STUDY ON A METHOD TO IDENTIFY EXTERNAL FORCES AND DAMPING PARAMETERS OF STRUCTURE FROM RANDOM RESPONSE VALUES

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1 STUDY ON A METHOD TO IDENTIFY ETERNAL FORCES AND DAMPING PARAMETERS OF STRUCTURE FROM RANDOM RESPONSE VALUES 114 M KANDA 1, E MARUTA, D INABA 3 And T MATSUYAMA 4 SUMMARY Proposed in his paper is a mehod of idenifying exernal forces and damping parameers of srucure from random response values. This mehod esimaes exernal forces acing on srucure from response values by means of he inverse procedure of response analysis in he frequency range. An appropriae value of he damping parameer is idenified by invesigaing he specrum of exernal force for various damping parameers. This is very effecive in cases of involving analyzing aerodynamic vibraion ess in wind unnels and field measuremens of real srucure vibraion. The advanage of his mehod over oher mehods is ha i can more accuraely idenify damping parameers even wih less daa. INTRODUCTION In order o accumulae basic daa o improve srucural design echnique, many experimens and field measuremens o idenify damping parameers and exernal forces of srucure are being carried ou. Various mehods of idenifying damping parameers from field measuremens were proposed, and go noeworhy resuls. Addiionally apparen damping effecs by exernal force such as aerodynamic damping were invesigaed by means of wind unnel ess. Furhermore, acing forces on srucures were also esimaed from experimens and field measuremens. However, hese mehods are necessary for measuring a large amoun of daa and for carrying ou a large scale analyses when damping parameers are esimaed. Moreover, he accuracy of idenified values is limied. In paricular, here migh no be an appropriae mehod of idenifying exernal wind force acing on srucure. These facs preven improvemen in design echnique of srucure. Thus, he echnique can be improved if he properies of he exernal forces and he damping parameers can be invesigaed easily. Based on he above, he purpose of his paper is o propose a new mehod of idenifying exernal forces and damping parameers from random response values of srucure. OUTLINE OF THIS METHOD The basic concep underlying his mehod is shown in Figure1. This mehod esimaes damping parameers and exernal forces by means of he inverse procedure of response analysis in he frequency range. The appropriae damping parameer is esimaed from he specra of exernal forces by rial and error. The basic principle and procedure are as follows: If i is assumed ha he orhogonaliy condiion applies no only o he mass and siffness bu also he damping, i is convenien o invoe reduce down o a-single-degree-of-freedom. Based on he propery, he equaion of moion for a-single-degree-of-freedom sysem is expressed as follows: Dep.of Arch. & Arch Eng.,College of Indus. Technology,Nihon Universiy 1--1 Izumicho,Narashino,Chiba, Japan Dep.of Arch. & Arch Eng.,College of Indus. Technology,Nihon Universiy 1--1 Izumicho,Narashino,Chiba, Japan Dep.of Arch. & Arch Eng.,College of Indus. Technology,Nihon Universiy 1--1 Izumicho,Narashino,Chiba, Japan Kumagaigumi Research Insiue, Onigaubo 143, Tsuuba, Ibaragi, Japan

2 + ξ + f / M = (1) where, M, and ξ represen mass, naural circular frequency and damping parameer respecively., and represen acceleraion, velociy and displacemen respecively. f represen exernal force. Generally, he response values derived from Equaion. (1) are expressed as he sum of he harmonic vibraion and he free vibraion. Wih regard o he response values, he displacemen is expressed as follows: = H + F () where, H represens he harmonic vibraions of displacemen, and F represens he free vibraion of displacemen. I is assumed ha exernal force f can be expressed by he summaion of many sine waves. The ampliudes and phases of he sine waves of naural circular frequency are expressed as α and ψ respecively. Exernal force f() a ime can be expressed as follows: f ( ) = n = 1 α e i( + ϕ ) (3) When he force acs on a vibraion sysem of mass M, naural circular ferquency and damping parameer ξ respecively, H () and F () of he vibraion sysem can be expressed as follows: H n ( ) = β e = 1 i( + ϕ ) (4) F ξ ( ) = A e cos( 1 ξ φ ) F F (5) where α = M β γ (6) ϕ = φ + Φ (7) γ = ( 1 ( / ) ) + 4ξ ( / ) (8) Φ = an 1 ξ ( / )/(1 ( / ) ) (9) A F and ψ F are ampliude and phase of free vibraion respecively. These are deermined by he iniial condiions. Real and imaginary number of specra are represened by A and B respecively. Then, β and φ are expressed as follows: β = A + B (1) φ = an 1 ( B / A ) (11) Based on he principle of vibraion sysem and equaions, he procedure for idenifying damping parameers and exernal forces from displacemen response values is as follows: 1. Response values are obained.. The free vibraion values which are generaed from he assumed iniial values and he damping parameer are aen from he response values o evaluae he harmonic vibraion values. 114

3 3. The harmonic vibraion values obained in are ransformed and he specrum of harmonic vibraion values is esimaed. 4. The specra of exernal force are derived from he specra of harmonic vibraion by means of he inverse ransfer funcion of a-single-degree-of-freedom vibraion sysem. 5. The apiude of free vibraion values is deermined from he idenified specra of exernal forces. 6. Processes o 5 are repeaed unil he appropriae free vibraion values are deermined. The damping parameer of he deermined free vibraion values becomes he appropriae damping parameer of he srucure. 7. The exernal force on ime hisory is evaluaed from he specra of exernal force by means of inverse ransformaion. Then, if apparen damping effecs are included, i is necessary o add he effec o he he value of evaluaed exernal forces. The concep for deerming he suiable value for he damping parameer in 5 is shown in Figure imiaively. The ordinary ampliude specrum of exernal force doesn have power in he high frequency range. On he oher hand, he ampliude specrum of exernal force, in he case when free vibraion isn reduced perfecly, increases as frequency increases. The free vibraion value for which he increase in he high frequency range is smalles is considered suiable, and is damping parameer is considered suiable, oo. THE VALIDITY OF THIS METHOD In order o verify he validiy of his mehod, he damping parameers and exernal forces are idenified from random response values of a-single-degree-of-freedom sysem by means of compuer simulaion and a wind unnel vibraion es. In he compuer simulaion, he naural circular frequency and he mass of his vibraion sysem are =66.57rad/sec and M=.19Kgcm respecively. The damping parameers areξ=.1,1.,. and 5.%. The number of sampling daa is 14 in he case of ξ=.1,1. and.%, and is 51 in he case of ξ=5.%. The sampling ime inerval is 1μsec. Sv Sx SF Sx x Mechanial Admiance Inverse Specra of Exernal Force Specra of Response Value Harmonic Vibraion Inverse V Time Hisory of Exernal Force Time Hisory of Response Value Figure 1: The Concep of This Mehod Free Vibraion 3 114

4 S f ( ) Specrum Including Free Vibraion Derived Specrum from Harmonic Vibraion Figure : The Differences beween he Specra of Exernal Forces The esimaed specrum and ime hisories of he exernal forces in he case of ξ=1. and.% are shown in Figure3 and Figure4 respecively. Every graphs in Figure3 indecas ha increase of specra in he high frequency ranges are suppresed as he assumed ξ approaches o inpu value. From Figure4, he esimaed values of exernal force on ime hisory correspond o he inpu values of exernal force very well in he case of ξ=.%. However, he esimaed values don correspond o he inpu values very well in he case of ξ=1.%. There are some differences beween he esimaed value in he case of ξ=1.% and.%. The difference is especially remarable a he beginning and end of he ime hisories. The damping parameer and exernal forces are idenified from he response values, which is obained from he wind unnel vibraion es, in across wind direcion. The naural circular frequency, mass, number of sampling and sampling ime inerval are similar o he previous case.ξ of he sysem is se a approximaely %. The idenified specra from he response values are shown in Figure5. If he assumed ξ approaches he inpu value, he increase of he specrum in high frequency ranges is suppressed as frequency increases in he previous case. So, he idenified value of he damping parameer is deermined o be.%. The ime hisory of oal response value, free vibraion, harmonic vibraion and exernal wind force when damping parameer and iniial parameer are considered as suiable is shown in Figure6. Especially, he ime hisory of oal response value is compared wih he response value obained from he esimeed exernal force by means of sep-by-sep inegraion analysis on ime hisory. The oal response value obained from wind unnel vibraion ess corresponds very well o he response value obained from he esimaed exernal force. CONCLUSION The resul obained hough his sudy can be summarized as follows: 1) A new mehod of idenifying damping parameers and exernal forces is proposed. ) The proposed mehod can idenify he values accuraely even if here is lile daa. 3) This mehod can be widely applied, for example, for analysis of he resuls of wind unnel ess and field measuremens of srucure vibraion

5 Ampliude of Exernal Momeng /sec ŠO Í U ikg m /sec j 1.E-3 Idenified Valueξ=.1% è lfh=.1% 8.E-4 ξ=.3% h=.3% 6.E-4 ξ=.% 4.E-4 h=.% ξ=.% h=.%.e-4 ξ=.1% h=.1% Inpu Force.E+ ^ Š O Í. f ³ŸŒ³U FfEB/v Reduced Frequency : fb/v } Q-a Œ Š l Ì èi Ý è lfh=.1% j Figure 3-a : Specrum of Exernal Force ( :ξ=.1%) Ampliude of Exernal Momeng /sec ŠO Í U ikg m /sec j 1.5E-3 1.E-3 9.E-4 6.E-4 Idenified Valueξ=1.% è lf1.% ƒ Ì=.1% ƒ Ì=.5% ƒ Ì=.% ƒ Ì=1.5% 3.E-4 ƒ Ì=1.% Inpu inpu Force value.e+. f Reduced ³ ŸŒ Frequency ³ FfEB/v : fb/v } Q-b Œ Š l ÌZ èiý è lfƒì=1.% j FiFigure 3-b : Specrum of Exernal Force ( :ξ=1.%) Ampliude of Exernal Momeng /sec Idenified Valueξ=.% ξ=1.% ξ=3.% ξ=1.5% ξ=.5% ξ=.% Inpu Force Reduced Frequency : fb/v Ampliude of Exernal Momeng /sec ŠO Í U ikg m /sec j.e-3 Idenified Valueξ=5.% è lf5.% 1.5E-3 ξ=3.% h=3.% 1.E-3 ξ=7.% h=7.% ξ=4.% 5.E-4 h=4.% ξ=6.% h=6.% ξ=5.% h=5.% Inpu.E+ INPUT Force ^ Š O Í f Reduced ³ ŸŒ Frequency ³ U Ff B : fb/v / v } Q-d Œ Š l Ì èi Ý è lfh=5.% j Figure 3-c : Specrum of Exernal Force ( :ξ=.%) Figure 3-d : Specrum of Exernal Force ( :ξ=5.%) Reduced Naural Frequency 5 114

6 Exernal Force (gm /sec ) Inpu Value Idenified Value Figure4-a : Comparison Idenified Value wih Inpu Value (ξ=1.%) } S-a ^ ŠO Í Æ èšo Í Ì äšriý è lfƒì=1.% j Exernal Force (gm /sec ) Inpu Value Idenified Value Figure4-b : Comparison Idenified Value wih Inpu Value (ξ=.%).e-3 Ampliude of Exernal Momeng ŠO Í U ikg m /sec j 1.5E-3 1.E-3 5.E-4 Idenified Value ξ=.% è lf.% ƒ Ì=1.% ƒ Ì=3.% ƒ Ì=.5% ƒ Ì= 1.5% ƒ Ì=.%.E f ³ ŸŒ ³ U Ff B / v Reduced Frequency : fb/v Figure } Q-d 5 Œ: Specrum Š l Ì of Exernal è( ÀŒ Force ±ƒ f [ƒ^ æ èj 6 114

7 MEASURED VALUE IDENTIFIED VALUE Displacemen Angle ψʊp Response Displacemen.6 Angle Displacemen Angle ψʊp Free Vibraion Displacemen Angle ψʊp Harmonic.6 Vibraion Exernal Momeng ŠO Í ikg m /sec /sec j Idenified Force Exernal Figure 6 : Idenificaion of Exernal Force from Wind Tunnel Tes REFERENCE TAMURA, Y. and JEARY, A. (1996), Meeing on Srucual Damping Inernaional Wind Engineering Fourum and Addiional Paper, Journal of W 7 114