Harmonic Oscillator Problem: Consideration & Solutions

[mattmatt]

Problem:

Consider a harmonic oscillator of undamped frequency ω₀ (= $\sqrt{k/m}$) and damping constant β (=b/(2m), where b is the coefficient of the viscous resistance force).

a) Write the general solution for the motion of the position x(t) in terms of two arbitrary constants assuming an underdamped oscillator (ω₀ > β).

b) Express the arbitrary constants in terms of the initial position of the oscillator x₀ = x(t=0) and its initial speed v₀ = $\dot{x}$(t=0).

c) Take the limit in the above solution where the oscillator becomes critically damped, β $\rightarrow$ ω₀, assuming the initial conditions remain the same. (Be careful that β appears also in the constants of the solutions once they have been expressed in terms of initial conditions.)

d) Repeat steps (a) and (b) but now assuming an overdamped oscillator (β > ω₀).

e) Take, again, the limit β $\rightarrow$ ω₀ in the solutions found in (d). What do you conclude from the results (c) and (e)?Solution:

a)
m$\ddot{x}$=-b$\dot{x}$-kx
m$\ddot{x}$+b$\dot{x}$+kx=0
$\ddot{x}$+(b/m)$\dot{x}$+(k/m)x=0
$\ddot{x}$+2β$\dot{x}$+(ω₀)²=0
where β=b/(2m) and $\sqrt{k/m}$=ω₀
r=(-2β$\pm$(4β²-4ω₀²)^1/2)/2
r=-β$\pm$(β²-ω₀²)^1/2
r=-β$\pm$iω_D
where ω_D²=ω₀²-β²
x(t)=e^-βt(c₁e^iω_Dt+c₂e^-iω_Dt)
c₁=Acos(δ)
c₂=Asin(δ)
x(t)=Ae^-βtcos(ω_Dt-δ) : General Solution

b)
$\dot{x}$(t)=-Ae^-βt(ω_Dsin(ω_Dt-δ)+βcos(ω_Dt-δ))
x₀=Acos(δ)
$\dot{x}$₀=-A(βcos(δ)-ω_Dsin(δ))
x₀β+$\dot{x}$₀=Aω_Dsin(δ)
(x₀β+$\dot{x}$₀)/x₀=ω_Dtan(δ)
δ=arctan((x₀β+$\dot{x}$₀)/(x₀ω_D))

I get stuck on part (c)
If I take the limit as β approaches ω₀ then ω_D goes to zero. And I get δ=∏/2. A turns out to be infinity. How can there be an infinite amplitude? Is this wrong?

 

[SteamKing]

Hint: look up resonance.

 

[mattmatt]

Correct me if I'm wrong but wouldn't resonance require an external driving force? This problem doesn't have an external driving force.

 

[vela]

Problem:

Consider a harmonic oscillator of undamped frequency ω₀ (= $\sqrt{k/m}$) and damping constant β (=b/(2m), where b is the coefficient of the viscous resistance force).

a) Write the general solution for the motion of the position x(t) in terms of two arbitrary constants assuming an underdamped oscillator (ω₀ > β).

b) Express the arbitrary constants in terms of the initial position of the oscillator x₀ = x(t=0) and its initial speed v₀ = $\dot{x}$(t=0).

c) Take the limit in the above solution where the oscillator becomes critically damped, β $\rightarrow$ ω₀, assuming the initial conditions remain the same. (Be careful that β appears also in the constants of the solutions once they have been expressed in terms of initial conditions.)

d) Repeat steps (a) and (b) but now assuming an overdamped oscillator (β > ω₀).

e) Take, again, the limit β $\rightarrow$ ω₀ in the solutions found in (d). What do you conclude from the results (c) and (e)?Solution:

a)
m$\ddot{x}$=-b$\dot{x}$-kx
m$\ddot{x}$+b$\dot{x}$+kx=0
$\ddot{x}$+(b/m)$\dot{x}$+(k/m)x=0
$\ddot{x}$+2β$\dot{x}$+(ω₀)²=0
where β=b/(2m) and $\sqrt{k/m}$=ω₀
r=(-2β$\pm$(4β²-4ω₀²)^1/2)/2
r=-β$\pm$(β²-ω₀²)^1/2
r=-β$\pm$iω_D
where ω_D²=ω₀²-β²
x(t)=e^-βt(c₁e^iω_Dt+c₂e^-iω_Dt)
c₁=Acos(δ)
c₂=Asin(δ)
x(t)=Ae^-βtcos(ω_Dt-δ) : General Solution

b)
$\dot{x}$(t)=-Ae^-βt(ω_Dsin(ω_Dt-δ)+βcos(ω_Dt-δ))
x₀=Acos(δ)
$\dot{x}$₀=-A(βcos(δ)-ω_Dsin(δ))
x₀β+$\dot{x}$₀=Aω_Dsin(δ)
(x₀β+$\dot{x}$₀)/x₀=ω_Dtan(δ)
δ=arctan((x₀β+$\dot{x}$₀)/(x₀ω_D))

I get stuck on part (c)
If I take the limit as β approaches ω₀ then ω_D goes to zero. And I get δ=∏/2. A turns out to be infinity. How can there be an infinite amplitude? Is this wrong?

No, it's not wrong, but what's happening is a little obscure. Rewriting your solution slightly, you get
$$x(t) = e^{-\beta t}[(A\cos \delta)\cos \omega_D t + (A\sin\delta)\sin \omega_D t].$$ When $\beta \to \omega_0$, $A$ indeed blows up and $\delta \to \pi/2$. The product $A\cos\delta$, however, remains finite. Similarly, $\sin\omega_D t \to 0$ because $\omega_D \to 0$, and the product $A \sin\omega_D t$ remains finite.

You'll find it more straightforward to work with the solution in the form $x(t) = e^{-\beta t}(A \cos \omega_D t + B \sin \omega_D t)$.

 

[paranormal]

c) Taking the limit as β approaches ω0, we have ωD=0 and δ=∏/2. The general solution then becomes:
x(t)=Ae-βtcost(ωDt-∏/2)
= Ae-βt(sin(ωDt)-(1/ωD)cos(ωDt))
= Ae-βt(-1+0)
= Ae-βt
where A is a constant determined by the initial conditions. This solution represents critically damped motion, where the oscillator returns to its equilibrium position without oscillating.

d) For an overdamped oscillator, ωD is imaginary and the general solution becomes:
x(t)=Ae-βt(cos(ωDt-δ)+iωDsin(ωDt-δ))
= Ae-βt(ei(ωDt-δ))
= Ae-βt(e-βt(cos(ωDt)+iωDsin(ωDt)))
= Ae-βt(e-βt(cos(ωDt)+βsin(ωDt)))
where A is a constant determined by the initial conditions. This solution represents a decaying exponential motion without any oscillations.

e) Taking the limit as β approaches ω0, we have δ=∏/2 and the general solution becomes:
x(t)=Ae-βt(cos(ωDt-∏/2)+iωDsin(ωDt-∏/2))
= Ae-βt(cos(ωDt)+iωDsin(ωDt))
= Ae-βt(e-βt(cos(ωDt)+0))
= Ae-βt(e-βt(cos(ωDt)))
= Ae-βt(cos(ωDt))
This solution is similar to the one in part (c), but with an imaginary component that results in a decaying exponential motion without oscillations. This indicates that as the damping constant approaches the undamped frequency, the motion of the oscillator becomes less oscillatory and more decaying.