On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

2001 ◽  
Vol 124 (1) ◽  
pp. 157-160 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Normal Modes ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

Interpreting Proper Orthogonal Modes in Vibrations

1997 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Distributed System ◽  
Normal Modes ◽  
Single Mode ◽  
Dominant Mode ◽  
Coordinate Space ◽  
Nonlinear Simulation ◽  
Principal Axes ◽  
Linear And Nonlinear ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

Abstract We investigate the interpretation of proper orthogonal modes (POMs) of displacements in both linear and nonlinear vibrations. The POMs in undamped linear symmetric systems can represent linear natural modes if the mass distribution is known. This is appoximately true in a distributed system if it is discretized uniformly. If a single mode dominates, the dominant POM approximates the dominant mode. This is also true if a distributed system is discretized arbitrarily. Generally, the POMs represent the principal axes of inertia of the data in the coordinate space. For synchronous nonlinear normal modes, the dominant POM represents a best fit of the nonlinear modal curve. Linear and nonlinear simulation examples are presented.

Experimental Application of a Damage Localisation Technique Based on Smoothed Proper Orthogonal Modes

2007 ◽  
Vol 347 ◽  
pp. 121-126 ◽  
Author(s):  
U. Galvanetto ◽  
L. Monopoli ◽  
Cecilia Surace ◽  
Alessandra Tassotti
Keyword(s):  
Damage Detection ◽  
Proper Orthogonal Decomposition ◽  
Real Time ◽  
Orthogonal Decomposition ◽  
Steel Beams ◽  
Acceleration Response ◽  
Experimental Application ◽  
Lab Experiments ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

The paper presents an experimental application of the Proper Orthogonal Decomposition (POD) to damage detection in steel beams. A damaged beam has been excited with a sinusoidal force, the acceleration response at points regularly spaced along the structure has been recorded and the relevant Proper Orthogonal Modes calculated. In this way it is possible to locate damage by comparing the measured dominant Proper Orthogonal Mode with a smoothed version of it which does not exhibit apparent peaks in correspondence with the damage. One of the principal advantages of the proposed damage detection technique is that it does not require vibration measurements to be performed on the undamaged structure. Moreover the ‘optimality’ of the proper orthogonal modes only requires the use of a few (one-two) of them which can be computed in real time during lab experiments or while the structure is functioning in the field.

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 Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

2019 ◽  
Author(s):  
Mikel Balmaseda ◽  
G. Jacquet-Richardet ◽  
A. Placzek ◽  
D.-M. Tran
Keyword(s):  
Normal Modes ◽  
Nonlinear Dynamic Analysis ◽  
Evaluation Procedure ◽  
Reduced Order Models ◽  
Reduced Basis ◽  
Reduced Order ◽  
Stiffness Evaluation ◽  
Fan Blade ◽  
Proper Orthogonal ◽  
Good Agreement

Abstract In the present work reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical non linearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the pre-stressed equilibrium induced by rotation enhancing the classical linearised approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the Stiffness Evaluation Procedure (STEP) and then corrected by means of a Proper Orthogonal Decomposition (POD) that filters the full order nonlinear forces (StepC ROM). The Linear Normal Modes (LNM) and Craig-Bampton (C-B) type reduced basis are considered here. The latter are parametrised with respect to the rotating velocity. The periodic solutions obtained with the StepC ROM are in good agreement with the solutions of the FOM and are more accurate than the linearised ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

Detection of Coherence in Collocated Experimental Acceleration Signals in a Flexible Aluminum Multi-Beam Structure

2011 ◽  
Author(s):  
Ioannis T. Georgiou
Keyword(s):  
State Of The Art ◽  
Normal Modes ◽  
Piezoelectric Sensors ◽  
Beam Structure ◽  
Original Result ◽  
Unit Space ◽  
Modes Of Vibration ◽  
Proper Orthogonal ◽  
The Impact ◽  
Acceleration Signals

The present work concerns the study of the experimental Proper Orthogonal Decomposition (POD) modes of three simultaneously acquired ensembles of collocated impulse-induced acceleration signals in a complex multi-beam aluminum structure. The impact-induced locally transverse acceleration of a three-beam structure is measured simultaneously at three fixed points with state-of-the-art piezoelectric sensors. Each ensemble of collocated databases is processed by the POD Transform to find out that it is underlined by strong coherence in space and time manifested by a small number of POD modes. It is found that the unit space modulations of the first-the dominant-POD modes of the three databases of experimental acceleration signals form an orthonormal set. The same is true for their companion unit time modulations. This original result leads to the identification of three normal modes of vibration for the complex beam structure.

scholarly journals Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Mikel Balmaseda ◽  
G. Jacquet-Richardet ◽  
A. Placzek ◽  
D.-M. Tran
Keyword(s):  
Normal Modes ◽  
Nonlinear Dynamic Analysis ◽  
Evaluation Procedure ◽  
Reduced Order Models ◽  
Reduced Basis ◽  
Reduced Order ◽  
Stiffness Evaluation ◽  
Fan Blade ◽  
Proper Orthogonal ◽  
Good Agreement

Abstract In this work, reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical nonlinearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the prestressed equilibrium induced by rotation enhancing the classical linearized approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the stiffness evaluation procedure (STEP) and then corrected by means of a proper orthogonal decomposition (POD) that filters the full order nonlinear forces (StepC ROM). The linear normal modes (LNM) and Craig-Bampton (C-B) type reduced basis are considered here. The latter are parameterized with respect to the rotating velocity. The periodic solutions obtained with the StepC ROM are in good agreement with the solutions of the FOM and are more accurate than the linearized ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

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