By Albert C. J. Luo
Nonlinear difficulties are of curiosity to engineers, physicists and mathematicians and plenty of different scientists simply because so much platforms are inherently nonlinear in nature. As nonlinear equations are tricky to unravel, nonlinear structures are in general approximated by way of 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 comparable to chaos and singularities are hidden via linearization and perturbation research. It follows that a few points of the habit of a nonlinear process seem more often than not to be chaotic, unpredictable or counterintuitive. even if any such chaotic habit could resemble a random habit, it's completely deterministic.
Analytical Routes to Chaos in Nonlinear Engineering discusses analytical recommendations of periodic motions to chaos or quasi-periodic motions in nonlinear dynamical structures in engineering and considers engineering functions, layout, and keep an eye on. It systematically discusses complicated nonlinear phenomena in engineering nonlinear platforms, together with the periodically compelled Duffing oscillator, nonlinear self-excited platforms, nonlinear parametric platforms and nonlinear rotor structures. Nonlinear versions utilized in engineering also are offered and a quick heritage of the subject is equipped.
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Extra resources for Analytical Routes to Chaos in Nonlinear Engineering
41) Analytical Routes to Chaos in Nonlinear Engineering 32 where u = (Δb1 , Δc1 )T , u̇ = (Δḃ 1 , Δċ 1 )T , ü = (Δb̈ 1 , Δ̈c1 )T ] [ ] [ ???? 2Ω K K C= , K = 11 12 ; −2Ω ???? K21 K22 ) ( 3 3 K11 = ???? − Ω2 + ???? 3b∗2 , K12 = ????Ω + ????b∗1 c∗1 , + c∗2 1 1 4 2 ( ) 3 ∗ ∗ 3 . 42) |????2 I + ????C + K| = ????. 43) |????2 + ???????? + K11 2Ω???? + K12 | | = 0. 44) From the eigenvalues, the stability and bifurcation of approximate symmetric period-1 motion are determined. For one harmonic term, the symmetric period-1 motion cannot be approximated well.
The period-m motions for the van der Pol-Duffing oscillator will be presented, and bifurcation tree of period-m motion will be discussed. For a better understanding of complex period-m motions in such a van der Pol-Duffing oscillator, trajectories and amplitude spectrums will be illustrated numerically. In Chapter 4, analytical solutions for period-m motions in parametrically forced, nonlinear oscillators are discussed. The bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity will be discussed analytically.
Nonlinear behaviors of such periodic motions will be characterized through frequency-amplitude curves. This investigation shows that period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well. In addition, analytical solutions for periodic motions in a Mathieu-Duffing oscillator are presented. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions will be discussed. Period-1 asymmetric and period-2 symmetric motions will be illustrated for a better understanding of periodic motions in the Mathieu-Duffing oscillator.