Appendix A: Physical meaning of proper orthogonal modes

2015 ◽  
pp. 227-230
Keyword(s):  
Physical Meaning ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

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.

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.

Reduced-Order P.O.D. Models for Nonlinear Vibrations of Cylindrical Shells

2003 ◽  
Author(s):  
M. Amabili ◽  
A. Sarkar ◽  
M. P. Pai¨doussis
Keyword(s):  
Time Series ◽  
Degrees Of Freedom ◽  
Cylindrical Shells ◽  
Harmonic Excitation ◽  
Computational Effort ◽  
Inviscid Fluid ◽  
Galerkin Approach ◽  
System Behaviour ◽  
Proper Orthogonal Modes ◽  
Proper Orthogonal

The nonlinear (large-amplitude) response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of their lowest natural frequencies is investigated. The shell is assumed to be completely filled with an incompressible and inviscid fluid at rest. Donnell’s nonlinear shallow-shell theory is used, and the solution is obtained by the Galerkin method. The proper orthogonal decomposition (POD) method is used to extract proper orthogonal modes that describe the system behaviour from time-series response data. These time-series have been obtained via the conventional Galerkin approach (using normal modes as a projection basis) with an accurate model involving 16 degrees of freedom, validated in previous studies. The POD method, in conjunction with the Galerkin approach, permits a lower-dimensional model with respect to the conventional Galerkin approach. Different proper orthogonal modes computed from time-series at different excitation frequencies are used and solutions are compared. Some of these modes are capable of describing the system behaviour over the whole frequency range around the fundamental resonance with good accuracy and with only three degrees of freedom. They allow a drastic reduction in the computational effort, as compared to using the 16 degrees-of-freedom model necessary when the conventional Galerkin approach is used.

A Cantilever Conveying Fluid: Coherent Modes Versus Beam Modes

2002 ◽  
Author(s):  
A. Sarkar ◽  
M. P. Paidoussis
Keyword(s):  
Eigenvalue Problem ◽  
Cantilever Beam ◽  
Large Scale ◽  
Galerkin Projection ◽  
Reduced Order ◽  
Non Linear ◽  
Proper Orthogonal Modes ◽  
Linear Oscillation ◽  
Proper Orthogonal ◽  
Conveying Fluid

A semianalytical approach to obtain the proper orthogonal modes (POMs) is described for the non-linear oscillation of a cantilevered pipe conveying fluid. Theoretically, while the spatial coherent structures are the eigenfunctions of the time-averaged spatial autocorrelation functions, it emerges that once the Galerkin projection of the proper orthogonal modes is carried out using the uniform cantilever-beam modes, the spatial dependency of the integral eigenvalue problem can be eliminated by analytical manipulation which avoids any spatial discretization error. As the solution of the integral equation is obtained semianalytically by linearly projecting the proper orthogonal modes on the cantilever-beam modes, any linear or non-linear operation can be carried out analytically on the proper orthogonal modes. Futhermore, the reduced-order eigenvalue problem minimizes the numerical pollution which leads to spurious eigenvectors, as may arise in the case of a large-scale eigenvalue problem (without the Galerkin projection of the eigenvectors on the cantilever-beam modes). This methodology can conveniently be used to study the convergence of the numerically calculated proper or-thogonal modes obtained from the full-scale eigenvalue problem.

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