Driven Coupled Oscillators

Equations of motion of driven damped coupled oscillators

𝑚𝑥 1 = −𝑘𝑥1 − 𝑘0 𝑥1 − 𝑥2 − 𝑟𝑥 1 + 𝐹0 cosΩ𝑡

𝑚𝑥 2 = −𝑘𝑥2 − 𝑘0 𝑥2 − 𝑥1 − 𝑟𝑥 2

𝑘 𝑘

m m F(t)

Equations of motion of driven damped coupled oscillators

𝑥 1 + 2𝛽𝑥 1 +𝑘 + 𝑘0𝑚

𝑥1 −𝑘0𝑚 𝑥2 = 𝑓0 cosΩ𝑡

𝑥 2 + 2𝛽𝑥 2 +𝑘 + 𝑘0𝑚

𝑥2 −𝑘0𝑚 𝑥1 = 0

⟹ 𝑥1= 𝑞1 + 𝑞2 , 𝑥2 = 𝑞1 − 𝑞2

𝑞1 = 𝑥1 + 𝑥2 /2, 𝑞2 = 𝑥1 − 𝑥2 /2

𝑞 1 + 2𝛽𝑞 1 +𝜔12𝑞1 =

𝑓02cosΩ𝑡 , 𝜔1

2 =𝑘

𝑚

𝑞 2 + 2𝛽𝑞 2 + 𝜔22𝑞2 =

𝑓02cosΩ𝑡 , 𝜔2

2 =𝑘 + 2𝑘0

𝑚

𝑞1 𝑡 = 𝑎1 cos Ω𝑡 − 𝜙1 , 𝑞2 𝑡 = 𝑎2 cos Ω𝑡 − 𝜙2

Steady state solution:

𝑎1 =𝑓02

1

𝜔12 − Ω2 2 + 4𝛽2Ω2

, tan𝜙1 =2𝛽Ω

𝜔12 − Ω2

𝑎2 =𝑓02

1

𝜔22 − Ω2 2 + 4𝛽2Ω2

, tan𝜙2 =2𝛽Ω

𝜔22 − Ω2

𝑥1 𝑡 = 𝑞1(𝑡) + 𝑞2(𝑡) = 𝐴1 cosΩ𝑡 + 𝐵1 sinΩ𝑡

𝑥2 𝑡 = 𝑞1(𝑡) − 𝑞2(𝑡) = 𝐴2 cosΩ𝑡 + 𝐵2 sinΩ𝑡

𝐵1,2 = 𝑎1 sin𝜙1 ±𝑎2 sin𝜙2

= 𝛽𝑓0Ω1

𝜔12 − Ω2 2 + 4𝛽2Ω2

±1

𝜔22 − Ω2 2 + 4𝛽2Ω2

𝐴1,2 = 𝑎1 cos𝜙1 ±𝑎2 cos𝜙2

=𝑓02

𝜔12 − Ω2

𝜔12 − Ω2 2 + 4𝛽2Ω2

±𝜔22 − Ω2

𝜔22 − Ω2 2 + 4𝛽2Ω2

For very small damping (𝛽 → 0) ,

𝐵1,2 → 0

𝐴1,2 →𝑓02

1

𝜔12 − Ω2

±1

𝜔22 − Ω2

𝑥2(𝑡)

𝑥1(𝑡)≈𝐴2 cosΩ𝑡

𝐴1 cosΩ𝑡≈

𝜔22 −𝜔1

2

𝜔22 +𝜔1

2 − 2Ω2

𝜔2 > 𝜔1

Special cases

𝑥2(𝑡)

𝑥1(𝑡)≈

𝜔22 −𝜔1

2

𝜔22 +𝜔1

2 − 2Ω2, 𝜔2> 𝜔1, 𝛽 → 0

Case1: Ω = 𝜔1, 𝑥2(𝑡)/𝑥1(𝑡) ≈ 1

Case3: Ω ≪ 𝜔1,𝑥2(𝑡)

𝑥1(𝑡)≈𝜔22 − 𝜔1

2

𝜔22 + 𝜔1

2 > 0

Case2: Ω = 𝜔2, 𝑥2(𝑡)/𝑥1(𝑡) ≈ −1

Case4: Ω ≫ 𝜔2,𝑥2(𝑡)

𝑥1(𝑡)≈ −

𝜔22 −𝜔1

2

2Ω2< 0

Mechanical filters Bandpass filter:

𝑥2(𝑡)

𝑥1(𝑡)≈

𝜔22 −𝜔1

2

𝜔22 +𝜔1

2 − 2Ω2

𝜔1 = 𝜔0 − ∆𝜔,𝜔2 = 𝜔0 + ∆𝜔, ∆𝜔 ≪ 𝜔0 For

𝑥2(𝑡)/𝑥1(𝑡) ≈ 2𝜔0∆𝜔/ 𝜔02 − Ω2

𝑥2(𝑡)/𝑥1(𝑡) → 0+ 𝜔0−1 𝑓𝑜𝑟 Ω ≪ 𝜔0 as

𝑥2(𝑡)/𝑥1(𝑡) → 0− Ω−2 𝑓𝑜𝑟 Ω ≫ 𝜔0 as

2∆𝜔 Band of pass frequencies: Cut-off frequencies: 𝜔1, 𝜔2

Low-pass filter: 𝜔1 → 0

𝑥2(𝑡)

𝑥1(𝑡)≈

𝜔22

𝜔22 − 2Ω2

Ω ≫ 𝜔2, 𝑥2(𝑡)/𝑥1(𝑡) → 0− For Ω−2 as

For Ω → 𝜔2/ 2, 𝑥2(𝑡)/𝑥1 𝑡 → ∞

For Ω → 0, 𝑥2(𝑡)/𝑥1 𝑡 = +1

For Ω = 𝜔2, 𝑥2(𝑡)/𝑥1 𝑡 = −1

Longitudinal waves

A chain of oscillators

ti 1

ti

ti 1

i-1 i i+1

Continuum limit

𝜉𝑖−1(𝑡) → 𝜉(𝑥 − ∆𝑥, 𝑡)

𝜉𝑖+1 𝑡 − 𝜉𝑖(𝑡) − 𝜉𝑖(𝑡) − 𝜉𝑖−1(𝑡) →

Waves in a rod

xx x x

xx x x

tx ,

txx ,

A A

x

𝜉𝑖+1 𝑡 − 𝜉𝑖(𝑡) − 𝜉𝑖(𝑡) − 𝜉𝑖−1(𝑡)

Longitudinal waves in thin rods

a

Young’s modulus of elasticity

a

x x

a a

x

x

F F F

Also,

Newton’s law:

As

Acoustic waves in a gaseous medium

Fractional change in volume

Change in pressure

compressibility

|ξ|<<|x|

x x+x

aP aP a

ξ ξ+ξ

a(P+p) a(P+p+p)

x +ξ

In an infinitely thin layer of gas,

Newton’s law:

As