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à 6.3èForced Oscillations - No Damplïg; Resonance
äèèSolve ê problem
âèèFïd ê resonance frequency for ê differential equation
y»» +è25y =è4sï[10t]èèè
The ståard form for undamped, forced oscillation is
y»» +èÜìyè=èF╠sï[ßt]
ThusèÜì = 25, so ê resonance frequency would be 5 rad súî
éSèè In addition ë ê maï external force (e.g. sprïg) ï a
simple harmonic oscillaër system, êre may be an EXTERNAL
OSCILLATOR affectïg ê motion ç ê system.èFor ê
UNDAMPED system ê DRIVEN simple harmonic oscillaër equation
will be
y»»è+èÜìyè=èF(t)
The situation where ê external oscillaër has a fixed
angular frequency ß occurs whenè F(t) = F╠cos[ßt]èorè
equivalentlyè F(t) = F╠sï[ßt]è Choosïg ê latter gives
y»»è+èÜìyè=èF╠sï[ßt]
èèThis lïear, constant coefficient, INHOMOGENEOUS, second
order differential equation is solved ï two parts.èThe
HOMOGENEOUS equation was solved ï Section 6.1 with ê
general solution
yè=èC¬cos[Üt] + C½sï[Üt]
The particular solution ë ê INHOMOGENEOUS equation is
found usïg ê method ç UNDETERMINED COEFFICIENTS (Section
4.3).èThe particular solution is assumed ë be
yè =èAcos[ßt] + Bsï[ßt]
where A å B are undetermïed.èDifferentiatïg twice gives
y»è=è- ßAsï[ßt] + ßBcos[ßt]
y»» =è-ßìAcos[ßt] - ßìBsï[ßt]
Substitutïg ïë ê driven simple harmonic oscillaër
differential equation yields
èèè-Aßìcos[ßt] - Bßìsï[ßt] + AÜìcos[Üt] + BÜìsï[Üt] = F╠sï[ßt]
Or
è [A(Üì - ßì)]cos[ßt] + [B(Üì - ßì)sï[ßt]è=èF╠sï[ßt]
Equatïg coefficients ç ê functions yields
A(Üì - ßì)è=è0è i.e.èA = 0
èèèèèèèèèèèèèèèèF╠
B(Üì - ßì)è=èF╠èorèBè=è─────────
èèèèèèèèèèèèèèèÜì - ßì
Thus ê general solution is
èèèèèèèèèèèèèèèèèF╠
yè=èC¬cos[Üt] + C½sï[Üt] +è──────── sï[ßt]
èèèèèèèèèèèèèèèèÜì - ßì
In situation where ê NATURAL FREQUENCY ç ê system, Ü ,
is not close ë ê EXTERNAL FREQUENCY ç ê external
oscillaër, ß, ê third term ç ê solution will have only a
small effect due ë ê presence ç êèÜì - ßì ï ê
denomïaër.èHowever, as ê external frequency ß approaches
ê natural frequency Ü, ê denomïaër gets closer ë zero
å ê coefficient becomes large so that ê third term ï
ê solution becomes ê DOMINANT term.èIn ê limit as
ß goes ë Ü, ê solution becomes unbounded.èThis phenomena
is known as RESONANCE.
èèèResonance manifests itself ï a number ç physical
phenomena as ê tunïg ç a radio signal å ê breakïg
ç a crystal goblet by a sustaïed high note ç an opera sïger.
The most famous resonance episode was ê collapse ç ê
bridge at Tacoma Narrows, Washïgën state ï 1939.èA wïd
blowïg at a constant speed ç about 38 ë 40 miles per hour
gave energy ë a twistïg ç ê road about ê center lïe.
After about a half hour ç this motion, ê straï caused
ê bridge ë collapse ïë Puget Sound.èThe cause was NOT
ê speed ç ê wïd as ê bridge had withsëod much higher
wïd speeds, but ê fact that ê wïds were blowïg stead-
ily at just ê right speed ë match ê resonance frequency
ç ê ërsional simple harmonic motion ç ê roadway.
1è Fïd ê resonance frequency ç ê driven simple
harmonic oscillaër described byèy»» + 9y = 4sï[2t]
A)è2 rad súîè B)è3 rad súîèC)è4 rad súîèD)è9 rad súî
üèèThe ståard driven simple harmonic oscillaër is given by
y»» +èÜìyè=èF╠sï[ßt]
Matchïg with ê given equation
y»» + 9y = 4sï[2t]
Thusè Üìè=è9è soèÜè=è3 rad súî
Ç B
2è Fïd ê general solution ç ê driven simple
harmonic oscillaër described byèy»» + 9y = 4sï[2t]
A)èC¬cos[3t] + C½sï[3t] + 4/5 sï[2t]è
B)èC¬cos[3t] + C½sï[3t] - 4/5 sï[2t]
C)èC¬cos[9t] + C½sï[9t] + 4/5 sï[2t]
D)èC¬cos[9t] + C½sï[9t] - 4/5 sï[2t]
üèèThe ståard driven simple harmonic oscillaër is given by
y»» +èÜìyè=èF╠sï[ßt]
The general solution is given by
èèèèèèèèèèèèèèèèèF╠
yè=èC¬cos[Üt] + C½sï[Üt] +è──────── sï[ßt]
èèèèèèèèèèèèèèèèÜì - ßì
The differential equation
y»» + 9y = 4sï[2t]
hasè Üè=è3 rad súî,èß = 2 rad súî,è
F╠ = 4è soèF╠/(Üì - ßì) = 4/(3ì - 2ì) = 4/5
Thus ê general equation is
yè=èC¬cos[3t] + C½sï[3t] + 4/5 sï[2t]
Ç A
3è Fïd ê solution ç ê driven simple harmonic oscilla-
ër described byèy»» + 9y = 4sï[2t]èy(0) = -4, y»(0) = 7
A)è4cos[3t] + 9/5 sï[3t] + 4/5 sï[2t]è
B)è4cos[3t] - 9/5 sï[3t] + 4/5 sï[2t]è
C)è-4cos[3t] + 9/5 sï[3t] + 4/5 sï[2t]è
D)è-4cos[3t] - 9/5 sï[3t] + 4/5 sï[2t]è
üèèAs found ï Problem 2, ê general equation is
yè=èC¬cos[3t] + C½sï[3t] + 4/5 sï[2t]
Substitutïg t = 0 gives
-4è=èC¬
Differentiatïg
y»è=è-3C¬sï[3t] + 3C½cos[3t] + 8/5 cos[2t]
Substitutïgèt = 0
7è=è3C½è+ 8/5èi.e.èC½ = 9/5
Thus ê solution is
yè=è-4cos[3t] + 9/5 sï[3t] + 4/5 sï[2t]
Ç C
4è Fïd ê resonance frequency ç ê driven simple
harmonic oscillaër described byèy»» + 4y = 5sï[2.1t]
A)è2 rad súîè B)è2.1 rad súîèC)è4 rad súîèD)è5 rad súî
üèèThe ståard driven simple harmonic oscillaër is given by
y»» +èÜìyè=èF╠cos[ßt]
Matchïg with ê given equation
y»» + 4y = 5cos[2.1t]
Thusè Üìè=è4è soèÜè=è2 rad súî
Ç A
5è Fïd ê general solution ç ê driven simple
harmonic oscillaër described byèy»» + 4y = 5sï[2.1t]
A)èC¬cos[2t] + C½sï[2t] + 500/41 sï[2.1t]è
B)èC¬cos[2t] + C½sï[2t] - 500/41 sï[2.1t]
C)èC¬cos[4t] + C½sï[4t] + 500/41 sï[2.1t]
D)èC¬cos[4t] + C½sï[4t] - 500/41 sï[2.1t]
üèèThe ståard driven simple harmonic oscillaër is given by
y»» +èÜìyè=èF╠sï[ßt]
The general solution is given by
èèèèèèèèèèèèèèèèèF╠
yè=èC¬cos[Üt] + C½sï[Üt] +è──────── sï[ßt]
èèèèèèèèèèèèèèèèÜì - ßì
The differential equation
y»» + 4y = 5sï[2.1t]
hasè Üè=è2 rad súî,èß = 2.1 rad súî,è
F╠ = 4èsoèF╠/(Üì-ßì) = 5/(2ì-2.1ì) = -5/[41/100] = -500/41
Thus ê general equation is
yè=èC¬cos[2t] + C½sï[2t] - 500/41 sï[2t]
Ç B
6è Fïd ê solution ç ê driven simple harmonic oscilla-
ër described byèy»» + 4y = 5sï[2.1t]èy(0) = 0, y»(0) = -4
A)è500/41 cos[2t] + 2 sï[2t] - 500/41 sï[2.1t]è
B)è500/41 cos[2t] - 2 sï[2t] - 500/41 sï[2.1t]è
C)è-500/41 cos[2t] + 2 sï[2t] + 500/41 sï[2.1t]èèè
D)è-500/41 cos[2t] - 2 sï[2t] + 500/41 sï[2.1t]èèè
üèèAs found ï Problem 5, ê general equation is
yè=èC¬cos[2t] + C½sï[2t] - 500/41 cos[2.1t]
Substitutïg t = 0 gives
0è=èC¬ - 500/41è i.e.èC¬ = 500/41
Differentiatïg
y»è=è-2C¬sï[2t] + 2C½cos[2t] - 336/41 cos[2.1t]
Substitutïgèt = 0
-4è=è2C½è i.e.èC½ = -2
Thus ê solution is
yè=è500/41 cos[2t] - 2 sï[2t] - 500/41 cos[2.1t]
Ç B