scholarly journals Nonlinear normal modes and their application in structural dynamics

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Christophe Pierre ◽  
Dongying Jiang ◽  
Steven Shaw
Keyword(s):  
Large Scale ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Realistic Model ◽  
Galerkin Projection ◽  
Engineering Structures ◽  
Nonlinear Modes ◽  
Nonlinear Modal Analysis ◽  
Nonlinear Normal

Recent progress in the area of nonlinear modal analysis for structural systems is reported. Systematic methods are developed for generating minimally sized reduced-order models that accurately describe the vibrations of large-scale nonlinear engineering structures. The general approach makes use of nonlinear normal modes that are defined in terms of invariant manifolds in the phase space of the system model. An efficient Galerkin projection method is developed, which allows for the construction of nonlinear modes that are accurate out to large amplitudes of vibration. This approach is successfully extended to the generation of nonlinear modes for systems that are internally resonant and for systems subject to external excitation. The effectiveness of the Galerkin-based construction of the nonlinear normal modes is also demonstrated for a realistic model of a rotating beam.

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

Nonlinear Modal Analysis and Energy Localization in a Bladed Disk Assembly

2008 ◽  
Author(s):  
F. Georgiades ◽  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval ◽  
M. Ruzzene
Keyword(s):  
Modal Analysis ◽  
Nonlinear Oscillations ◽  
Periodic Structures ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Energy Localization ◽  
Bladed Disk ◽  
Bladed Disk Assembly ◽  
Nonlinear Modal Analysis ◽  
Nonlinear Normal

The objective of this study is to carry out modal analysis of nonlinear periodic structures using nonlinear normal modes (NNMs). The NNMs are computed numerically with a method developed in [18] that is using a combination of two techniques: a shooting procedure and a method for the continuation of periodic motion. The proposed methodology is applied to a simplified model of a perfectly cyclic bladed disk assembly with 30 sectors. The analysis shows that the considered model structure features NNMs characterized by strong energy localization in a few sectors. This feature has no linear counterpart, and its occurrence is associated with the frequency-energy dependence of nonlinear oscillations.

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.

scholarly journals Numerical Computation of Nonlinear Normal Modes of Nonconservative Systems

2013 ◽  
Author(s):  
L. Renson ◽  
G. Kerschen
Keyword(s):  
Periodic Solutions ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Techniques ◽  
Nonlinear Beam ◽  
Conservative Systems ◽  
Linear Damping ◽  
Nonlinear Normal ◽  
Element Technique

Since linear modal analysis fails in the presence of non-linear dynamical phenomena, the concept of nonlinear normal modes (NNMs) was introduced with the aim of providing a rigorous generalization of linear normal modes to nonlinear systems. Initially defined as periodic solutions, numerical techniques such as the continuation of periodic solutions were used to compute NNMs. Because these methods are limited to conservative systems, the present study targets the computation of NNMs for non-conservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, the partial differential equations governing the manifold geometry are considered as transport equations and an adequate finite element technique is proposed to solve them. The method is first demonstrated on a conservative nonlinear beam and the results are compared to standard continuation techniques. Then, linear damping is introduced in the system and the applicability of the method is demonstrated.

Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation

2003 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Beam Model ◽  
Periodic Excitation ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to periodic excitation. The approach is an extension of the nonlinear normal mode formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree of freedom, whose response is known. A reduced order model for the forced system is then determined by the usual nonlinear normal mode procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple two-degree-off-reedom mass-spring system with cubic nonlinearities, and for a discretized beam model with 12 degrees of freedom. The results show that this method provides very accurate responses over a range of frequencies near resonances.

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.

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.

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