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.

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.

Constructing the Frequency-Energy Plot of Nonlinear Vibratory Systems via the Modified Lindstedt-Poincaré Method

2013 ◽  
Vol 655-657 ◽  
pp. 547-550
Author(s):  
Xin Hua Zhang
Keyword(s):  
Energy Transfer ◽  
Nonlinear Vibration ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Results ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Vibrating Systems

The frequency-energy plot(FEP) of nonlinear vibration systems is a powerful tool for investigating the energy transfer phenomena related wiht the internal resonances occured in multi-digreeof- freedom(Multi-DOF) nonlinear vibration systems. In this paper, the modified Lindstedt-Poincare method is employed for constructing the FEP of a two-DOF nonlinear vibrating systems. First, the original vibartion equations are modified for the application of the modified Linstedt-Poincaré method. Then, by using the modified Linstedt-Poincaé method, the nonlinear normal modes(NNMs) of the system are obtained. Finally, the frequency-energy plot of the system is constructed analytically. Numerical results show that the method adopted in this paper is effective and accurate.

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

2002 ◽  
Vol 124 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Eric Pesheck ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Coupling ◽  
Rotating Beam ◽  
Internal Resonances ◽  
Modal Expansion ◽  
Nonlinear Normal

A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses.

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.

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

2018 ◽  
Vol 13 (4) ◽  
Author(s):  
S. Mahmoudkhani
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Oscillatory Systems ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

Numerical Computation of Nonlinear Normal Modes in Multi-Mode Models of an Inertially Coupled Elastic Structure

2009 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Small Amplitude ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Elastic Structure ◽  
Internal Resonances ◽  
Reduced Order ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom ◽  
Quadratic And Cubic Nonlinearities

There are many techniques available for the construction of nonlinear normal modes. Most studies for systems with more than one degree of freedom utilize asymptotic techniques or the method of multiple time scales, which are valid only for small amplitude motions. Previous works of the authors have investigated nonlinear normal modes in elastic structures with essential inertial nonlinearities, and considered two degree-of-freedom reduced-order models that exhibit 1:2 resonance. For small amplitude oscillations with low energy, this reduced analysis is acceptable, while for higher energy vibrations and vibrations that are away from internal resonances, this may not provide an accurate representation of NNMs. For high energy vibration and vibrations away from internal resonances, two natural issues to be addressed are the dimension of the reduced-order model used for constructing NNMs, and the order of nonlinearities retained in the truncated models. To address these issues, a comparison of NNMs computed for three different reduced degree of freedom models for the elastic structure is reported here. The reduced models considered are: (i) A two degree-of-freedom reduced model with only quadratic nonlinearities; (ii) A two degree-of-freedom reduced model with both quadratic and cubic nonlinearities; (iii) A five degrees-of-freedom model with both quadratic and cubic nonlinearities. A numerical method based on shooting technique is used for constructing the NNMs and results for system near 1:2 internal resonances between the two lowest modes and away from any internal resonance are compared.

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