Longitudinal vibrations of rheological rod with variable cross section

Hedrih, Katica; Filipovski, Aleksandar

doi:10.1016/s1007-5704(99)90005-9

Cited by 9 publications

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“…Introducing relation (33) in (18), multiplying by R 0 , the transposed modal matrix, considering (29) and (30) and the relation:…”

Section: Transformation Of Matrix Fo Differential Equationmentioning

confidence: 99%

“…The type of each of these FODE was addressed in [15,23]. The solutions of each FODE (39) along the eigen main FO system coordinate ξ s , s ¼ 1; …; n, are possible in the form of a polynomial expansion along time [29,30]:…”

Section: Solution Of Matrix Fo Differential Equation For a Special Famentioning

confidence: 99%

“…On the basis of the analytical approximation of solution of (40), of the FODE (39) for free FO oscillations along each eigen main FO system coordinate ξ s , s ¼ 1; …; n, is possible to separate two fractional modes ξ s ðtÞ ¼ ξ s ð0ÞT s; cos ðt; αÞþ _ ξ s ð0ÞT s; sin ðt; αÞ "similar" to sine T s; sin ðt; αÞ, or to cosine T s; cos ðt; αÞ, s ¼ 1; 2; …; n, with difference in phase as follows [29,30]: In conclusion we can formulate the following Theorem.…”

Section: Solution Of Matrix Fo Differential Equation For a Special Famentioning

confidence: 99%

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Discrete fractional order system vibrations

Hedrih

Machado

2015

International Journal of Non-Linear Mechanics

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Keyw Q4ords: Fractional order derivative Fractional order element Generalized function of fractional order dissipation of system energy Matrix fractional order differential equation Eigen fractional order mode Fractional order oscillator Mechanical to electrical analogy a b s t r a c t A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage-current is formulated. Also a function of thermal energy FO dissipation of a FO electrical resistor is discussed.

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“…Introducing relation (33) in (18), multiplying by R 0 , the transposed modal matrix, considering (29) and (30) and the relation:…”

Section: Transformation Of Matrix Fo Differential Equationmentioning

confidence: 99%

Section: Solution Of Matrix Fo Differential Equation For a Special Famentioning

confidence: 99%

Section: Solution Of Matrix Fo Differential Equation For a Special Famentioning

confidence: 99%

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Discrete fractional order system vibrations

Hedrih

Machado

2015

International Journal of Non-Linear Mechanics

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“…Some special cases of this problem may be found in [24][25][26][27][28]. They model some phenomena with hereditary properties.…”

Section: (S))(x)dη(s)mentioning

confidence: 99%

Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

Tatar

2011

Adv Differ Equ

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“…We mention here that fractional derivatives appear in many engineering problems. They arise naturally in many fields such as in probability, physics, astrophysics, chemical physics, anomalous diffusion, seismic analysis, finance, optic and signal processing, robust control, electromagnetism, biology, viscoelasticity, acoustics (see [12,15,21,22,23] to cite but a few). In mechanics, for instance, fractional derivatives are more adequate to model the damping than the usual frictional or structural dampings (described by the time derivative of order one or the time derivative of the Laplacian, respectively).…”

Section: Introductionmentioning

confidence: 99%

On a Second-Order Differential Problem With Fractional Derivatives of Order Greater Than One

Tatar

2013

Mathematical Modelling and Analysis

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A second-order abstract problem with derivatives of non-integer order is investigated. The nonlinearity involves fractional derivatives between 1 and 2. Existence and uniqueness of mild and classical solutions are established in appropriate spaces. This work extends similar works with or without a derivative of first order and also a work of the present author, where the order of the derivatives were between 0 and 1.

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Longitudinal vibrations of rheological rod with variable cross section

Cited by 9 publications

References 0 publications

Discrete fractional order system vibrations

Discrete fractional order system vibrations

Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

On a Second-Order Differential Problem With Fractional Derivatives of Order Greater Than One

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