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

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## 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.

### Keywords

chaos differential-algebraic system Poincaré’s sections recurrence analysis; bifurcation diagram implicit constitutive relations Duffing oscillator Bingham dashpot rigid-elastic spring### MSC 2010

34C28 70K55 34A09## Preview

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