Identifying the Shapes of Coupled Vibrations and Deriving Reduced Order Models for Nonlinear Shafts: A Finite Element-Proper Orthogonal Decomposition Approach

2004 ◽  
Author(s):  
I. T. Georgiou ◽  
M. A. Bani-Khaled
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Rotating Shaft ◽  
Decomposition Approach ◽  
Reduced Order ◽  
Coupled Vibrations ◽  
Orthogonal Decompositions ◽  
Proper Orthogonal Decompositions ◽  
Proper Orthogonal ◽  
Two Degree Of Freedom

The spatial structure of the dynamics of a rotating nonlinear shaft is identified by processing its finite element dynamics by the method of Proper Orthogonal Decompositions. The Proper Orthogonal modes furnish characteristic signatures for the rigid body and the whirling modes of a motion. The pattern of energy distribution over the components of a mode reveals the strength of coupling between rigid body rotations and coupled vibrations. These modes are used to derive a two-degree-of freedom reduced model for the whirling motion of the rotating shaft.

scholarly journals The Reduced-Order Extrapolating Method about the Crank-Nicolson Finite Element Solution Coefficient Vectors for Parabolic Type Equation

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1261 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Matrix Theory ◽  
Type Equation ◽  
Parabolic Type ◽  
Theoretical Development ◽  
Reduced Order ◽  
Proper Orthogonal ◽  
Unknown Solution ◽  
Crank Nicolson

This study is mainly concerned with the reduced-order extrapolating technique about the unknown solution coefficient vectors in the Crank-Nicolson finite element (CNFE) method for the parabolic type partial differential equation (PDE). For this purpose, the CNFE method and the existence, stability, and error estimates about the CNFE solutions for the parabolic type PDE are first derived. Next, a reduced-order extrapolating CNFE (ROECNFE) model in matrix-form is established with a proper orthogonal decomposition (POD) method, and the existence, stability, and error estimates of the ROECNFE solutions are proved by matrix theory, resulting in an graceful theoretical development. Specially, our study exposes that the ROECNFE method has the same basis functions and the same accuracy as the CNFE method. Lastly, some numeric tests are shown to computationally verify the validity and correctness about the ROECNFE method.

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.

scholarly journals Reduced-order modeling by POD-multiphase approach for fluid-structure interaction

2010 ◽  
pp. 41-52
Author(s):  
Erwan Liberge ◽  
Marie Pomarede ◽  
Aziz Hamdouni
Keyword(s):  
Rigid Body ◽  
High Viscosity ◽  
Reduced Order Modeling ◽  
Weak Formulation ◽  
Reduced Order ◽  
Fixed Domain ◽  
Rigidity Constraint ◽  
Moving Domains ◽  
Reference Domain ◽  
Proper Orthogonal

This paper describes the Reduced Order Modeling (ROM) for fluid rigid body interaction problem and discusses Proper Orthogonal Decomposition (POD) utilisation. The principal difficulty for using POD being the moving domains, a referenced fixed domain has been introduced. The POD has been applied for the velocity field obtained on the fixed domain. Then a method to reduce dynamical system for rigid body fluid interaction has been developed. This method consists in treating the entire fluid-solid domain as a fluid. The rigid body has then been considered as a fluid, by using a high viscosity which can play the role of a penalisation factor of the rigidity constraint. The fluid flow problem is then formulated on the reference domain and POD modes have been used in the weak formulation.

Developing a Reduced-Order Model of Non-Synchronous Vibration in Turbomachinery Using Proper Orthogonal Decomposition Methods

2014 ◽  
Author(s):  
Stephen T. Clark ◽  
Fanny M. Besem ◽  
Robert E. Kielb ◽  
Jeffrey P. Thomas
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Cross Flow ◽  
Initial Study ◽  
Orthogonal Decomposition ◽  
Reduced Order Model ◽  
Two Dimensional ◽  
Order Model ◽  
Decomposition Approach ◽  
Reduced Order ◽  
Proper Orthogonal

The paper develops a reduced-order model of non-synchronous vibration (NSV) using proper orthogonal decomposition (POD) methods. The approach was successfully developed and implemented, requiring between two and six POD modes to accurately predict CFD solutions that are experiencing non-synchronous vibration. This POD method was first developed and demonstrated for a transversely-moving, two-dimensional cylinder in cross-flow. Later, the method was used for the prediction of CFD solutions for a two-dimensional compressor blade. This research is the first to offer a proper orthogonal decomposition approach to the reduced-order modeling of non-synchronous vibration in turbomachinery. Modeling non-synchronous vibration is especially challenging because NSV is caused by complicated, unsteady flow dynamics; this initial study helps researchers understand the causes of NSV, and aids in the future development of predictive tools for aeromechanical design engineers.

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