Section 3.8, Mechanical and Electrical Vibrations

Transcription

1 Secion 3.8, Mechanical and Elecrical Vibraions Mechanical Unis in he U.S. Cusomary and Meric Sysems Disance Mass Time Force g (Earh) Uni U.S. Cusomary MKS Sysem CGS Sysem fee f slugs seconds sec pounds lb 3 f/sec meers m kilograms kg seconds sec Newons N 9.81 m/sec cenimeers cm grams g seconds sec dynes dyn 981 cm/sec Consider he spring of lengh l wih a weigh aached of mass m. When a mass aached o he spring is allowed o hang under he influence of graviy The spring becomes sreched by a disance L beyond is naural lengh. The lengh l + L is called he equilibrium lengh of he spring-mass sysem. There are wo forces acing where he mass is aached o he spring. The graviaional force, or weigh of he mass, acs downward and has magniude mg, where g is he acceleraion due o graviy. Example 1 illusraions The force F s, he resoraive force of he spring, acs upward. Example 1. Illusrae he lenghs l and L and he forces mg and F s. Hooke s law (he linear spring assumpion): When sreched, a spring exers a resoring force opposie o he direcion of is elongaion wih a magniude direcly proporional o he amoun of elongaion. Hooke s law is a reasonable approximaion unil he spring is sreched o near is elasic limi. F s kl, where k, he spring consan, is always >. The minus sign is due o he fac ha he spring force acs in a direcion opposie he sreching direcion (one direcion is posiive, he oher negaive). Example. Find he spring consan (in he f-lb sysem) if a -lb weigh sreches a spring 6 in. Sec. 3.8, Boyce & DiPrima, p. 1

2 Le u( be he displacemen of a mass from is equilibrium posiion a ime. According o Newon s second law (when mass remains consan mu ''( ma f ( where f( represens he sum of forces acing on he mass a ime. The Forces Acing on he Mass m Graviy. The force of graviy, w, is a downward force wih magniude mg, where g is he acceleraion due o graviy. Hence, w = mg. Spring (resoring) Force. The spring force is proporional o he oal elongaion L + u of he spring and always acs o resore he spring o is naural posiion. Since his force always acs in he direcion opposie of u we have F s = k(l + u). Damping Force. This is a fricional or resisive force acing on he mass. For example, his force may be air resisance or fricion due o a shock absorber. In eiher case we assume (an assumpion ha may be open o quesion, see p. 194 of ex ha he damping force is proporional o he magniude of he velociy of he mass, bu opposie in direcion. Tha is, F d = u (, where > is he damping consan. Exernal Forces. Any exernal forces acing on he mass (for example, a magneic force or he forces exered on a car by bumps in he pavemen will be denoed by F(. Ofen an exernal force is periodic. These forces considered as a single funcion are ofen referred o as he forcing funcion. Mass-Spring-Damper Sysem Equaions A he equilibrium posiion he only forces acing on he spring-mass sysem are w and F s. Since here is no acceleraion here, he oal force is zero. Thus a equilibrium ma Fs w kl mg. When you recall ha he spring was sreched an amoun L under he influence of graviy, i makes sense ha w and F s should cancel each oher ou a he equilibrium posiion. For a spring no a equilibrium, ha is, for a sreched or compressed spring, we can use Newon s second law o ge: mu' '( mg Fs ( Fd ( F( or mu ''( mg k( u( L) u'( F(. Rearranging his las equaion we ge mu '' mg ku kl u' F(. Recalling ha kl mg, we can simplify his equaion o obain mu ''( u'( ku( F(, he mass-spring-damper sysem equaion of moion When =, he sysem is said o be undamped; oherwise i is damped. When F( =, he moion is said o be free; oherwise he moion is forced. Sec. 3.8, Boyce & DiPrima, p.

3 I. Undamped Free Vibraions (Simple Harmonic Moion) In he case where = and F( = he sysem equaion above reduces o k mu + ku =. If we divide by m and le, we obain '' m u x. The characerisic equaion, r always has complex roos + i. The general soluion is of he form u Acos Bsin (. I is cusomary o simplify he general soluion by employing a rig ideniy for he cosine of a difference * as follows: Le A Rcos and B Rsin. Then u( Acos Bsin = Rcos cos Rsin sin u( = R cos( ) B Noe ha R A B and = arcan. A (We mus be very careful o choose he correc quadran for cos and sin.) according o he signs of R, he maximum displacemen of he mass from he equilibrium, is called he ampliude of he moion., a dimensionless parameer ha measures he displacemen of he wave from is normal posiion corresponding o =, is referred o as he phase angle. Since cosine is a periodic funcion wih period, cos( ) goes hrough one period when < <. If we solve for, we obain = m k 1/. which ells us he period is T = k The circular frequency, measured in radians per uni ime, is called he naural m frequency of he spring-mass sysem. * cos( a b) cos acosb sinasinb Sec. 3.8, Boyce & DiPrima, p. 3

4 Example 3. A mass weighing 4 lb sreches a spring 3 in. afer coming o res a equilibrium. Neglecing any damping or exernal forces ha may be presen, deermine he general soluion of he equaion of moion. Now find specific equaions for he moion of he mass along wih is ampliude, period, and naural frequency under he following iniial condiions: a) The mass is pulled 6 in. down below he equilibrium poin and given a downward velociy of f/sec. Sec. 3.8, Boyce & DiPrima, p. 4

5 b) The mass is lifed 6 in. above he equilibrium poin (compressing he spring) and given a downward velociy of f/sec. c) The mass is lifed 6 in. above he equilibrium poin (compressing he spring) and given an upward velociy of f/sec. Sec. 3.8, Boyce & DiPrima, p. 5

6 Maple skeches of Example 3 soluions The phase shif,, measures he horizonal disance beween he wo graphs. Sec. 3.8, Boyce & DiPrima, p. 6

7 Damped Free Vibraions Here we ineresed in he various effecs of damping, while assuming ha f( =. The moion of he sysem is governed by mu + u + ku = 1. Underdamped or Oscillaory Moion. This is he case where he characerisic equaion mr r k has a complex pair of roos, + i. (noe ha he real par of he roo is always negaive). The general soluion is u c e cos c e sin ( 1. As wih harmonic moion, he rig ideniy for he cosine of a difference can be employed o express u( in an alernae form: B u( Re cos( ), where, again, R A B and = arcan. A Since he cosine akes on values beween 1 and 1, we can wrie 1 cos( ) 1 or Re Re cos( ) Re or Re u( Re In oher words, he soluion u( is sandwiched beween wo exponenial envelopes which boh approach zero. Re is called he damping facor of he soluion. The sysem is called underdamped because he damping facor is oo small o preven he sysem from oscillaing.. Criically Damped Moion This is he case where he characerisic equaion mr r k has one repeaed roo (of r r mulipliciy ). In his case he general soluion is u( c1e ce. All soluions of criically damped moion end o zero as increases and he graph of he soluion will cross he -axis exacly once (Where?). This special case is called criically damped moion because if (in mr r k ) were any smaller he roos would be complex and oscillaion would occur. 3. Overdamped Moion This is he case where he characerisic equaion mr r k has wo disinc real (negaive) roos, r 1 and r. The roos mus be negaive because is always posiive. In his case he general r soluion is r 1 u( c1e ce. Using calculus we can show ha u( has a mos one local max or min. Thus, he moion is nonoscillaory. Sec. 3.8, Boyce & DiPrima, p. 7

8 Example 4. Assume ha he moion of a spring-mass sysem wih damping is governed by u + u + 5u = ; u() = 1, u () =. Find he equaion of moion and skech is graph for he hree cases where = 8 (Case 1), 1 (Case ), and 1 (Case 3). Sec. 3.8, Boyce & DiPrima, p. 8

9 Maple skeches of Example 4 soluions Case 1 Soluion Case soluion (solid line) compared wih case 3 soluion (poin line). Sec. 3.8, Boyce & DiPrima, p. 9

10 Circuis The quaniies used o describe he sae of a simple series elecrical circui are: The consans which are all posiive and assumed o be known. o The impressed volage E, measured in vols o The resisance R, measured in ohms o The capaciance C, measured in farads o The inducance L, measured in henrys. The curren I, a funcion of ime, measured in amperes The oal charge Q, measured in coulombs, on he capacior a ime dq The relaion beween Q and I is I Example 5. Skech a picure of a simple LRC series elecrical circui. d LCR series circui Using he fundamenal laws firs formulaed in he mid-nineeenh cenury by he German physicis G.R. Kirchhoff, * a second-order linear model of an LCR circui can be obained: d Q dq 1 L R Q E(. d d C 1 If we compare he circui model LQ '' RQ' Q E( wih he mechanical model, C mu + u + ku = F( we can see he following correlaions: Mechanical Sysem Mass, m Damping Consan, γ Spring Consan, k Impressed Force, F( Displacemen, u( Velociy, u ( Correlaions Elecrical Sysem Inducance, L Resisance, R Reciprocal of Capaciance, 1/C Impressed Volage, E( Charge, Q( Curren, I = Q ( Thus, we can compare he behavior of charge on an LRC circui wih harmonic, free, damped, ec. mechanical vibraions. * Kirchhoff s curren law saes ha he oal curren enering any poin of a circui equals he oal curren leaving i. Kirchhoff s volage law saes ha he sum of he volage changes around any loop in a circui is zero. Sec. 3.8, Boyce & DiPrima, p. 1

11 Example 6. Find he charge Q on he capacior in an L-R-C series circui when L =.5 henry, R = 1 ohms, C =.1 farad, E( =, wih iniial condiions Q() = Q o coulombs, and I() =. Discuss he behavior of his charge as ime increases. Wih wha kind of mechanical vibraion does his circui compare? Sec. 3.8, Boyce & DiPrima, p. 11