Abstract
Each Duffing equation has an unstable solution area with a boundary, which is also a line of bifurcation. Generally, in a system that can be modeled by the Duffing equation, bifurcations can occur at frequencies lower than the origin point frequency of the unstable solution area for a softening system and at higher frequencies for a hardening system. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account. To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. For each system, for three combinations of linear and nonlinear stiffness coefficients with the same nonlinearity factor, bistability areas and unstable solution areas were determined for a series of damping coefficient values. For each case, curves determined for different damping values were grouped to obtain the origin point curve of the unstable solution, ultimately developing the target formulas.
Highlights
The Duffing equation was introduced by Georg Duffing in1 and is considered one of the prototype systems of nonlinear dynamics[2]
Books devoted to nonlinear systems present a comprehensive description of the issue, covering the knowledge that is already k nown[2,25–27], while studies published in scientific journals usually concern a well-defined p roblem[14,28,29] or propose a new solution m ethod[30], new analytical derivations[31,32] or a new tool that may be useful in the analysis of nonlinear dynamic systems[33]
Knowing the position of the origin point, it should be possible to determine the analytical formulas for the curves describing bistability area and unstable solution area charts of a nonlinear system modeled by a Duffing equation with any value of damping
Summary
Each Duffing equation has an unstable solution area with a boundary, which is a line of bifurcation. The main goal of this research is to determine the analytical formulas for the origin point of the unstable solution area of a system described by a forced Duffing oscillator with softening stiffness, taking damping into account To achieve this goal, two systems of softening Duffing oscillators that differ strongly in their nonlinearity factor value have been selected and tested. The main goal of this research was to determine analytical formulas for the origin point of the unstable solution area of a forced Duffing oscillator with softening stiffness, taking into account damping. Knowing the position of the origin point, it should be possible to determine the analytical formulas for the curves describing bistability area and unstable solution area charts of a nonlinear system modeled by a Duffing equation with any value of damping. It will not be necessary to perform tedious and time-consuming numerical simulations to identify these areas
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