sin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0

Transcription

1 Q1. The free vibrtio of the plte is give by By ssumig h w w D t, si cos w W x y t B t Substitutig the deflectio ito the goverig equtio yields For the plte give, the mode shpe W hs the form h D W W W si m F y x where F(y) stisfies the ODE The geerl solutio of the bove equtio is with d F m d F m h F dy dy D F c cosh y c si y c cos y c si y h m h m, D D The symmetric coditio bout x = yields c = c 4 =, d t edges b w y, w, y b b c1cosh c3cos b b c1sih c3si The frequecy equtio of the plte is give by

2 b b cosh cos det b b sih si or sih b cos b cosh b si b For squre plte, b =, the smllest root of the frequecy equtio occurs whe m = 1, d Below is the grphic represettio of f m 8.95 D h h 1 1 Here, m, m. D b b b b, sih cos cosh si whe m = 1.

3 Q. The free vibrtio of circulr plte is govered by h w w D t where 1 1 w r r r r w The hrmoic motio of the plte is The modl shpe stisfies the followig equtio, si cos w W r t B t Tke the form of the solutio s 1 1 h W W r r r r D W r, F r cos Here = is the xisymmetric motio. The equtio relted to the mplitude F(r) is The geerl solutio of the bove equtio is d F 1 df h F dr r dr D r F c J r c I r c Y r c K r h where 4, d J, I, Y,d K re the Bessel fuctios of the first d secod kid of the rel D d imgiry rgumets, respectively. For solid plte, due to the fiite deflectio t the cetre of the plte, c 3 = c 4 =, d therefore The clmped boudry coditio requests Therefore F c J r c I r 1 df F t r dr

4 c J c I 1 The frequecy equtio is the give by or The first few roots of the frequecy re c1 J J c I I J I det J J I I J I J I 1 1 = = = = Thus the first three turl frequecies of the plte re 1 D h 1 D 1 D 1 D 1.16, 1.6, h h h

5 Q3. By the ssumed modl shpe of the plte W x y csi 1cos b The mximum stri eergy is 1 W W W W W Umx D 1 d x y x y xy The mximum kietic eergy is D 16 8m b 3m b 3 3 c 8 b K mx 1 hw d 3 hbc 8 By usig the Ryleigh s priciple, Umx Kmx, we hve the lowest turl frequecy s Umx D 16 8m b 3m b c 3 hbc hbc 8 b D 16 8m b 3m b 4 4 3h b or D 16 8m b 3m b 4 h 3b To compre with the results from Q1, cosiderig the cse of squre plte, i.e., = b, the lowest turl frequecy occurs t m = 1 d is give by D 16 8m b 3m b 3 D 4 m1 h 3b b h It is bout.3% lrge th the lyticl solutio from Q1.

6 Q4. Summry of Glerki s method: Tke the modl shpe s W cw i ix, y The ukow coefficiets ci will be determied by the followig weighted itegrtio s i1 Dci Wi x, y hcw i i x, ywk dxdy, k 1,,..., i1 i1 The bove equtio c be writte s k1c1 k c... kc, k 1,,..., with For the questio, =, D W x, y hw x, y W dxdy, k 1,,..., ki i i k Thus x y W, W 1 x y b b,, D W x y hw x y W dxdy D bh b b 1575,, D W x y hw x y W dxdy D 3b bh b b 1395,, D W x y hw x y W dxdy D 3b bh b b 1395

7 ,, D W x y hw x y W dxdy 584D 3b bh b b The frequecy equtio is give by 11 1 det 1 Or 4 b.468bh Dh b b b b b D The roots of the bove frequecy equtio re The lowest turl frequecy is the give by D h b b D h b b k D h with k b b 4 4 The k vlue of differet rtio b/ is show i the tble below: b/ k

8 Q5. For cocetrted mss, the differetil equtio of the free vibrtio is w t D w h Mi x y where (, ) is the positio hvig the cocetrted mss. The Glerki s equtio gives Dci Wi x, y h M x y cw i i x, ywk dxdy, k 1,,..., i1 i1 or Dci Wi x, y hcw i i x, ywk dxdy McW i i,, k 1,,..., i1 i1 i1 Retiig oe term, D W1 x, y hw1 x, y W1dxdy MW1, c1 Tke x y W1 si si b We hve b b 1 D hb M si si c b 4 b The fudmetl turl frequecy is b b D b hb M si si 4 b Whe (, ) = (/, b/), we hve

9 b b D b D M b hb M h 4 b Or 1 1 D b 4M h b

10 Q6. The forced vibrtio of plte is give by w,, t D w h p x y t Its solutio will be expressed by the modl summtio s m1 1, w T t W x y where W is the modl shpe of the plte, T is the mplitude of the modl W. Substitutig the solutio ito the goverig equtio of motio, t d T m1 1 m1 1 dt D T t W x, y h W x, y p x, y, t Cosiderig the reltioship for ech mode D W h W, the bove equtio will hve the form m1 1 d T dt t T t hw x y p x y t,,, Multiply W kl (x,y) o both sides of the bove equtio, d use the orthogolity of the modl shpes the bove equtio of motio c be decoupled s where the modl force where the modl mss,, or hw x y W x y d m k l kl d T t P t T t dt M,,, P t p x y t W x y d M hw x, y d The solutio of the sigle-degree-freedom vibrtio is give by

11 1 T t t b t P t d t cos si si M The itegrtio from the lst equtio is clled Duhmel s itegrl. The costts d b re determied by the iitil coditios. b v x y For the questio give, the movig lod hs the form b px, y, t P x vt y 4 Therefore the modl force is give by,,, P t p x y t W x y d b P x vt y W x, yd 4 b, 4 PW vt For the simply supported plte, the modl shpe d correspodig turl frequecy re d the modl mss is W mx y m D x, y si si, b b h

12 The modl respose is the give by M hw x, y d b 4 hsi mx si y d b 1 T t t b t P t d t cos si si M 1 t m v cos t b si t P si si si t d M 4 mv mvt 4P si sit si cos si 4 t b t, t b m v v The ukow costt d b re determied ccordig to the zero iitil coditios d re give by b Therefore the forced respose of the plte due to movig lod, whe the time less th /v, is m1 1, w T t W x y 4P si 4 si si, m1 1 m v mv mvt si t si m x y t b b v Whe the time t > /v, the plte is i free vibrtio with iitil deflectio d velocity of w(x,y, /v) = w 1 w x, y, v d v1, respectively. t