scholarly journals A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS

2002 ◽  
Vol 249 (5) ◽  
pp. 971-993 ◽  
Author(s):  
E. PESHECK ◽  
C. PIERRE ◽  
S.W. SHAW
Keyword(s):  
Invariant Manifolds ◽  
Normal Modes ◽  
Non Linear

Non-Linear Dynamics of a Rotor-Bearing System Based on the Invariant Manifold Approach

2003 ◽  
Author(s):  
Cristiano Viana Serra Villa ◽  
Jean-Jacques Sinou ◽  
Fabrice Thouverez
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Initial Condition ◽  
Order Model ◽  
Linear Dynamics ◽  
Reduced Order ◽  
Spin Speed ◽  
Non Linear ◽  
Bearing System

The invariant manifold approach is used to explore the dynamics of a non-linear rotor, by determining the non-linear normal modes, constructing a reduced order model and evaluating its performance in the case of response to an initial condition. The procedure to determine the approximation of the invariant manifolds is discussed and a strategy to retain the speed dependent effects on the manifolds without solving the eigenvalue problem for each spin speed is presented. The performance of the reduced system is analysed in function of the spin speed.

scholarly journals Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems

1993 ◽  
Author(s):  
Nicolas Boivin ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Linear Systems ◽  
Invariant Manifolds ◽  
Computational Cost ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
High Order Accuracy ◽  
Continuous Systems ◽  
Continuous Approach ◽  
Non Linear ◽  
Non Linear Systems

Abstract Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by denning proper normal modes of motion. These are defined in terms of invariant manifolds in the system’s phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied — are simultaneously applied to the same study case — an Euler-Bernoulli beam constrained by a non-linear spring —, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the nonlinear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differentia] equation.

scholarly journals Non-linear normal modes and invariant manifolds

1991 ◽  
Vol 150 (1) ◽  
pp. 170-173 ◽  
Author(s):  
S.W. Shaw ◽  
C. Pierre
Keyword(s):  
Invariant Manifolds ◽  
Normal Modes ◽  
Non Linear

NORMAL MODES OF A NON-LINEAR CLAMPED–CLAMPED BEAM

2002 ◽  
Vol 250 (2) ◽  
pp. 339-349 ◽  
Author(s):  
W.C. XIE ◽  
H.P. LEE ◽  
S.P. LIM
Keyword(s):  
Normal Modes ◽  
Non Linear
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