Module 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II

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1 Module 3: The Damped Oscillaor-II Lecure 3: The Damped Oscillaor-II 3. Over-damped Oscillaions. This refers o he siuaion where β > ω (3.) The wo roos are and α = β + α 2 = β β 2 ω 2 = (3.2) β 2 ω 2 = 2 (3.3) where boh, 2 > and 2 >. The wo roos give rise o exponenially decaying soluions, one of which decays faser han he oher x() = A e + A 2 e 2. (3.4) The consans A and A 2 are deermined by he iniial condiions. For iniial posiion x and velociy v we have x() = v + 2 x 2 e v + x 2 e 2 (3.5) The overdamped oscillaor does no oscillae. Figure 3. shows a ypical siuaion. In he siuaion where β ω [ β 2 ω 2 = β ω2 β β ω 2 ] 2 2 β 2 and we have = ω 2 /2β and 2 = 2β. (3.6) 7

2 8 CHAPTER 3. THE DAMPED OSCILLATOR-II x Figure 3.: x Figure 3.2: 3.2 Criical Damping. This corresponds o a siuaion where β = ω and he wo roos are equal. The governing equaion is second order and here sill are wo independen soluions. The general soluion is The soluion is for an oscillaor saring from res a x while x() = e β [A + A 2 ] (3.7) x() = x e β [ + β] (3.8) x() = v e β (3.9) is for a paricle saring from x = wih speed v. Figure 3.2 shows he laer siuaion. Figure 3.3 shows a ypical comparision of he hree ypes of damping viz, underdamped, overdamped and criically damped. Figure 3.4 shows he comparision of a criically damped oscillaor wih an over damped oscillaor for differen values of β. One observes ha he criically damped oscillaor reaches

3 3.3. SUMMARY 9 x() Underdamped Criical Figure 3.3: he mean posiion in he smalles possible ime. This is he reason ha he resisances in sea-shock absorber of vehicles, in sliding doors or in Galvanomeers are adjused o he criically damped condiion so ha when hey are disurbed hey come back o he mean posiion quickly. x() Criically damped Figure 3.4: 3.3 Summary There are wo physical effecs a play in a damped oscillaor. The firs is he damping which ries o bring any moion o a sop. This operaes on a ime-scale T d /β. The resoring force exered by he spring ries o make he sysem oscillae and his operaes on a ime-scale T = /ω. We have overdamped oscillaions if he damping operaes on a shorer ime-scale compared o he oscillaions ie. T d < T which compleely desroys he oscillaory behaviour. Figure 3.5 shows he behaviour of a damped oscillaor under differen combinaions of damping and resoring force. The plo is for ω =, i can be used for any oher value of he naural frequency by suiably scaling he values of β.

4 2 CHAPTER 3. THE DAMPED OSCILLATOR-II Underdamped 2 2 β Criical Figure 3.5: I shows how he decay rae for he wo exponenially decaying overdamped soluions varies wih β. Noe ha for one of he modes he decay rae ends o zero as β is increased. This indicaes ha for very large damping a paricle may ge suck a a posiion away from equilibrium. Problems. Obain soluion (3.7) for criical damping as a limiing case (β ω ) of overdamped soluion (3.5). 2. Find ou he condiions for he iniial displacemen x() and he iniial velociy ẋ() a = such ha an overdamped oscillaor crosses he mean posiion once in a finie ime. 3. A door-shuer has a spring which, in he absence of damping, shus he door in.5s. The problem is ha he door bangs wih a speed m/s a he insan ha i shus. A damper wih damping coefficien β is inroduced o ensure ha he door shus gradually. Wha are he ime required for he door o shu and he velociy of he door a he insan i shus if β =.5π and β =.9π? Noe ha he spring is unsreched when he door is shu. (.57s, 4.67 m/s;.4s, m/s) 4. A highly damped oscillaor wih ω = 2 s and β = 4 s is given an iniial displacemen of 2 m and lef a res. Wha is he oscillaor s posiion a = 2 s and = 4 s? (2. m, 2.7 m) 5. A criically damped oscillaor wih β = 2 s is iniially a x = wih velociy 6 m s. Wha is he furhes disance he oscillaor moves from he origin? (. m) 6. A criically damped oscillaor is iniially a x = wih velociy v. Wha is he raio of he maximum kineic energy o he maximum poenial energy of his oscillaor? (e 2 )

5 3.3. SUMMARY 2 7. An overdamped oscillaor is iniially a x = x. Wha iniial velociy, v, should be given o he oscillaor ha i reaches he mean posiion (x=) in he minimum possible ime. 8. We have shown ha he general soluion, x(), wih wo consans can describe he moion of damped oscillaor saisfying given iniial condiions. Show ha here does no exis any oher soluion saisfying he same iniial condiions.