Forced Response Analysis of Pipes Conveying Fluid by Nonlinear Normal Modes Method and Iterative Approach

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Feng Liang ◽  
Xiao-Dong Yang ◽  
Ying-Jing Qian ◽  
Wei Zhang
Keyword(s):  
Normal Modes ◽  
Harmonic Balance Method ◽  
Nonlinear Normal Modes ◽  
Forced Response ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Iterative Approach ◽  
Pipes Conveying Fluid ◽  
Nonlinear Normal ◽  
Conveying Fluid

The forced vibration of gyroscopic continua is investigated by taking the pipes conveying fluid as an example. The nonlinear normal modes and a numerical iterative approach are used to perform numerical response analysis. The nonlinear nonautonomous governing equations are transformed into a set of pseudo-autonomous ones by using the harmonic balance method. Based on the pseudo-autonomous system, the nonlinear normal modes are constructed by the invariant manifold method on the state space and substituted back into the original discrete equations. By repeating the above mentioned steps, the dynamic responses can be numerically obtained asymptotically using such iterative approach. Quadrature phase difference between the general coordinates is verified for the gyroscopic system and traveling waves instead of standing waves are found in the time-domain complex modal analysis.

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.

Bifurcation of Nonlinear Normal Modes by Means of Synge’s Stability

2011 ◽  
Author(s):  
Young S. Lee ◽  
Heng Chen
Keyword(s):  
Harmonic Balance ◽  
Coupled Oscillators ◽  
Normal Modes ◽  
Harmonic Balance Method ◽  
Orbital Stability ◽  
Nonlinear Normal Modes ◽  
Energy Transfers ◽  
Wide Range ◽  
Energy Domain ◽  
Nonlinear Normal

We study bifurcation of fundamental nonlinear normal modes (FNNMs) in 2-degree-of-freedom coupled oscillators by utilizing geometric mechanics approach based on Synges concept, which dictates orbital stability rather than Lyapunovs classical asymptotic stability. Use of harmonic balance method provides reasonably accurate approximation for NNMs over wide range of energy; and Floquet theory incorporated into Synges stability analysis predicts the respective bifurcation points as well as their types. Constructing NNMs in the frequency-energy domain, we seek applications to study of efficient targeted energy transfers.

scholarly journals Comparison of different methodologies for the computation of damped nonlinear normal modes and resonance prediction of systems with non-conservative nonlinearities

2021 ◽  
Author(s):  
Yekai Sun ◽  
Jie Yuan ◽  
Alessandra Vizzaccaro ◽  
Loïc Salles
Keyword(s):  
Normal Modes ◽  
Engineering Application ◽  
Nonlinear Normal Modes ◽  
Forced Response ◽  
Energy Balance Method ◽  
Periodic Problems ◽  
Modal Synthesis ◽  
Comprehensive Comparison ◽  
Detailed Assessment ◽  
Nonlinear Normal

AbstractThe nonlinear modes of a non-conservative nonlinear system are sometimes referred to as damped nonlinear normal modes (dNNMs). Because of the non-conservative characteristics, the dNNMs are no longer periodic. To compute non-periodic dNNMs using classic methods for periodic problems, two concepts have been developed in the last two decades: complex nonlinear mode (CNM) and extended periodic motion concept (EPMC). A critical assessment of these two concepts applied to different types of non-conservative nonlinearities and industrial full-scale structures has not been thoroughly investigated yet. Furthermore, there exist two emerging techniques which aim at predicting the resonant solutions of a nonlinear forced response using the dNNMs: extended energy balance method (E-EBM) and nonlinear modal synthesis (NMS). A detailed assessment between these two techniques has been rarely attempted in the literature. Therefore, in this work, a comprehensive comparison between CNM and EPMC is provided through two illustrative systems and one engineering application. The EPMC with an alternative damping assumption is also derived and compared with the original EPMC and CNM. The advantages and limitations of the CNM and EPMC are critically discussed. In addition, the resonant solutions are predicted based on the dNNMs using both E-EBM and NMS. The accuracies of the predicted resonances are also discussed in detail.

Persistent and ghost nonlinear normal modes in the forced response of non-smooth systems

2012 ◽  
Vol 241 (22) ◽  
pp. 2058-2067 ◽  
Author(s):  
Paolo Casini ◽  
Oliviero Giannini ◽  
Fabrizio Vestroni
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Forced Response ◽  
Nonlinear Normal

scholarly journals On the Use of the Proper Generalised Decomposition for Solving Nonlinear Vibration Problems

2012 ◽  
Author(s):  
Aurelien Grolet ◽  
Fabrice Thouverez
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
A Priori ◽  
Geometrical Nonlinearity ◽  
Normal Modes ◽  
Harmonic Balance Method ◽  
Nonlinear Normal Modes ◽  
Vibration Problems ◽  
Nonlinear Normal ◽  
Proper Orthogonal

This paper presents the use of the so called Proper Generalized Decomposition method (PGD) for solving nonlinear vibration problems. PGD is often presented as an a priori reduction technique meaning that the reduction basis for expressing the solution is computed during the computation of the solution itself. In this paper, the PGD is applied in addition with the Harmonic Balance Method (HBM) in order to find periodic solutions of nonlinear dynamic systems. Several algorithms are presented in order to compute nonlinear normal modes and forced solutions. Application is carried out on systems containing geometrical nonlinearity and/or friction damping. We show that the PGD is able to compute a good approximation of the solutions event with a projection basis of small size. Results are compared with a Proper Orthogonal Decomposition (POD) method showing that the PGD can sometimes provide an optimal reduction basis relative to the number of basis components.

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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