Experimental Study for Extraction of Normal Modes

1995 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Frequency Response ◽  
Normal Mode ◽  
Normal Modes ◽  
Measurement Data ◽  
Mode Shapes ◽  
Complex Frequency ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Incomplete System ◽  
Coupled Structures

Abstract A frequency-domain technique to extract the normal mode from the measurement data for highly coupled structures is developed. The relation between the complex frequency response functions and the normal frequency response functions is derived. An algorithm is developed to calculate the normal modes from the complex frequency response functions. In this algorithm, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. In addition, the developed technique is independent of the damping types. It is only dependent on the model of analysis. Two experimental examples are employed to illustrate the applicability of the technique. The effects due to different measurement locations are addressed. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled incomplete system.

Extraction of Normal Modes from Contaminated Measurement With Noise for Highly Coupled Structures

1996 ◽  
Vol 118 (3) ◽  
pp. 430-435 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Normal Mode ◽  
Mode Shape ◽  
Present Method ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Mode Shapes ◽  
Response Functions ◽  
Exact Relation ◽  
Phase Data ◽  
Coupled Structures

A frequency-domain technique to extract the normal mode shapes from the contaminated FRF measurements for highly coupled structures is developed. The relation between the complex FRFs and the normal mode shapes is derived. It is found that the normal mode shape cannot be extracted exactly from the complex mode shape. However, an exact relation between the normal mode shape and the complex FRF does exist. In the present method, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. Hence, the effects of measurement noise can be reduced. A numerical example is employed to illustrate the applicability of the technique. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled structure.

scholarly journals Bias Errors due to Leakage Effects When Estimating Frequency Response Functions

2012 ◽  
Vol 19 (6) ◽  
pp. 1257-1266 ◽  
Author(s):  
Andreas Josefsson ◽  
Kjell Ahlin ◽  
Göran Broman
Keyword(s):  
Frequency Response ◽  
Measurement Data ◽  
Approximate Expression ◽  
Bias Error ◽  
Systematic Bias ◽  
Response Functions ◽  
Random Errors ◽  
Frequency Response Functions ◽  
Rectangular Window ◽  
Wide Range

Frequency response functions are often utilized to characterize a system's dynamic response. For a wide range of engineering applications, it is desirable to determine frequency response functions for a system under stochastic excitation. In practice, the measurement data is contaminated by noise and some form of averaging is needed in order to obtain a consistent estimator. With Welch's method, the discrete Fourier transform is used and the data is segmented into smaller blocks so that averaging can be performed when estimating the spectrum. However, this segmentation introduces leakage effects. As a result, the estimated frequency response function suffers from both systematic (bias) and random errors due to leakage. In this paper the bias error in theH1andH2-estimate is studied and a new method is proposed to derive an approximate expression for the relative bias error at the resonance frequency with different window functions. The method is based on using a sum of real exponentials to describe the window's deterministic autocorrelation function. Simple expressions are derived for a rectangular window and a Hanning window. The theoretical expressions are verified with numerical simulations and a very good agreement is found between the results from the proposed bias expressions and the empirical results.

Hydraulic Modal Analysis in Theory and Practice

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Gudrun Mikota ◽  
Bernhard Manhartsgruber ◽  
Franz Hammerle ◽  
Andreas Brandl
Keyword(s):  
Frequency Response ◽  
Theory And Practice ◽  
Mode Shapes ◽  
Pipeline System ◽  
Response Functions ◽  
Hydraulic Systems ◽  
Modal Assurance Criterion ◽  
Frequency Response Functions ◽  
Practical Applications ◽  
The Impact

Theoretical and experimental modal analyses are treated for hydraulic systems modeled by discrete capacities, inductances, resistances, and fluid lines with dynamic laminar flow. Based on an approximate multi-degrees-of-freedom description, it is shown how hydraulic natural frequencies, damping ratios, and mode shapes can be identified from measured frequency response functions between flow rate excitation and pressure response. Experiments are presented for a pipeline system that includes three side branches and an accumulator. In view of practical applications, two different types of servovalve excitation as well as impact hammer excitation are considered. Pressure is measured by 19 sensors throughout the system. Results are compared in terms of frequency response functions between 50 and 850 Hz, the first five hydraulic modes, and weighted auto modal assurance criteria of experimental mode shapes. Out of the tested excitation devices, the servovalve is clearly preferred; if valves cannot be used, the impact hammer offers a reasonable workaround. For a reduced number of five sensors, different sensor arrangements are assessed by the respective weighted auto modal assurance criteria of experimental mode shapes. A theoretical hydraulic modal model provides a similar assessment. The quality of the theoretical model is confirmed by the weighted modal assurance criterion of theoretical and experimental mode shapes from servovalve excitation.

scholarly journals On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Simon Jones ◽  
Mathias Legrand
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Frequency Response ◽  
Natural Frequencies ◽  
Basis Functions ◽  
Mode Shapes ◽  
Response Functions ◽  
Partial Differential ◽  
Frequency Response Functions

The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler–Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.

Damage Detection of Shear Connectors Based on Power Spectral Density Transmissibility

2013 ◽  
Vol 569-570 ◽  
pp. 1241-1248 ◽  
Author(s):  
Jun Li ◽  
Hong Hao
Keyword(s):  
Spectral Density ◽  
Damage Detection ◽  
Frequency Response ◽  
Power Spectral Density ◽  
Experimental Studies ◽  
Measurement Data ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Shear Connectors ◽  
Power Spectral

Damage of shear connectors in slab-on-girder structures will result in shear slippage between slab and girder, which significantly reduces the load-carrying capacity of the bridge. This paper proposes a dynamic damage detection approach to identify the damage of shear connectors in slab-on-girder bridges with power spectral density transmissibility (PSDT). PSDT formulates the relationship between the auto-spectral density functions of two responses. Measured impact force and acceleration responses from hammer tests are analyzed to obtain the frequency response functions at the slab and girder sensor locations by experimental modal analysis. When measurement data from the undamaged structure are available, PSDT from the slab response to the girder response is derived with the obtained frequency response functions. PSDT matrices in the undamaged and damaged states are directly compared to identify the damage of shear connectors. When the measurement data from the undamaged structure are not available, PSDT matrices from measured response at a reference sensor response to those of the slab and girder in the damaged state can also be used to detect the damage of shear connectors. Experimental studies with a concrete slab supported by two steel girders are conducted to investigate the accuracy and efficiency of the proposed approach. Identification results demonstrated that damage of shear connectors can be identified accurately and efficiently with and without measurement data from the undamaged structure.

Modal Analysis of Gas Turbine Buckets Using a Digital Test System

1980 ◽  
Vol 102 (2) ◽  
pp. 357-368
Author(s):  
H. A. Nied
Keyword(s):  
Frequency Response ◽  
Modal Analysis ◽  
Gas Turbine ◽  
Test System ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Impulse Excitation ◽  
Digital Test

A modal analysis was conducted on gas turbine buckets using a digital Fourier analyzer. This digital test/computer system measures a set of frequency response functions for broadband impulse excitation at successive locations on the bucket airfoil. From the set of frequency response functions, the analyzer computes the modal parameters used to determine the natural frequencies, critical damping ratio and mode shapes of the turbine buckets. An animated display of the mode shapes for a discrete experimental model graphically revealed compound modes due to coupling. The test has shown that the digital modal analysis using the impulse excitation technique is a rapid and precise experimental method to determine the modal parameters of turbine buckets with a high degree of repeatability.

scholarly journals Sensitivity Analysis of Viscoelastic Structures

2006 ◽  
Vol 13 (4-5) ◽  
pp. 545-558 ◽  
Author(s):  
A.M.G. de Lima ◽  
M.H. Stoppa ◽  
D.A. Rade ◽  
V. Steffen Jr.
Keyword(s):  
Sensitivity Analysis ◽  
Frequency Response ◽  
Numerical Models ◽  
Complex Frequency ◽  
Response Functions ◽  
Viscoelastic Materials ◽  
Frequency Response Functions ◽  
Element Discretization ◽  
Analysis And Design ◽  
First Order

In the context of control of sound and vibration of mechanical systems, the use of viscoelastic materials has been regarded as a convenient strategy in many types of industrial applications. Numerical models based on finite element discretization have been frequently used in the analysis and design of complex structural systems incorporating viscoelastic materials. Such models must account for the typical dependence of the viscoelastic characteristics on operational and environmental parameters, such as frequency and temperature. In many applications, including optimal design and model updating, sensitivity analysis based on numerical models is a very usefull tool. In this paper, the formulation of first-order sensitivity analysis of complex frequency response functions is developed for plates treated with passive constraining damping layers, considering geometrical characteristics, such as the thicknesses of the multi-layer components, as design variables. Also, the sensitivity of the frequency response functions with respect to temperature is introduced. As an example, response derivatives are calculated for a three-layer sandwich plate and the results obtained are compared with first-order finite-difference approximations.

The Effect of Stringer Width and Damping on the Response of Skin—Stringer Structures

1972 ◽  
Vol 94 (1) ◽  
pp. 159-166 ◽  
Author(s):  
J. P. Henderson ◽  
A. D. Nashif
Keyword(s):  
Frequency Response ◽  
Transfer Matrix ◽  
Forced Response ◽  
Experimental Results ◽  
Mode Shapes ◽  
Response Functions ◽  
Resonant Frequencies ◽  
Frequency Response Functions

Analytical and experimental results for a five-span skin-stringer structure are presented. The analysis uses a transfer matrix technique which considers the effects of stringer dimensions including finite stringer width and damping on the computed forced response. Resonant frequencies, frequency response functions, damping and mode shapes for the first group of modes are compared for theoretical and experimental results. This agreement is found to be good.

A Simple Method for Extracting Normal Modes

1993 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Frequency Response ◽  
Normal Mode ◽  
Response Function ◽  
Frequency Response Function ◽  
Normal Modes ◽  
Mode Shapes ◽  
Phase Information ◽  
Simple Method ◽  
Resonant Frequencies ◽  
Complex Modes

Abstract A simple method for extracting the normal modes of structures is developed. The frequency response function relation between the complex and the normal modes is derived and a technique is developed to calculate the normal modes from the identified (damped) complex modes. In this method, only the magnitude and phase information at resonant frequencies are needed for extracting the normal mode shapes. A numerical example is employed to illustrate the theory. The results indicate that this method is more robust than other methods when the frequency response measurements are contaminated with noise.

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