69 research outputs found
A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions
Combinig the harmonic balance method (HBM) and a continuation method is a
well-known technique to follow the periodic solutions of dynamical systems when
a control parameter is varied. However, since deriving the algebraic system
containing the Fourier coefficients can be a highly cumbersome procedure, the
classical HBM is often limited to polynomial (quadratic and cubic)
nonlinearities and/or a few harmonics. Several variations on the classical HBM,
such as the incremental HBM or the alternating frequency/time domain HBM, have
been presented in the literature to overcome this shortcoming. Here, we present
an alternative approach that can be applied to a very large class of dynamical
systems (autonomous or forced) with smooth equations. The main idea is to
systematically recast the dynamical system in quadratic polynomial form before
applying the HBM. Once the equations have been rendered quadratic, it becomes
obvious to derive the algebraic system and solve it by the so-called ANM
continuation technique. Several classical examples are presented to illustrate
the use of this numerical approach.Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-
Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators
In this set of lectures, we review briefly some of the recent developments in
the study of the chaotic dynamics of nonlinear oscillators, particularly of
damped and driven type. By taking a representative set of examples such as the
Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain
the various bifurcations and chaos phenomena associated with these systems. We
use numerical and analytical as well as analogue simulation methods to study
these systems. Then we point out how controlling of chaotic motions can be
effected by algorithmic procedures requiring minimal perturbations. Finally we
briefly discuss how synchronization of identically evolving chaotic systems can
be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in
Physics Please Lakshmanan for figures (e-mail: [email protected]
Nonlinear dynamics of a spinning shaft with non-constant rotating speed
Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales perturbation method. The system in first-order approximation takes the form of two coupled sets of paired equations. The first pair describes the torsional and the rigid body rotation, whilst the second consists of the equations describing the two lateral bending motions. Notably, equations of the lateral bending motions of first-order approximation coincide with the system in case of constant rotating speed, and considering the amplitude modulation equations, as it is shown, there are detuning frequencies from the Campbell diagram. The nonlinear normal modes of the system have been determined analytically up to the second-order approximation. The comparison of the analytical solutions with direct numerical simulations shows good agreement up to the validity of the performed analysis. Finally, it is shown that the Campbell diagram in the case of spin-up or spin-down operation cannot describe the critical situations of the shaft. This work paves the way, for new safe operational ‘modes’ of rotating structures bypassing critical situations, and also it is essential to identify the validity of the tools for defining critical situations in rotating structures with non-constant rotating speeds, which can be applied not only in spinning shafts but in all rotating structures
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