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.
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This work was initiated and mostly performed during the research stay of P. Suková at the Department of Mechanical Engineering at Texas A&M University. The research has been supported in part by GAUK-428011 and by DEC-2012-/05/E/ST9/03914 from the Polish National Science Center (P.S.), J.Málek acknowledges the support of the ERCCZ project LL1202 financed by MŠMT (Ministry of Education, Youth and Sports of the Czech Republic).
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Málek, J., Rajagopal, K.R. & Suková, P. Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations. Appl Math 61, 79–102 (2016). https://doi.org/10.1007/s10492-016-0123-0
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DOI: https://doi.org/10.1007/s10492-016-0123-0
Keywords
- chaos
- differential-algebraic system
- Poincaré’s sections
- recurrence analysis; bifurcation diagram
- implicit constitutive relations
- Duffing oscillator
- Bingham dashpot
- rigid-elastic spring