Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc xapls and discussion usic structural and chanical nginring wavs Sapl probls Oscillations suary chart apd Oscillations nglct gravity Non-consrvativ forcs ay b prsnt Friction is a coon nonconsrvativ forc No longr an idal syst (such as thos dalt with so far) Th chanical nrgy of th syst diinishs in ti, otion is said to b dapd Th otion of th syst can b dcaying oscillations if th daping is wa. If daping is strong, otion ay di away without oscillating. Still no driving forc, onc syst has bn startd
Add aping: E ch not constant, oscillations not sipl nglct gravity Spring oscillator as bfor, but with dissipativ forc F dap such as th syst in th figur, with van oving in fluid. F dap viscous drag forc, proportional to vlocity F dap = bv Prvious forc quation gts a nw, daping forc tr F nt d d = = () b d d nw tr b d + = Solution for apd oscillator quation d nw tr b d + = Solution: odifid oscillations = x xponntially dcaying nvlop cos( ω' altrd frquncy ω can b ral or iaginary ω' b 4 ω = : natural frquncy ω' ω ( b/ ) Rcovr undapd solution for b
apd physical systs can b of thr typs Solution: dapd oscillations = x cos( ω' ω' b 4 Undrdapd: d d sall b< b <, for which ω is positiv. 4 Critically dapd: b= b ω for which ω' 4 Ovrdapd: b > ω for which ω' 4 x x Math Rviw: cos( ix) = cosh( x) = ( + ) / x x sin( ix) = sinh( x) = ( ) / is iaginary cos( ix + y) = cos( ix)cos( y) sin( ix)sin( y) Typs of aping, cont (Lin to Activ Fig.) a) an undrdapd oscillator b) a critically dapd oscillator c) an ovrdapd oscillator For critically dapd and ovrdapd oscillators thr is no priodic otion and th angular frquncy ω has a diffrnt aning 3
Waly dapd oscillator : b << ω 4 b ' 4 ω ω xt () = x cos( ω t+ ϕ) X = x - () (t) x slow dcay of aplitud nvlop cos( ωt + φ) sall fractional chang in aplitud during on coplt cycl Waly dapd oscillator : b ' 4 x (t) x - b << ω 4 ω ω xt () = x cos( ω t+ ϕ) Aplitud : X = A slow dcay of aplitud nvlop sall fractional chang in aplitud during on coplt cycl cos( ωt + φ) Vlocity with wa daping: find drivativ d v(t) = v axiu vlocity sin( ω't + φ) v = ωx xponntially dcaying nvlop altrd frquncy ~ ω 4
Mchanical nrgy dcays xponntially in an waly dapd oscillator (sall b) Ech = K(t) + U(t) = v (t) + x (t) () = cos( ω + ϕ) xt x t Vlocity with wa daping: find drivativ d v(t) = v axiu vlocity sin( ω't + φ) v = ωx xponntially dcaying nvlop altrd frquncy ~ ω b x cos( ωt+ ϕ) tr is ngligibl, bcaus b is sall.. Mchanical nrgy dcays xponntially in an waly dapd oscillator (sall b) Ech = K(t) + U(t) = v (t) + Substitut prvious solutions: E = x ch = cos( ω' ω x / As always: cos (x) + sin (x) = Also: ω E ch + (t) = x v(t) x ω x sin ( ω' x / (t) cos ( ω' / sin( ω' Initial chanical nrgy xponntial dcay at twic th rat of aplitud dcay 5
apd physical systs can b of thr typs Solution: dapd oscillations = x xponntially dcaying nvlop cos( ω' altrd frquncy ω can b ral or iaginary ω' b 4 b << ω for which ω' ω 4 Th rstoring forc is larg copard to th daping forc. Th syst oscillats with dcaying aplitud Undrdapd: Critically dapd: b ω for which ω' 4 Th rstoring forc and daping forc ar coparabl in ffct. Th syst can not oscillat; th aplitud dis away xponntially Ovrdapd: b 4 > ω for which ω' is iaginary Th daping forc is uch strongr than th rstoring forc. Th aplitud dis away as a odifid xponntial Not: Cos( ix ) = Cosh( x ) Forcd (rivn) Oscillations and Rsonanc An xtrnal driving forc starts oscillations in a stationary syst Th aplitud rains constant (or grows) if th nrgy input pr cycl xactly quals (or xcds) th nrgy loss fro daping Evntually, E driving = E lost and a stady-stat condition is rachd Oscillations thn continu with constant aplitud Oscillations ar at th driving frquncy ω F (t) = F cos( ω t + φ') Oscillating driving forc applid to a dapd d oscillator F (t) 6
Equation for Forcd (rivn) Oscillations ω = natural frquncy ω = ω = driving frquncy of xtrnal forc Extrnal driving forc function: F (t) = F cos( ω t + φ') dx t Fnt = F ( t) -b - = () d x() t F (t) Solution for Forcd (rivn) Oscillations dx t Fnt = F ( t) -b - = F (t) = F cos( ω () d x() t t + φ') Solution (stady stat solution): = Acos( ωt + φ) whr A = ( ω F ω ) / bω + ( ) F (t) Th syst always oscillats at th driving frquncy ω in stady-stat Th aplitud A dpnds on how clos ω is to natural frquncy ω rsonanc ω = 7
Aplitud of th drivn oscillations: Th largst aplitud oscillations occur at or nar RESONANCE (ω ~ ω ) As daping bcos war rsonanc sharpns & aplitud at rsonanc incrass. A = ( ω F ω ) / bω + ( rsonanc ) Rsonanc At rsonanc, th applid forc is in phas with th vlocity and th powr Fov transfrrd to th oscillator is a axiu. Th aplitud of rsonant oscillations can bco norous whn th daping is wa, storing norous aounts of nrgy Applications: buildings drivn by arthquas bridgs undr wind load all inds of radio dvics, icrowav othr nurous applications 8
Forcd rsonant torsional oscillations du to wind - Tacoa Narrows Bridg Roadway collaps - Tacoa Narrows Bridg 9
Twisting bridg at rsonanc frquncy Braing glass with voic