Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators

Yuan, Jian; Zhang, Youan; Liu, Jingmao; Shi, Bao; Gai, Mingjiu; Yang, Shujie

doi:10.1016/j.jsv.2017.02.050

Cited by 16 publications

(8 citation statements)

References 34 publications

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“…where C eq is the equivalent constant damping coefficient. The issue of replacing the dissipated energy associated to a nonlinear damper by that of an equivalent linear damper when the excitation is periodic has been studied in [21]. This enables to state the equivalence of dissipated energy while providing the possibility of replacing the nonlinear Eq.…”

Section: Equivalent Viscous Dampingmentioning

confidence: 99%

“…This has a considerable numerical advantage in practice when dealing with multiple loading cases as those occurring in the case of high speed train bridges which are periodic of nature. According to [21], the equivalent viscous damping coefficient can be calculated as follows…”

Section: Equivalent Viscous Dampingmentioning

confidence: 99%

“…Lewandowski and Pawlak [20] investigated the fractional derivatives model by using the continuation method to solve the nonlinear eigenvalue problem. To overcome numerical difficulties resulting from nonlinearity, equivalent linear VEDs were introduced through linearization as indicated in [21].…”

Section: Introductionmentioning

confidence: 99%

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Application of viscoelastic dampers for reducing dynamic response of high-speed railway bridges

Tahiri

Khamlichi

Bezzazi

2020

International Review of Applied Sciences and Engineering

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Due to the extensive development of high-speed railway lines which are operating at increasing velocities, the dynamic performance of railway bridges has become an important issue of scientific research. The aim of this study is to investigate the possibility of reducing the vertical acceleration and displacement of pre-stressed reinforced concrete bridges beams by using passive nonlinear viscoelastic dampers to retrofit them. The proposed solution is based on connecting the dampers directly to the abutments and the bottom surface of the bridge deck with an eccentricity between the neutral axis of the bridge and the contact point of the viscoelastic dampers. First, the dampers are modeled through the concept of linearized fractional derivatives to obtain energetic equivalent linear viscoelastic dampers. Optimization of the configuration of these dampers was performed then as function of the orientation angle and the eccentricity. Considering two bridges having different length that were studied in the literature with other systems of damping, it was found that the best orientation angle of dampers is close to 60°. It was found also that, in order to satisfy Eurocode 1 requirements, the total equivalent damping coefficient for the actual damping system is less than half of that required for systems using auxiliary beam to fix dampers, which indicates higher efficiency of the proposed solution.

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Section: Equivalent Viscous Dampingmentioning

confidence: 99%

Section: Equivalent Viscous Dampingmentioning

confidence: 99%

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Application of viscoelastic dampers for reducing dynamic response of high-speed railway bridges

Tahiri

Khamlichi

Bezzazi

2020

International Review of Applied Sciences and Engineering

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“…Since Scott-Blair 6 introduced Riemann-Liouville fractional derivative to construct the constitutive equation of viscoelastic material, by replacing the damping term with fractional derivatives in the linearly damped oscillator equations, the fractional damped oscillator equations are obtained and discussed from numerous perspectives [21][22][23][24][25][26][27] in Riemann-Liouville or Caputo sense. And in Bagley, 7 Bagley pointed out that Caputo and Riemann-Liouville fractional derivatives are identical in describing the linear viscoelastic material just under two minimal restrictions.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness and iterative formula for fractional oscillator equations with delays

Pang

Wei

Song

et al. 2019

Math Methods in App Sciences

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This paper investigates the well-posedness and iterative formula for fractional oscillator equations with two different fractional orders and time delays. By the mathematical induction, a new generalized Grönwall's inequality in terms of a multivariate Mittag-Leffler function is derived with an iteration argument. Subsequently, using the inequality, a generalized Mittag-Leffler estimation of the solution is proposed. Furthermore, the explicit solution and iterative formula, as well as the unique existence, are presented on the basis of the method of steps.Finally, an example is given to illustrate the validity of our main results.

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“…In 2017, the authors proposed some new fractional calculus to model the rheological phenomena [14]. Many research works about the applications of the viscoelastic models in mechanical systems can be found in [33][34][35][36][37][38][39]. For example, the single degree of freedom oscillator with a springpot [39] is in Fig.…”

Section: Introductionmentioning

confidence: 99%

Analytical solution of the generalized Bagley–Torvik equation

Pang

Wei

et al. 2019

Adv Differ Equ

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In this paper, we investigate the generalized Bagley-Torvik equation with the fractional order (0, 2). With a novel max-metric containing a Caputo derivative, the existence and uniqueness of the solution to the initial value problem are derived. We obtain the analytical solutions in terms of the Prabhakar function and the Wiman function, and they expand the well-known results about the general Bagley-Torvik equation. Two examples are presented to illustrate the validity of our main results.

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Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators

Cited by 16 publications

References 34 publications

Application of viscoelastic dampers for reducing dynamic response of high-speed railway bridges

Application of viscoelastic dampers for reducing dynamic response of high-speed railway bridges

Well‐posedness and iterative formula for fractional oscillator equations with delays

Analytical solution of the generalized Bagley–Torvik equation

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