Non-linear normal forced vibration modes in systems with internal resonance

Perepelkin, Nikolay V.; Mikhlin, Yuri V.; Pierre, Christophe

doi:10.1016/j.ijnonlinmec.2013.06.002

Cited by 13 publications

(13 citation statements)

References 25 publications

(60 reference statements)

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“…48b) the explicit form of amplitudes defined in equations (26)(27) which lead to elimination of the second term in left side was considered. Also, with averaging of the rest secular terms of Eq.…”

Section: Solution Of First-order Approximation For Rigid Body With Tomentioning

confidence: 99%

“…The multiharmonic balance technique has been used in [25], to examine stability and vibrations analysis of a complex flexible rotor bearing system. Based on invariant manifolds approach, in [26] a new method has been developed for the determination of NNMs, incorporating Rausher and harmonic balance methods, and it has been applied to determine the dynamics of a spinning shaft with constant rotating speed with nonlinearities and inertia forces in the supports. It should be noted that in [22][23][24][25][26] the models were considering constant rotating speed and nonlinearities in bearings, without examination of non-constant rotating speed.…”

Section: Introductionmentioning

confidence: 99%

“…Based on invariant manifolds approach, in [26] a new method has been developed for the determination of NNMs, incorporating Rausher and harmonic balance methods, and it has been applied to determine the dynamics of a spinning shaft with constant rotating speed with nonlinearities and inertia forces in the supports. It should be noted that in [22][23][24][25][26] the models were considering constant rotating speed and nonlinearities in bearings, without examination of non-constant rotating speed. In [27], non-constant rotating speed for a rotating composite blade was considered, whereas a nonlinear system of partial differential equations (PDEs) coupled with one integro-differential equation (IDE) describing the rigid body angular motion, which is a similar model with the spinning shaft in non-constant rotating speed derived, herein, is derived.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Georgiades

2017

Nonlinear Dyn

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Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spindown 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 firstorder 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 F. Georgiades (B) School of Engineering, College of Science, University of Lincoln, Brayford Pool, Lincoln LN6 7TS, UK e-mail: fgeorgiadis@lincoln.ac.uk 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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Section: Solution Of First-order Approximation For Rigid Body With Tomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Georgiades

2017

Nonlinear Dyn

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“…This expression is not changed if the phase of external load is varied by an arbitrary constant. This equivalent system of a new type can be also considered as the result of substitution (14) into (13) and division of the latter by Q2 . Due to the considerations above the system 20 The equivalent system (20) will be called the equivalent system o f 3rd type.…”

Section: Equivalent System Investigation Via Invariant Manifolds Methmentioning

confidence: 99%

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Perepelkin

2017

Nonlinear Dyn

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In the paper a new non-iterative variant o f Rauscher method is considered. In its current statement the method can be used in analysis o f forced harmonic oscillations in 1-DOF nonlinear system. It is shown that three different types o f equivalent authonomous dynamical systems can be built fo r a given 1-DOF non-autonomous one. Two o f them (1st and 2nd type) have wider set o f solutions than that o f the initial system. These solutions correspond to various values o f amplitude and phase o f external excitation. Solutions o f the equivalent system o f 3rd type are exclusively periodic ones. Based on the equivalent system of 3rd type such a function W(x,x') can be constructed that its level curves correspond to periodic orbits o f the initial non-autonomous system. This function can be built a priori via computation o f the invariant manifold o f the equivalent system o f 1st type. Using the same approach the Rauscher expansions cos(Qt)=C(x,x'), sin(Qt)=S(x,x') can also be constructed. It is also shown that equivalent systems can be investigated by means o f harmonic balance method which allows construction o f W(x,x'), C(x,x') andS(x,x') in semi-analytical manner.

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“…In [2], the Duffing type equation was obtained by expanding the restoring and damping forces of a nonlinear isotropic Jeffcott rotor in Taylor series, and it was shown that there are ranges of rotational speed in which the periodic solution is unstable and some complicated solutions and even chaotic solutions exist. In [3], the nonlinear normal modes of the single disk rotor system with the isotropic elastic shaft and the nonlinear supports of Duffing type were analyzed considering gyroscopic effects, asymmetrical disposition of the disk on the shaft and internal resonance, in which, two different kinds of resonance regimes were summarized. In [4], a bladed rotor system considering the Duffing type nonlinearity for the stiffness of the blades and bearings was analyzed analytically, it reveals that two simple Hopf bifurcation points exist around the upper and lower critical speeds and both subcritical and supercritical Hopf bifurcations are expected due to the nonlinearity of the blade or bearing stiffness.…”

Section: Introductionmentioning

confidence: 99%