TRANSMISSIBILITY MATRIX IN HARMONIC AND RANDOM PROCESSES

Transcription

1 RANMIIBILIY MARIX IN ARMONIC AND RANDOM PROCEE Ribeio, A.M.R., Fotul, M., ilva,.m.m., Maia, N.M.M. Istituto upeio écico, Av. Rovisco Pais, Lisboa, Potugal UMMARY: he tasmissibility cocept may be geealised to multi-degee-of-feedom systems with multiple adom ecitatios. his geealizatio ivolves the defiitio of a tasmissibility mati, elatig two sets of esposes whe the stuctue is subjected to ecitatio at a give set of coodiates. Applyig this cocept to a theoetical eample is the easiest way to illustate this method. EYWORD: tasmissibility, adom vibatio, spectal desity, modal aalysis. INRODUCION he cocept of tasmissibility fo MDOF systems, ca be foud i pevious papes by the authos [ - 4, whee the tasmissibility mati was calculated usig eithe the FRF mati of the stuctue, o the measued esposes oly. I the fist case, we ca calculate the tasmissibility mati usig the FRFs of the system, whee we ca elate the ukow esposes with the kow oes usig such a tasmissibility mati. I the secod case we ca calculate the tasmissibility mati usig oly measued esposes, buildig a mati of kow esposes ad aothe mati of ukow esposes fo seveal tests. I ay case, it was always assumed that the applied foces wee peiodic. his pape is a cotiuatio of those pevious woks, poposig a method fo evaluatig the tasmissibility mati whe the applied foces ae of a adom kid. We will ty, theefoe, to evaluate the tasmissibility mati by elatig the spectal desities of the esposes ad will compae the esult with the oe obtaied whe usig the tasmissibility mati comig fom the FRFs. I fact, it will be pove that the tasmissibility mati is the same, idepedet of the applied foces. DEVELOPMEN f a f b f N 2 Figue Cosideig a stable liea system, as show i Figue, which is subjected to adom ecitatio at cetai locatios, f a (t), f b (t),, f (t) ad cosideig the esposes of the system as (t), 2 (t),, N (t), the FRFs ω,..., ω,..., ω,..., ω. elatig the outputs (esposes) to the iputs (foces) ae ( ) ( ) ( ) ( ) fa f Nfa Nf

2 he fist espose of the system may be epessed as [5: (t) + = ( h (θ) f ( t θ) + h (θ) f ( t θ) h ( θ) f ( t θ) ) dθ fa a f b b f he autocoelatio fuctio of the output pocess (t), R () τ, is defied by: he auto-spectal desity fo the same output pocess (t), () τ = E[ () t ( t + τ) R 2π + = () τ iωτ R e dτ, ca by epessed as: () (2) (3) ubstitutig elatios () ad (2) i (3) ad afte some mathematic calculatios, the auto-spectal desity of espose (t) becomes: = f a fa f af a f f f f (4) I elatio (4), (*) deotes comple cojugate. Now, we ca wite epessio (4) i the followig fom: = f f s f f s = s= (5) Cosideig ow two abitay esposes (t) ad (t), the spectal desity fuctio ca be epessed as: = = s= f fs ffs (6) he coss-coelatio fuctio betwee the iput pocess f (t) ad the output pocess (t), R () τ is defied as: he coss-spectal desity betwee the same iput pocess f (t) ad the output pocess (t), give by: R f f () τ = E[ f () t ( t + τ) 2π + f f (7) ca be Usig elatios () ad (7) i elatio (8), ad makig same mathematic calculatios, the coss-spectal desity betwee the iput pocess f (t) ad the output pocess (t) becomes: = () τ iωτ R e dτ f (8) f = f a f f a f f f (9) o, i a moe compact fom, f = s= fs f fs (0) Let us compae ow epessios (6) ad (0). It ca easily be obseved that the last pat of elatio (6) epesets i fact elatio (0). hus, the spectal desity betwee two abitay esposes (t) ad (t), ca be epessed as:

3 ( ) = ω = f f () I mati fom, epessios (0) ad () become, espectively: [ [ f f [ [ f ff f (2) (3) Let us ow biefly ecall the theoetical developmet of the tasmissibility cocept i MDOF system with peiodical ecitatio. Coside the followig sets of co-odiates: A fo foces, fo kow esposes ad U fo ukow esposes; the, we ca wite the followig equatios: { = [ { f { = [ { f U A A (4) (5) It is assumed that the umbe of kow espose co-odiates is at least equal to the umbe of applied foce coodiates, i.e., A (6) I the case of = A, fom (4), it tus out that: { f = [ { A If > A, the ivese i equatio (7) becomes the pseudo-ivese. Combiig ow the elatios (5) ad (7), we ca wite a equatio betwee ukow ad kow esposes: (7) he tasmissibility mati was the defied as: { = [ [ { U [ [ AU (8) (9) ad cosequetly, we ca elate the ukow esposes to kow esposes usig the tasmissibility mati defied by elatio (9): { = [ { U AU (20) Let us ow ecall elatio (3), ad coside the same sets of co-odiates: A fo foces, fo kow esposes ad U fo ukow esposes. We ca wite the followig epessios: [ [ A [ [ U A (2) (22) whee the followig simplificatios wee used: =; U =U; f A =A. It ca be show that (2) ad (22) may be epessed i the followig fom: [ [ [ [ U (23) (24) Cosideig that the system satisfies coditio (6), fom (23) we have: [ [ (25)

4 Combiig (24) ad (25) leads to [ [ [ U Epessio (26) elates the spectal desities amog ukow ad kow esposes with spectal desities amog kow esposes. akig ito accout elatio (9), we ca coclude that the same tasmissibility mati appeas i this case too: (26) [ = [ [ U AU (27) Fom (27), we ca calculate the spectal desities betwee ukow ad kow esposes, cosideig the tasmissibility mati ad spectal desities amog the kow esposes. EXAMPLE 3 2 k4 k3 m3 m2 c4 c3 f3 f2 m=m2=m3= [N; c=c2=0 [Ns/m; c3=30 [Ns/m; k= [N/m; k2= [N/m; k3= [N/m; k4=4.000 [N/m; k2 m c2 k c Figue 2 he easiest way to illustate the method is though a umeical eample. I Figue 2 we coside a simple thee DOF system, compose of thee masses m, m2,ad m3, fou spigs k, k2, k3, ad k4, ad fou viscous dampig elemets c, c2, c3, ad c4. he applied foces f2 ad f3 ae adom. he coodiates fo applied foces ae 2 ad 3, the kow esposes ae 2 ad 3 too ad the ukow espose is. Fom elatio (9), we ca calculate the tasmissibility mati, because we kow all the FRFs of the system: [ = [ (28) Usig fiite diffeeces we ca calculate the esposes (t), 2 (t), ad 3 (t). he coespodig Fouie tasfom of those esposes ae X (ω), X 2 (ω), ad X 3 (ω), espectively. With these Fouie tasfoms, we ca calculate all the spectal desities betwee esposes: PQ = X P I elatio (29) P= 3, Q= 3. O the othe had, fom elatio (27), we ca calculate the spectal desities betwee ukow ad kow esposes cosideig that we kow the spectal desities betwee kow esposes: X Q (29)

5 22 32 [ = [ (30) Replacig ow the tasmissibility mati epessio fom (28) i (30): [ = [ (3) Now, we ca compae the calculated spectal desities 2 ad 3 fom elatio (3) with the theoetical spectal desities 2 ad 3 fom elatio (29) (Figues 3 ad 4). Figue 3 As we ca obseve fom Figues 3 ad 4, the theoetical ad calculated spectal desities betwee ukow ad kow esposes have the same shape. CONCLUION It has bee pove that the same tasmissibility mati appeas i both peiodical ad adom ecitatio cases. It has bee cofimed oce moe that the tasmissibility mati does ot deped o the applied type of foces ecitig the system, but oly o the chose sets of co-odiates fo the applied foces ad fo the ukow ad kow esposes. A simple method to calculate the spectal desities betwee the ukow ad kow esposes usig the tasmissibility mati ad spectal desities betwee kow esposes was peseted. A simple eample illustated the goodess of the pocedue.

6 Figue 4 REFERENCE. Ribeio, A.M.R., O he Geealizatio Of he asmissibility Cocept, Poceedigs of NAO/AI Cofeece o Modal Aalysis ad estig, esimba, Potugal, pp , May Maia, N.M.M., ilva,.m.m., Ribeio, A.M.R., Epeimetal Evaluatio of the asmissibility Mati, Poceedigs of the 7 th Iteatioal Modal Aalysis Cofeece (IMAC XVII), Olado, Floida, pp.26-29, Febuay Ribeio, A.M.R., Maia, N.M.M., ilva,.m.m., O the Geealisatio of he asmissibility Cocept, Mechaical ystems ad igal Pocessig, 4(), pp , Maia, N.M.M., ilva,.m.m., Ribeio, A.M.R., "he asmissibility Cocept i Multi-Degee-of-Feedom ystems", Mechaical ystems ad igal Pocessig (200), 5(), 29-37, Newlad, D.E., Radom vibatios ad spectal aalysis, ecod editio, Logma, 984.