Modal Testing of Nonlinear Vibrating Structures Based on a Nonlinear Extension of Force Appropriation

2009 ◽  
Author(s):  
M. Peeters ◽  
G. Kerschen ◽  
J. C. Golinval
Keyword(s):  
Normal Modes ◽  
Ground Vibration ◽  
Nonlinear Normal Modes ◽  
Modal Testing ◽  
Mathematical Tool ◽  
Nonlinear Dynamical ◽  
Time Frequency ◽  
Analysis Methodology ◽  
Conceptual Relation ◽  
Nonlinear Normal

Modal testing and analysis is well-established for linear systems. The objective of this paper is to progress toward a practical experimental modal analysis methodology of nonlinear mechanical structures. In this context, nonlinear normal modes (NNMs) offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear and simple conceptual relation to the classical linear normal modes (LNMs). A nonlinear extension of force appropriation techniques is investigated in this study in order to isolate one single NNM during the experiments, similarly to what is carried out for ground vibration testing. With the help of time-frequency analysis, the modal curves and the corresponding backbones are then extracted from the time series. The proposed methodology is demonstrated using a numerical benchmark, which consists of a planar cantilever beam with a cubic spring at its free end.

scholarly journals The use of normal forms for analysing nonlinear mechanical vibrations

2015 ◽  
Vol 373 (2051) ◽  
pp. 20140404 ◽  
Author(s):  
Simon A. Neild ◽  
Alan R. Champneys ◽  
David J. Wagg ◽  
Thomas L. Hill ◽  
Andrea Cammarano
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Nonlinear Dynamical Systems ◽  
Mechanical Vibration ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Alternative Theory ◽  
Nonlinear Dynamical ◽  
Normal Form Method ◽  
Nonlinear Normal

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

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.

DETERMINING THE CHARACTERISTICS OF A SELF-EXCITED OSCILLATION IN ROTOR/STATOR SYSTEMS FROM THE INTERACTION OF LINEAR AND NONLINEAR NORMAL MODES

2010 ◽  
Vol 20 (12) ◽  
pp. 4137-4150 ◽  
Author(s):  
JUN JIANG ◽  
ZHIQIANG WU
Keyword(s):  
Dry Friction ◽  
Dominant Role ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Dynamical ◽  
Backward Whirl ◽  
Excited Oscillation ◽  
Linear And Nonlinear ◽  
Contact Surfaces ◽  
Nonlinear Normal

In this paper, the linear and nonlinear modes of the unforced coupled rotor/stator system from a general rotor/stator model, which accounts for both the dynamics of the rotor and the stator as well as the friction and the deformation at the contact surfaces, are derived. The bifurcations of the nonlinear normal modes are analyzed based on the constrained bifurcation theory with the linear normal modes as the constraints. Then, the existence boundaries and the backward whirl frequencies of dry friction backward whirl — a hazardous self-excited oscillation in rotor/stator systems — of this model are derived. It is found by analysis that many inherent characteristics of the dry friction backward whirl can be derived from the information of the interaction of the linear and the nonlinear normal modes of the coupled rotor/stator system, such as the number of existence regions and their position relationship, the minimal friction on the contact surfaces that may induce the self-excited oscillation, the upper limits of the backward whirl frequencies of the response, and more. This study has well demonstrated the dominant role of the interaction of the linear and the nonlinear normal modes in deciding the characteristics of some nonlinear dynamical behaviors.

Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

2015 ◽  
Author(s):  
Sean J. Kelly ◽  
Matthew S. Allen ◽  
Hamid Ardeh
Keyword(s):  
Normal Modes ◽  
Nonlinear Behavior ◽  
Nonlinear Normal Modes ◽  
Structural Systems ◽  
Arc Length ◽  
Frequency Method ◽  
Mass System ◽  
Nonlinear Modes ◽  
Time Frequency ◽  
Nonlinear Normal

Fast numerical approaches have recently been proposed to compute the undamped nonlinear normal modes (NNMs) of discrete and structural systems, and these have enabled remarkable insights into the nonlinear behavior of more complicated systems than are tractable analytically. Ardeh et al. recently proposed the multi-point, multi-harmonic collocation (MMC) method, which finds a truncated harmonic description of a structure’s NNMs. The MMC algorithm resembles harmonic balance but doesn’t require any analytical pre-processing nor the computational expense of the alternating time-frequency method. In a previous work this method showed significantly faster performance than the shooting method and it also allows the possibility of truncating the description of the NNM to avoid having to compute every internal resonance. This work presents a pseudo-arc length continuation version of the MMC algorithm which can compute a branch of NNMs and compares its performance with the shooting/pseudo-arc length continuation algorithm presented by Peeters and Kerschen in 2009. The algorithms are compared for two systems, a two degree-of-freedom (DOF) spring-mass system and a 10-DOF model of a simply-supported beam.

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.

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