Differenial Equaions (MTH40) Lecure Daped Moion In he previous lecure, we discussed he free haronic oion ha assues no rearding forces acing on he oving ass. However No rearding forces acing on he oving body is no realisic, because There always eiss a leas a resising force due o surrounding ediu. For eaple a ass can be suspended in a viscous ediu. Hence, he daping forces need o be included in a realisic analysis. Daping Force In he sudy of echanics, he daping forces acing on a body are considered o be d proporional o a power of he insananeous velociy. In he hydro dynaical probles, he daping force is proporional o Daping fo rce ( d / ) d -β. So ha in hese probles Where β is a posiive daping consan and negaive sign indicaes ha he daping force acs in a direcion opposie o he direcion of oion. In he presen discussion, we shall assue ha he daping force is proporional o he d insananeous velociy. Thus for us The Differenial Equaion Suppose Tha Daping fo rce d -β A body of ass is aached o a spring. The spring sreches by an aoun s o aain he equilibriu posiion. The ass is furher displaced by an aoun and hen released. No eernal forces are ipressed on he syse. Therefore, here are hree forces acing on he ass, naely: a) Weigh g of he body b) Resoring force k ( s + ) d c) Daping force -β 7
Differenial Equaions (MTH40) Therefore, oal force acing on he ass g k is d ( s + ) β So ha by Newon s second law of oion, we have d g k Since in he equilibriu posiion Therefore g ks 0 d ( s + ) β d d k β Dividing wih, we obain he differenial equaion of free daped oion d β d k + + 0 For algebraic convenience, we suppose ha Then he equaion becoes: β λ, ω k d d + λ + ω 0 Soluion of he Differenial Equaion Consider he equaion of he free daped oion Pu e, Then he auiliary equaion is: d d + λ + ω 0 d d e, e + λ + ω 0 Solving by use of quadraic forula, we obain λ± λ ω Thus he roos of he auiliary equaion are λ + λ ω, λ λ ω 8
Differenial Equaions (MTH40) Depending upon he sign of he quaniy λ ω, we can now disinguish hree possible cases of he roos of he auiliary equaion. Case Real and disinc roos If λ ω > 0 hen β > k and he syse is said o be over-daped. The soluion of he equaion of free daped oion is () c e c e + λ λ ω λ ω or () e [ ce + ce ] This equaion represens sooh and non oscillaory oion. Case Real and equal roos If λ ω 0, hen β k and he syse is said o be criically daped, because any sligh decrease in he daping force would resul in oscillaory oion. The general soluion of he differenial equaion of free daped force is () c e c e + or () e λ ( c c ) Case Cople roos + If λ w < 0, hen β < k and he syse is said o be under-daped. We need o rewrie he roos of he auiliary equaion as: λ + ω λ i, λ ω λ i Thus, he general soluion of he equaion of free daped oion is () λ e c λ + c λ cos ω sin ω This represens an oscillaory oion; bu apliude of vibraion 0 as because of e λ he coefficien. Noe ha Each of he hree soluions conain he daping facor e he ass becoe negligible for larger ies. λ, λ > 0, he displaceens of 9
Differenial Equaions (MTH40) Alernaive for of he Soluion When λ ω < is () 0, he soluion of he differenial equaion of free daped oion d d + λ + ω 0 λ e c λ + c λ cos ω sin ω Suppose ha A and φ are wo real nubers such ha So ha c c sin φ, cosφ A A c A c + c, anφ c The nuber φ is known as he phase angle. Then he soluion of he equaion becoes: () Ae λ sin ω λ cosφ + cos λ or () Ae sin( ω λ + φ ) Noe ha ω λ sinφ The coefficien Ae λ is called he daped apliude of vibraions. The ie inerval beween wo successive aia of ( ) is called quasi period, and is given by he nuber π ω λ The following nuber is known as he quasi frequency. ω λ π The graph of he soluion λ () Ae sin ( ω λ +φ ) crosses posiive -ais, i.e he line 0, a ies ha are given by ω λ + φ nπ Where n,,,. For eaple, if we have () 0.5 π e sin 0
Differenial Equaions (MTH40) Then π nπ or π π π 0, π, π, or π 4π 7π,,, 6 6 6 We noice ha difference beween wo successive roos is π k k quasi period π Since quasi period π. Therefore π k k quasi period λ Since () Ae when sin ω λ + φ, he graph of he soluion λ () Ae sin( ω λ + φ ) ouches he graphs of he eponenial funcions ± Ae λ a he values of for which sin( ω λ + φ ) ± This eans hose values of for which ( n ) π ω λ + φ + ( n + ) or ( π / ) φ where n 0,,,, ω λ Again, if we consider () 0.5 π e sin Then * π π * π π * π 5π,,, Or * 5 π * π 7,, π, Again, we noice ha he difference beween successive values is Eaple * * π k k The values of for which he graph of he soluion λ () Ae sin( ω λ + φ ) ouches he eponenial graph are no he values for which he funcion aains is relaive ereu.
Differenial Equaions (MTH40) Inerpre and solve he iniial value proble d d + 5 + 4 0 ( 0 ), ( 0) Find eree values of he soluion and check wheher he graph crosses he equilibriu posiion. Inerpreaion Coparing he given differenial equaion d d + 5 + 4 0 wih he general equaion of he free daped oion we see ha d d + λ + ω 0 λ 5, ω 4 so ha λ ω > 0 Therefore, he proble represens he over-daped oion of a ass on a spring. Inspecion of he boundary condiions ( 0 ), ( 0) reveals ha he ass sars uni below he equilibriu posiion wih a downward velociy of f/sec. Soluion To solve he differenial equaion d d + 5 + 4 0
Differenial Equaions (MTH40) d d We pu e, e, Then he auiliary equaion is + 5 + 4 0 ( + 4 )( + ) 0, 4, Therefore, he auiliary equaion has disinc real roos, 4 Thus he soluion of he differenial equaion is: 4 () c e + c e 4 So ha () c e c e 4 Now, we apply he boundary condiions Thus ( ) c.+ c. 0 ( ) c 4 c 0 c c + c 4c Solving hese wo equaions, we have. c 5, c Therefore, soluion of he iniial value proble is 4 () 5 e e Ereu 4 Since () 5 e e Therefore d 5 8 e + e 4 So ha () 0 or e e 5 8 4 e + e 0 8 5 or 0. 57 ln 8 5
Differenial Equaions (MTH40) Since Therefore a 0.57, d 4 5 e e we have () 5 0.57 0.68 d e e.45 5.69 4.67 < 0 So ha he soluion has a aiu a 0. 57 and aiu value of is: ( 0.57). 069 Hence he ass aains an eree displaceen of posiion. Check ().069 f below he equilibriu Suppose ha he graph of does cross he ais, ha is, he ass passes hrough he equilibriu posiion. Then a value of eiss for which () 0 5 i.e 4 e e 0 e or ln 0. 05 5 5 This value of is physically irrelevan because ie can never be negaive. Hence, he ass never passes hrough he equilibriu posiion. Eaple An 8-lb weigh sreches a spring f. Assuing ha a daping force nuerically equals o wo ies he insananeous velociy acs on he syse. Deerine he equaion of oion if he weigh is released fro he equilibriu posiion wih an upward velociy of f / sec. Soluion Since Therefore, by Hook s law Weigh 8 lbs, Srech s f 8 k 4 k lb / f 4
Differenial Equaions (MTH40) Since Therefore β Also d Daping force Weigh ass g 8 4 slugs Thus, he differenial equaion of oion of he free daped oion is given by or d d k β d d 4 4 d d or + 8 + 6 0 Since he ass is released fro equilibriu posiion wih an upward velociy f / s. Therefore he iniial condiions are: ( 0) 0, ( 0) Thus we need o solve he iniial value proble d d Solve + 8 + 6 0 Subjec o ( 0) 0, ( 0) Pu Thus he auiliary equaion is e, d e + 8 + 6 0, d e or ( + 4) 0 4, 4 So ha roos of he auiliary equaion are real and equal. 4 Hence he syse is criically daped and he soluion of he governing differenial equaion is 4 4 () c e + c e Moreover, he syse is criically daped. 5
Differenial Equaions (MTH40) We now apply he boundary condiions. ( ) 0 c.+ c.0 0 0 c 0 Thus () c e d 4 c e c e 4 4 c So ha ( 0). 0 4 c Thus soluion of he iniial value proble is Ereu () e 4 Since () e 4 Therefore Thus d 4 4 e ( ) e 4 4 d 0 The corresponding eree displaceen is e 4 4 Thus he weigh reaches a aiu heigh of Eaple + e 4 0.76 f 0.76 f above he equilibriu posiion. A 6-lb weigh is aached o a 5 - f long spring. A equilibriu he spring easures 8.f.If he weigh is pushed up and released fro res a a poin - f above he equilibriu posiion. Find he displaceen ( ) if i is furher known ha he surrounding ediu offers a resisance nuerically equal o he insananeous velociy. Soluion Lengh of un - sreched spring 5 f Lengh of spring a equilibriu 8. f Thus Elongaion of spring s By Hook s law, we have. f 6
Differenial Equaions (MTH40) Furher Since (.) 5 lb / f 6 k k Weigh ass g Daping force Therefore β d 6 slugs Thus he differenial equaion of he free daped oion is given by or d k d 5 d β d d d or + + 0 0 Since he spring is released fro res a a poin f above he equilibriu posiion. The iniial condiions are: ( 0 ), ( 0) 0 Hence we need o solve he iniial value proble d d + + 0 0 ( 0 ), ( 0) 0 To solve he differenial equaion, we pu e Then he auiliary equaion is or, d e + + 0 0 ± i So ha he auiliary equaion has cople roos, d + i, i The syse is under-daped and he soluion of he differenial equaion is: () e ( c c sin ) cos + e. 7
Differenial Equaions (MTH40) Now we apply he boundary condiions ( ) c.+ c.0 0 c Thus () e ( cos + c sin ) e ( sin + c cos ) e ( cos c sin d Therefore ( 0) 0 + 0 6 + ) c c Hence, soluion of he iniial value proble is () e cos sin Eaple 4 Wrie he soluion of he iniial value proble d d + + 0 0 ( 0 ), ( 0) 0 in he alernaive for () Ae sin ( + φ ) Soluion We know fro previous eaple ha he soluion of he iniial value proble is () e cos sin Suppose ha A and φ are real nubers such ha / sinφ, cosφ A A 4 Then A 4 + 0 9 Also an φ / Therefore an ( ).49radian Since sin φ < 0, cosφ < 0, he phase angle φ us be in rd quadran. Therefore φ π +.49 4.9 radians Hence 8
Differenial Equaions (MTH40) () 0e sin( + 4.9) The values of where he graph of he soluion crosses posiive - ais and he values * γ γ given in he following able. where he graph of he soluion ouches he graphs of ± 0e are Quasi Period γ γ * ( ) * γ γ.6.54 0.665.678.0-0..75.49 0.08 4.77 4.96-0.09 Since () 0e sin( + 4.9) Therefore λ ω So ha he quasi period is given by π π seconds λ ω Hence, difference beween he successive * γ and γ is π unis. 9
Differenial Equaions (MTH40) Eercise Give a possible inerpreaion of he given iniial value probles.. + + 0, ( 0) 0, ( 0). 5 6 6. + + 0, ( 0), ( 0). A 4-lb weigh is aached o a spring whose consan is lb /f. The ediu offers a resisance o he oion of he weigh nuerically equal o he insananeous velociy. If he weigh is released fro a poin f above he equilibriu posiion wih a downward velociy of 8 f / s, deerine he ie ha he weigh passes hrough he equilibriu posiion. Find he ie for which he weigh aains is eree displaceen fro he equilibriu posiion. Wha is he posiion of he weigh a his insan? 4. A 4-f spring easures 8 f long afer an 8-lb weigh is aached o i. The ediu hrough which he weigh oves offers a resisance nuerically equal o ies he insananeous velociy. Find he equaion of oion if he weigh is released fro he equilibriu posiion wih a downward velociy of 5 f / s. Find he ie for which he weigh aains is eree displaceen fro he equilibriu posiion. Wha is he posiion of he weigh a his insan? 5. A -kg ass is aached o a spring whose consan is 6 N / and he enire syse is hen suberged in o a liquid ha ipars a daping force nuerically equal o 0 ies he insananeous velociy. Deerine he equaions of oion if a. The weigh is released fro res below he equilibriu posiion; and b. The weigh is released below he equilibriu posiion wih and upward velociy of /s. 6. A force of -lb sreches a spring f. A.-lb weigh is aached o he spring and he syse is hen iersed in a ediu ha ipars daping force nuerically equal o 0.4 ies he insananeous velociy. a. Find he equaion of oion if he weigh is released fro res f above he equilibriu posiion. λ Ae sin ω λ + φ b. Epress he equaion of oion in he for () ( ) c. Find he firs ies for which he weigh passes hrough he equilibriu posiion heading upward. 0
Differenial Equaions (MTH40) 7. Afer a 0-lb weigh is aached o a 5-f spring, he spring easures 7-f long. The 0-lb weigh is reoved and replaced wih an 8-lb weigh and he enire syse is placed in a ediu offering a resisance nuerically equal o he insananeous velociy. a. Find he equaion of oion if he weigh is released / f below he equilibriu posiion wih a downward velociy of f / s. λ Ae sin ω λ + φ b. Epress he equaion of oion in he for () ( ) c. Find he ie for which he weigh passes hrough he equilibriu posiion heading downward. 8. A 0-lb weigh aached o a spring sreches i f. The weigh is aached o a dashpo-daping device ha offers a resisance nuerically equal o β ( β > 0) ies he insananeous velociy. Deerine he values of he daping consan β so ha he subsequen oion is a. Over-daped b. Criically daped c. Under-daped 9. A ass of 40 g. sreches a spring 0c. A daping device ipars a resisance o oion nuerically equal o 560 (easured in dynes /(c / s)) ies he insananeous velociy. Find he equaion of oion if he ass is released fro he equilibriu posiion wih downward velociy of c / s. 0. The quasi period of an under-daped, vibraing -slugs ass of a spring isπ / seconds. If he spring consan is 5 lb / f, find he daping consan β.