Damped Harmonic Motion Closing Doors and Bumpy Rides

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1 Prerequisies and Goal Damped Harmonic Moion Closing Doors and Bumpy Rides Andrew Forreser May 4, 21 Assuming you are familiar wih simple harmonic moion, is equaion of moion, and is soluions, we will now proceed o damped harmonic moion. In his analysis we ll find ha among all he possible levels of damping, from zero o infiniy, ha here is a special swee spo level of damping called criical damping, ha will allow you o auomaically shu a door mos quickly (and smoohly) wihou slamming i, for example. This principle can apply o shocks in vehicle suspension as well, o quickly absorb impacs from bumps in he road. We ll examine hese opics again near he end of his paper. Qualiaive Analysis of Moion Before geing mahemaical, we should examine some real sysems and ge an inuiive feel for hem. Well, on paper, we ll jus examine some graphs represening damped harmonic moion wih differen levels damping and discuss why hey make sense. In his secion we ll preend ha he graphs are creaed using daa from a real oscillaor, bu we should realize ha real daa would acually be a bi more complicaed. You can ignore he graphs s for now and refer o hem once you ve examined he mahemaics laer in his paper. Damped Harmonic Moion: iniial condiions 1 x.4ω n r = 1ω n x -x 2π/ω n 4π/ω n Figure 1: Damped harmonic moion wih differen levels of damping, wih iniial condiions 1. We ll jus examine one-dimensional moion in his paper. For concreeness, we ll imagine he oscillaor is an objec wih mass siing on a fricionless able, aached o a spring (which is aached o a wall) and a damper (which conains some viscous fluid). The variable x will denoe is displacemen from is equilibrium posiion. We ll wach he moion of he objec under several condiions, where he spring and mass are kep he same bu he damping fluid in he damper is made progressively more viscous, saring 1

2 from zero viscosiy. We ll examine wo ses of iniial condiions; 1) one where he objec is released from res a x (see Figure 1), and 2) one where he objec sars a he equilibrium posiion wih a velociy v, afer being smacked wih he appropriae impulse (see Figure 2). The firs noiceable fac from Figure 1 is ha here are wo disinc kinds of moion: oscillaions wih some rae of decay and decay or relaxaion wih no oscillaions. Explaining he exreme cases in Figure 1 should be mos obvious. Wih no damping, we simply have simple harmonic moion, a sinusoidal moion which is given by an unshifed cosine because of he iniial condiions. Wih infinie damping, he objec does no move a all, being frozen by an infiniely viscous fluid (ha is, a solid) a is iniial posiion, and so is posiion-versus-ime curve is a sraigh horizonal line. Wha abou he inermediae cases? We migh firs noe ha we know inuiively ha a damping force ends o ake he energy ou of a sysem, evenually leaving i wihou moion and wih an unsressed spring. As he damping is increased, we see ha he objec moves more slowly owards he equilibrium posiion, and when oscillaing, is period of oscillaion is increased since i s moving more slowly. I seems ha as he damping increases, he period increases more and more rapidly, apparenly going o infiniy a some poin, a which poin he curve becomes pure relaxaion. Then, as he damping coninues o increase, he relaxaion jus akes longer and longer o reach equilibrium. Damped Harmonic Moion: iniial condiions 2 v /ω n.4ω n r = 1ω n x -v /ω n 2π/ω n 4π/ω n Figure 2: Damped harmonic moion wih differen levels of damping, wih iniial condiions 2. Wih he second se of iniial condiions in Figure 2, where he iniial posiion is he equilibrium posiion and he iniial velociy is v, some oher facs become more obvious. We see from he highly damped cases ha energy is aken ou of he sysem mosly when he objec is a high velociy. So we ve found ha he damping acs in wo qualiaive ways: 1) i slows he sysem down (akes ou kineic energy) and 2) i acs more srongly when he sysem is a higher velociy. Modelling he damping force as negaively proporional o he velociy would ake care of boh of hese qualiies (and jus so happens o reproduce hese graphs exacly, as we ll find ou), since he magniude of he force would be proporional o speed and he work i does would always be negaive. Slowing he sysem down also causes he rae of change of poenial energy o slow as well, since he poenial energy is proporional o he square of he displacemen. Afer we ve done he mahemaics o find he formulae for hese curves, we ll be able o plo he energy o see exacly wha s going on in all of hese scenarios. 2

3 To finish up wih Figure 2 we should jus look a he exreme case of infinie damping. Essenially, ha s equivalen o smacking a solid sysem. The sysem s componen pars will no move wih respec o each oher, so he mass will no move wih respec o he damping apparaus (and he spring will jus say unsressed). Tha produces a fla posiion-versus-ime curve a x =. Where d he energy of your smack go? Maybe i wen ino huring or deforming he smacker (your hand?), or maybe i wen ino hea in he damper and noise due o body vibraions of he objec iself, or maybe you jus smacked your whole objec-spring-damper sysem off of your lab able and ono he floor! Equaion of Moion and Soluions Now le s go from qualiaive o quaniaive. Given ha an objec of mass m has a Hooke resoring force, which is negaively proporional o displacemen x from an equilibrium posiion, and a damping force ha is negaively proporional o he objec s velociy, he equaion of moion is, in hree differen noaions, Rearranging, we have F ne,x = kx bv x = ma x, F ne,x = kx b dx d = m d2 x d 2, F ne,x = kx bẋ = mẍ. ( ) ( ) ẍ + 2 b 2m ẋ + k m x =. I ve muliplied and divided he second erm by wo because we ll find ha b/2m is an imporan consan in describing he resuling moion. Le s call i r for now: r = b/2m. We already know ha k/m is he square of he angular frequency of oscillaion for an undamped (b = ) oscillaor. We ll find ha his sysem exhibis a differen frequency of moion han k/m, so we will give a special name o he undamped frequency o disinguish i from he damped oscillaion frequency. We ll call i he naural frequency : ω n = k/m. This name is a lile srange, since he absence of damping is a bi unnaural, bu his is he common name anyway. So he equaion can be wrien his way: ẍ + 2rẋ + ω 2 nx =. The general soluions 1 o his equaion come in hree differen forms depending on how r and ω n relae o each oher. Here is one way o express hese general soluions: Underdamped (ligh damping) ( ) r < ω n x() = A exp( r) cos ω 2 n r 2 + φ (1) Criically damped r = ω n x() = (B + x ) exp( r) (2) Overdamped (heavy damping) r > ω n x() = C + exp [ ( r + r 2 ω 2 n ) ] [ ( + C exp r ) ] r 2 ωn 2 (3) [ ( ) ( = exp( r) D c cosh r 2 ωn 2 + D s sinh r 2 ωn ) ] 2 (4) 1 To see a fairly good explanaion of he derivaion of hese soluions, and o see some very ineresing informaion abou (non-linear) pendulum moion, see <hp:// 3

4 Noe ha each of he hree cases has in is soluion wo free parameers (A and φ, or x and B, or C + and C ). If he iniial condiions, ha is he iniial posiion and velociy of he oscillaor, and he values of r and ω n are specified, hen he values for hese parameers will be se. Noe also ha if you leave he iniial condiions and ω n he same bu change r, he parameers will change heir values! We ll see ha explicily in a momen. I will be convenien for his discussion and for memorizaion o define a couple more variables: = ω 2 n r 2 β = r 2 ω 2 n We ll discuss heir meaning and ge around o naming r,, and β below. If we enforce he iniial condiions sipulaed in he previous secion and ake ω n and r o be given, we find he soluions ake he forms given in Table 1 below. Damping Soluion wih Iniial Condiions 1 Soluion wih Iniial Condiions 2 Condiion x() = x, v x () = x() =, v x () = v Under. r < ω n Cri. r = ω n Over. r = ω n x() = x ω n e r cos [ arcan(r/ ) ] x() = v e r sin( ) x() = x (r + 1)e r x() = x { 1 2 (1 + r β )e( r+β) (1 r β )e( r β) } } = x e {cosh(β) r + r β sinh(β) x() = (v )e r x() = v β { 1 2 e( r+β) 1 2 e( r β) } = v β e r sinh(β) Table 1: Soluions ha conform o he wo ses of iniial condiions. These are he equaions for he graphs in Figures 1 and 2. Quaniaive Analysis of Moion I ll now discuss each of he hree cases of soluions and relae hem o he wo ypes of moion observed earlier, and you should refer o Equaions 1 4, Table 1, and Figures 1 and 2 above. The soluions should confirm and even build up our inuiion abou damped harmonic moion. For he underdamped case, we can firs noe ha he sub-case of (when b = ) is simply he case wihou damping and yields simple harmonic moion wih naural frequency ω n. We see ha for r, he moion is an exponenially decaying oscillaion, where r is he exponenial decay rae of he ampliude for he oscillaions. We also see ha he frequency of he oscillaions, or he frequency of he cyclic par of he soluion, has changed from he naural frequency o a lower value. We ll call his new frequency he (under)damped angular frequency = ω 2 n r 2, where < ω n. 2 This means ha when damping is increased, he oscillaion becomes slower, wih a longer period. As we approach he criical case (as r increases o almos reach ω n ), approaches zero and he period of he oscillaion approaches infiniy, jus as we observed earlier. If we look a he soluions for iniial condiions 1 in Table 1, we can see ha as r goes o ω n, he facor x ω n cos [ arcan(r/ ) ] in he firs soluion becomes x (r + 1) in he second soluion. The cosine 2 Don confuse he (under)damping angular frequency wih a frequency encounered in sinusoidally forced harmonic moion, he forcing angular frequency ω f, which is ofen called he driving frequency. 4

5 ges sreched horizonally (wih a longer period) and verically (wih a larger ampliude) and shifed (where he shif approaches π/2, nearing he unshifed sine) in such a way ha i urns ino a sraigh line inersecing x = x wih a slope of x r. The slope-facor r compees wih he exponenial decay (which iniially has a slope of r) o allow he resuling funcion o sar off wih zero slope and hus zero velociy and saisfy he iniial condiions. (The produc rule for derivaives dicaes ha he compeing slopes are weighed by he values of he oher funcion: given f = gh, f = hġ + gḣ, and in our case h = 1 and g = 1 a =.) Wih his effec he curve looks more and more like an exponenial decay wih some iniial hins of a cosine a he beginning. Noe ha wih hese soluions we can calculae exacly how he firs inersecion of he curve of moion wih he ime axis shoos off o infiniy as we approach he criical case. Of course, i is much easier o see how, wih he second se of soluions in Table 1, he facor v sin( ) becomes (v ) as r goes o ω n. For he criically damped case, he moion is no oscillaory and neiher is i purely exponenial, unless B happens o be zero. (Neiher of our iniial condiions cause B o be zero, bu B could be zero under differen condiions, where he iniial velociy is negaive.) The parameer r sill gives he decay rae for he exponenial par of he soluion, bu he overall value of x has wo ime-dependen facors: (B + x ) and exp( r). Afer enough ime, however, he exponenial par of he soluion will be dominan, meaning ha evenually i will be hard o ell he difference beween he B = case and he B case, since he exponenial exp( r) evenually shrinks a lo faser han grows. So, alhough his soluion isn necessarily purely exponenial decay, you could say ha in he long run (or asympoically) i is exponenial. For he overdamped case, we see wo compeing pure exponenially decaying erms; boh exponens have negaive decay raes. However, he resul of adding hese erms can someimes look (for a shor while) disincly non-exponenial. (See, for example, he overdamped bumps in Figure 2.) In he argumen for each exponenial erm we see somehing ha looks like bu is differen; we ve given i he symbol β: β = r 2 ω 2 n. Noe ha β < r (ha s why he decay raes are boh negaive), and β is closer o r for larger values of r. We could call β he decay raes offse (from r) or perhaps he hyperbolic phase velociy. We can also now name r. For under- and criical damping, r can definiely be called he exponenial decay rae, bu for overdamping i s a lile more complicaed and harder o come up wih a name ha encompasses i s funcion in all he ways you can express he soluion. Maybe if we call i an exponenial decay rae raher han he exponenial decay rae, ha would be saisfacory. I ll do a quick analysis now and clean his up laer. (Explain how he criically damped case is equivalen o he overdamped case when r is slighly greaer han ω n. Explain how he wo erms allow here o be a hin of a cosine wih he firs iniial condiions.) I appears ha one of he exponenial erms does he job of eiher easing he objec ino a long exponenial decay (firs iniial condiions) or geing rid of he kineic energy and hen easing i ino a long exponenial decay (second iniial condiions). As r goes o infiniy, one of he decays becomes incredibly fas and one becomes incredibly slow. A he exreme of infinie damping, he infiniely-fas decay immediaely rids he sysem of kineic energy and he infiniely-slow decay (no decay) holds he objec a i s iniial posiion. 5

6 Energy of Damped Harmonic Moion Damped Harmonic Moion Energy: iniial condiions 1 (1/2)mx 2 ωn 2.4ω n r = 1ω n E Figure 3: Energy of harmonic moion wih differen levels of damping, wih iniial condiions 1. Damped Harmonic Moion Energy: iniial condiions 2 (1/2)mv 2.4ω n r = 1ω n E Figure 4: Energy of harmonic moion wih differen levels of damping, wih iniial condiions 2. 6

7 Doors, Shocks Search online for criically damped or criical damping and shocks, and you can find some ineresing aricles, such as he one I found here: <hp:// ediorial/index.hml> Summary of Equaion and Soluions 2 x x + 2r 2 + ω2 nx =. Underdamped (ligh damping) r < ω n x() = A() cos( + φ ) Criically damped r = ω n x() = (B + x ) e r Overdamped (heavy damping) r > ω n x() = C + e ( r+β) + C e ( r β) = e r [ D c cosh(β) + D s sinh(β) ] A() ime-dependen (exponenially decaying) ampliude A() = A e r r an exponenial decay rae damped exponenial phase velociy r = Examples: (damping consan) 2 (inerial consan) b 2m, Γ 2I, ec. ω n naural angular frequency undamped angular phase velociy ω n = Examples: (resoring consan) (inerial consan) k m, κ I, ec. (under)damped angular frequency (under)damped angular phase velociy = ω 2 n r 2 β hyperbolic phase velociy (over)damped hyperbolic phase velociy β = r 2 ω 2 n 7