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Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations. (English) Zbl 1389.34045

The paper studies vibration of one-degree-of-freedom lumped parameter systems whose constituents are described through novel implicit constitutive relations. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables (displacement and velocity) which, when substituted into the balance of linear momentum, lead to a single governing ordinary differential equation for the system as a whole. However, many important physical problems cannot specify the forces explicitly in terms of the kinematic variables, but the kinematic parameters are functions of the force. There is an implicit relation between the forces acting on the system and appropriate kinematical variables. In that case the mathematical model of the vibration system is a differential-algebraic system which has to be solved simultaneously. Standard techniques such Poincaré sections, bifurcation diagrams and Lyapunov exponents are used for study. Four examples are investigated: oscillator with a spring with non-standard cubic constitutive relation, inverse Duffing oscillator, oscillator with rigid-elastic spring with linear dashpot and Bingham fluid which does not flow unless a certain threshold in the force is reached in the dashpot. The obtained results give interesting new physical insight in the problems.

MSC:

34A09 Implicit ordinary differential equations, differential-algebraic equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
70H45 Constrained dynamics, Dirac’s theory of constraints
34C23 Bifurcation theory for ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

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