Pedersen, Ivar Chr. Bjerg; Hansen, Søren Mosegaard; Brincker, Rune; Aenlle, Manuel López

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1 Downloaded from vbn.aau.dk on: Aprl 2, 209 Aalborg Unverstet Load Estmaton by Frequency Doman Decomposton Pedersen, Ivar Chr. Bjerg; Hansen, Søren Mosegaard; Brncker, Rune; Aenlle, Manuel López Publshed n: Proceedngs of the 2nd Internatonal Operatonal Modal Analyss Conference Publcaton date: 2007 Document Verson Publsher's PDF, also known as Verson of record Lnk to publcaton from Aalborg Unversty Ctaton for publshed verson (APA): Pedersen, I. C. B., Hansen, S. M., Brncker, R., & Aenlle, M. L. (2007). Load Estmaton by Frequency Doman Decomposton. In R. Brncker, & N. Møller (Eds.), Proceedngs of the 2nd Internatonal Operatonal Modal Analyss Conference (Vol. Vol. 2, pp ). Aalborg Unverstet. General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts.? Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research.? You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan? You may freely dstrbute the URL dentfyng the publcaton n the publc portal? Take down polcy If you beleve that ths document breaches copyrght please contact us at vbn@aub.aau.dk provdng detals, and we wll remove access to the work mmedately and nvestgate your clam.

2 LOAD ESTIMATION BY FREQUENCY DOMAIN DECOMPOSITION Ivar Chr. Bjerg Pedersen Søren Mosegaard Hansen Rune Brncker Manuel López Aenlle Department of Buldng Technology and Structural Engneerng. Aalborg Unversty. Department of Buldng Technology and Structural Engneerng. Aalborg Unversty. Department of Buldng Technology and Structural Engneerng. Aalborg Unversty. Department of Constructon and Manufacturng Engneerng. Unversty of Ovedo Denmark Denmark Denmark Span / Abstract When performng operatonal modal analyss the dynamc loadng s unknown, however, once the modal propertes of the structure have been estmated, the transfer matrx can be obtaned, and the loadng can be estmated by nverse flterng. In ths paper loads n frequency doman are estmated by analyss of smulated responses of a 4 DOF system, for whch the exact modal parameters are known. Ths estmaton approach entals modal dentfcaton of the natural egenfrequences, mode shapes and dampng ratos by the frequency doman decomposton technque. Scaled mode shapes are determned by use of the mass change method. The problem of nvertng the often sngular or nearly sngular transfer functon matrx s solved by the sngular value decomposton technque usng a lmted number of sngular values. The dependence of the egenfrequences on the accuracy of the scalng factors s nvestgated and the errors on the estmated loads are determned. Nomenclature Estmated natural egenfrequency f Estmated dampng rato ζ Estmated unscaled mode shape matrx Φ Scalng factor dagonal matrx α Estmated scaled mode shape matrx Ψ Frequency response functon (FRF) H ( ω) Smulated response Y ( ω) Re-estmated load X ( ω) Correlaton coeffcent functon Frequency resoluton ρ YY Samplng frequency f s f Power spectral densty S ( ω ) Mean value µ Coeffcent of varaton δ

3 Introducton Operatonal Modal Analyss (OMA) s manly concentrated on analyss of measured responses. The nput load s thus n far the most cases unknown. Ths unmeasured nput s, however, of great nterest snce valuable nformaton on the magntude and dstrbuton of such ambent exctatons as wnd, wave and traffc loads etc. can be estmated. The process of estmatng nput loads from measured responses nvolves at frst dentfcaton of the modal parameters ω, Φ and ζ, from whch the frequency response functon (FRF) can be formulated, see (2) and (3). Scaled mode shapes Ψ are n ths relatonshp needed n order to estmate the FRF correctly, see (4). The modal parameters are dentfed wth the frequency doman decomposton (FDD) technque, see [], whle the scaled mode shapes are determned by use of the mass change method, see [2]. The governng equatons are: X H Y () ( ω ) = ( ω) ( ω) where the FRF s calculated by: H H ( ω) 0 H ω = ψ ψ m ψ ( ) ψ ( m) n n 0 H n n m ( ω) ( ) ( ) ( ) (2) H k ( ) ω = 2 2 ω + 2 ζ ω ω + ω k k k, k ;m The scalng factor α, whch here s formulated on dagonal matrx form, s defned by [2]: α = kk ( ω k ) ( ω k ) 0 ( ω k ) ( φ k ) M ( φ k ) 2 H Ψ = Φα The mass change method s based on repeated egenvbraton tests performed pror and post to structural modfcatons of the structure beng consdered. The structural modfcaton s performed by applyng addtonal masses to the structure, whch result n changed egenfrequences, and possbly changed mode shapes. The superscrpt 0 ndcates modal parameters of the unmodfed structure, whle superscrpt ndcates modal parameters of the modfed structure. M s the devaton mass matrx between the unmodfed and modfed structure. [2] suggests a homogenous mass change of 5% of the total mass. The load estmaton procedure nvolves furthermore an nverson of the FRF, see (). Problems wth ths nverson are solved by use of the sngular value decomposton (SVD) technque, snce the FRF matrx s often sngular or close to sngular. In ths case the nverson s performed by the followng actons [3]: ( ) ( ) H H ω = U S ω V (5) (3) (4) H V U 0 0 ( ) D( ω) 0 H ω = (6)

4 where U, S and V are matrces obtaned from the sngular value decomposton of the FRF. D s the non-sngular part of S. (6) s the also referred to as the Moore-Penrose pseudo nverse. The results presented n ths paper are based on smulated responses, for whch the nput loads are known. In ths way the re-estmated loads can be compared wth the exact nput loads. The modal parameters are dentfed from the smulated responses, and by use of the scalng factors and the modal parameters, the FRF s estmated, see (2) and (3). An estmaton overvew s gven n Fgure. FFT x ( t) X ( ω) Y ( ω) = H ( ω) X ( ω) IFFT FFT ym ( t) = y ( t) + y n ( t) Y m ( ω ) S ( ω) vs. S ( ω) X ( ω) = H ( ω) Ym ( ω) Moore-Penrose H ( ω) ( ) α, H ω f, Φ,ζ FDD S yy ( ω) Fgure. Estmaton overvew. y n s addtonal nose added to the generated response. The system on whch the responses are smulated s a 4 DOF system, whch s shown n Fgure 2. f f 2 f 3 f 4 x m m x 2 m 2 m 2 x 3 m 3 m 3 x 4 m 4 k c k 2 c 2 k 3 c 3 k 4 c 4 Fgure 2. 4 DOF system for load estmaton analyss. Sample records of two dfferent lengths are smulated; the second one ten tmes longer than the frst one. The smulated responses are splt up n 00 and 000 data segments, respectvely (no wndowng has been used). The nput load s a Gaussan whte nose load generated segment wse n MATLAB. Ths elmnates the effect of leakage, whch otherwse would occur when no wndowng s used. The samplng frequency f s s chosen to 0.52 Hz. whle the frequency resoluton f s equal to 0.00 Hz. Ths gves 52 frequency lnes n frequency doman. The followng load estmaton analyss s partly concentrated on the nfluence of the estmated egenfrequences on the accuracy of the scalng factors, and partly on the error on the re-estmated loads compared to the exact loads (dstnctons are made between the exact loads, the realzed loads and the re-estmated loads). For ths purpose 00 smulatons are performed for each sngle calculaton n order to obtan a representatve sample quantty. The load comparson s made upon the spectral denstes n a bandwdth of a /6 decade wth /3 of the Nyqust frequency f ν as the centre frequency. The error ε s thus defned by:

5 f 3 νβ f 3 νβ exact S ( ω) dω S ( ω) dω f f 3 ν 3 ν exact ( S ( ),S β β ε ω ( )) ω = 00 [%], f 3 νβ exact S ( ω) dω 3 fν β β= 0 2 f fs ν = 2 In Fgure 4, Fgure 5 and Fgure 6 the centre frequency and the band wdth are shown, respectvely. The centre frequency s chosen n the centre of the most modal domnated area, as t s seen from Fgure 4. 2 Enhanced Egenfrequency Estmaton From prelmnary studes t has been concluded that the accuracy of the scalng factor s strongly dependent on the accuracy on the estmated egenfrequences, and to a lesser extent on the estmated mode shapes. The accuracy of the peak pcked egenfrequences s dependent on the frequency resoluton f, whch means that long measurements mght be requred n order to obtan a satsfactory resoluton of the frequency axs. The estmated auto correlaton coeffcent functon ρ YY, whch s also used for dampng estmaton [4], may alternatvely be used to perform an enhanced egenfrequency estmaton wthout ncreasng the frequency resoluton. At frst, all postve and negatve extreme values of ρ YY are dentfed, as t s also the case when estmatng dampng ratos [4]. Next, all crossngs wth the tme decay axs are dentfed. A lnear regresson of all the dentfed ponts of tme provdes an estmate of the damped egenperod T d, see Fgure 3. The enhanced estmate of the egenfrequency s fnally determned upon the followng well-known relatonshp: (7) 2π ω = 0 2 T d ζ (8) Fgure 3. Extreme values and zero crossngs of the estmated auto correlaton coeffcent functon for enhanced egenfrequency estmaton.

6 For the 4 DOF system n Fgure 2 the followng exact, peak pcked and enhanced egenfrequences are determned. For the enhanced egenfrequences t s the mean values µ and coeffcents of varaton δ over 00 smulatons that are shown. Mean values and coeffcents of varaton of the errors on the correspondng scalng factors are shown n Table 2. Table. Exact, peak pcked and enhanced egenfrequences. µ = mean value of enhanced egenfrequency, δ = coeffcent of varaton of enhanced egenfrequency [%]. f / f Exact f [Hz] µ and δ [%] on estmated and enhanced f [Hz] Peak pcked f [Hz] Enhanced f [Hz] (00 seg.) Enhanced f [Hz] (000 seg.) = µ = / δ = 0.3 µ = / δ = 0. = µ = / δ = 0.3 µ = / δ = 0. = µ = / δ = 0.3 µ = / δ = 0. = µ = / δ = 0.3 µ = / δ = 0. Table 2. Mean values and coeffcents of varaton of the errors on the estmated scalng factors n accordance wth the peak pcked and enhanced egenfrequences, respectvely. µ and δ [%] on error of (, ) 2 α f φ [ mass ] for mass changes of 2% and 5%, respectvely. Peak pcked Enhanced Enhanced φ exact = = = = f [Hz] f [Hz] (00 seg.) f [Hz] (000 seg.) α α 2% M α 5% M α 2% M α 5% M α 2% M α 5% M µ = 55.9 δ = 64.7 µ = 28.8 δ = 0.6 µ = 46.9 δ = 00. µ = 57.2 δ = 8.4 µ = 6.5 δ = 4.3 µ = 8.2 δ = 87. µ = 4.5 δ = 89.5 µ = 6.0 δ = 75.3 µ = 4.5 δ = 95.6 µ = 6. δ = 76.6 µ = 7.8 δ = 70. µ = 7.9 δ = 86.0 µ = 6.0 δ = 83.3 µ = 5.6 δ = 82.5 µ = 6.7 δ = 78.3 µ = 6.7 δ = 70.9 µ = 6.0 δ = 62.0 µ = 5.0 δ = 70.7 µ = 5.0 δ = 70.5 µ = 4.5 δ = 77.7 µ = 2. δ = 74.7 µ =.9 δ = 8.5 µ = 2. δ = 73.3 µ = 2.6 δ = 73.9 The accuracy of the egenfrequences s very decsve for the error on the scalng factor, whch s seen from Table and Table 2. In the end, the magntude of the FRF s very dependent on the scalng factor, whch agan s decsve for the best possble re-estmate of the load. It s furthermore seen from Table 2 that the best results are obtaned wth a mass change of 5% and 000 segments n preference to 00 segments.

7 3 Load Estmaton In the overall perspectve, three dfferent cases are present n reference to (2). In the frst case the response s known n all four degrees of freedom and four mode shapes are known. Ths means that m = n n (2). In practce, however, one may often have ether more responses than mode shapes (m < n), or more mode shapes than responses (m > n). Ths load estmaton analyss s however lmted to the frst case,.e. 4 responses and 4 mode shapes. The load estmaton analyses are performed wth scalng factors determned wth 5% mass changes upon the conclusons drawn from Table 2. Calculatons are performed wth no addtonal nose, % nose and 5% nose added to the generated response, respectvely. The results, presented n the form of the errors defned by (7), are shown n Table 3 and Table 4. Table 3. Mean values and coeffcents of varaton of the errors on the spectral denstes of the reestmated loads. DOF S exact ( ω realzed ) ( ) µ and δ [%] on error of S ( ω ) [N 2 /Hz] (00 seg.) S ω Wthout nose % nose 5% nose = µ = 4.5 δ = 97. µ = 7.0 δ = 9.3 µ = 8.9 δ = 07.9 = µ = 7.2 δ = 83.5 µ = 7.6 δ = 96.4 µ = 20.5 δ = 06.8 = µ = 6.3 δ = 86. µ = 8.0 δ = 96.0 µ = 20.8 δ = 04.6 = µ = 4.5 δ = 93.7 µ = 6.5 δ =92.7 µ = 20.4 δ = 07.8 Table 4. Mean values and coeffcents of varaton of the errors on the spectral denstes of the reestmated loads. µ and δ [%] on error of S ( ω ) [N 2 /Hz] (000 seg.) DOF S exact ( ω realzed ) ( ) S ω Wthout nose % nose 5% nose = µ = 4.7 δ = 77.7 µ = 4.8 δ = 76.7 µ = 5.8 δ = 76.0 = µ = 4.8 δ = 72.0 µ = 5. δ = 79.7 µ = 5.6 δ = 78.6 = µ = 4.9 δ = 74.2 µ = 5.3 δ = 77.2 µ = 5.6 δ = 82.0 = µ = 5.2 δ = 70.6 µ = 5. δ = 77.5 µ = 5.7 δ = 79.3

8 The dfference between the exact spectral denstes and the realzed spectral denstes s less than 0.5%. Ths error s due to the fact that the measurement seres have fnte lengths, and the error must thus be conceved as a random error. The errors on the estmated spectral denstes are conceved as random errors too, snce the estmated spectral denstes converge towards the realzed spectral denstes as the number of data segments ncreases. In Fgure 5 and Fgure 6 examples of the realzed and re-estmated loads are shown for the two cases (00 segments / 000 segments). 4 Concluson It s seen from Table 2 that the error on the scalng factor spans from.9% to 57.2%. The best result s obtaned by splttng up of the long measurement n 000 segments, enhanced egenfrequences and a homogeneous mass change of 5%, whle the most naccurate results s obtaned wth the peak pcked egenfrequences and a mass change of 2%. Generally seen, the best results are obtaned wth a 5% mass change and by use of the response averaged over 000 segments. The errors on the spectral denstes of the loads span from 4.8% to 20.8%. Agan, the best result s obtaned wth 000 segments. The addtonal nose has only a lttle nfluence on the end result, even wth 5% nose added. Snce the centre frequency for the nvestgated frequency band s centred n the modal domnated area the addtonal added nose consttute a lmted part of the total response. The errors on the estmated spectral denstes are larger n the areas, where the nose s somewhat more domnatng compared to the orgnal generated response. The errors on the estmated spectral denstes seem to converge towards zero as the number of data segments ncreases. Whether the estmate s based or unbased can, however, not be answered wthout performng calculatons wth longer measurements and more data segments. When estmatng the loads usng the exact modal parameters, the errors are equal to zero, whch ndcates that the estmate s unbased. 5 References [] Brncker, R., Zhang, L. and Andersen, P.: Modal Identfcaton from Ambent Responses usng Frequency Doman Decomposton, Proc. of the 8 th Internatonal Modal Analyss Conference [2] Aenlle, M. L., Brncker, R., Cantel, A. F., and García, L. M. V.: Scalng Factor Estmaton by the Mass Change Method, Proc. of the st Internatonal Operatonal Modal Analyss Conference [3] Aenlle, M. L., Brncker, R. and Cantel, A. F., Load Estmaton from Natural Input Modal Analyss, Proc. of the 23 rd Internatonal Modal Analyss Conference [4] Brncker, R., Ventura, C. E., and Andersen, P., Dampng Estmaton by Frequency Doman Decomposton, Proc. of the 9 th Internatonal Modal Analyss Conference

9 6 Graphcs Fgure 4. Spectral densty S YY (ω) of response. Note the centre frequency /3 f ν. Fgure 5. Example of realzed and estmated spectral densty S (ω) of load for 00 segments. Fgure 6. Example of realzed and estmated spectral densty S (ω) of load for 000 segments.