Reading: Main 5.1, 6.1 Taylor 5.5, 5.6

THE DRIVEN, DAMPED HARMONIC OSCILLATOR

1

Natural motion of damped, driven harmonic oscillator

€

Force = m˙ ̇ x

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restoring + resistive + driving force = m˙ ̇ x

x

m

m k

k viscous medium

F0cosωt

−kx − b x + F0 cos ωt( ) = mx

mx +ω02x + 2β x+ = F0 cos ωt( )

Note ω and ω0 are not the same thing! ω  is driving frequency ω0 is natural frequency

ω0 =km

ω1 =ω0 1− β2

ω02

2

Natural motion of damped, driven harmonic oscillator

q + 2β q +ω 0

2q = V0Leiω t

V0 cos ωt( ) − Lq − q

C− R q = 0

L

R

C I

Vocosωt

Apply Kirkoff’s laws

http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/accircuit.htm

3

What is the response of the system? x(t), q(t), or in general, ψ(t)? Qualitative questions first: • What is the basic form of the system response after long times? Sinusoidal. ψ(t) = ψmaxcos(ωt+φ) (after times longer than 1/β) • Is the frequency of the system response the same, smaller, or larger than the driving frequency? The same - it must be! • How does the magnitude of the response depend on the driving frequency? It is large close to the natural frequency ω0, and small at lower and at higher frequencies (this is called resonance) • How does the phase of the response depend on the driving frequency? We’ll have to see. It changes, certainly. But it depends on what you mean by “response”. Is it displacement/charge? Velocity/current? 4

V0 real, constant, and known

But now q0 is complex: q0 = q0 eiφq

This solution makes sure q(t) is oscillatory (and at the same frequency as Fext), but may not be in phase with the driving force. Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and φq in terms of the known quantities. Note that these constants depend on driving frequency ω (but not on t - that’s why they’re “constants”). How does the shape vary with ω?

Vext = Re V0eiω t⎡⎣ ⎤⎦

q t( ) = Re q0eiω t⎡⎣ ⎤⎦

5

Let’s assume this form for q(t)

Assume V0 real, and constant

Task #2: In the lab, you’ll actually measure I (current) or dq/dt. So let’s look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency.

Vext = Re V0eiω t⎡⎣ ⎤⎦

q t( ) = Re q0eiω t⎡⎣ ⎤⎦ I t( ) = ?

6

Assume V0 real, and constant Vext = Re V0eiω t⎡⎣ ⎤⎦

q t( ) = Re q0eiω t⎡⎣ ⎤⎦

q0 =V0 L

ω02 −ω 2( )2 + 4β 2ω 2⎡

⎣⎤⎦

1/2

€

tanφ =−2βωω02 −ω 2

Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and φ in terms of the known quantities. Note that these constants depend on ω (but not on t - that’s why they’re “constants”). How does the shape vary with ω?

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Charge Amplitude

|q0|

Charge Phase φq

Driving Frequency------>

“Resonance”

0

-π

-π/2

8

Vext = Re V0eiω t⎡⎣ ⎤⎦ q t( ) = Re q0e

iω t⎡⎣ ⎤⎦

I0 =ωV0 L

ω02 −ω 2( )2 + 4β 2ω 2⎡

⎣⎤⎦

1/2

φI =π2+ arctan −2βω

ω02 −ω 2

I(t) = q t( ) = Re ω q0 e

i φq +π /2( )eiω t⎡⎣

⎤⎦

Task #2: In the lab, you’ll actually measure I (current) or dq/dt. So let’s look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency.

9

Current Amplitude |I0|

Current Phase

Driving Frequency------>

“Resonance”

10

0

π/2

-π/2

Charge Amplitude

|q0|

Driving Frequency------>

“Resonance”

11

Current Amplitude |I0|

ω0

ω0

Current Phase

Driving Frequency------>

12

Charge Phase φq

0

-π

-π/2

0

π/2

-π/2

ω0

ω0