Dynamics of a backlash chain

Machado, J. A. Tenreiro

doi:10.2478/s11534-013-0234-0

Cited by 2 publications

(1 citation statement)

References 43 publications

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“…In the last chapter he defined two types of analogies so that results and phenomena in one system could be used for the study of analogous phenomena in other system. Based on these ideas we consider two FO chain system oscillations: a mechanical chain [19] and a electrical ladder [41][42][43].…”

Section: Qualitative and Mathematical Analogies Between Mechanical Anmentioning

confidence: 99%

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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Section: Qualitative and Mathematical Analogies Between Mechanical Anmentioning

confidence: 99%

Discrete fractional order system vibrations

Hedrih

Machado

2015

International Journal of Non-Linear Mechanics

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The Bouncing Ball and the Grünwald-Letnikov Definition of Fractional Derivative

Machado

2021

Fract Calc Appl Anal

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This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

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Dynamics of a backlash chain

Cited by 2 publications

References 43 publications

Discrete fractional order system vibrations

Discrete fractional order system vibrations

The Bouncing Ball and the Grünwald-Letnikov Definition of Fractional Derivative

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