Singular characteristics of nonlinear normal modes in a two degrees of freedom asymmetric systems with cubic nonlinearities

2001 ◽  
Vol 22 (8) ◽  
pp. 972-982
Author(s):  
Xu Jian ◽  
Lu Qi-shao ◽  
Huang Ke-lei
Keyword(s):  
Degrees Of Freedom ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Two Degrees Of Freedom ◽  
Nonlinear Normal ◽  
Singular Characteristics

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

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.

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.

Nonlinear Normal Modes of Real-World Structures: Application to a Full-Scale Aircraft

2011 ◽  
Author(s):  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval ◽  
C. Stephan
Keyword(s):  
Real World ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Ground Vibration ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Full Scale ◽  
Modal Interactions ◽  
Wing Tips ◽  
Nonlinear Normal

The objective of this paper is to demonstrate that the numerical computation of the nonlinear normal modes (NNMs) of complex real-world structures is now within reach. The application considered in this study is the airframe of the Morane-Saulnier Paris aircraft, whose ground vibration tests have exhibited some nonlinear structural behaviors. The finite element model of this aircraft, elaborated from drawings, has more than 80000 degrees of freedom, and softening nonlinearities exist in the connection between the external fuel tanks and the wing tips. From this model, a reduced-order model, which is accurate in the [0–100Hz] range, is constructed using the Craig-Bampton technique. The NNMs of the reduced model are then computed using a numerical algorithm combining shooting and pseudo-arclength continuation. The results show that the NNMs of this full-scale structure can be computed accurately even in strongly nonlinear regimes and with a reasonable computational burden. Nonlinear modal interactions are also highlighted by the algorithm and are discussed.

scholarly journals Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

2004 ◽  
Vol 10 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Mathias Legrand ◽  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotating Shaft ◽  
Nonlinear Mode ◽  
Large Amplitude Motions ◽  
Nonlinear Normal ◽  
The Individual

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig–Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig–Bampton model.

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