ModalID Modal Identi cation and Diagnosis User Guide

1 Institut Sup erieurde l A eronautique et de l EspaceResearch ProjectModalIDModal Identification and DiagnosisUser GuideAuthor:ElisaBoscoAnkitChiplunkarSup ervisor:Dr. JosephMorlierJune 27, 2012 June 27, 2012 ModalID - user GuideContents1 Getting started22 About the ModalID toolbox23 Installing ModalID34 Terms of use45 Theoretical LSCE - Least Square Complex Esponential .. UMPA - Unified Matrix Complex Polynomial Approach .. Order Frequency Domain Algorithm .. Order Frequency Domain Algorithm ..86 Functions - By format87 Functions - By General functions .. LSCE Method .. UMPA Method .. Validation method .. 128 Example121 June 27, 2012 ModalID - user Guide1 Getting startedThis document describes how to start using theModalIDtoolbox for provide an overview of all the functions used inModalIDand a tutorial,presenting an example to illustrate the use of this graphical this Guide contains a brief introduction on the two Modal analysis meth-ods that are implemented in the toolbox.

2 Anyway this document should not beconsidered as a textbook on Modal analysis hence, for interested users, more in-depth examination is presented in the works cited in the bibliography, [1, 2].2 About the ModalID toolboxThe past three decades have seen the development of several Modal analysis soft-ware packages, starting from SDOF methods and leading to more efficient andgeneral MDOF development of this toolbox aims at an easy tool that allows to determinethe Modal parameters of a simple structure before and after its damage. This isfollowed by an analysis of the results in order to relate the change in the modalparameter to the level of degradation of the structure till this date two MIMO (multiple input/multiple output) identification meth-ods have been implemented: theUnified Matrix Ploynomial MethodThis method makes the analysis in frequency domain on MIMO system; thismethod has been the major part of our contribution in this toolbox theLeast-Square Complex Exponential;This is a time domain method.

3 The data analysis achieved by this method isdone making use of the codes available in theEasyMod/EasyAnimsoftwarepackage [3].Several Matlab functions have been developed and used for various applicationsin structural dynamics: reading and writing of UFF (uniuversal file format) files, mode indicators (sum of FRFs, sum of FRF real part and sum of FRFsimaginary part) and their visualisation,2 June 27, 2012 ModalID - user Guide MAC ( Modal assurance criterion) and Modal collinearity for a comparisonof two sets of 1: Schematic operating diagram of the toolbox ModalID3 Installing ModalIDModalIDcan be found as a RAR archive is written to work on Matlab, therefore the archive should be ex-tracted to a directory on the hard disk, e.

4 G. on Windows OS:C:\Programs\MATLAB\R2010a\toolbox\Aft er extracting the RAR archive the directory will contain different the toolbox by launching must be addes to the MatLab path to make the toolboxfunctions available in MatLab: In MatLab, click onFile, Set Path ..3 June 27, 2012 ModalID - user Guide Click onAdd with Subfoldersand select theModalIDdirectory. Save the path and close the dialog Terms of useModalIDis a free software; you can redistribute it and/or modify publications presenting results obtained withModaIDmust include aproper reference:[1] G. Kouroussis, L. Ben Fekih, C. Conti, O. Verlinden, EasyMod: A Mat-Lab/SciLab toolbox for teaching Modal analysis ,Proceedings of the 19th Interna-tional Congress on Sound and Vibration, Vilnius (Lithuania), July 9-12, Theoretical IntroductionThis section deals with a quick overview on the two methods of analysis utilizedby the LSCE - Least Square Complex EsponentialLeast Square Complex Exponential is a time domain Modal analysis method.

5 Itexplores the relationship between the IRF of a MDOF system and its complexpoles and residues through a complex exponential. By establishing the analyticallinks between the two, we can construct an AR model. The solution of this modelleads to the establishment of a polynomial whose roots are the complex roots of thesystem. Having estimated the roots (alias the natural frequencies and dampingratios), the residues can be derived from the AR model for mode shapes. TheIRF can be derived from the inverse Fourier transform of an FRF or from randomdecrement process. The LSCE method begins with the transfer function of aMDoF system, follows its inverse Laplace transform to get the (t) =2N k=1(Aij)reskt(1)The IRF may be sampled at a series of equally spaced time 27, 2012 ModalID - user Guidehk= 2Nk=1(Aij)rzkr(k= 0,1.)

6 ,2N)zkr=esrk All these samples are real value data, although the residues (Aij)rand the rootssrare complex quantities. The next step is to estimate the roots and residues fromthe sampled data. This solution is aided by the conjugacy of the roots. Mathe-matically, this means thatzrare roots of a polynomial with only real coefficients: 0+ 1zr+ 2z2r+ + 2N 1z2N 1r+ 2Nz2Nr= 0(2)The coefficients can be estimated from the samples of the IRF data. since thereare 2N+ 1 equalities in the IRF equation, we can multiply each equality with acorresponding coefficient and add all equalities together to form the followingequation:2N k=0 khk=2N r=1(Aij)r2N k=0 kzkr(3)We know that the right hand side is going to be zero whenzris a root ofthe polynomial equation 2.

7 This will lead us to a simple relationship between thecoefficients and the IRF samples, namely:2N k=0 khk= 0(4)This equation offers a numerical way of estimating the coefficients. In equa-tion 2 we can assign 2 Nto be one. Taking a set of 2 Nsamples of IRF, one linearequation is formed 4. Taking 2 Nsets of 2 Nsamples of IRF, a set of 2 Nlinearequations is drawn: h0h1h2 h2N 1h1h2h3 1h2Nh2N+1 h4N 2 0 2N 1 = h2Nh2N+ 1 (5)The selection of IRF data samples can vary provided that thehelements ineach row are evenly spaced in sampling and sequentially arranged. No two rowshave identicalhelements. The number of rows in equation 5 can exceed thenumber of coefficients for the least-square the known coefficients, equation2 can be solved to yield thezrroots.

8 Theseroots are related to the system complex natural frequenciessr. Since the complexnatural frequenciessrare determined by the undamped natural frequencies randdamping ratios r, as shown below:5 June 27, 2012 ModalID - user Guidesr= r r+j r 1 2r(6)s r= r r j r 1 2r(7)we can derive the natural frequency and damping ratio of the rth mode as: r=1 lnzrlnz r(8) r= ln(zrz r)2 r (9)To determine the mode shapes of the system from the IRF data, we can write: 11 1z1z2z3 11z2N 12 z2N 12N (Aij)1(Aij) (Aij)2N = 1 (10)The solution to this set of linear equations will yield the residues. The aboveanalysis describes the main thrust of the LSCE method and its UMPA - Unified Matrix Complex Polynomial ApproachThe Unied Matrix Polynomial Approach (UMPA) is an historical attempt to placemost commonly used Modal parameter estimation algorithms within a single ed-ucational framework.

9 It is a frequency domain MDOF mthod for extracting themodal parameters of a system. To understand its formulation, the polynomialmodel used for frequency response functions is ( i) =Xp( i)Fq( i)= nk=0 k(j )k mk=0 k(j )k(11)Rewriting this model for a general multiple input, multiple output case andstating it in terms of frequency response functions:[m k=0(j )k[ k]][Hpq( i)] = [n k=0(j )k[ k]](12)This model in the frequency domain is the AutoRegressive with eXogenousinputs (ARX(m,n)) model that corresponds to the AutoRegressive (AR) model intime domain for the case of free decay or impulse response data:6 June 27, 2012 ModalID - user Guidem k=0[ k]hpq(ti+k) = 0(13)The general matrix polynomial model concept recognizes that both time andfrequency domain models generate functionally similar matrix polynomial mod-els.

10 This model which describes both domains is thus termed as Unified MatrixPolynomial Approach (UMPA). Low Order Frequency Domain AlgorithmLower order, frequency domain algorithms are basically UMPA based models thatgenerate first or second order matrix coefficient polynomials. Starting with themultiple input, multiple output frequency response model a second order matrixpolynomial model is formed.[m k=0(j )k[ k]][Hpq( i)] = [n k=0(j )k[ k]](14)for orderm= 2[[ 2](j i)2+ [ 1(j i) + [ 0][H( i] = [ 1(j i)] + [ 0](15)This basic equation can be repeated for several frequencies and the matrixpolynomial coefficients can be obtained using either [ 2] or [ 0] normalization.)]]