Computing Nonlinear Normal Modes of Aerospace Structures Using the Multi-harmonic Balance Method

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.

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

Volume 4: Dynamics, Control and Uncertainty, Parts A and B ◽

10.1115/imece2012-87538 ◽

2012 ◽

Cited By ~ 2

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.

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Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2015-46106 ◽

2015 ◽

Cited By ~ 10

Author(s):

J. P. Noël ◽

T. Detroux ◽

L. Masset ◽

G. Kerschen ◽

L. N. Virgin

Keyword(s):

Nonlinear System ◽

Harmonic Balance ◽

Global Analysis ◽

Normal Modes ◽

Nonlinear Normal Modes ◽

Basins Of Attraction ◽

Response Curves ◽

Restoring Force ◽

Nonlinear Normal ◽

Two Degree Of Freedom

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4037594 ◽

2017 ◽

Vol 13 (1) ◽

Cited By ~ 2

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.

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