A Method for Damping Estimation Based On Least Square Fit

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1 Amerca Joural of Egeerg Research (AJER) 5 Amerca Joural of Egeerg Research (AJER) e-issn: p-issn : Volume-4, Issue-7, pp Research Paper Ope Access A Method for Dampg Estmato Based O Least Square Ft Jtao Gu, Mepg Sheg School of Mare Scece ad Techology, Northwester Polytechcal Uversty, P. R. Cha Abstract: A ew approach based o least square ft method s proposed to estmate dampg. Nose resstace of the proposed method ad half-power badwdth method are aalyzed ad compared by plety of smulatos wth dfferet sgal-to-ose ratos (SNR). The proposed method s more accurate ad stable tha half-power badwdth method all SNRs, especally whe the ose level s hgh. If SN R 3dB, the proposed method should be used for dampg estmato stead of half-power badwdth method. A dampg estmato expermet s carred out wth both methods, ad the results dcate ad verfy that there s smaller varablty for the proposed method. Keywords: Least square ft, dampg estmato, half-power badwdth I. INTRODUCTION Sce dampg s a valuable parameter for resoat respose of structures or systems, dampg s of great sgfcace for structural dyamcs. It s dffcult to obta dampg through a theoretcal method so that the uversal way for dampg estmates s expermetal vestgato[,]. alf-power badwdth method[3,4], whch calculates resoat dampg wth frequecy badwdth for vbrato eergy decreases 3dB ad the resoat frequecy, s a wdely used approach for dampg estmato. alf-power badwdth method s troduced as ma method for dampg estmates the Amerca test stadard ASTM E Bertha [5] obtaed dampg of a system that does ot possess real modes wth half-power badwdth method. Guo [6] mproved the half-power badwdth method ad proposed a ew method based o tegral opo. Badsar [7] determed the materal dampg rato shallow sol layers wth the half-power badwdth method. As the half-power badwdth method uses oly three data pots, ts accuracy wll be affected by the sgal-to-ose rato (SNR). Whe some of the three data pots are serously affected by ose, a large error wll appear ad caot be eglected. The am of ths work s to put forward a ew method whch has hgh accuracy ad strog ose resstace by usg the ampltude of frequecy respose. II. ALF-POWER BANDWIDT METOD A wdely used method, amely half-power badwdth method, s troduced as follows. The dampg loss factor ca be obtaed by the quotet of half-power badwdth ad the resoat frequecy, as show equato (). The dagrammatc sketch of the method s show Fg.. f alf-power Badwdth f () f Fg.. alf-power badwdth method w w w. a j e r. o r g Page 5

2 Dmesoless ampltude Amerca Joural of Egeerg Research (AJER) 5 The half-power badwdth method s a classcal ad wdely used method, but there s a fatal short comg that f the sgal-to-ose rato (SNR) s ot hgh eough. It s dffcult to recogze the half-power badwdth, see Fg Fg.. alf-power badwdth method wth low SNR It ca be see Fg. that the half-power badwdth s hard to decde because there are more tha two frequeces related to half-power respose. III. DAMPING ESTIMATION BASED ON LEAST SQUARE FIT The parameter of dampg s ofte estmated wth data ear the resoat frequecy. The wdely used half-power badwdth method for dampg vestgato oly uses three data pots of frequecy respose so that t wll cause a large error whe the expermetal data s affected by ose. It s expected to obta a more accurate dampg f more data pots of frequecy fucto are used. Therefore, a ew approach based o least square ft method s proposed whch a umbers of data pots ear resoat frequecy are used. The ampltude of frequecy fucto s F K () where F s exctato force, K the stffess, atural crcle frequecy, ad the loss factor. It s obvous that the maxmum ampltude ca be obtaed m ax F (3) K The dmesoless ampltude s defed as = = m ax Rewrte equato (4) as = Make that x ad y f x x (6), (7) (5) (4) A umber of data pots x, y,,, ca be obtaed accordg to the expermetal frequecy respose. The resdual error s defed as Dmesoless frequecy w w w. a j e r. o r g Page 6

3 Dmesoless ampltude Amerca Joural of Egeerg Research (AJER) 5, (8) err y f x The dampg detfcato ca be trasferred to gettg a proper value of to make the resdual error the most mmum. Gauss-Newto teratve method s used to fd the optmal parameter of dampg. Expad f x, to Taylor seres at pot of tal value as,, ', f x f x f x (9) where f ' x, d f Rewrte equato (8) as x,. d, ', () err y f x f x y g x where y y f x, g x,, g x, f ' x,. Equato () s a typcal least square optmzato problem. The estmated ca be obtaed for the frst geerato by solvg equato (). T T () A A A b where,,,,,, T, b y, y,..., y A g x g x g x. After s obtaed, wll be replaced by ad repeat the above step to get the estmated value for the secod geerato. The terato wll be fshed f the dfferece betwee the eghbor geeratos of s small eough for satsfactory demad. The last geerato of s the estmated dampg. The expermetal ad fttg respose are show Fg. 3 ad dampg ca be obtaed the fttg process expermetal respose fttg respose Fg. 3. Expermetal ad fttg respose IV. RESISTANCE TO NOISE Ths smulato s coducted to llustrate the ose sestvty of the proposed methods. A sgledegree-of-freedom system s adopted to verfy the valdty of least square ft method. A comparso s made betwee least square ft method ad half-power badwdth method. Gaussa radom ose s appled to the smulated frequecy respose, ad the sgal-to-ose rato (SNR) s defed as where S N R lo g sgal ose s ampltude of effectve sgal ad s ampltude of ose. s g a l o s e The smulated sgal parameters are., f 5 z, = f,ad f. z. It s smulated o samples wth the SNR the rage from db to 5 db. Errors of estmated dampg wth the two methods are show Fg. 4. I Fg. 5, each method s tested for 4 ose levels, whch s from db to 5dB wth db terval, wth samples each ose level. The average error s calculated by equato (3). a v e rag e erro r N N Dmesoless frequecy ex act ex act (3) where s the detfed dampg, s the exact loss factor set the smulato, N s the umber e x a c t of samples. w w w. a j e r. o r g Page 7 ()

4 Error% error% error% Amerca Joural of Egeerg Research (AJER) (a) (b) Fg. 4. Error of the estmated dampg versus SNR: (a) proposed method, (b) half-power badwdth method. 4 alf-power badwdth Proposed method Fg. 5. Average error for the two methods It ca be see Fg. 4 that detfed dampg by the two methods are more ad more accurate wth the creased SNR. The accuracy of proposed method s more tha half-power badwdth method at all SNRs. It may also be otced that there s cosderable varablty of half-power badwdth method especally the lower SNR. As see Fg. 5, both methods are of satsfed accuracy f SN R 3dB. But f SN R 3dB, error of half-power badwdth s much larger tha proposed method. It s clear that the proposed method should be used to estmate dampg stead of half-power badwdth f SN R 3dB V. EXPERIMENT The expermet s carred out o a alloy beam wth the dmesos of.3m.m. m, as show Fg. 6. Fg. 7 s the dmesoless respose of the thrd mode. Fg. 5 s the estmated dampg obtaed from sx repeat expermet results by the two methods. Mult-aalyzer Vbrato excter Electroc amplfer Accelerometer specme Power amplfer Fg. 6. Expermetal setup w w w. a j e r. o r g Page 8

5 Dampg Normalzed Ampltude Amerca Joural of Egeerg Research (AJER) Frequecy (z) Fg. 7. Dmesoless respose alf-power Proposed method Fg. 8. Dampg estmated by the two methods As see Fg. 8, t s clear that there s smaller varablty for the proposed method tha half-power badwdth method. Thus, dampg estmated by the proposed method s more accurate tha half-power badwdth method. VI. CONCLUSION A oval method based o least square ft s proposed to estmate dampg ths paper. There s smaller varablty for the proposed method tha half-power badwdth method. The proposed method s more accurate tha half-power badwdth method, especally the lower SNR crcumstaces. The proposed method should be used stead of half-power badwdth method for dampg estmato f SN R 3dB. REFERENCE [] O. Guasch. A drect trasmssblty formulato for expermetal statstcal eergy aalyss wth o put power measuremets. Joural of Soud ad Vbrato, 33(5),, [] M. Rak, M. Ichchou, ad J. olck-szulc. Idetfcato of structural loss factor from spatally dstrbuted measuremets o beams wth vsoelastc layer. Joural of Soud ad Vbrato, 3(4), 8, 8-8. [3] G. A. Papagaopoulos, ad G. D. atzgeorgou. O the use of the half-power badwdth method to estmate dampg buldg structures. Sol Dyamcs ad Earthquake Egeerg, 3(7),, [4] P. W. Wag, W. W. Zhog, ad J. F. su. Ivestgato of mult-layer sadwch beams through sgle degree-of-freedom trasformato. Appled acoustcs, 74(4), 3, [5] B. A. Olmos, ad J. M. Roesset. Evaluato of the half-power badwdth method to estmate dampg systems wthout real modes. Earthquake Egeerg & Structural Dyamcs. 39(4),, [6] Z. W. Guo, M. P. Sheg, J. G. Ma, ad W. L. Zhag. Dampg detfcato frequecy doma usg tegral method. Joural of Soud ad Vbrato, 338(3), 5, [7] S. A. Badsar, M. Scheveels, W. aegema, ad G. Degrade. Determato of the materal dampg rato the sol from SASW tests usg the half-power badwdth method. Geophyscal Joural teratoal, 8(3),, w w w. a j e r. o r g Page 9

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