Nomenclature j : response (or output) degree of freedom k : reference (or input) degree of freedom ... opbmlzation technique called âGoal Programmingâ 121 in.
MODAL PARAMETERS ESTIMATION USING AN OPTIMIZATION TECHNIQUE
L.F.L. Rosa; C. Magluta and No Roitman COPPE / UFRJ Civil Enaineerina DeDartment
POBOX &3506
ZIP 21945.970 Rio de Janeiro - RJ Brazil
ABSTRACT The estimabon of modal parameters using the optim;zat;on technique called “Goal Proarammino” is “resented. The estimaf;on is performed ;n thk frequency d&i”, trying to minimize the total squared error between the experimental and the theoretical values of the response functions. /ts purpose ;s to obta;n greater effinency ;!I complex cases (highly damped systems, high modal density systems and nase on exper;mental data), where most of the classical methods present low accuracy. Numerical simulations that reproduce complex cases adsing in practice were done with a set of FRFs. The resu/ts obtained through the developed algor;thm were compared to those obtained through a classkal modal parameters est;matlon method the Rational Fract;on Po/ynomia/s~
1. Introduction After performing an experimental test, the next stage in Modal Analysis is the estimation of modal parameters : natural frequencies, damping ratios and residues. The different existent modal parameters estimation methods have advantages and limitations and the choice must be done according to the specific case to be solved Some of the most complex cases that arise in practice are high modal density structures, heavily damped structures and experimental signals contaminated with noise. In these cases. most of the classical methods doesn’t present satisfactory accuracy~ “The increased use of highly damped materials, non-homogeneous materials, and active damping methodology requires a reevaluation of modal parameter estimation algorithms. One obvious technical need is the development and evaluation of the performance of modal parameter estimation algorithms capable of identifying systems with high modal density and high damping”, says Allemang [I].
Nomenclature j : response (or output) degree of freedom k :reference (or input) degree of freedom II /L”(w) : theoretical value of the FRF Il~~(a)j : experimental (measured) value of the FRF r : mode number N : total number of modes , ~ A ,L : residue value for mode r i., : pale value for mode r (* designates complex conjugate) 4 ,r : modal vector value at point j for mode r m, : modal mass for mode r Q, damped natural frequency for mode I (rad/s) n, : damping factor for mode r ;, : damping ratio for mode r (:l,,., : undamped natural frequency for mode r (radls) i undamped natural frequency for mode r (Hz) No : number of output degree of freedoms Ni : number of input degree of freedoms i=fi
Based on successfully experiences obtained with the opbmlzation technique called “Goal Programming” 121 in other areas [3-51, the authors decided to use it for estlmatlng modal parameters. So, the subject of this paper is the development of a modal parameter estimation technique using -Goal Programming”. “Goal Programming’ IS an optimization algorithm whose purpose is to minimize or maximize one or more functions subjected to some constraints. In the case of modal estimation, this function is the squared error between the experimental and the theoretical frequency response functions (FRFs). It is an iterative procedure, so an initial point is required~ The theory of the technique is presented and some numerical simulations that represent complex cases are performed to test it. To allow comparisons, the simulations were also analyzed using a classical method: the Rational Fraction Polynomials Method (6). in the frequency domain. 2 -Theory In Modal Analysis, there are several methods, in frequency domain, that estimate the modal parameters based on an
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error minimization approach~ Usually, they are multiple degrees of freedom and global methods that use, as error function to be minimized. the squared error between the experimental and the theoretical values of the response function, defined by Eq. (1) : 1,. I:],, = -&,,ico,j c;i(“,:) r-i,
The conventional formulation of a non linear programming problem is stated as: Minimize a scalar objective function of the problem variables \- : r(T):
(1)
; c II”
(7)
subjected to a set of constrants
where :
,~ g,j,/zo:
i=l?r,....I
(8.a)
I, = 1.X .ti
(8.b)
‘,r(“‘, = W(0,) ( 1 Ivy- lr’;y ia,,)
c-1
l,,(Lj =o.
e;,irt,i= \h.i~n’,(ll~~~(ari~II~~i~io,))
(2.b)
and to upper and lower bounds imposed to ;
F, < 1, _;I 16) The process involves the determination of the unknown coefficients in eq~ (3) (i,e.. the modal parameters) such that the total squared error is minimized. From this point, the several algorithms choose their individual procedures, making different simplifications and assumptions, So, the authors decided to try to make this error minimization using the optimization technique “Goal Programming”~ The “0” linear programming algorithm “Goal Programming’. developed and presented by lgnlao [2]. allows the search of multiple goals under rigid or flexible con&ants. Additional flexibility in the formulation allows for consideration of a compromised goal that can satisfy approximately two or more simultaneous criteria, each one measured by one respectwe performance Index.
Illill~llli/C
F[di\j] = $; /d, +J( j: 1
k _ II~~...K
(11)
where the actual variables (T,k) are indirectly Included since the deviations are determined through the difference between the sought values (h, ) and those obtamed on the problems In the modal identification problem, the goal is to obtain a group of variables (modal parametersl that minimize the squared error. For each vibration mode, the variables are the natural frequency, the damping ratio and some residues (one for each measured FRF)~ The effects of out-of-band modes can be taken into account by means of residual terms (residual mass and residual flexibility). increasing. in this case, the number of variables of the problems These variables must be limited within lower and upper bounds. This is considered by the algorithm as a goal of higher
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priority than the squared error minimization. So, the constraints that must be achieved are lower and upper bounds for each natural frequency; lower and upper bounds for each damping ratio -there are no constraints related to residues and to residual terms Some considerations about the modal parameters u s i n g t h e o p t i m i z a t i o n t e c h n i q u e ~Goal estlmatlon Programming’ must be dew To start the process, each variable must have a” initial estimate and a” initial increment. The initial estimates may be obtained through some other method available by the user. In other words, the “Goal Programming” algorithm will try to improve the accuracy of the modal parameters estimated through another modal parameter estimation methods The user must define the “umber of FRFs and the number of modes to be identified withln the frequency range of analysis~ In some cases. espeaally when the system presents high modal density, the user must use some tool to choose the correct number of modes [7]~ Other possibilities are : to give different weights to different parts of the FRFs and to exclude some parts of the FRFs. The first one allows that the resonance peaks neighborhoods have more influence on the error function and the second one allows the elimination of certain parts with a high level of noise, for example A tolerance value may be given by the user. so if all parameters differences, between two consecutive iterations, are lower than this tolerance, the program stops running~ 3
-Numerical
values and the real ones (for natural frequencies, damping ratios and residues). In tables 2 and 3 these errors are show” for the results obtained through RFP and GP respectively. The RFP’s results were used as starting values for the GP optimization algorithms In cases A and B, both methods yield very good results for all levels of noise. The efficiency of the estimation using GP is better than that obtained using RFP. mainly in case B with 10% of “we. In case C without noise, the parameters estimated through the two algorithms are exactly the real ones. When some noise is added, the quality of the results obtained through GP is higher than that of RFP, mainly with respect to damping and residues. Case 0 represents a heavily damped system with high modal density, So, it was the most difficult of the studied cases, and the accuracy of the damping and residues esbmated through GP was always greater than that of RFP, When 10% of noise is added, RFP was not able to identify the three existent modes Finally, in case E the modal parameters estimated using GP were better than those estimated using RFP once mow Figure 1 shows, for case C with 10% of noise, a comparlso” between the theoretical FRF and the FRF synthesized through RFP. Figure 2 shows. for the same case, a comparison between the theoretical FRF and the FRF synthesized through GP, One can see in these figures that the curve fitted through GP is mc~re accurate than the one fitted through RFP.
Simulations
In order to study the behavior of the developed technique for estimating modal parameters, some simulations were done with a set of response functions~ Each simulated FRF has three vibration modes. To allow a comparison. the parameters were also estimated through the Rational Fraction Polynomials Method [6], called here simply RFP. The algorithm that estimates modal parameters using the ~Goal Programming” technique is called here simply GP. The frequency response functions used are based on those proposed by Formenti [8] and they try to represent some different practical cases A Lightly damped, lightly coupled modes B Heavily damped, lightly coupled modes C Lightly damped, high modal density D Heavily damped, high modal density E Coincident frequencies Table 1 shows the values of modal parameters for the five cases abow The FRFs were synthesized according to expression (3). In order to simulate experimental functions, three levels of noise (2, 5 and 10%) were added to the theoretical FRFs. The level of noise is the relation between the RMS value of the noise and the RMS value of the theoretical response function. The frequency ranges also appear in table 1~ The quality of the identification is expressed through the relative error between the estimated modal parameter
Based on the comparison above, the authors conclude that the efficiency of the modal estimation using the ~Goal Programming~ technique in complex cases is higher than that of the Rational Fraction Polynomials Method, mainly with respect to damping ratios and residues. Concerning to natural frequencies, both algorithms yield good results in all cases~ In the simulations. only one random signal was added to each response functlon~ A statistical treatment should be done to guarantee the reliability of the results~ However, the expressive difference of accuracy obtained through the two estimation algorithms are e n o u g h t o a s s u r e t h e conclusions, An important topic in the modal parameters estimation using GP is the initial point, which includes initial estimates for the modal parameters and their initial increments. Some tests were performed to study the influence of the initial point on the final results. The authors concluded that in the most difficult cases. where the influence of other modes on the neighborhood of a certain resonance peak is great (as cases C. D and E). the results may be dependent on the initial point. So, in these cases the initial increments for damping ratios and residues must be high in relation to their initial estimates for avoiding the iterative procedure of being “bounded” to a local minimum. For example, case D with 2% of noise can be mentioned. The damping ratio and Imaginary part of the residue of the second vibration mode
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estimated through RFP were, respectively, 0.10% (real = 5.0%) and -6.55E.7 [s/kg] (real = -l.OOE-3 [s/kg]). Using these values as initial estimates on GP, with little initial Increments, the optimization algorithm didn’t yield satisfactory results. However using the same initial estimates with high initial increments (0,8% for damping ratio and l.e-4 [s/kg] for residuei, the results obtained through GP were much better damping ratio = 4~62% (relative error of 7.6%) and residue = -804e-4 [s/kg] (relative error of 19,6%)~ The modal parameters estimation using GP in complex cases will depend on the us&s experience and, in most of the cases. the procedure must be repeated a few times. modifying the initial point until the final results converge to the same point Usually, the algorithms used for yielding initial estimates have low efficiency on the damping ratios and residues estimation in complex cases~ So, the authors advise, as mentioned. the use of high initial increments for these modal paiameters~ 4 Concluding Remarks The estimation of modal parameters using the optimization algorithm called Goal Programming was developed and its application to solve several complex problems showed its good performance, Heavily damped systems and high modal density are some of the most difficult problems that arise I” practice and the comparison between the results obtained through the developed algorithm and those obtained through another classical method (the Rational Fraction Polynomials Method) proved the efficiency of the first one, The neediness of initial values for the parameters is not a great problem and results from other estimateon methods can be used The quality of these initial values ~miay accelerate the convergence of the process, but doesn’t have influence on the flnal results since high initial increments are used, A disadvantage of the developed algorithm is its running time_ mainly when a global estimation is done
The main advantage of the modal parameters estimation using GP algorithm is its ability for obtaining results with high accuracy even using ill initial estimates, 5 - References [I] Allemang. R, J.; ~Modal Analysis . Where Do We Go From Here 7’. Proc, of the 11th InternatIonal Modal Analysis Conference, 1993, [Z] Ignizio, J,P~. ~Goal Lexington Books, 1976
Programming and Extensions”,
13) Vasconcelos. Jo M . A~: Batista. R C . ; Cyrino, Jo C~ ‘Design of Semisubmersible Columns Using Multiple Criteria Methods and Expert Systems~. PRADS’92 Practical Design of Ships and Moblle Units. Volt 2. pp 981. 994, Elsevier Applied Saence. U K May 1992. [4] Magluta. C~: “Paswe Dynamic Absorbers to Attenuate Structural Vibrations’. D. Sc~ Dissertation (I” Portuguese), COPPE/UFRJ. Rio de Janetro. Brazil, 1993, [5] Neves. F A,. “Multi-Objective Programming Applied to the Optimization of Cable-Stayed Bridges Project”, D SC Dissertation (in Portuguese), to be presented. COPPEIUFRJ. Rio de janaro,Brazil [6] The Star System Structural Measurement System, GenRad Structural Tests Products. November 1994~ [i] Cafeo. J A~, D e Clerck. Jo Pi, “ A n A s s e s s m e n t Function for Improving the Accuracy of Estimated Modal Parameters”. The International Journal of AnalytIcal and Experimental Modal Analysis October 1995, Vol 10~ Number 4 181 Formenti, D.L. “A Study of Parameter Esttlmation Errors A Curve Fitbng Challenge”, Structural Measurement Systems, 1982~
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TABLE 2 Relative errors between the modal parameters estimated through RFP and the real ones (%) LCYCL VT NOISE
0 %
I
2%
I
5 Q/n
I
I” %
TABLE 3 Relative errors between the modal parameters estimated through GP and the real ones (%)~
I
