Identifying VIV Vibration Modes by Use of the Empirical Orthogonal Functions Technique

2002 ◽  
Author(s):  
Gudmund Kleiven
Keyword(s):  
Model Test ◽  
Vibration Mode ◽  
Multivariate Data ◽  
Empirical Orthogonal Functions ◽  
Mode Shapes ◽  
Data Sets ◽  
Vibration Modes ◽  
Ocean Engineering ◽  
Orthogonal Functions ◽  
Related Technique

The Empirical Orthogonal Functions (EOF) technique has widely being used by oceanographers and meteorologists, while the Singular Value Decomposition (SVD being a related technique is frequently used in the statistics community. Another related technique called Principal Component Analysis (PCA) is observed being used for instance in pattern recognition. The predominant applications of these techniques are data compression of multivariate data sets which also facilitates subsequent statistical analysis of such data sets. Within Ocean Engineering the EOF technique is not yet widely in use, although there are several areas where multivariate data sets occur and where the EOF technique could represent a supplementary analysis technique. Examples are oceanographic data, in particular current data. Furthermore data sets of model- or full-scale data of loads and responses of slender bodies, such as pipelines and risers are relevant examples. One attractive property of the EOF technique is that it does not require any a priori information on the physical system by which the data is generated. In the present paper a description of the EOF technique is given. Thereafter an example on use of the EOF technique is presented. The example is analysis of response data from a model test of a pipeline in a long free span exposed to current. The model test program was carried out in order to identify the occurrence of multi-mode vibrations and vibration mode amplitudes. In the present example the EOF technique demonstrates the capability of identifying predominant vibration modes of inline as well as cross-flow vibrations. Vibration mode shapes together with mode amplitudes and frequencies are also estimated. Although the present example is not sufficient for concluding on the applicability of the EOF technique on a general basis, the results of the present example demonstrate some of the potential of the technique.

Empirical Methods in Short-Term Climate Prediction

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Climate Prediction ◽  
Empirical Orthogonal Functions ◽  
Data Sets ◽  
Seasonal Forecasts ◽  
Short Term ◽  
Orthogonal Functions ◽  
Ocean Atmosphere Interaction ◽  
Global Coverage ◽  
Atmosphere Interaction ◽  
Global Data

This clear and accessible text describes the methods underlying short-term climate prediction at time scales of 2 weeks to a year. Although a difficult range to forecast accurately, there have been several important advances in the last ten years, most notably in understanding ocean-atmosphere interaction (El Nino for example), the release of global coverage data sets, and in prediction methods themselves. With an emphasis on the empirical approach, the text covers in detail empirical wave propagation, teleconnections, empirical orthogonal functions, and constructed analogue. It also provides a detailed description of nearly all methods used operationally in long-lead seasonal forecasts, with new examples and illustrations. The challenges of making a real time forecast are discussed, including protocol, format, and perceptions about users. Based where possible on global data sets, illustrations are not limited to the Northern Hemisphere, but include several examples from the Southern Hemisphere.

Modal Analysis for Swing-Arm of LED Die Bonder

2011 ◽  
Vol 422 ◽  
pp. 379-382
Author(s):  
Wei Chuang Quan ◽  
Mei Fa Huang ◽  
Zhi Yue Wang ◽  
Da Wei Zhang
Keyword(s):  
Modal Analysis ◽  
Vibration Mode ◽  
Three Dimensional ◽  
Element Model ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Lead Frame ◽  
Swing Arm ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

Led die bonder used for bond lead frame and chip is one of the key equipment of led production line. The swing-arm is an important component of led die bonder and its dynamic characteristics will directly affect the piece accuracy. At present, the accuracy and efficiency of led die bonder are limited because of the vibration of the swing-arm. In solving this problem, a three-dimensional finite-element model for swing-arm is built to provide analytical frequencies and vibration modes. Then the modal distribution and vibration mode shapes for swing-arm are obtained after analyzing the modal by ansys10.0. Finally the dynamics effects of this structure by modal frequency and vibration mode are analyzed. The modal analysis of structural would provide the reference to dynamics analysis and structural optimization for swing-arm in practical use.

Mode Evolution of Cyclic Symmetric Rotors Assembled to Flexible Bearings and Housing

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
Hyunchul Kim ◽  
Nick Theodore Khalid Colonnese ◽  
I. Y. Shen
Keyword(s):  
Vibration Mode ◽  
Fluid Dynamic ◽  
Travelling Wave ◽  
Modal Testing ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Phase Index ◽  
Symmetric Rotor ◽  
Cyclic Symmetric ◽  
Inertia Forces

This paper is to study how the vibration modes of a cyclic symmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to assembly, the vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as unbalanced modes. By applying fundamental theorems of continuum mechanics, we conclude that balanced modes will present vanishing inertia forces and moments as they vibrate. Since each vibration mode of a cyclic symmetric rotor can be characterized in terms of a phase index (Chang and Wickert, “Response of Modulated Doublet Modes to Travelling Wave Excitation,” J. Sound Vib., 242, pp. 69–83; Chang and Wickert, 2002, “Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks,” J. Sound Vib., 250, pp. 379–400; Kim and Shen, 2009, “Ground-Based Vibration Response of a Spinning Cyclic Symmetric Rotor With Gyroscopic and Centrifugal Softening Effects,” ASME J. Vibr. Acoust. (in press)), the criterion of vanishing inertia forces and moments implies that the phase index by itself can uniquely determine whether or not a vibration mode is a balanced mode as follows. Let N be the order of cyclic symmetry of the rotor and n be the phase index of a vibration mode. Vanishing inertia forces and moments indicate that a vibration mode will be a balanced mode if n≠1,N−1,N. When n=N, the vibration mode will be balanced if its leading Fourier coefficient vanishes. To validate the mathematical predictions, modal testing was conducted on a disk with four pairs of brackets mounted on an air-bearing spindle and a fluid-dynamic bearing spindle at various spin speeds. Measured Campbell diagrams agree well with the theoretical predictions.

Mode Evolution of Asymmetric Rotors Assembled to Flexible Bearings and Housing

2009 ◽  
Author(s):  
Hyunchul Kim ◽  
I. Y. Shen
Keyword(s):  
Boundary Conditions ◽  
Vibration Mode ◽  
Fluid Dynamic ◽  
Modal Testing ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Fixed Boundary ◽  
Practical Applications ◽  
Asymmetric Rotor ◽  
Surface Integrals

This paper is to study how vibration modes of a stationary asymmetric rotor evolve when it is assembled to a flexible housing via multiple bearing supports. Prior to the assembly, vibration modes of the rotor are classified as “balanced modes” and “unbalanced modes.” Balanced modes are those modes whose natural frequencies and mode shapes remain unchanged after the rotor is assembled to the housing via bearings. Otherwise, the vibration modes are classified as “unbalanced modes.” In this paper, we first develop two mathematical criteria to identify balanced modes. For the first criteria, the rotor is subjected to fixed boundary conditions at the bearings prior to assembly. In this case, a vibration mode will be a balanced mode if the reactions at the fixed boundary vanish. For the second criterion, the rotor is subjected to free boundary conditions (including the bearing points) prior to assembly. In this case, a vibration mode will be a balanced mode if the bearing locations are nodal points of the vibration mode. These mathematical criteria are then applied to a rotor consisting of a rigid hub supporting a flexible structure, which appears in many practical applications. For balanced modes, the criteria lead to a conclusion that surface integrals of modal forces and moments at the flexible-rigid rotor interface will vanish. Moreover, these surface integrals can be conveniently calculated using finite element methods. To validate the mathematical criteria, modal testing was conducted on a disk with 4 pairs of brackets mounted on a rigid spindle, a ball-bearing spindle and a fluid-dynamic bearing spindle.

Measurement of Fan Vibration Using Double Pulse Holography

1978 ◽  
Vol 100 (4) ◽  
pp. 655-663 ◽  
Author(s):  
B. S. Hockley ◽  
R. A. J. Ford ◽  
C. A. Foord
Keyword(s):  
Vibration Mode ◽  
Mechanical Vibration ◽  
Propagation Speed ◽  
Double Pulse ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Television System ◽  
Temporal Phase ◽  
Frequency Split ◽  
Laser Holography

Supersonic unstalled flutter in gas turbine fans is a self-excited instability in which mechanical vibrations give rise to unsteady aerodynamic forces which drive the mechanical vibration. The phenomenon is very sensitive to the deflected shapes of the blades and to the spatial and temporal phases of the blades’ responses. This paper is concerned with the measurement of vibrational behavior on static fans and relating it to flutter. Accurate detailed data on the blade and disk vibration mode shapes of fans up to 2.2 m diameter has been measured using double pulse laser holography. Both axial and tangential components of the blade mode shape are obtained by taking holograms from two directions. The analysis of the holograms is performed with the aid of a computer linked television system which generates the required blade mode shapes directly from the photographs of the hologram reconstructions. The disk mode measurements on real fans have shown the existence of pairs of spatially orthogonal vibration modes which have similar shapes (e.g. both 4D) but slightly different natural frequencies. This frequency split between modes means that the flutter wave will experience a cyclic variation in amplitude and propagation speed as it travels round the fan. In addition, the temporal phase angle between twist and flap in a single blade, which is generally assumed to be 90 deg, will vary from blade to blade.

Empirical Orthogonal Functions

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Missing Values ◽  
Original Data ◽  
Empirical Orthogonal Functions ◽  
Basis Set ◽  
Data Matrix ◽  
Multidimensional Data ◽  
Data Sets ◽  
Eof Analysis ◽  
Orthogonal Functions ◽  
Multidimensional Data Sets

This chapter focuses on empirical orthogonal functions (EOFs). One of the most useful and common eigen-techniques in data analysis is the construction of EOFs. EOFs are a transform of the data; the original set of numbers is transformed into a different set with some desirable properties. In this sense the EOF transform is similar to other transforms, such as the Fourier or Laplace transforms. In all these cases, we project the original data onto a set of functions, thus replacing the original data with the set of projection coefficients on the chosen new set of basis vectors. However, the choice of the specific basis set varies from case to case. The discussions cover data matrix structure convention, reshaping multidimensional data sets for EOF analysis, forming anomalies and removing time mean, missing values, choosing and interpreting the covariability matrix, calculating the EOFs, projection time series, and extended EOF analysis.

scholarly journals Identification of Vibration Modes of Quartz Crystal Plates with Proportion of Strain and Kinetic Energies

2020 ◽  
Vol 25 (3) ◽  
pp. 392-407
Author(s):  
Qi Huang ◽  
Rongxing Wu ◽  
Lihong Wang ◽  
Longtao Xie ◽  
Jianke Du ◽  
...  
Keyword(s):  
Quartz Crystal ◽  
Vibration Mode ◽  
Plate Theory ◽  
Traditional Approach ◽  
Mode Shapes ◽  
Energy Concentration ◽  
Mindlin Plate ◽  
Vibration Modes ◽  
Coupled Modes ◽  
Mode Identification

For the design of quartz crystal resonators, finding and determining the vibration modes have always been very important and cumbersome. Vibration modes are usually identified through plotting displacement patterns of each coupled modes and making comparisons. Over the years, there is not much improvement in the identification procedure while tremendous efforts have been made in refining the equations of the Mindlin plate theory to obtain more accurate results, such as the adoption of the Finite Element Method (FEM) by implementing the high-order Mindlin plate equations for efficient analysis. However, due to the old fashioned mode identification method, the FEM application is still inadequate and cannot be fully automated for this purpose. To have this situation improved, a method using the proportions of strain and kinetic energies to characterize the energy level of each vibration mode is proposed. With solutions of displacements, the energy distribution of each vibration mode is calculated and the mode with the highest energy concentration at a specific frequency is designated as the dominant mode. The results have been validated with the traditional approach by plotting mode shapes at each frequency. Clearly, this energy approach will be advantageous with the FEM analysis for vibration mode identification automatically.

On the Vibration of Bolted Plate and Flange Assemblies

1994 ◽  
Vol 116 (4) ◽  
pp. 468-473 ◽  
Author(s):  
J.-G. Tseng ◽  
J. A. Wickert
Keyword(s):  
Data Storage ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Annular Plate ◽  
Approximate Model ◽  
Outer Edge ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Laboratory Measurements ◽  
Structural Imperfections

The vibration of an annular plate that is free along its outer edge, and that is connected to a flange along its inner edge by bolts that are equally spaced in the circumferential direction, is studied. A disk with this geometry, or a stacked array of such disks, is common in applications involving data storage, rotating machinery, or brake systems. The periodic structural imperfections that are associated with the bolt pattern can have interesting implications for the plate’s dynamic response. Changes that occur in the natural frequencies and mode shapes as a result of such deviations from an ideally clamped inner edge are studied through laboratory measurements, and through an approximate model that captures the rotationally periodic character of the bolted plate and flange system. In the axisymmetric case, the natural frequencies of the plate’s “sine” and “cosine” vibration modes are repeated for a specified number of nodal diameters. Under the influence of a regular bolt pattern, and the resulting local variations of the stiffness and compression at the plate/flange interface, some natural frequencies are repeated and others split. This process depends on the number of bolts used to mount the plate, and on the number of nodal diameters present in a specific vibration mode. A straightforward criterion to predict the split and repeated modes is discussed.

scholarly journals Common metrics of calibration for continuous Gaussian data and exceedance probabilities

2018 ◽  
Author(s):  
Rita Glowienka-Hense ◽  
Andreas Hense ◽  
Thomas Spangehl ◽  
Marc Schröder
Keyword(s):  
Gaussian Processes ◽  
Low Frequency ◽  
Empirical Orthogonal Functions ◽  
Brier Score ◽  
Data Sets ◽  
Orthogonal Functions ◽  
Leibler Divergence ◽  
The Mean ◽  
Decadal Climate ◽  
Concept Of Information

Abstract. A framework of ensemble forecast verification tools is discussed which is founded on the concept of information entropy. It can be based on a common yardstick namely that of "correlation". With these measures calibration is deduced from the balance between ensemble sharpness and resolution. With the same units these features can be put into one diagram for continuous time series from Gaussian processes and exceedance probabilities, the latter usually tested with the reliability term from the Brier score. The sharpness and resolution terms allow to use the same vocabulary of over- and underdispersion which is established for frequency histograms. The concept is based on the fact that mutual information (MI) of two Gaussian processes is directly related to Pearson's anomaly correlation. Further MI can be written as the Kullback-Leibler divergence of the conditional probability of observations given the model forecasts and the unconditioned observations. Thus the MI is a measure of resolution. The mean of the UTILITY defined by (Kleeman, 2002) is the corresponding measure of sharpness. For Gaussian processes the mean UTILITY is very close to the ratio of ensemble mean variance to mean ensemble variance (ANOVA) which is the analysis of variance factor when time is taken as treatment. The ensemble spread score (ESS) (Palmer et al., 2006) is shown to be a measure of calibration if model and observed data are scaled with their respective means and standard deviations. For exceedance probabilities the resolution term of the divergence score (Weijs et al., 2010) is already defined as a MI term and it is here complemented with a mean UTILITY formed similarly to the resolution term but with forecasts only. The entropy terms are then rescaled to the "correlation" yardstick. The concept is applied to temperature data from the German project on decadal climate prediction, Mittelfristige Klimaprognose (MiKlip). It is shown that both over – and underdispersion can be found for the 2m temperature forecasts. Increasing ensemble sharpness of surface ocean temperature with lead year in the southern ocean hints at model-data inconsistencies at some locations in the ocean. Finally empirical orthogonal functions (EOF) of northern hemisphere annual mean surface temperature for ERA-40/ERA-Interim and MiKlip retrospective hindcasts are determined. For both data sets the respective first EOF represents the low frequency temperature development. The time coefficients of the EOF are used to compare resolution and sharpness of continuous data and exceedance probabilities in one diagram.

Sign in / Sign up

Export Citation Format

Share Document