5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.


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1 5.8 Resonance Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) () + ω 2 () = F cos(ω) is he eisence of a soluion ha is unbounded as. We already know (page 224) ha for ω ω, he general soluion of (1) is he sum of wo harmonic oscillaions, hence i is bounded. Equaion (1) for ω = ω has by he mehod of undeermined coefficiens he unbounded oscillaory soluion () = F 2ω sin(ω ). To summarize: Pure resonance occurs eacly when he naural inernal frequency ω maches he naural eernal frequency ω, in which case all soluions of he differenial equaion are unbounded. In Figure 2, his is illusraed for () + 16() = 8cos 4, which in (1) corresponds o ω = ω = 4 and F = 8. Figure 2. Pure resonance for () + 16() = 8 cosω. Graphed are he soluion () = sin 4 for ω = 4 and he envelope curves = ±. Resonance and undeermined coefficiens. An eplanaion of resonance can be based upon he heory of undeermined coefficiens. An iniial rial soluion of () + 16() = 8cos ω is = d 1 cos ω + d 2 sin ω. The homogeneous soluion h = c 1 cos 4 + c 2 sin4 considered in he fiup rule has duplicae erms eacly when he naural frequencies mach: ω = 4. Then he final rial soluion is (2) () = { d1 cos ω + d 2 sin ω ω 4, (d 1 cos ω + d 2 sin ω) ω = 4. Even before he undeermined coefficiens d 1, d 2 are evaluaed, we can decide ha unbounded soluions occur eacly when frequency maching
2 232 ω = 4 occurs, because of he ampliude facor. If ω 4, hen p () is a pure harmonic oscillaion, hence bounded. If ω = 4, hen p () equals a ime varying ampliude C imes a pure harmonic oscillaion, hence i is unbounded. The Wine Glass Eperimen. Equaion (1) is adverised as he basis for a physics eperimen which has appeared ofen on Public Television, called he wine glass eperimen. A famous physicis, in fron of an audience of physics sudens, equips a lab able wih a frequency generaor, an amplifier and an audio speaker. The valuable wine glass is replaced by a glass beaker. The frequency generaor is uned o he naural frequency of he glass beaker (ω ω ), hen he volume knob on he amplifier is suddenly urned up (F adjused larger), whereupon he sound waves emied from he speaker break he glass beaker. The glass iself will vibrae a a cerain frequency, as can be deermined eperimenally by pinging he glass rim. This vibraion operaes wihin elasic limis of he glass and he glass will no break under hese circumsances. A physical eplanaion for he breakage is ha an incoming sound wave from he speaker is imed o add o he glass rim ecursion. Afer enough ampliude addiions, he glass rim moves beyond he elasic limi and he glass breaks. The eplanaion implies ha he eernal frequency from he speaker has o mach he naural frequency of he glass. Bu here is more o i: he glass has some naural damping ha nullifies feeble aemps o increase he glass rim ampliude. The physicis uses o grea advanage his naural damping o une he eernal frequency o he glass. The reason for urning up he volume on he amplifier is o nullify he damping effecs of he glass. The ampliude addiions hen build rapidly and he glass breaks. Soldiers Breaking Cadence. The collapse of he Broughon bridge near Mancheser, England in 1831 is blamed for he now sandard pracise of breaking cadence when soldiers cross a bridge. Bridges like he Broughon bridge have many naural low frequencies of vibraion, so i is possible for a column of soldiers o vibrae he bridge a one of he bridge s naural frequencies. The bridge locks ono he frequency while he soldiers coninue o add o he ecursions wih every sep, causing larger and larger bridge oscillaions. Pracical Resonance The noion of pure resonance is easy o undersand boh mahemaically and physically, because frequency maching characerizes he even. This ideal siuaion never happens in he physical world, because damping is
3 5.8 Resonance 233 always presen. In he presence of damping c >, i will be esablished below ha only bounded soluions eis for he forced springmass sysem (3) m () + c () + k() = F cos ω. Our inuiion abou resonance seems o vaporize in he presence of damping effecs. Bu no compleely. Mos would agree ha he undamped inuiion is correc when he damping effecs are nearly zero. Pracical resonance is said o occur when he eernal frequency ω has been uned o produce he larges possible soluion (a more precise definiion appears below). I will be shown ha his happens for he condiion (4) ω = k/m c 2 /(2m 2 ), k/m c 2 /(2m 2 ) >. Pure resonance ω = ω k/m is he limiing case obained by seing he damping consan c o zero in condiion (4). This srange bu predicable ineracion eiss beween he damping consan c and he size of soluions, relaive o he eernal frequency ω, even hough all soluions remain bounded. The decomposiion of () ino homogeneous soluion h () and paricular soluion p () gives some inuiion ino he comple relaionship beween he inpu frequency ω and he size of he soluion (). The homogeneous soluion. For posiive damping, c >, equaion (3) has homogeneous soluion h () = c 1 1 () + c 2 2 () where according o he recipe he basis elemens 1 and 2 are given in erms of he roos of he characerisic equaion mr 2 + cr + k =, as classified by he discriminan D = c 2 4mk, as follows: Case 1, D > Case 2, D = 1 = e r 1, 2 = e r 2 wih r 1 and r 2 negaive. 1 = e r 1, 2 = e r 1 wih r 1 negaive. Case 3, D < 1 = e α cos β, 2 = e α sin β wih β > and α negaive. I follows ha h () conains a negaive eponenial facor, regardless of he posiive values of m, c, k. A soluion () is called a ransien soluion provided i saisfies he relaion lim () =. The conclusion: The homogeneous soluion h () of he equaion m () + c () + k() = is a ransien soluion for all posiive values of m, c, k. A ransien soluion graph () for large lies aop he ais =, as in Figure 21, because lim () =.
4 π Figure 21. Transien oscillaory soluion = 2e (cos + sin ) of he differenial equaion =. The paricular soluion. The mehod of undeermined coefficiens gives a rial soluion of he form () = Acos ω + B sinω wih coefficiens A, B saisfying he equaions (5) (k mω 2 )A + (cω)b = F, ( cω)a + (k mω 2 )B =. Solving (5) wih Cramer s rule or eliminaion produces he soluion (6) A = (k mω 2 )F (k mω 2 ) 2 + (cω) 2, B = cωf (k mω 2 ) 2 + (cω) 2. The seady sae soluion, periodic of period 2π/ω, is given by (7) p () = = F ( ) (k mω 2 ) 2 + (cω) 2 (k mω 2 )cos ω + (cω)sin ω F cos(ω α), (k mω 2 ) 2 + (cω) 2 where α is defined by he phase ampliude relaions (see page 216) (8) C cos α = k mω 2, C sin α = cω, C = F / (k mω 2 ) 2 + (cω) 2. The erminology seady sae refers o ha par ss() of he soluion () ha remains when he ransien porion is removed, ha is, when all erms conaining negaive eponenials are removed. As a resul, for large T, he graphs of () and ss() on T are he same. This feaure of ss() allows us o find is graph direcly from he graph of (). We say ha ss() is observable, because i is he soluion visible in he graph afer he ransiens (negaive eponenial erms) die ou. Readers may be mislead by he mehod of undeermined coefficiens, in which i urns ou ha p () and ss() are he same. Alernaively, a paricular soluion p () can be calculaed by variaion of parameers, a mehod which produces in p () era erms conaining negaive eponenials. These era erms come from he homogeneous soluion heir appearance canno always be avoided. This jusifies he careful definiion of seady sae soluion, in which he ransien erms are removed from p () o produce ss(). Pracical resonance is said o occur when he eernal frequency ω has been uned o produce he larges possible seady sae ampliude.
5 5.8 Resonance 235 Mahemaically, his happens eacly when he ampliude funcion C = C(ω) defined in (8) has a maimum. If a maimum eiss on < ω <, hen C (ω) = a he maimum. The derivaive is compued by he power rule: (9) C (ω) = F 2(k mω 2 )( 2mω) + 2c 2 ω 2 ((k mω 2 ) 2 + (cω) 2 ) 3/2 = ω (2mk c 2 2m 2 ω 2) C(ω) 3 F 2 If 2km c 2, hen C (ω) does no vanish for < ω < and hence here is no maimum. If 2km c 2 >, hen 2km c 2 2m 2 ω 2 = has eacly one roo ω = k/m c 2 /(2m 2 ) in < ω < and by C( ) = i follows ha C(ω) is a maimum. In summary: Pracical resonance for m ()+c ()+k() = F cosω occurs precisely when he eernal frequency ω is uned o ω = k/m c 2 /(2m 2 ) and k/m c 2 /(2m 2 ) >. In Figure 22, he ampliude of he seady sae periodic soluion is graphed agains he eernal naural frequency ω, for he differenial equaion + c + 26 = 1cos ω and damping consans c = 1,2,3. The pracical resonance condiion is ω = 26 c 2 /2. As c increases from 1 o 3, he maimum poin (ω,c(ω)) saisfies a monooniciy condiion: boh ω and C(ω) decrease as c increases. The maima for he hree curves in he figure occur a ω = 25.5, 24, Pure resonance occurs when c = and ω = 26. C c = 1 c = 2 c = 3 ω Figure 22. Pracical resonance for + c + 26 = 1 cosω: ampliude C = 1/ (26 ω 2 ) 2 + (cω) 2 versus eernal frequency ω for c = 1, 2, 3. Uniqueness of he Seady Sae Periodic Soluion. Any wo soluions of he nonhomogeneous differenial equaion (3) which are periodic of period 2π/ω mus be idenical. The vehicle of proof is o show ha heir difference () is zero. The difference () is a soluion of he homogeneous equaion, i is 2π/ω periodic and i has limi zero a infiniy. A periodic funcion wih limi zero mus be zero, herefore he wo soluions are idenical. A more general saemen is rue: Consider he equaion m () + c () + k() = f() wih f( + T) = f() and m, c, k posiive. Then a T periodic soluion is unique.
6 236 In Figure 23, he unique seady sae periodic soluion is graphed for he differenial equaion = sin +2cos. The ransien soluion of he homogeneous equaion and he seady sae soluion appear in Figure 24. In Figure 25, several soluions are shown for he differenial equaion = sin + 2cos, all of which reproduce evenually he seady sae soluion = sin π 6π π Figure 23. Seadysae periodic soluion () = sin of he differenial equaion = sin + 2 cos. Figure 24. Transien soluion of = and he seadysae soluion of = sin + 2 cos. Figure 25. Soluions of = sin + 2 cos wih () = 1 and () = 1, 2, 3, all of which graphically coincide wih he seadysae soluion = sin for π. Pseudo Periodic Soluion. Resonance gives rise o soluions of he form () = A()sin(ω α) where A() is a ime varying ampliude. Figure 26 shows such a soluion, which is called a pseudo periodic soluion because i has a naural period 2π/ω arising from he rigonomeric facor sin(ω α). The only requiremen on A() is ha i be non vanishing, so ha i acs like an ampliude. The pseudo period of a pseudo periodic soluion can be deermined graphically, by compuing he lengh of ime i akes for () o vanish hree imes. 4 e Eercises Figure 26. The pseudoperiodic soluion = e /4 sin(3) of = 96e /4 cos3 and is envelope curves = ±e /4. Resonance and Beas. Classify for resonance or beas. In he case of resonance, find an unbounded soluion. In he case of beas, find he soluion for () =, () =, hen graph i hrough a full period of he slowlyvarying envelope = 1 sin = 5 sin2
7 5.8 Resonance = 1 sin = 5 sin = 5 sin = 5 cos = 5 sin = 1 sin4 Resonan Frequency and Ampliude. Consider m + c + k = F cos(ω). Compue he seadysae oscillaion A cos(ω) + B sin(ω), is ampliude C = A 2 + B 2, he uned pracical resonance frequency ω, he resonan ampliude C and he raio 1C/C. 9. m = 1, β = 2, k = 17, F = 1, ω = m = 1, β = 2, k = 1, F = 1, ω = m = 1, β = 4, k = 5, F = 1, ω = m = 1, β = 2, k = 6, F = 1, ω = m = 1, β = 4, k = 5, F = 5, ω = m = 1, β = 2, k = 5, F = 5, ω = 3/2.
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