Some Observations of Mode Localizations in Flexibly Coupled Two Rotors

2005 ◽  
Author(s):  
Yohta Kunitoh ◽  
Hiroshi Yabuno ◽  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotor System ◽  
The Other ◽  
Whirling Motion ◽  
Averaged Equations ◽  
Weakly Coupled ◽  
Nonlinear Normal

Mode localizations in a weakly coupled two-span rotor system are theoretically and experimentally discussed. One rotor has a slight unbalance and the other one is well-assembled. First, the equations governing the whirling motions of the coupled rotors are expressed due to nonlinearity in each span and the weakness of the coupling. The averaged equations are obtained by the method of multiple scales and it is shown that the nonlinear normal modes are bifurcated from the linear normal modes. It results from this bifurcation that the number of nonlinear normal modes exceeds the equivalent degree of freedom of the two-span rotor system, i.e., 2-degree under the assumption that the trajectory of the whirling motion is circle. Also, it is theoretically clarified that whirling motion caused by the unbalance in the rotor is localized in the rotor with unbalance or in one without unbalance depending on the rotational speed. Furthermore, these mode localizations are experimentally confirmed.

Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

1995 ◽  
Vol 117 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Quadratic Nonlinearities ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

1995 ◽  
Vol 1 (4) ◽  
pp. 389-430 ◽  
Author(s):  
Ali H. Nayfeh
Keyword(s):  
Normal Forms ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Direct Method ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Internal Resonances ◽  
Relief Valve ◽  
Nonlinear Normal

A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal resonances.

Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

2003 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Richard H. Rand
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Resonance Capture ◽  
Periodic Motions ◽  
Internal Resonances ◽  
Dynamical Mechanism ◽  
Elliptic Orbits ◽  
Weakly Coupled ◽  
Resonant Dynamics ◽  
Nonlinear Normal

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes — NNMs), as well as, asynchronous periodic motions (elliptic orbits — EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Nonlinear Normal Modes of a Cantilever Beam

1995 ◽  
Vol 117 (4) ◽  
pp. 477-481 ◽  
Author(s):  
A. H. Nayfeh ◽  
C. Chin ◽  
S. A. Nayfeh
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Mode Shapes ◽  
Nonlinear Modes ◽  
Linear Mode ◽  
Complete Set ◽  
Nonlinear Normal

Two approaches for determination of the nonlinear planar modes of a cantilever beam are compared. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.

Degenerate Bifurcation Scenarios in the Dynamics of Coupled Oscillators With Symmetric Nonlinearities

2005 ◽  
Author(s):  
O. V. Gendelman
Keyword(s):  
Coupling Parameter ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Weakly Coupled ◽  
Nonnegative Coefficients ◽  
Nonlinear Normal ◽  
Infinite Energy ◽  
Good Agreement

We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. Both the potential of the oscillator and of the coupling spring are adopted to be even-power polynomials with nonnegative coefficients. By defining the coupling parameter ε, the dynamics of this system at the limit ε → 0 and for finite ε is investigated. Bifurcation scenario of the nonlinear normal modes is revealed. The degeneracy in the dynamics is manifested by a ‘bifurcation from infinity’ where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by mechanism of successive cusp catastrophes with growth of the coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincare sections). For particular case of pure cubic nonlinearity of the oscillator and the coupling spring good agreement between quantitative analytical predictions and numerical results is observed.

Internally Resonant Nonlinear Free Vibrations of Horizontal/Inclined Sagged Cables

2005 ◽  
Author(s):  
G. Rega ◽  
N. Srinil ◽  
S. Chucheepsakul
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Original System ◽  
Nonlinear Normal Modes ◽  
Free Vibrations ◽  
Order Effects ◽  
Internal Resonances ◽  
Closed Form Solutions ◽  
Nonlinear Normal

Internally resonant dynamics in the nonlinear free vibrations of suspended cables are analytically investigated by means of a multi-mode Galerkin-based discretization and second-order multiple scales. Emphasis is placed on planar 2:1 internal resonances. The equations of motion of a general inclined cable model, which account for the dynamic extensibility effects and the system asymmetry due to inclined equilibrium, are considered. By considering higher-order effects due to quadratic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations associated with the resonant nonlinear normal modes reveal the dependence of cable nonlinear response on different resonant and non-resonant modes. Based on the modal convergence properties performed on the resonantly activated cables, the illustrative results provide hints for proper reduced-order model selections from the asymptotic solution. The underlying effects of cable inclination and cable sag are presented. The theoretical predictions are validated by finite difference numerical time laws of the original system equations of motion.

Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span Beam With Geometric Nonlinearities

1996 ◽  
Vol 118 (4) ◽  
pp. 533-542 ◽  
Author(s):  
J. Aubrecht ◽  
A. F. Vakakis
Keyword(s):  
Coupling Parameter ◽  
Galerkin Approximation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Finite Amplitude ◽  
Torsional Stiffness ◽  
Mode Localization ◽  
Weakly Nonlinear ◽  
Weakly Coupled ◽  
Nonlinear Normal

The nonlinear normal modes of a geometrically nonlinear multi-span beam consisting of n segments, coupled by means of torsional stiffeners are examined. Assuming that the stiffeners possess large torsional stiffness, the beam displacements are decomposed into static and flexible components. It is shown that the static components are much smaller in magnitude than the flexible ones. A Galerkin approximation is subsequently employed to discretize the problem, whereby the computation of the nonlinear normal modes of the multi-span beam is reduced to the study of the periodic solutions of a set of weakly coupled, weakly nonlinear ordinary differential equations. Numerous stable and unstable, localized and non-localized nonlinear normal modes of the multi-span beam are detected. Assemblies consisting of n = 2, 3, and 4 beam segments are examined, and are found to possess stable, strongly localized nonlinear normal modes. These are free synchronous oscillations during which only one segment of the assembly vibrates with finite amplitude. As the number of periodic segments increases, the structure of the nonlinear normal modes becomes increasingly more complicated. In the multi-span beams examined, nonlinear mode localization is generated through two distinct mechanisms: through Pitchfork or Saddle-node mode bifurcations, or as the limit of a continuous mode branch when a coupling parameter tends to zero.

DEGENERATE BIFURCATION SCENARIOS IN THE DYNAMICS OF COUPLED OSCILLATORS WITH SYMMETRIC NONLINEARITIES

2006 ◽  
Vol 16 (01) ◽  
pp. 169-178 ◽  
Author(s):  
O. V. GENDELMAN
Keyword(s):  
Coupled Oscillators ◽  
Coupling Parameter ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Saddle Node Bifurcation ◽  
High Energies ◽  
Saddle Node ◽  
Weakly Coupled ◽  
Nonlinear Normal ◽  
Infinite Energy

We study the degenerate bifurcations of nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. The potentials of both the oscillator and the coupling spring are adopted to be even-power polynomials with non-negative coefficients. Coupling parameter ε is defined and the bifurcations of the nonlinear normal modes structure with change of this coupling parameter are revealed. The degeneracy in the dynamics is manifested by a "bifurcation from infinity" where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Other (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1 : 1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by the mechanism of successive cusp catastrophes with the growth of coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincaré sections). In the particular case of pure cubic nonlinearity of the oscillator and the coupling spring, an agreement between quantitative analytical predictions and numerical results is observed.

LIMITING PHASE TRAJECTORIES AND TRANSIENT RESONANCE OSCILLATIONS IN 1 AND 2 DOF ASYMMETRIC NONLINEAR SYSTEMS

2011 ◽  
Vol 21 (10) ◽  
pp. 2919-2928
Author(s):  
E. L. MANEVITCH ◽  
L. I. MANEVITCH
Keyword(s):  
Energy Exchange ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Weakly Coupled ◽  
Limiting Phase Trajectories ◽  
Nonlinear Normal ◽  
Nonstationary Oscillations ◽  
Resonance Oscillations

The concept of limiting phase trajectories (LPT) has been introduced by one of the authors to describe intensive energy exchange between weakly coupled oscillators or oscillatory chains. It turns out that LPT can be considered as an alternative to nonlinear normal modes (NNMs), which are characterized by conservation of energy. LPT (in the introduced coordinates) describes the vibroimpact-type process with saw-tooth amplitude and a discontinuous derivative. It was shown earlier that this concept could also be extended to systems with one degree of freedom (DoF). In this case energy exchange between the oscillator and the source of energy can occur. In this paper, we generalize the above results in several ways, namely: (1) a consideration of the asymmetry of elastic potential; (2) a detailed description of the origin of vibroimpact-type behavior and the transition from nonresonant nonstationary oscillations to resonant ones (3) a direct application of obtained results to transient vibrations in strongly asymmetric 2DoF systems.

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