill-posed problem, inverse problem, nonparametric regression, sideways problem MSC CLASSIFICATION 35J92; 35R06; 46E30; 35R06; 46E30 INTRODUCTIONUntil recent years, the classical partial differential equations (PDEs) are a tool used to model many spatial-time systems in nature (see Kumar et al 1 ). However, the integer-order derivatives are defined locally so the mathematical model does not seem to be quite suitable for reflecting the mutual spatial-influence and time-memory in the systems. To improve this inadequacy, nonlocal (space-fractional or time-fractional) derivatives were used in modeling of natural and engineering systems. Fractional derivatives have been found to be quite flexible in describing viscoelastic, viscoplastic flow and anomalous diffusion in fluid mechanic, quantum mechanic, etc (superdiffusion, subdiffusion), which might be inconsistent with the classical Brownian motion model. [2][3][4][5] Hence, in the last decades, the investigation of fractional heat Math Meth Appl Sci. 2020;43:5314-5338. wileyonlinelibrary.com/journal/mma

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Section: Data Simulationmentioning

confidence: 99%

“…Then, we have ||g − g n || 2 = ||h − h n || 2 → 0 as n → +∞. From Equation (7) and choosing x > 2, we obtain…”

Section: Ill-posedness Of the Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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A fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Trong

Nhung

Minh

et al. 2020

Math Methods in App Sciences

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“…Building structures occasionally suffer from unpredictable earthquakes, strong winds, or other natural hazards that may cause severe damage and threaten human lives. Thus, effective control methods are needed to protect against structural vibration in buildings [1,2]. During the past few decades, a variety of control techniques, including linear quadratic regulator (LQR) [3], sliding-mode [4], neural network [5], fuzzy [6], neural terminal sliding-mode [7], disturbance rejection [8], and proportional-derivative (PD) [9,10] algorithms were analyzed.…”

Section: Introductionmentioning

confidence: 99%

Improved Decentralized Fractional PD Control of Structure Vibrations

Chen

Wang

et al. 2020

Mathematics

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This paper presents a new strategy for the control of large displacements in structures under earthquake excitation. Firstly, an improved fractional order proportional-derivative (FOPD) controller is proposed. Secondly, a decentralized strategy is designed by adding a regulator and fault self-regulation to a standard decentralized controller. A new control architecture is obtained by combining the improved FOPD and the decentralized strategy. The parameters of the control system are tuned using an intelligent optimization algorithm. Simulation results demonstrate the performance and reliability of the proposed method.

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“…Influences of aspect ratio, nonlocal parameter, non-homogeneous index and plate thickness were studied on the responses. Thermoelastic damping in-plane vibrations of FG nanoplates was presented by Rahimi et al 6 using the Eringen nonlocal theory. Galerkin-based method was used to solve nonlocal governing equations of motion.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal vibration analysis of the three-layered FG nanoplates subjected to applied electric potential considering thickness stretching effect

Arefi

Arani

2020

Proceedings of the Institution of Mechanical Engineers, Part L:

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Comprehensive nonlocal piezoelasticity relations are developed in this paper for a sandwich functionally graded nanoplate subjected to applied electric potential based on higher-order shear and normal deformation theory. To account thickness stretching effect, the higher-order shear and normal deformation theory is developed. Based on this theory, the transverse deflection is decomposed into bending, shear and stretching portions in which the third term is reflected variation of transverse deflection along the thickness direction. Size dependency is accounted in governing equations based on nonlocal elasticity theory. The sandwich nanoplate is made of a functionally graded core integrated with two piezoelectric layers. Distribution of material properties are assumed according to the power-law function in the thickness direction. The Hamilton’s principle is used to derive governing equations of motion. Navier’s technique is implemented to solve partial differential equation of motion. Accuracy and efficiency of the presented technique are verified by a comparison between obtained results and existing results in literature for two cases including and excluding thickness stretching effect. The comparison between the results with and without thickness stretching effect can justify necessity of present work. Large parametric analysis is organized to investigate effect of significant parameters such as external applied voltage, nonlocal parameter, non-homogeneous index, stretching effect, length-to-thickness, length-to-width and core-to-face sheet thickness ratios on the vibrational behavior of the system. As an important result of this study, one can conclude that accounting thickness stretching effect leads to decrease of natural frequencies in comparison with cases disregards thickness stretching.

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Study and control of thermoelastic damping of in-plane vibration of the functionally graded nano-plate

Cited by 25 publications

References 36 publications

A fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

A fractional sideways problem in a one‐dimensional finite‐slab with deterministic and random interior perturbed data

Improved Decentralized Fractional PD Control of Structure Vibrations

Nonlocal vibration analysis of the three-layered FG nanoplates subjected to applied electric potential considering thickness stretching effect

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