By Albert C. J. Luo
Nonlinear difficulties are of curiosity to engineers, physicists and mathematicians and plenty of different scientists simply because so much structures are inherently nonlinear in nature. As nonlinear equations are tough to resolve, nonlinear platforms are as a rule approximated through linear equations. This works good as much as a few accuracy and a few variety for the enter values, yet a few attention-grabbing phenomena similar to chaos and singularities are hidden through linearization and perturbation research. It follows that a few features of the habit of a nonlinear procedure seem as a rule to be chaotic, unpredictable or counterintuitive. even though one of these chaotic habit may possibly resemble a random habit, it really is totally deterministic.
Analytical Routes to Chaos in Nonlinear Engineering discusses analytical options of periodic motions to chaos or quasi-periodic motions in nonlinear dynamical structures in engineering and considers engineering purposes, layout, and regulate. It systematically discusses complicated nonlinear phenomena in engineering nonlinear structures, together with the periodically compelled Duffing oscillator, nonlinear self-excited structures, nonlinear parametric structures and nonlinear rotor structures. Nonlinear types utilized in engineering also are offered and a short background of the subject is supplied.
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Additional resources for Analytical Routes to Chaos in Nonlinear Engineering
The detailed mathematical theory of the generalized harmonic balance method with the vigorous proof was presented in Luo (2012a, 2013, 2014). In fact, this method provides a finite-harmonic-term transformation with different time scales to obtain an autonomous nonlinear system of coefficients in the Fourier series form with finite harmonic terms. The dynamical behaviors of such an autonomous nonlinear system will determine the solution behaviors of original dynamical systems. For periodic solutions, the Fourier series forms of the finite harmonic terms are convergent.
The bifurcation trees of period-1 motions to chaos for the Duffing oscillators will be discussed and the bifurcation trees of period-3 motions to chaos will also be presented for the Duffing oscillators. Different types of Duffing oscillators possess completely different bifurcation trees. In Chapter 3, analytical solutions for period-m motions in periodically forced, self-excited oscillators will be presented in the Fourier series form with finite harmonic terms, and the stability and bifurcation of the corresponding period-m motions will be completed.
The upper and lower stable branches of solutions exist. The unstable solution is in the middle branch, which is similar to the traditional analysis. 52). 745, 35). 1(b), the higher order harmonics contribution to the upper and lower branches are less than 10% and 1% for Ω > 5, respectively. However, for Ω < 5, the higher order harmonic contribution to the upper branch solution are the same quantity level. So many higher order harmonic terms should be taken into account. 1(c) and (d). 2(i)–(vii).