.4 Base Exiaio Ipoa lass of vibaio aalysis Peveig exiaios fo passig fo a vibaig base hough is ou io a suue Vibaio isolaio Vibaios i you a Saellie opeaio Dis dives, e. FBD of SDOF Base Exiaio x() y() Syse Seh base Syse FBD ( x y) ( x y) F-(-)-(-) x y x y x x + x + x y + y (.61) 1 SDOF Base Exiaio (o) Fo a a, Assue: y Ysi( ω) ad plug io Equaio(.61) x + x + x ωy os( ω ) + Y si( ω ) (.63) haoi foig fuios π πv ω τ λ The seady-sae soluio is jus he supeposiio of he wo idividual paiula soluios (syse is liea). f 0 f 0 s x x x Y Y + ζω + ω ζωω os( ω ) + ω si( ω ) (.64) 3 Paiula Soluio (sie e) Wih a sie fo he foig fuio, x+ ζω x + ω x f siω 0s x A osω + B siω X si( ω φ ) ps s s s s whee ζω ωf 0s A s ( ω ω ) + ( ζω ω ) ( ω ω ) f 0s B s ( ω ω ) + ζω ω Use eagula fo o ae i easie o add he os e 4
Paiula Soluio (os e) Wih a osie fo he foig fuio, we showed x ζωx ω x f ω + + 0 os x A osω+ B siω X os( ω φ ) p whee ω ω A ( ω ω ) ζωω B ( ) f0 + ζω ω f 0 ( ) + ω ω ζωω 5 Magiude X/Y Now add he si ad os es o ge he agiude of he full paiula soluio X f 0 + f 0s (ω ω ) + ζω ω ω Y (ζω) whee f 0 ζω ωy ad f 0s ω Y if we defie ω ω his beoes X Y X Y 1+ (ζ) (1 ) + ζ +ω (ω ω ) + ζω ω 1+ (ζ) (1 ) + ζ (.71) (.70) 6 X/Y (db) The elaive agiude plo of X/Y vesus fequey aio: Called he Displaee Tasissibiliy 40 30 0 10 0-10 -0 0 0.5 1 1.5.5 3 Fequey aio Figue.13 ζ 0.01 ζ 0.1 ζ 0.3 ζ 0.7 7 Fo he plo of elaive Displaee Tasissibiliy obseve ha: X/Y is alled Displaee Tasissibiliy Raio Poeially sevee aplifiaio a esoae Aeuaio fo > sq() Isolaio Zoe If < sq() asissibiliy deeases wih dapig aio Aplifiaio Zoe If >> 1 he asissibiliy ieases wih dapig aio X p ~Yζ/ 8
Nex exaie he Foe Tasied o he ass as a fuio of he fequey aio FT ( x y) ( x y) x A seady sae, x Xos( ω φ ), ω ω φ so x- X os( ) ω FT X X y() x() base Fo FBD F T 9 Plo of Foe Tasissibiliy (i db) vesus fequey aio F/Y (db) 40 30 0 10 0-10 ζ 0.01 ζ 0.1 ζ 0.3 ζ 0.7-0 0 0.5 1 1.5.5 3 Fequey aio Figue.14 10 Figue.15 Copaiso bewee foe ad displaee asissibiliy Foe Tasissibiliy Exaple.4.1: Effe of speed o he apliude of a vibaio Displaee Tasissibiliy 11 1
Model he oad as a siusoidal ipu o base oio of he a odel Appoxiaio of oad sufae: y() (0.01 )siω b 1 ω b v(/h) 0.006 hou 3600 s π ad yle 0.909v ad/s ω b (0/h) 5.818 ad/s Fo he daa give, deeie he fequey ad dapig aio of he a suspesio: ω 4 104 N/ 1007 g ζ 000 Ns/ ( 4 10 4 N/)1007 g 6.303 ad/s ( 1 Hz) Fo he ipu fequey, ipu apliude, aual fequey ad dapig aio use equaio (.70) o opue he apliude of he espose: X Y ( 0.01 ) 1 + (ζ) (1 ) + (ζ) ω b ω 5.818 6.303 1 + [ (0.158)(0.93) ] 1 ( 0.93) ( ( 0.93) ) 0.0319 + 0.158 Wha happes as he a goes fase? See Table.1. 0.158 13 14 Exaple.4.: Copue he foe asied o a ahie hough base oio a esoae F T Y Fo (.77) a 1: 1+ (ζ ) (ζ) 1/ F T Y ζ Fo give,, ad : 1+ 4ζ Fo easued exiaio Y 0.001 : F T Y ζ 1 + 4ζ (40,000 N/)(0.001 ) (0.04) 900 ζ 0.04 40, 000i3000 1 + 4(0.04) 501.6 N.5 Roaig Ubalae Gyos Cyo-ooles Ties Washig ahies Mahie of oal ass i.e. 0 iluded i e eeiiy o ass ubalae ω oaio fequey e 0 ω 15 16
R x e θ Roaig Ubalae (o) ω 0 R y Fo sophooe dyais, Wha foe is ipaed o he suue? Noe i oaes wih x opoe: a esiω x eω siω R x 0 a x o eω siθ o eω siω x R y 0 a y o eω osθ o eω osω x 17 x() Roaig Ubalae (o) The poble is ow jus lie ay ohe SDOF syse wih a haoi exiaio 0 eω si(ω ) x + x + x oeω si ω (.8) o o x+ ζωx + ωx eω siω Noe he ifluees o he foig fuio (we ae assuig ha he ass is held i plae i he y dieio as idiaed i Figue.18) 18 Roaig Ubalae (o) Jus aohe SDOF osillao wih a haoi foig fuio Expessed i es of fequey aio Figue.0: Displaee agiude vs fequey aused by oaig ubalae x p () Xsi(ω φ) (.83) X o e (1 ) + ζ (.84) ζ φ a 1 1 (.85) 19 0
Exaple.5.1:Give he defleio a esoae (0.1), ζ 0.05 ad a 10% ou of balae, opue e ad he aou of added ass eeded o edue he axiu apliude o 0.01. A esoae 1 ad X 0 e 1 ζ 1 (0.05) 10 0.1 e 1 10 e 0.1 ζ Now o opue he added ass, agai a esoae; 0 X 0.1 10 Use his o fid Δ so ha X is 0.01: +Δ 0 0.01 0.1 10 +Δ 100 Δ 9 (0.1) Hee 0 is 10% o 0.1 1 Exaple.5. Heliope oo ubalae Give 1 10 5 N/ ail 60 g o 0 g 0 0.5 g ζ 0.01 Fig.1 Fig. Copue he defleio a 1500 p ad fid he oo speed a whih he defleio is axiu Exaple.5. Soluio The oaig ass is 0 + 0.5 o 0.5. The siffess is povided by he Tail seio ad he oespodig ass is ha deeied i Exaple 1.4.4. So he syse aual fequey is ω + ail 3 The fequey of oaio is 10 5 N/ 46.69 ad/s 60 g 0.5 + 3 ω 1500 p 1500 ev i π ad 157 ad/s i 60 s ev 157 ad/s 49.49 ad/s 3.16 3 Now opue he defleio a 3.16 ad ζ 0.01 usig eq (.84) X 0e (1 ) + (ζ) 0.5 g 0.15 0.5 g ( 3.16) 0.004 ( 1 (3.16) ) (0.01)(3.16) A aoud 1, he ax defleio ous: 1 ω 49.69 ad/s 49.69 ad ev 60 s 474.5 p s π ad i A 1: 0.5 g X 0.15 0.5 g 1 (0.01) 0.183 o 18.3 4
.6 Measuee Devies A basi asdue used i vibaio easuee is he aeleoee. This devie a be odeled usig he base equaios developed i he pevious seio F- ( x- y)- ( x - y) x x - ( x y ) - ( x y) (.86) ad (.61) Hee, y() is he easued espose of he suue 5 Base oio applied o easuee devies Le z() x() y() (.87): z + z + z Y Z (.90) Y (1 ) + ( ζ ) ad 1 ζ θ a (.91) 1 ωb os ωb (.88) These equaios should be failia fo base oio. Hee hey desibe easuee! Aeleoee Sai Gauge 6 Magiude ad sesiiviy plos fo aeleoees. Fig.6 Magiude plo showig Regios of easuee I he ael egio, oupu volage is ealy popoioal o displaee Effe of dapig o popoioaliy osa Fig.7 Hoe Wo-Chape.6,.7,.9.11,.15,.19.4,.8.30,.35.40,.44.51,.5,.53,.57,.58 7 8