Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations

Josef Málek; Kumbakonam R. Rajagopal; Petra Suková

Applications of Mathematics (2016)

  • Volume: 61, Issue: 1, page 79-102
  • ISSN: 0862-7940

Abstract

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We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary differential equation for the system as a whole. However, in the case considered we obtain a system of equations: the balance of linear momentum, and the implicit constitutive relation for each constituent, that has to be solved simultaneously. From the mathematical perspective, we have to deal with a differential-algebraic system. We study the vibration of several specific systems using standard techniques such as Poincaré's surface of section, bifurcation diagrams, and Lyapunov exponents. We also perform recurrence analysis on the trajectories obtained.

How to cite

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Málek, Josef, Rajagopal, Kumbakonam R., and Suková, Petra. "Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations." Applications of Mathematics 61.1 (2016): 79-102. <http://eudml.org/doc/276283>.

@article{Málek2016,
abstract = {We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary differential equation for the system as a whole. However, in the case considered we obtain a system of equations: the balance of linear momentum, and the implicit constitutive relation for each constituent, that has to be solved simultaneously. From the mathematical perspective, we have to deal with a differential-algebraic system. We study the vibration of several specific systems using standard techniques such as Poincaré's surface of section, bifurcation diagrams, and Lyapunov exponents. We also perform recurrence analysis on the trajectories obtained.},
author = {Málek, Josef, Rajagopal, Kumbakonam R., Suková, Petra},
journal = {Applications of Mathematics},
keywords = {chaos; differential-algebraic system; Poincaré's sections; recurrence analysis; bifurcation diagram; implicit constitutive relations; Duffing oscillator; Bingham dashpot; rigid-elastic spring; chaos; differential-algebraic system; Poincaré’s sections; recurrence analysis; bifurcation diagram; implicit constitutive relations; Duffing oscillator; Bingham dashpot; rigid-elastic spring},
language = {eng},
number = {1},
pages = {79-102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations},
url = {http://eudml.org/doc/276283},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Málek, Josef
AU - Rajagopal, Kumbakonam R.
AU - Suková, Petra
TI - Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 79
EP - 102
AB - We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary differential equation for the system as a whole. However, in the case considered we obtain a system of equations: the balance of linear momentum, and the implicit constitutive relation for each constituent, that has to be solved simultaneously. From the mathematical perspective, we have to deal with a differential-algebraic system. We study the vibration of several specific systems using standard techniques such as Poincaré's surface of section, bifurcation diagrams, and Lyapunov exponents. We also perform recurrence analysis on the trajectories obtained.
LA - eng
KW - chaos; differential-algebraic system; Poincaré's sections; recurrence analysis; bifurcation diagram; implicit constitutive relations; Duffing oscillator; Bingham dashpot; rigid-elastic spring; chaos; differential-algebraic system; Poincaré’s sections; recurrence analysis; bifurcation diagram; implicit constitutive relations; Duffing oscillator; Bingham dashpot; rigid-elastic spring
UR - http://eudml.org/doc/276283
ER -

References

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