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.

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 Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

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

2015 ◽  
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.

Constructing the Frequency-Energy Plot of Nonlinear Vibratory Systems via the Modified Lindstedt-Poincaré Method

2013 ◽  
Vol 655-657 ◽  
pp. 547-550
Author(s):  
Xin Hua Zhang
Keyword(s):  
Energy Transfer ◽  
Nonlinear Vibration ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Results ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Vibrating Systems

The frequency-energy plot(FEP) of nonlinear vibration systems is a powerful tool for investigating the energy transfer phenomena related wiht the internal resonances occured in multi-digreeof- freedom(Multi-DOF) nonlinear vibration systems. In this paper, the modified Lindstedt-Poincare method is employed for constructing the FEP of a two-DOF nonlinear vibrating systems. First, the original vibartion equations are modified for the application of the modified Linstedt-Poincaré method. Then, by using the modified Linstedt-Poincaé method, the nonlinear normal modes(NNMs) of the system are obtained. Finally, the frequency-energy plot of the system is constructed analytically. Numerical results show that the method adopted in this paper is effective and accurate.

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.

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 Analytical and Dynamic study of Pulled Mass Nonlinear Vibration by Two Cables using Newton's Harmonic Balance Method

2020 ◽  
Vol 2 (2) ◽  
pp. 79-86
Author(s):  
Masoud Rahmani ◽  
Ionut Cristian Scurtu ◽  
Amin Moslemi Petrudi
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Vibrations ◽  
Nonlinear Differential Equations ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Good Match ◽  
Engineering Research ◽  
Solution Methods ◽  
Vibration Problems

In recent years, much research has been done on nonlinear vibrations, and analytical and numerical methods have been used to solve complex nonlinear equations. The behavior of nonlinear oscillating equations is discussed until the second order is approximated. Harmonic balance method, which itself has limitations in application. This method continues to be able to study a wider range of nonlinear differential equations. In general, nonlinear vibration problems are of great importance in physics, mechanical structures, and other engineering research. First, the equation of nonlinear vibrations governing the mass of the particle mass connect to the drawn cable is calculated and then the Newton Harmonic Balance Method is used to study the nonlinear vibrations of the set and obtain the answer and its frequency. The method (NHBM) is done with Maple software and a comparison between the results of this method with the solution methods used by other researchers is shown to be a good match.  

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