Oscillations CHAPTER 3. ν = = 3-1. gram cm 4 E= = sec. or, (1) or, 0.63 sec (2) so that (3)

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1 CHAPTER 3 Oscillaios 3-. a) gram cm 4 k dye/cm sec cm ν sec π m π gram π gram π or, ν.6 Hz () or, π τ sec ν τ.63 sec () b) so ha 4 3 ka dye-cm E 4 E 4.5 erg c) The maximum velociy is aaied whe he oal eergy of he oscillaor is equal o he kieic eergy. Therefore, 4 mv max 4.5 erg v max

2 8 CHAPTER 3 or, v 3 cm/sec (4) max 3-. a) The saeme ha a a cerai ime he maximum ampliude has decreased o oehalf he iiial value meas ha or, so ha Sice sec, () β xe Ae A β e β () β l.69 b) Accordig o Eq. (3.38), he agular frequecy is 6.9 sec (4) β (5) where, from Problem 3-, sec. Therefore, so ha ( 6.9 ) ( 6.9) sec 6 (6) which ca be wrie as where Tha is, ν is oly slighly differe from ν. ( 4 5 ) sec ν. (7) π ν ν δ (8) δ 5.4 (9)

3 OSCILLATIONS 8 βτ c) The decreme of he moio is defied o be e e βτ.445 where τ ν. The, 3-3. The iiial kieic eergy (equal o he oal eergy) of he oscillaor is m g ad v cm/sec. mv, where a) Maximum displaceme is achieved whe he oal eergy is equal o he poeial eergy. Therefore, mv kx or, x m v 4 cm k x cm () b) The maximum poeial eergy is or, U kx 4 max U 5 ergs () max 3-4. a) Time average: The posiio ad velociy for a simple harmoic oscillaor are give by where km The ime average of he kieic eergy is x Asi () x Acos () T + τ mx d τ π where τ is he period of oscillaio.

4 8 CHAPTER 3 By iserig () io, we obai or, + τ cos T ma τ d (4) ma T (5) 4 I he same way, he ime average of he poeial eergy is U + τ τ kx d ad sice km, (6) reduces o ka τ ka 4 + τ si d (6) From (5) ad (7) we see ha U ma (7) 4 T U (8) The resul saed i (8) is reasoable o expec from he coservaio of he oal eergy. E T + U (9) This equaliy is valid isaaeously, as well as i he average. O he oher had, whe T ad U are expressed by () ad (), we oice ha hey are described by exacly he same fucio, displaced by a ime τ : T ma cos ma U si Therefore, he ime averages of T ad U mus be equal. The, by akig ime average of (9), we fid b) Space average: The space averages of he kieic ad poeial eergies are T () E U ()

5 OSCILLATIONS 83 ad is readily iegraed o give T A A mx dx () A m U kx dx x A dx A A m A U (4) 6 To iegrae (), we oice ha from () ad () we ca wrie ( A x ) x A cos A si The, subsiuig (5) io (), we fid A m T A x A (5) dx or, 3 m 3 A A A 3 (6) From he compariso of (4) ad (7), we see ha To see ha his resul is reasoable, we plo T T(x) ad U U(x): U U(x) T T(x) m A T (7) 6 T U (8) x T m A A (9) U m x ma A O Eergy E cos. A ma Ad he area bewee T(x) ad he x-axis is jus wice ha bewee U(x) ad he x-axis. x

6 84 CHAPTER Differeiaig he equaio of moio for a simple harmoic oscillaor, we obai Bu from () Therefore, ad subsiuio io () yields x Asi () x A cos () x si A ( x ) cos A (4) x A x The, he fracio of a complee period ha a simple harmoic oscillaor speds wihi a small ierval x a posiio x is give by x x τ τ π A x A x τ (5) (6) A 3 A A A A 3 A x This resul implies ha he harmoic oscillaor speds mos of is ime ear x ±A, which is obviously rue. O he oher had, we obai a sigulariy for τ a x ±A. This occurs because a hese pois x, ad () is o valid k m m x x x Suppose he coordiaes of m ad m are x ad x ad he legh of he sprig a equilibrium is. The he equaios of moio for m ad m are mx k x x + () mx k x x + ()

7 OSCILLATIONS 85 From (), we have Subsiuig his expressio io (), we fid from which Therefore, x d d oscillaes wih he frequecy We obai he same resul for x ( m x + kx k ) k mmx + ( m + m) kx m + m (4) x kx (5) mm m + m k mm (6). If we oice ha he reduced mass of he sysem is defied as x we ca rewrie (6) as µ m + m (7) k (8) µ k µ This meas he sysem oscillaes i he same way as a sysem cosisig of a sigle mass µ. Iserig he give values, we obai µ 66.7 g ad.74 rad s A h s h b Le A be he cross-secioal area of he floaig body, h is heigh, h he heigh of is submerged par; ad le ρ ad ρ deoe he mass desiies of he body ad he fluid, respecively. The volume of displaced fluid is herefore V Ahs. The mass of he body is M ρahb. b s

8 86 CHAPTER 3 There are wo forces acig o he body: ha due o graviy (Mg), ad ha due o he fluid, pushig he body up ( ρgv ρghsa ). The equilibrium siuaio occurs whe he oal force vaishes: which gives he relaio bewee h ad h : s Mg ρ gv b ρgah ρ gh A () b s hs h ρ b ρ For a small displaceme abou he equilibrium posiio ( h Upo subsiuio of () io, we have or, () b b s Thus, he moio is oscillaory, wih a agular frequecy s h +x), () becomes Mx ρah x ρgah ρ g h + x A b s ρah x ρ gxa (4) ρ x + g x (5) ρh b b g ρ g ga (6) ρ h h V where use has bee made of (), ad i he las sep we have muliplied ad divided by A. The period of he oscillaios is, herefore, Subsiuig he give values, τ 8. s. s π V τ π (7) ga 3-8. y O l m s The force resposible for he moio of he pedulum bob is he compoe of he graviaioal force o m ha acs perpedicular o he sraigh porio of he suspesio srig. This compoe is see, from he figure (a) below, o be a a x F ma mv mg cos α ()

9 OSCILLATIONS 87 where α is he agle bewee he verical ad he age o he cycloidal pah a he posiio of m. The cosie of α is expressed i erms of he differeials show i he figure (b) as where dy cos α () ds ds dx + dy m mg α F α dy ds dx S (a) (b) The differeials, dx ad dy, ca be compued from he defiig equaios for x(φ) ad y(φ) above: ( cosφ) dx a dφ dy asi φ dφ (4) Therefore, ds dx + dy ( cos ) si ( cos ) a φ φ dφ a φ dφ + φ 4a si d φ (5) so ha Thus, The velociy of he pedulum bob is φ ds a si d φ (6) dy a si φ dφ ds φ asi d φ φ cos cos α (7)

10 88 CHAPTER 3 ds φ dφ v asi d d d φ 4a cos d (8) from which d φ v a d 4 cos (9) φ Leig z cos be he ew variable, ad subsiuig (7) ad (9) io (), we have 4maz mgz () or, g z+ z () 4a which is he sadard equaio for simple harmoic moio, If we ideify z+ () z g where we have used he fac ha 4a. Thus, he moio is exacly isochroous, idepede of he ampliude of he oscillaios. This fac was discovered by Chrisia Huygee (673) The equaio of moio for is while for, he equaio is I is coveie o defie which rasforms () ad () io mx k x x + F kx + F + kx () mx k x x kx + kx () ξ x x m ξ kξ+ F ; m ξ kξ ; (4)

11 OSCILLATIONS 89 The homogeeous soluios for boh ad (4) are of familiar form ξ Ae Be i i +, where km. A paricular soluio for is ξ F k. The he geeral soluios for ad (4) are F Ae i Be i ξ k + + ; (5) ξ Ce De i + i ; (6) + To deermie he cosas, we use he iiial codiios: The codiios give wo equaios for A ad B: ξ ( ) ξ ( ) x x ad x( ). Thus, (7) F + A + B k i ( A B) (8) The ad, from (5), we have Sice for ay physical moio, x ad are he iiial codiios for ξ The equaios i () ca be rewrie as: A B F k F ξ x x ( cos ) k ; (9) x mus be coiuous, he values of ξ ( ) ad ξ + which are eeded o deermie C ad D: F ξ+ ( ) ( cos) Ce + De k i i F ξ+ ( ) si i Ce De k F + k i i ( cos ) i i Ce De if k i i Ce De si The, by addig ad subracig oe from he oher, we obai ( ) F C e e k i i ( ) F D e e k i i () () ()

12 9 CHAPTER 3 Subsiuio of () io (6) yields F ξ+ e e + e e k i i i i Thus, F i( ) i ( e e e ) + e k i i F cos ( ) cos k x F x cos ( ) cos ; k (4) 3-. The ampliude of a damped oscillaor is expressed by β cos( δ ) Sice he ampliude decreases o e afer periods, we have x Ae + () βt β π () Subsiuig his relaio io he equaio coecig ad (he frequecy of udamped oscillaios), β, we have Therefore, so ha + + π 4π 4 + π 8 π (4) 3-. The oal eergy of a damped oscillaor is where E mx kx () () + () β cos( δ ) x Ae () β Ae cos( ) si ( ) x β δ δ ()

13 OSCILLATIONS 9 β, Subsiuig () ad io (), we have k m A E e m k m β () ( β + ) cos ( δ) + si ( δ) Rewriig (4), we fid he expressio for E(): ( ) mβ si δ cos δ + (4) ma e β δ β β δ + β () cos( ) + si E (5) Takig he derivaive of (5), we fid he expressio for de d : de ma e β 3 ( β 4β ) cos( δ ) d (6) 4β β si( δ) β The above formulas for E ad de d reproduce he curves show i Figure 3-7 of he ex. To fid he average rae of eergy loss for a lighly damped oscillaor, le us ake β. This meas ha he oscillaor has ime o complee some umber of periods before is ampliude decreases cosiderably, i.e. he erm e β does o chage much i he ime i akes o complee oe period. The cosie ad sie erms will average o early zero compared o he cosa erm i de d, ad we obai i his limi de β mβ A e (7) d 3-. θ l mg si θ mg The equaio of moio is m θ mgsi θ () g θ si θ If θ is sufficiely small, we ca approximae si θ θ, ad () becomes ()

14 9 CHAPTER 3 which has he oscillaory soluio g θ θ ( ) cos θ θ (4) where g ad where θ is he ampliude. If here is he reardig force m equaio of moio becomes or seig si θ θ ad rewriig, we have g θ, he m θ mgsi θ + m g θ (5) θ + θ + θ (6) Comparig his equaio wih he sadard equaio for damped moio [Eq. (3.35)], x + βx + x (7) we ideify β. This is jus he case of criical dampig, so he soluio for θ() is [see Eq. (3.43)] For he iiial codiios θ θ θ ( A B) e ad θ(), we fid + (8) () ( + ) e θ θ 3-3. For he case of criical dampig, β. Therefore, he equaio of moio becomes If we assume a soluio of he form we have x + βx + β x () x ye β () β β x ye βye β β β x ye βye + β ye Subsiuig io (), we fid β β β β β β ye βye + β ye + βye β ye + β ye (4) or, Therefore, y (5) y A+ B (6)

15 OSCILLATIONS 93 ad which is jus Eq. (3.43). () x A+ Be β (7) 3-4. For he case of overdamped oscillaios, x() ad x are expressed by β x e Ae Ae + () () e ( Ae Ae ) ( A e β ++ + Ae ) x β where β. Hyperbolic fucios are defied as () or, y y y y e + e e e cosh y, sih y e e y y cosh y+ sih y cosh y sih y (4) Usig (4) o rewrie () ad (), we have () ( ) ( + ) + ( ) x cosh β sih β A A cosh A A sih (5) ad () ( cosh β sih β ) ( β)( cosh + sih ) ( Aβ A)( cosh sih ) x A A + (6) 3-5. We are asked o simply plo he followig equaios from Example 3.: β Ae cos( ) x δ () v () Ae β cos δ + si δ β () wih he values A cm, rad s, β. s, ad δ π rad. The posiio goes hrough x a oal of 5 imes before droppig o. of is iiial ampliude. A exploded (or zoomed) view of figure (b), show here as figure (B), is he bes for deermiig his umber, as is easily show.

16 94 CHAPTER 3 (b) x() (cm) v() (cm/s) (s) (c).5 v (cm/s) x (cm) (B). x (cm) (s) 3-6. If he dampig resisace b is egaive, he equaio of moio is x βx + x () where β b m> because b <. The geeral soluio is jus Eq. (3.4) wih β chaged o β: β () e Aexp( β ) + Aexp( β ) x () From his equaio, we see ha he moio is o bouded, irrespecive of he relaive values of β ad. The hree cases disiguished i Secio 3.5 ow become: a) If > β, he moio cosiss of a oscillaory soluio of frequecy β, muliplied by a ever-icreasig expoeial:

17 OSCILLATIONS 95 b) If β, he soluio is which agai is ever-icreasig. β i i x e Ae Ae + x ( A+ Be ) β (4) c) If < β, he soluio is: where β x e Ae Ae + (5) β β (6) This soluio also icreases coiuously wih ime. The ree cases describe moios i which he paricle is eiher always movig away from is iiial posiio, as i cases b) or c), or i is oscillaig aroud is iiial posiio, bu wih a ampliude ha grows wih he ime, as i a). Because b <, he medium i which he paricle moves coiually gives eergy o he paricle ad he moio grows wihou boud For a damped, drive oscillaor, he equaio of moio is ad he average kieic eergy is expressed as T x βx + x Acos () ma β ( ) Le he frequecy ocaves above be labeled ad le he frequecy ocaves below be labeled ; ha is () The average kieic eergy for each case is T T ma 4 + (4) β ma 4 + (4) β (4) (5) Muliplyig he umeraor ad deomiaor of (5) by 4, we have

18 96 CHAPTER 3 Hece, we fid T ma 4 + (4) β T T (6) ad he proposiio is prove Sice we are ear resoace ad here is oly ligh dampig, we have R, where is he drivig frequecy. This gives Q β. To obai he oal eergy, we use he soluio o he drive oscillaor, eglecig he rasies: We he have cos( δ ) x D () md E mx + kx si ( δ) + cos ( δ) m D () The eergy los over oe period is where T π. Sice, we have which proves he asserio. T ( m β x) ( xd) π m β D E Q eergy los over oe period 4πβ π (4) 3-9. The ampliude of a damped oscillaor is [Eq. (3.59)] D A + β 4 () A he resoace frequecy, β, D becomes R D R A β β Le us fid he frequecy,, a which he ampliude is ( ) D R : A A DR β β + 4 β () Solvig his equaio for, we fid

19 OSCILLATIONS 97 β β ± β (4) For a lighly damped oscillaor, β is small ad he erms i β ca be egleced. Therefore, or, ± β (5) which gives β ± + β β β (7) We also ca approximae R for a lighly damped oscillaor: β (8) R Therefore, Q for a lighly damped oscillaor becomes Q β (9) (6) 3-. From Eq. (3.66), A () x si( δ ) ( ) + 4 β Therfore, he absolue value of he velociy ampliude v is give by v A 4 + β The value of for v a maximum, which is labeled v, is obaied from ad he value is v. v v Sice he Q of he oscillaor is equal o 6, we ca use Eqs. (3.63) ad (3.64) o express β i erms of : We eed o fid wo frequecies, ad, for which v vmax We fid β (4) 46, where v v ( ). max ()

20 98 CHAPTER 3 vmax A A β + 4 β ( ) (5) Subsiuig for β i erms of from (4), ad by squarig ad rearragig erms i (5), we obai from which (6) 73 ( ) (,, ) Solvig for, we obai, ±, ±, (7) 73 6 ± ±, I is sufficie for our purposes o cosider, posiive: he (8) so ha A graph of v vs. for Q 6 is show. + ; + (9) 6 () A v max β A β v We wa o plo Equaio (3.43), ad is derivaive: x A+ Be β () [ ] β where A ad B ca be foud i erms of he iiial codiios v B A+ B e β () A x B v βx + (4)

21 OSCILLATIONS 99 The iiial codiios used o produce figure (a) were ( x v ),,, (, 4), (4, ), (, 4), 4 (, 4), ad (, 4 ), where we ake all x o be i cm, all v i cm s, ad β s. Figure (b) is a magified view of figure (a). The dashed lie is he pah ha all pahs go o asympoically as. This ca be foud by akig he limis. so ha i his limi, v βx, as required. (a) 4 3 lim v βbe β (5) lim x Be β (6) v (cm/s) 3 (b) x (cm).4. v (cm/s) x (cm) 3-. For overdamped moio, he posiio is give by Equaio (3.44) x Ae + Ae β β ()

22 CHAPTER 3 The ime derivaive of he above equaio is, of course, he velociy: a) A : The iiial codiios A. b) Whe A, we have v β x have e β β β β v A e Aβ e x A + A v Aβ Aβ (4) x ad v ca ow be used o solve for he iegraio cosas A ad ad v β x x v β A β as sice β < β. quie easily. For A, however, we () 3-3. Firsly, we oe ha all he δ π soluios are jus he egaive of he δ soluios. The δ π soluios do make i all he way up o he iiial ampliude, A, due o he reardig force. Higher β meas more dampig, as oe migh expec. Whe dampig is high, less oscillaio is observable. I paricular, β 9. would be much beer for a kiche door ha a smaller β, e.g. he door closig (δ ), or he closed door beig bumped by someoe who he chages his/her mid ad does o go hrough he door ( δ π ).

23 OSCILLATIONS β., δ β.5, δ β.9, δ.5.5 β., δ π/ β.5, δ π/ β.9, δ π/.5.5 β., δ π β.5, δ π β.9, δ π As requesed, we use Equaios (3.4), (3.57), ad (3.6) wih he give values o evaluae he complemeary ad paricular soluios o he drive oscillaor. The ampliude of he complemeary fucio is cosa as we vary, bu he ampliude of he paricular soluio becomes larger as goes hrough he resoace ear 96. rad s, ad decreases as is icreased furher. The plo closes o resoace here has., which shows he leas disorio due o rasies. These figures are show i figure (a). I figure (b), he 6 plo from figure (a) is reproduced alog wih a ew plo wih A p m s.

24 CHAPTER 3 / /9 / /3 /. 3 / / 6.5 (s) (s).5 3 (s) Leged: xc xp x (a) A p A p (b) 3-5. This problem is early ideical o he previous problem, wih he excepio ha ow Equaio (3.43) is used isead of (3.4) as he complemeary soluio. The disorio due o he rasie icreases as icreases, mosly because he complemeary soluio has a fixed ampliude whereas he ampliude due o he paricular soluio oly decreases as icreases. The laer fac is because here is o resoace i his case.

25 OSCILLATIONS 3 / /9 / /3 / / 3 5 / 6 5 / 6, A p Leged: xc xp x 3-6. The equaios of moio of his sysem are mx kx b x x + Fcos mx bx b( x x ) The elecrical aalog of his sysem ca be cosruced if we subsiue i () he followig equivale quaiies: m L ; m L ; F ε k ; b R ; x q C ; b R The he equaios of he equivale elecrical circui are give by Lq + R( q q ) + qεcos C Lq + Rq + R( q q ) () () Usig he mahemaical device of wriig exp(i) isead of cos i (), wih he udersadig ha i he resuls oly he real par is o be cosidered, ad differeiaig wih respec o ime, we have

26 4 CHAPTER 3 I i LI + R I I + iεe C LI + R( I ) + R( I I ) The, he equivale elecrical circui is as show i he figure: L The impedace of he sysem Z is I () ε cos C I () R R L Z il i + Z (4) C where Z is give by The, ad subsiuig (6) io (4), we obai Z + + Z R R il R R R + R + L + il R R + R + L R R ( R + R ) + L + i RL + L ( R + R ) + L C Z R R L + + (5) (6) (7) 3-7. From Eq. (3.89), We wrie F () a + ( a cos + b si ) () () F () a + c cos ( φ ) which ca also be wrie usig rigoomeric relaios as F() a + c cos cos φ + si si φ Comparig wih (), we oice ha if here exiss a se of coefficies c such ha

27 OSCILLATIONS 5 c c cos φ a si φ b (4) he () is equivale o (). I fac, from (4), c a + b b a φ a (5) wih a ad b as give by Eqs. (3.9) Sice F() is a odd fucio, F( ) F(), accordig o Eq. (3.9) all he coefficies vaish ideically, ad he b are give by a π b π F( ) si d π π π si d si d π + cos + cos π π cos cos π ( π ) π Thus, 4 for odd π () for eve b ( + ) b ( ) 4 ( + ) π,,, () The, we have F () si + si 3+ si 5+ π 3π 5π

28 6 CHAPTER 3 π F() π π π Terms +.99 π π Terms π π Terms I order o Fourier aalyze a fucio of arbirary period, say τ P isead of π, proporioal chage of scale is ecessary. Aalyically, such a chage of scale ca be represeed by he subsiuio for whe, he x, ad whe Thus, whe he subsiuio fucio π x or P τ P, he x π. Px () π Px π is made i a fucio F() of period P, we obai he Px F f x π ad his, as a fucio of x, has a period of π. Now, f(x) ca, of course, be expaded accordig o he sadard formula, Eq. (3.9): where f ( x) a + ( a cos x+ b si x) ()

29 OSCILLATIONS 7 π a f ( x ) cos x d π x π b f ( x ) si x dx π If, i he above expressios, we make he iverse subsiuios he expasio becomes π π x ad dx d (5) P P π P π a π π f F F() a cos b si P π P + + P P (6) ad he coefficies i (4) become (4) P π a F( ) cos P d P P π b F( ) si d P P (7) For he case correspodig o his problem, he period of F() is 4π, so ha P π. The, subsiuig io (7) ad replacig he iegral limis ad τ by he limis obai τ ad τ +, we ad subsiuig io (6), he expasio for F() is a b π π F( ) cos π d π π F( ) si d π (8) a F () + a cos b si + (9) Subsiuig F() io (8) yields Evaluaio of he iegrals gives a b π si cos π d π si si d π ()

30 8 CHAPTER 3 ad he resulig Fourier expasio is b ; b for eve a a a ( ) 4 odd π ( 4) () F () si + cos cos cos cos + () 3π 5π π 45π 3-3. The oupu of a full-wave recifier is a periodic fucio F() of he form π si ; < F () π si ; < < The coefficies i he Fourier represeaio are give by π a π ( si ) cos d si cos π + d π b π ( si ) si d + si si d π Performig he iegraios, we obai a b 4 ; if eve ( or ) π ( ) ; if odd for all () () The expasio for F() is 4 4 F () cos cos 4 (4) π 3π 5π The exac fucio ad he sum of he firs hree erms of (4) are show below.

31 OSCILLATIONS 9 Sum of firs hree erms F().5 si π π π π 3-3. We ca rewrie he forcig fucio so ha i cosiss of wo forcig fucios for > τ: () F m < ( τ ) a a ( τ) ( τ ) a > τ τ < < τ () Durig he ierval < < τ, he differeial equaio which describes he moio is The paricular soluio is xp a x + βx + x () τ C+D, ad subsiuig his io (), we fid from which Therefore, we have which gives a βc+ C+ D τ βc + D (4) a C τ βa a D, C τ τ 4 a βa xp 4 τ τ (5) (6) Thus, he geeral soluio for < < τ is ad he, a β a τ (7) β () e A cos + Bsi + 4 τ x

32 CHAPTER 3 β β a () β cos + si + si + cos + x e A B e A B (8) τ The iiial codiios, x(), x, implies Therefore, he respose fucio is βa A 4 τ a β B τ (9) () x β a β β e β β e cos + si + () τ ( τ ) a For he forcig fucio i (), we have a respose similar o (). Thus, we add hese τ wo equaios o obai he oal respose fucio: β a β β β e β x () e ( cos e cos ( τ) ) + τ β si + e si τ + τ () Whe τ, we ca approximae e βτ as + βτ, ad also si τ τ, cos τ. The, β a β β e β x () e cos ( βτ)( cos τ si ) τ τ si + βτ si τ cos + τ 3 a β β β β β e cos e + si () If we use β, he coefficie of e β si becomes β. Therefore, a β β β x () e cos e si τ This is jus he respose for a sep fucio a) Respose o a Sep Fucio: From Eq. (3.) H ( ) is defied as

33 OSCILLATIONS H Wih iiial codiios x ( ) ad x ( ), < a, > moio of a damped liear oscillaor) is give by Eq. (3.5):, he geeral soluio o Eq. (3.) (equaio of β( ) a β( ) βe x () cos e ( ) si ( ) for > x () for < () () where β. For he case of overdampig, < β, ad cosequely β is a pure imagiary umber. Hece, cos ( ) ad ( ) i si are o loger oscillaory fucios; isead, hey are rasformed io hyperbolic fucios. Thus, if we wrie β (where is real), The respose is give by [see Eq. (3.5)] ( ) i ( ) ( ) cos cos cosh si ( ) si i( ) isih ( ) β( ) a β( ) βe x () cosh e ( ) sih ( ) for > x () for < (4) For simpliciy, we choose, ad he soluio becomes β H () β βe x e cosh sih (5) This respose is show i (a) below for he case β 5. b) Respose o a Impulse Fucio (i he limi τ ): From Eq. (3.) he impulse fucio (, ) I is defied as < I (, ) a < < > For τ i such a way ha aτ is cosa b, he respose fucio is give by Eq. (3.): (6)

34 CHAPTER 3 x b e (7) β( ) () si for > Agai akig he spike o be a for simpliciy, we have b β x () e si () for > (8) For i i β (overdamped oscillaor), he soluio is This respose is show i (b) below for he case β 5. b β x () e sih ; > (9) (a) x H (b) x b a) I order o fid he maximum ampliude of he respose fucio show i Fig. 3-, we look for such ha x give by Eq. (3.5) is maximum; ha is, ( x ()) () From Eq. (3.6) we have ( x ) H β β e si + ()

35 OSCILLATIONS 3 For β., β.98. Evidely, π makes () vaish. (This is he absolue maximum, as ca be see from Fig. 3-.) The, subsiuig io Eq. (3.5), he maximum ampliude is give by or, a x x max e βπ () + x.53 a (4) b) I he same way we fid he maximum ampliude of he respose fucio show i Fig. 3-4 by usig x() give i Eq. (3.); he, ( x ()) β( ) β be cos ( ) si ( ) (5) If (5) is o vaish, is give by.37 β a a 4.9 Subsiuig (6) io Eq. (3.), we obai (for β. ) (6) or, β.37 b x () x e si(.37) (7) max.98 x.76 a τ (8) The respose fucio of a udamped (β ) liear oscillaor for a impulse fucio π I(,τ), wih τ, ca be obaied from Eqs. (3.5) ad (3.8) if we make he followig subsiuios: β ; π () ; τ (For coveiece we have assumed ha he impulse forcig fucio is applied a.) Hece, afer subsiuig we have

36 4 CHAPTER 3 x < a π x () cos < < a π x () cos ( w π) cosw τ > This respose fucio is show below. Sice he oscillaor is udamped, ad sice he impulse lass exacly oe period of he oscillaor, he oscillaor is reured o is equilibrium codiio a he ermiaio of he impulse. a () a π π The equaio for a drive liear oscillaor is x+ βx + w x f where f() is he siusoid show i he diagram. f() a I II III Regio I: x () Regio II: x+ βx + x asi () The soluio of () is i which Regio III: x+ βx + x ( si cos ) β x e A B x P + + (4) x Da (5) P si ( δ ) where D ( ) + 4 β (6)

37 OSCILLATIONS 5 Thus, The iiial codiio x() gives ad x δ β a ( Asi Bcos ) Dasi ( ) β x e + + δ (8) B adsi δ (9) (7) or, The soluio of is βb+ A+ Da cos δ ad A ( β si δ cos δ) () We require ha x ad x π π x gives II III β si x e A + B cos () x for regios II ad III mach a π. The codiio ha βπ ( si + Bcos ) + Dasi ( ) e ( A si + B cos φ ) βπ e A φ φ π δ φ where φ π or, ad si δ βπ A + B co φ A+ Bcoφ+ e () si φ The codiio ha x π π II xiii gives or, or, βπ ( si cos ) cos ( cos si φ) βπ βe A φ+ B φ + ad π δ + e A φ B βπ ( si cos ) ( cos si ) βπ βe A φ+ B φ + e A φ B φ ( cos φ β si φ) B ( si φ+ β cos φ) A ( ) ( ) A β si φ+ cos φ B si φ+ β cos φ e βπ ad cos δ si φ+ β cos φ si φ+ β cos φ Da cos δ cosφ β si φ cos φ β si φ cos φ β si φ A B A B e βπ

38 6 CHAPTER 3 Subsiuig io from (), we have B ( ) + ( + ) ( cos φ β si φ) si φ cos φ β si φ cos φ si φ β cos φ si φ from which (4) ( cos φ β si φ) cos φ+ ( si φ+ β cos φ) si φ si δ cos δ + ade ( cos si ) βπ + φ β φ si φ si φ cos φ β si φ Usig (), we ca fid A : a B adsi δ + De βπ si δ ( cos φ βsi φ) + cos δsi φ (5) βπ ad si δe A A+ Bco φ + B co φ (6) si φ Subsiuig for A, B, ad B from (), (9), ad (5), we have β βπ β ad cos δ βπ A adsi δ + e cos φ si φ ( e cos ) + + φ (7) Thus, we obaied all cosas givig us he respose fucios explicily Wih he iiial codiios, x x ad > give by Eq. (3.3) yields a x βx + A x ; A x x, he soluio for a sep fucio for Therefore, he respose o H ( ) for he iiial codiios above ca be expressed as βa x βx x e x β( ) () cos ( ) + + si ( ) a β( ) β β( ) + cos e ( ) e si ( ) for > The respose o a impulse fucio I (, ) H give by () for < < ad by a superposiio of soluios for H ( ) ad for, for he above iiial codiios will he be H ake idividually for >. We mus be careful, however, because he soluio for > mus be equal ha give by () for. This ca be isured by usig as a soluio for H ( ) Eq. (3.3) wih iiial codiios x, x, ad usig isead of i he expressio. The soluio for > is he () ()

39 OSCILLATIONS 7 where β( ) x βx x () e x cos + + si + x() β( ) ae βτ x() e cos ( τ) cos ( ) + βτ βe β + si ( τ) si ( ) for > βe βτ (4) We ow allow a as τ i such a way ha aτ b cosa; expadig for his paricular case, we obai β( ) x βx b x () e x cos si > (5) which is aalogous o Eq. (3.9) bu for iiial codiios give above Ay fucio F m ca be expaded i erms of sep fucios, as show i he figure below where he curve is he sum of he various (posiive ad egaive) sep fucios. I geeral, we have x + βx + x F m H () () where H () a > τ < τ The, sice () is a liear equaio, he soluio o a superposiio of fucios of he form give by () is he superposiio of he soluios for each of hose fucios. Accordig o Eq. (3.5), he soluio for H for > is () he, for β( ) a β( ) βe x() e cos ( ) si ( ) F m () H (4) he soluio is

40 8 CHAPTER 3 β( ) β( ) βe x () H() e cos( ) si () () () () mh G F G (5) where G () or, comparig wih β( ) β( ) βe cos e ( ) si ( ) ; m < G (), x ma < Therefore, he Gree s fucio is he respose o he ui sep. (6) (7) () F m H () The soluio for x() accordig o Gree s mehod is () (, ) x F G d F m γ β( ) e si e si ( ) d () Usig he rigoomeric ideiy, si si ( ) cos ( + ) cos ( ) + () we have

41 OSCILLATIONS 9 β Fe β γ β γ x () de cos ( ) de cos ( ) m + + z + Makig he chage of variable, for he secod iegral, we fid, for he firs iegral ad ( β γ) ( β γ) β + ( β γ) z ( β γ) y + dze cos z dye cos y y + Fe e e x () m + (4) Afer evaluaig he iegrals ad rearragig erms, we obai F x () m ( β γ) + ( + ) ( β γ) + ( ) γ si e ( γ β) cos+ [ β γ ] + β si + e ( β γ) cos+ [ β γ ] + (5) si < < π F () π < < π From Equaios 3.89, 3.9, ad 3.9, we have F a a b () + ( cos + si ) τ a F( ) cos τ d τ b F( ) si τ d π a si cos d π a π si d π π π a si cos d π

42 CHAPTER 3 ( ) ( ) π cos cos + a ( ) si cos d π π + Upo evaluaig ad simplifyig, he resul is π a eve π ( ) odd,,, b by ispecio b π π si d ( ) ( ) π si si + b ( ) si si d π π + π So or, leig F + + () si π π,4,6, ( ),, cos F () + si + cos π π 4 ( ) The followig plo shows how well he firs four erms i he series approximae he fucio.. Sum of firs four erms The equaio describig he car s moio is π π dy m k y asi d where y is he verical displaceme of he car from is equilibrium posiio o a fla road, a is he ampliude of sie-curve road, ad k elasic coefficie dm g N/m dy.

43 OSCILLATIONS πv λ The soluio of he moio equaio ca be cas i he form 74 rad/s wih v ad λ beig he car s speed ad wavelegh of sie-curve road. a y B + β + si cos wih k 9.9 rad/s m We see ha he oscillaio wih agular frequecy has ampliude a A.6 mm The mius sig jus implies ha he sprig is compressed a) The geeral soluio of he give differeial equaio is (see Equaio (3.37)) ad exp ( β ) exp β + exp β x A A β ( β ) ( β ) ( β ) v () x exp Aexp + A exp a, x () x, v () v ( β) β A ( β ) A ( β ) + exp exp exp A x + v + βx β ad A v + βx x β () b) i) Uderdamped, β I his case, isead of usig above parameerizaio, i is more coveie o work wih he followig parameerizaio ( β ) ( β δ) x () Aexp cos () ( β ) β ( β δ) β ( β δ) v () Aexp cos + si Usig iiial codiios of x() ad v(), we fid x v A + β β x + β ad a ( δ) v x + β β

44 CHAPTER 3 I he case β, ad usig (6) below we have so fially a δ v 3 x x v v A + + x 3 3 x x δ 3 3 () exp cos 3 3 x x + (4) ii) Criically damped, β, usig he same parameerizaio as i i) we have from () ad : ad v () x() xexp( v x x () Aexp x exp (5) β (6) iii) Overdamped, β, reurig o he origial parameerizaio () we have (always usig relaio (6)), exp( β ) exp( β ) + exp( β ) x A A ( x ) + ( ) 3 + x 3 exp 3 exp 3 + (7) 3 3 Below we show skeches for equaios (4), (5), (7) x Uderdamped Criically damped Overdamped 3-4. a) The mos geeral soluio is ( ) where he las erm is a paricular soluio. mx + x Fsi () x asi + bcos + Asi

45 OSCILLATIONS 3 To fid A we pu his paricular soluio (he las erm) io () ad fid F A m A, x, so we fid b, ad he we have ( ) x () asi + Asi v () a cos + A cos A, v A a F m x () si si + b) I he limi oe ca see ha 3 F x () 6m The skech of his fucio is show below. x a) Poeial eergy is he elasic eergy: Ur () kr ( a), where m is movig i a ceral force field. The he effecive poeial is (see for example, Chaper ad Equaio (8.4)): l l Ueff () r U() r + k( r a) + mr mr where l mvr m r is he agular momeum of m ad is a coserved quaiy i his problem. The solid lie below is U () r ; a low values of r, he dashed lie represes Ur () kr a, ad he solid lie is domiaed by l mr Ueff () r U() r k( r a). eff. A large values of r,

46 4 CHAPTER 3 Poeial eergy l mr Ur () kr ( a) b) I equilibrium circular moio of radius r, we have kr a m r U ( a) kr mr r r eff () c) For give (ad fixed) agular momeum l, V(r) is miimal a r, because V () r, so we make a Taylor expasio of V(r) abou r ; r r where 3 m ( r r) K( r r) Vr Vr ( ) + ( r r) V ( r) + ( r r) V ( r) +... K 3m, so he frequecy of oscillaio is K 3 m 3( kr a) mr This oscillaio mus be uderdamped oscillaio (oherwise o period is prese). From Equaio (3.4) we have so he iiial ampliude (a ) is A. 8π Now a 4T ( β ) ( δ) x () Aexp cos 8π x(4 T) Aexp β cos (8 π δ) The ampliude ow is 8π A exp β, so we have

47 OSCILLATIONS 5 ad because β, we fially fid 8π A exp β A e 8π π Eergy of a simple pedulum is For a slighly damped oscillaio θ θexp( β). mgl θ where θ is he ampliude. Iiial eergy of pedulum is mgl θ. Eergy of pedulum afer oe period, T π So eergy los i oe period is So eergy los afer 7 days is l g, is mgl mgl θ T θ βt exp( ) mgl mgl θ ( exp( βt) ) θ βt mglθ β T (7 days) mglθβt mglθβ(7 days) T This eergy mus be compesaed by poeial eergy of he mass M as i falls h meers: Mh Mgh mglθβ(7 days) β. s mlθ (7 days) Kowig β we ca easily fid he coefficie Q (see Equaio (3.64)) g β β R Q l 78 β β β

48 6 CHAPTER 3