Transient State Analysis of a damped & forced oscillator

Transcription

1 1 DSRC Transien Sae Analysis of a daped & forced oscillaor P.K.Shara 1* 1. M-56, Fla No-3, Madhusudan Nagar, Jagannah coplex, Uni-4, Odisha, India. Asrac: This paper deals wih he ehaviour of an oscillaor in is iniial sage of oscillaion. How are energy, displaceen ec of he oscillaor change wih ie? How does he phase difference eween he driving force and velociy change wih ie in forced oscillaion? Can we conserve energy of free daped and forced oscillaion during firs few seconds? How does he oscillaor asor energy fro he supply when we drive i? How does he viraing oscillaor aain is seady sae? How does an oscillaion in seady sae decay when he driving force is urned off? These all quesions are answered in his paper. Forced Oscillaion wih F ex By a ransien is ean a soluion of he differenial equaion when here is no force presen[1]. When we wihdraw he oscillaing driving force fro he forced oscillaor and do no neglec he fricion, i drains he sored energy and daps he oscillaion. For any deforaion x of he spring of siffness, he ne force acing on he loc is, where and are he spring force and viscous (fricion) force respecively. where A () = apliude of he oscillaing loc which decreases exponenially wih ie shown as he envelope of he displaceen-ie graph. We will see ha he frequency of oscillaion will e less han he undaped frequency. Or, d x x v d Incidenally alhough he concep of a definie frequency can e sricly applied only o a pure sine or cosine funcion, is coonly called he frequency of oscillaion. The zero crossings of he funcion. d x dx Or, x d d If he daping consan is saller han he spring force, he soluion of he aove differenial equaion is given as x Ae cos( x), where A axiu apliude of oscillaion and arirary phase consan. The coefficien of he haronic er cos( ) is given as A() Ae. Then, he aove equaion can e wrien as x A( )cos( ), ( ) cos( ) A Ae are separaed y equal ie inervals T, u he peas (P and Q) do no lie half way eween he[]. This eans ha 1 u ' ' ' ' as shown in he graph. 1 1 Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

2 P.K.Shara 1 DSRC The loc will oscillae wih a frequency lesser han he free undaped frequency due o he fricion given as. Since, for any value of 4, he loc will oscillae wih decreasing apliude so any ies efore coing o res. This is called under daping. If, he equaion of oion of he loc is given as x ( A B) e For, due o heavy (over) daping he oion of he paricle (loc) of he spring-ass syse is given y 1 x Ae Be, where A and B are posiive consans and 1 4 and 4 A, x AB and if, x as shown in he graph. This equaion ells us ha, he oion of he loc is no oscillaory. This physically signifies ha when displaced, he loc will coe ac o res wihou execuing any oscillaion. This condiion of over daping is also called dead ea or a periodic displaceen[4]. Energy of a daped oscillaor: There is an iniial rise in he displaceen due o he facor ( A B ), where A and B are posiive consans u susequenly he exponenial er doinaes as i increases furher. The displaceen can ecoe zero for one finie value of ie. This siuaion is called criical daping[3]. The oal energy of he daped oscillaor is E K U 1 1 v x d 1 d 1 Ae cos( x) Ae cos( x) d d 1 1 Ae sin( x) A e cos( ) A e cos ( ) For wea daping. Then he of he aove expression will ecoe zero. Hence, we can wrie 1 A E e sin ( ) A e cos ( ) Since is very sall 4. Then, Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

3 Transien Sae Analysis of a daped & forced oscillaor 1 DSRC 1 A E e sin ( ) A e cos ( ) Puing and aing A e coon in 1 oh ers we have, E A e, where 1 A E oal energy of he oscillaor a. Then, we can wrie he variaion of energy of he daped oscillaor wih ie as, E Ee Decay ie & relaxaion ie: This ells us ha he oal echanical energy decays exponenially wih a ie consan due o he fricion presen in he oscillaor. Tie consan (or daping ie or characerisic ie of he oscillaor) is defined as he ie during which he energy of he daped oscillaor decreases y e ies, ha is E E e. This us no e isinerpreed wih he ie ' during which displaceen apliude decreases y e ies, ecause ', which is called relaxaion ie or odulus of decay[5]. Boh & ' will e ore if daping is less and vice versa. Q-facor: Qualiy facor (or Q-facor) of a daped oscillaor is a easure of he rae a which he oscillaor losses energy. The energy los y he oscillaor per second, ha is, he power dissipaed y fricion is de P E e d Then, he energy loss during one ie period of oscillaion, ha is, one cycle is E de T d. E e T E ET E or, E T E Muliplying oh sides y, we have E T The LHS er is defined as he Q-facor of he oscillaor as ies he raio of energy of he oscillaor a any insan and he energy los (easured fro ha insan) during one cycle of oscillaion[6]. Puing and T, we have, Q ' ' For lighly daped oscillaor, is very less, hence is very large. So, Q is very large which physically signifies ha he oscillaor execues large nuer of oscillaions o decay is energy (and apliude) y a facor e. For wea daping, and Q. We can Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

4 P.K.Shara 1 DSRC also enion he rae of energy loss, de E. If Q d Q is ore, energy decays less rapidly and vice versa. Forced Oscillaion wih F : Afer we discussed he ransiens in a daped oscillaion, le us apply his idea of ransiens in he iniial sage of a forced oscillaor efore i aains a seady sae. When a haronic force F F Cos acs on a daped oscillaor of naural frequency o, he ne force acing on i is or criical ie of he daped oscillaion), all ransien oscillaions (eas) will vanish. Evenually he oscillaion apliude will e seady afer a long ie. As, a, he loc was a res a he equiliriu posiion and in he due course of ie he oscillaor asors energy fro he driving agen, is displaceen apliude will increase fro zero o a axiu value; efore i aains a seady sae, he apliude will oscillae several ies wih a frequency which is equal o half of he ea frequency f Since daping and a driving forces are here, he apliude of oscillaion will decrease o a consan value insead of going o zero. Fne F x v Or, d x dx F x cos d d This differenial equaion has general soluion x xp xc, where x p = paricular inegral = ACos( ) and x c =copleenary soluion of he hoogeneous equaion d x dx, x d d given as x Be Cos( ) ; discussed in earlier secion. c for wea daping, as, Then, x ACos Be Cos where AB,, and are consans o e calculaed y using iniial condiions[7]. Afer a long ie he second er, ha is, ransiens will vanish and he seady sae equaion is esalished, given as x ACos( ) Iniially he ransiens (shor lived oscillaions) given y he second expression of he general soluion will e presen afer a long ie (a ie uch greaer han he ean ie [8] For wea daping and he driving frequency near, he energy of he oscillaor can e given as he funcion of ie as 1 E E e e cos The energy graph ells us ha during few ean ies fro saring (=), he energy of he oscillaor increases fro zero, reaches he axiu and coes down o is iniu value alernaingly ainaining a ie period called ea iet. Furher ore, we can see fro he graph ha in each oscillaion he axiu energy decreases wih ie o a unifor average value Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

5 Transien Sae Analysis of a daped & forced oscillaor 1 DSRC afer a long ie (pracically few ea periods ecause T is large as ) Le us analyze he siuaion as following. A ie, he loc was a res. Hence, iniially he ineic energy is zero. Since he spring was undefored iniially, he iniial poenial energy sored in he spring is zero. This eans ha he oal echanical energy of he oscillaor is zero iniially. For he sae of sipliciy le us ignore any fricion. As he applied force coninues o ac oward righ (say) for a ie inerval (fro ), i sreches he 4 spring y a disance x, say. Afer a ie, he phase difference eween hese wo forces is, 1, where 1 and are he phases of he driving force and he spring force respecively afer a ie. Since, 1 and, we have ( ) Iniially, for soeie, he speed of he loc will e sall as i gains lile ineic energy fro he exernal agen. Moreover, we assued a sall daping consan. Hence he effec of fricion can e ignored for a sall ie inernal (few ie consans). Then he exernal force F will e added verorially wih he spring force o yield he ne force. Then he spring pulls he loc ac in he a spring force which is linear wih x, gives as Fs x Since x changes siple haronically, he spring force will also change siple haronically lie he applied (driving) force F. Force F F is ahead of F y 9 S F S This eans ha wo siple haronic forces ac on he loc siulaneously u hey do no ecoe axiu (or iniu) a a ie. Iniially he driving force F is axiu (ha is F ) and he spring force F was zero. Thus F heads Fs y 9 a. This is he iniial phase difference eween hese wo forces;. s F F F F F cos, sp ex sp ex where F F ex, Fsp A and This is quie eviden ha he ne force will flucuae eween axiu F F and iniu F sp ex sp ex F depending on he relaive phase difference. The ie period of flucuaion (aleraion) of he ne force acing on he loc is (ecause a 1 n ies, where n =1,,3., he forces are in phase or ne force is axiu). Then he eaing ie (he ie period of flucuaion of energy) is wice he period of aleraion of force ecause energy is Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

6 P.K.Shara 1 DSRC direcly proporional o he square of he force, which is given as T. ( ) We have over siplified he arguen y neglecing he fricion for soe ie fro saring. However, as he special increases, fricion f ( v) ecoes ore which will adjus he phase as we will discuss i laer on. The oion will e ore when he forces ac in phase which increases he speed ( KE) of he loc in he firs half of he ea period.this eans ha The applied or driving force soe ies (for firs half of T ) is pushing wih a relaive phase " " which helps uild up he oscillaion apliude, u i soeies (second half of he ea ie T ) pushing wih he opposie phase, hus diinishing he oscillaions[9]. 1 Kax vax. Here, we canno equae U ax wih K ax due o wo reasons; firs of all he oscillaing syse is no purely conservaive as fricion is presen, and secondly he driving force is eing aced on he syse fro ou side (exernal force). However, we can wrie he wor energy heore, as Wfricion WF Esyse U K As he fricion increases wih he increasing speed, wor done y fricion increases. When he speed is axiu, axiu fricional wor is done. Hence he axiu energy decreases since Wdriving Fv and W Fv v ( f v), for sall value of fricion v, Fv v. Hence he energy increases fro zero o soe seady value (via ups and down of he oal echanical energy of he oscillaor). A seady sae, he energy E flucuaes eween is axiu and iniu value hrough an average value. As he ie goes on, he wor done y he driving force will e ore, and hence he ineic and poenial energy of he oscillaor increases fro zero (hrough several axiu and iniu). When he displaceen of he loc will e axiu, he ineic energy will e zero. A ha ie he oal echanical energy of he oscillaing syse is 1 ax. A Siilarly when he speed of he loc is axiu, deforaion of he spring will e zero and he ineic energy ecoes axiu given as However, for soe ie fro saring, during which he de oal echanical energy increase, and hence d as he driving force is perforing a posiive wor. During each ea period he oal echanical energy pea decreases exponenially as displaceen apliude decreases exponenially due o fricional effec. As a resul he energy eas will decay. In oher words, he iniial oscillaion due o he spring force will dap due o fricion. In his sense we can say ha, ecause of daping he oscillaor gradually adjuss is phase wih respec o he driving force. Afer a sufficienly long ie he oscillaor seles ino a seady sae of viraion wih no eas, oscillaing exacly a driving frequency. This ells us ha during firs few seconds Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN:

7 Transien Sae Analysis of a daped & forced oscillaor 1 DSRC ( s ) he oscillaor asors energy which flucuaes eween decreasing peas and valleys. However, he average echanical energy increases fro zero o an 1 average value E ( av )A afer a long 4 ie, which is nown as seady sae. In seady sae he relaive phase difference eween he driving force and velociy (or displaceen or spring force) does no vary wih ie. Hence, we can say ha in seady sae. as he speed increases whose average value, ha is, average power dissipaed y fricion will e nuerically equal o he average rae a which he driving agen is feeding he energy o he oscillaior. Maheaically, in seady sae he average inpu power will e equal o he average power loss due o he fricion and he average echanical energy ha he oscillaor accuulaes since he eginning ( ) will reain consan. he aoun of energy delivered o he oscillaor in each push(each cycle) of he driving force is equal o he energy los y he oscillaor in one cycle due o fricional drag.then he oscillaor energy reains consan and he relaive phase of he oscillaor and driving force reains consan[11]. When he frequency of he driving force is equal o he naural or free undaped frequency of oscillaion, ha is, he energy will uild up (wihou any eas) soohly o a seady value given y he expression. E E 1e. Recapiulaing, Beas are daped due o he doinaing effec of fricion a higher speed. We can see ha he apliude daps o aou one half of he axiu value A i reaches a firs when he eas are presen[1]. The driving force F ay do soe posiive wor when he loc oves in he direcion of F in he firs 1 seconds 1 and hen does less negaive wor during ie. As a whole i does a posiive wor increasing he oal echanical energy of he oscillaing syse inspie of he less negaive fricional wor. However, in seady sae he wor done y fricion ecoes significan REFERENCES: 1. The Feynen Lecurers on Physics Vol-II, page - 9. An Inroducion o echanics; Kleppner.e.al -page no Universiy Physics-1, nd ediion Anwar Kaal -page no Universiy Physics-1, nd ediion Anwar Kaal -page no Universiy Physics-1, nd ediion Anwar Kaal -page no- 369 & An inroducion o Mechanics;Kleppner.a.al page no An inroducion o echanics; Kleppner.e.al. page no Mechanics 3 rd ediion; Course in heoreical Physics vol I L.D. Landau.e.al. page no Bereley Physics course vol-iii, waves; Fran 9. Bereley Physics course vol-iii, waves; Fran 1. Bereley Physics course vol-iii, waves; Fran 11. Bereley Physics course vol-iii, waves; Fran 1. Bereley Physics course vol-iii, waves; Fran Crawfore Jr. page no -11 Inernaional Journal of Copuers Elecrical and Advanced Counicaions Engineering Vol.1 (), ISSN: