Nonlinear Normal Modes in the Presence of Internal Resonances: An Energy-Based Approach

1995 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this work, modifications to existing energy-based nonlinear normal mode (NNM) methodologies are developed in order to investigate internal resonances. A formulation for computing resonant NNMs is developed for discrete, or discretized for continuous systems, sets of weakly nonlinear equations with uncoupled linear terms (i.e systems in modal, or canonical, form). By considering a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies. Additionally, the canonical formulation allows for a single (linearized modal) coordinate to parameterize all other (modal) coordinates during a resonant modal response. Energy-based NNM methodologies are then applied to the canonical equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered. Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the application of the resonant NNM methodology. Resonant normal mode solutions are obtained and the stability characteristics of these computed modes are considered. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus the transformation to canonical coordinates is necessary in order to define appropriate NNM relations.

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

1996 ◽  
Vol 63 (3) ◽  
pp. 810-819 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Internal Resonance ◽  
Natural Frequencies ◽  
Linear Expansion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 332-340 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Conservation Of Energy ◽  
Nonlinear Modes ◽  
One Dimensional ◽  
Linear Differential ◽  
Nonlinear Normal ◽  
The Stability ◽  
Nonlinear Elastic

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.

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.

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.

scholarly journals Multiharmonic Resonance Control Testing of an Internally Resonant Structure

Vibration ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 217-234
Author(s):  
Alexander D. Shaw ◽  
Thomas L. Hill ◽  
Simon A. Neild ◽  
Michael I. Friswell
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Testing Procedure ◽  
Modal Interactions ◽  
Nonlinear Structure ◽  
Testing Method ◽  
Passive Testing ◽  
Nonlinear Normal ◽  
Resonance Control ◽  
Experimental Characterisation

The experimental characterisation of a nonlinear structure is a challenging process, particularly for multiple degree of freedom and continuous structures. Despite attracting much attention from academia, there is much work needed to create processes that can achieve characterisation in timescales suitable for industry, and a key to this is the design of the testing procedure itself. This work proposes a passive testing method that seeks a desired degree of resonance between forcing and response. In this manner, the process automatically seeks data that reveals greater detail of the underlying nonlinear normal modes than a traditional stepped sine method. Furthermore, the method can target multiple harmonics of the fundamental forcing frequency, and is therefore suitable for structures with complex modal interactions. The method is presented with some experimental examples, using a structure with a 3:1 internal resonance.

Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

2007 ◽  
Author(s):  
David Wagg
Keyword(s):  
Mathematical Formulation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Lumped Mass ◽  
Localized Modes ◽  
Impact Oscillator ◽  
Mass System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

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