A study on the crack detection in beams using linear and nonlinear normal modes

2019 ◽  
Vol 23 (7) ◽  
pp. 1305-1321
Author(s):  
Yildirim Serhat Erdogan
Keyword(s):  
Normal Mode ◽  
Crack Detection ◽  
Normal Modes ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Crack Identification ◽  
Crack Location ◽  
Linear And Nonlinear ◽  
Nonlinear Normal ◽  
Cracked Beams

Linear and nonlinear normal mode motions may provide promising information about the condition of mechanical structures under small and large amplitude vibrations, respectively. In this view, this study investigates the nonlinear dynamics of cracked beams through use of the nonlinear mode motion and extends the crack identification methods that utilize the linear characteristics to nonlinear vibrating structures. At first, the nonlinear normal modes of the intact and cracked beams are calculated by a continuation algorithm. A finite element model of a geometrically nonlinear prismatic beam was created based on crack stress intensity. Subsequently, a method based on normal mode motion and minimization of strain energy, which is valid for linear and nonlinear vibrating beams, was developed as an optimization problem. To this end, hybrid optimization was also used due to its capability in finding global minimum along with its computational efficiency. It was shown that the proposed crack detection technique is applicable to beams vibrating in linear and/or nonlinear regimes and well capable of detecting both crack location and severity.

Characterization of Nonlinear Coupled Dynamics in Sandwich Structures

2008 ◽  
Author(s):  
Ioannis T. Georgiou
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Geometric Nonlinearity ◽  
Sandwich Structures ◽  
Normal Modes ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
High Fidelity ◽  
Coupled Dynamics ◽  
Nonlinear Normal

In this work, the nonlinear coupled dynamics of a sandwich structure with hexagonal honeycomb core are characterized in terms of Proper Orthogonal Decomposition modes. A high fidelity nonlinear finite element model is derived to describe geometric nonlinearity and displacement and rotation fields that govern the coupled dynamics. Contrary to equivalent continuum models used to predict vibration properties of lattice and sandwich structures, a high fidelity finite element model allows for a quite detailed description of the distributed complicated geometric nonlinearity of the core. It was found that the free dynamics excited by a blast load and the forced dynamics excited by a harmonic force posses POD modes which are localized in space and time. The processing of the simulated dynamics by the Time Discrete Proper Transform forms a means to study the nonlinear coupled dynamics of sandwich structures in the context of nonlinear normal modes of vibration and reduced order models.

scholarly journals Prediction of Isolated Resonance Curves Using Nonlinear Normal Modes

2015 ◽  
Author(s):  
R. J. Kuether ◽  
L. Renson ◽  
T. Detroux ◽  
C. Grappasonni ◽  
G. Kerschen ◽  
...  
Keyword(s):  
Normal Modes ◽  
Resonance Curve ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Initial Guess ◽  
Forced Response ◽  
Numerical Continuation ◽  
Continuation Algorithm ◽  
Resonance Curves ◽  
Nonlinear Normal

Isolated resonance curves are separate from the main nonlinear forced-response branch, so they can easily be missed by a continuation algorithm and the resonant response might be underpredicted. The present work explores the connection between these isolated resonances and the nonlinear normal modes of the system and adapts an energy balance criterion to connect the two. This approach provides new insights into the occurrence of isolated resonances as well as a method to find an initial guess to compute the isolated resonance curve using numerical continuation. The concepts are illustrated on a finite element model of a cantilever beam with a nonlinear spring at its tip. This system presents jumps in both frequency and amplitude in its response to a swept sinusoidal excitation. The jumps are found to be the result of a modal interaction that creates an isolated resonance curve that eventually merges with the main resonance branch as the excitation force increases. Excellent insight into the observed dynamics is provided with the NNM theory, which supports that NNMs can also be a useful tool for predicting isolated resonance curves and other behaviors in the damped, forced response.

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.

Characterizing the Dynamics of 3-D Elastic Nonlinear Continua Using the Method of Proper Orthogonal Decompositions

2005 ◽  
Author(s):  
Ioannis Georgiou ◽  
Dimitris Servis
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Normal Modes ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Modes Of Vibration ◽  
Nonlinear Normal ◽  
Proper Orthogonal Decompositions ◽  
Nonlinear Finite Element Model ◽  
Proper Orthogonal

A novel and systematic way is presented to characterize the modal structure of the free dynamics of three-dimensional elastic continua. In particular, the method of Proper Orthogonal Decomposition (POD) for multi-field dynamics is applied to analyze the dynamics of prisms and moderately thick beams. A nonlinear finite element model is used to compute accurate approximations to free motions which in turn are processed by POD. The extension of POD to analyze the dynamics of three-dimensional elastic continua, which are multi-field coupled dynamical system, is carried out by vector and matrix quantization of the finite element dynamics. An important outcome of this study is the fact that POD provides the means to systematically identify the shapes of nonlinear normal modes of vibration of three-dimensional structures from high resolution finite element simulations.

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.

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