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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.
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 -
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