Haddow s Experiment:

Transcription

1 schemtc drwng of Hddow's expermentl set-up movng pston non-contctng moton sensor bems of sprng steel poston vres to djust frequences blocks of sold steel shker Hddow s Experment: terr frm Theoretcl nd Expermentl Study of Modl Intercton n Two-Degree-of-Freedom Structure J. of Sound nd Vbrton 984, 97(3 p expermentl verfcton of the sturton phenomenon prt I

2 schemtc drwng of Hddow's expermentl set-up sprng steel bems m poston cn be vred to tune frequences from "THEORETICAL AND EXPERIMENTAL STUDY OF MODAL INTERACTION IN A TWO-DEGREE-OF- FREEDOM STRUCTURE" J. Sound nd Vbrton, 984, 97(3 p # bem sold steel blocks m W s # bem # nd # bems re joned so tht they re lwys perpendculr t jont B lwys rght ngle the msses of the bems re pproxmtely tken nto ccount by ddng porton of ther msses to m nd m the governng equtons for the bems become 4 4 w w y ( x, t 4 ( y, t 4 = 0 for 0 x l = 0 for 0 y l

3 3 w x, t = A t x + B t x + C t x+ D t for 0 x l 3 ( ( ( ( w y, t = A t y + B t y + C t y + D t for 0 y l where the tme-dependent coeffcents re the generlzed coordntes of the structure the boundry nd mtchng condtons for the nfntesml free vbrtons ws t = 0 : ( 0, t w w( 0, t = 0 D( t = 0 = 0 C ( t = 0 ( by neglectng the rotry nert of m (, w l t ( 0, = 0 = 0 = 0 3 w t D t 4 y by requrng the sme rotton by neglectng the rotry nert of m so tht the n ech bem t jont B moments n the two bems re equl t jont B w l, t w 0, t w l, t w 0, t 5 = 6 = y y

4 w x, t = A t x + B t x for 0 x l 3 3 ( ( ( w y, t = A t y + B t y + C t y for 0 y l 4 (, w l t y ( ( = 6A t l + B t = ( l, t w l, t w 0, t = 3A ( t l + B ( t l = = C ( t C = 3Al + Bl y w EI EI A t l B t EI = 6 ( + ( = w ( 0, t EI = EI B t B = 3A l + B y from 4 6Al + B = 0 A = B = 3Al + B 3l 3l EI summry : D = C = D = 0 3 ( 3 A = Al + B B = Al + B 3l EI EI C = 3Al + Bl there re two degrees of freedom, nd two unknowns: A nd fter A nd B re known, A, B, nd C cn redly be found B

5 the equtons of moton becuse we hve pproxmted the structure by plcng ll the mss n the two blocks nd re consderng nfntesml motons (so tht the only sgnfcnt motons re trnsverse to the bems, the knetc energy s gven by { } T t mw l t m w l t w l t ( = (, + (, + (, 3 3 (, = + (, = ( + w x t Ax Bx w l t Al Bl 3 3 (, = + + (, = ( + + w x t A x B x C x w l t A l B l C l { 3 ( } 3 3 l + m Al + Bl + A l+ B l+ C l T( t = m Al + B

6 the potentl energy (elstc strn energy n the two bems s gven by x= l w ( x, t Ut ( = ( EI dx = x = 0 ( x, t x= l x= l x= l w x= 0 x= 0 x= 0 ( x, t ( 6 ( dx = A x + B dx = A x + A B x + B dx = Al + ABl + Bl x= l x= l w 3 dx = ( 6Ax + Bx dx = Al + ABl + Bl x= 0 x= 0 U t EI A l ABl B l EI A l AB l B l 3 3 ( = ( + + ( + + recll ( 3 EI A = Al + B 3l EI B = 3Al + B C = 3Al + Bl the Lgrngn d L L d L L L A( t, B( t = T U = 0 = 0 dt A A dt B B

7 Lgrnge's method equtons of moton: m m q k k q 0 m m + = q k k q 0 where ( λ λ q τ = l A t q τ = l B t τ = t 3 l ml m = + R+ R 3 + ( EI ( λ l = 3+ m = + R+ R( 3λ + λ = m k = ( 3+ λ = k m = + R+ 4 R λ + λ /3 k = + λ /3 l λ m λ = λ = λ λ3 = R = λ m k lner equtons wth constnt coeffcents; ts soluton hs the form q = c e nd q = c e where c nd c re constnts ωτ ωτ

8 q = c e nd q = c e q = ω c e nd q = ω c e ωτ ωτ ωτ ωτ m m q k k q 0 k ω m k ω m c 0 m m + q k k = = q 0 k c ω m k ω m 0 k ω m k ω m for non-trvl soluton to exst, Det k ω m k ω m = 0 ( k ω m( k ω m ( k ω m ( k ω m = 0 4 ( m m m m ω ( k m k m k m k m ω ( k k k k + + = 0 k m ω = b± b c where b= + km km km kk kk c = mm mm mm mm ( ω ' s the egenvlues the re the nturl frequences of the structure the egenvectors re the nturl modes, or mode shpes, of the structure

9 the egenvectors re the nturl modes, or mode shpes, of the structure k ω m k ω m c 0 k ω m k m k m = ( k ω m c + ( k ω m c = 0 c = c ω c 0 k ω ω m recll q τ = l A t q τ = l B t τ = t 3 l ml ω t l ml ( EI ωτ c ωt ω ( τ = e = = e = 3 e where ω = l l ml q c l A t A t c 3 3 ωτ k ω m ωτ ωt ωt ( τ = e = e = lb ( t B ( t = e = c e k ω m k ω m l k ω m l q c c ( ω ( ω k m c k m l mode shpes: for convenence chose c 3 (, = ( + ( = ( x mode shpes for the horzontl bem, # bem, Φ : w x t A t x B t x ( ω ( ω k m l 3 ωt 3 = e Φ ( x Φ = x x k m l

10 ( ω ( ω 3 ω t ωt ( = e ( = e k m l ( y mode shpes for the vertcl bem, # bem, Φ : k m l recll A t B t nd A Al B l ( ω ( ω k m l 3 ωt = 3 + = 3 e 3l EI 3l EI k m l B Al B l ( ω ( k ω m l ( ω ( ω 3 ωt = 3 + = 3 e EI EI k m l C Al Bl l 4 ωt = 3 + = 3 e 3 ω (, ( ( ( e t = + + = Φ k m l k m l w y t A t y B t y C t y y k ω m l EI k ω 3 m l k ω m l Φ ( y = 3l y + 3l y + 3l y 3l EI ( k ω ( EI m l ( k ω m l ( k ω m l

11 frst nd second modes from Hddow: mode mode where l m λ = λ = λ R = l EI m comprson of expermentl nd theoretcl nturl frequences (Hz mode mode mode 3 exp the

12 prevew of comng ttrctons

13 Hddow s expermentl dt showng sturton: frequency content of the response, FFTs s the mpltude of the drectly excted mode, s the mpltude of the mode excted by the nternl (uto-prmetrc resonnce = 0 lner response = 0 f = 0 ncresng n equl ncrements = 0 modl mpltudes lner response = 0 > 0 lner response becomes unstble drectly excted mode sturtes: becomes constnt = ω 4μ + σ σ hgher hrmoncs 8Hz 6Hz 8Hz 6Hz frequency (Hz 8Hz 6Hz