Power curves and coupled oscillations

Question 1

power of a forced damped oscillator equation

Answer 1

P(t) = bv²
b: friction factor
v: velocity

Question 2

the equation for the maximum power of a forced oscillator

Answer 2

P = F₀² / 2mγ
P is actually P-bar
happens when ω = ω₀

Question 3

The normal frequency equations for a coupled pendulum

Answer 3

These are the frequency of the normal coordinates (q₁ and q₂ of the uncoupled equations of motion)
ω₁ = (g/L)^½
L: pendulum length

ω₂ = (g/L + 2k/m)^½
k: spring constant (of the spring joining them)

Question 4

how would you set up a coupled oscillator equation of motion
(assume xₕ > xₐ)

Answer 4

F = k(xₕ - xₐ) (tension in the spring connecting the two pendula)

restoring forces: (uses sin(θ)=x/l )
mẍₐ = -mg/l * xₐ + k(xₕ - xₐ) (the + or - F depends on the question)
mẍₕ = -mg/l * xₕ - k(xₕ - xₐ)

ẍₐ + g/l xₐ - k/m *(xₕ - xₐ) = 0 (equation 1)
ẍₕ + g/l xₕ + k/m *(xₕ - xₐ) = 0 ( equation 2)

add equations 1 and 2. Define xₕ + xₐ = q₁
subtract equation 2 from 1. Define xₐ - xₕ= q₂
this gives two uncoupled 2nd order ODEs.

Question 5

SHM equation for a ball held horizontally by two springs

Answer 5

|–O–| (like this)

ẍ = -2T/ml * x
it oscillated up and down

Question 6

trial solution for any coupled oscillator

Answer 6

x(t) = X₀ * e^(iΩt)

Question 7

how would you set up equations for a coupled oscillator with 3 springs at 2 particles

Answer 7

|–O—O–| (like this. A is on the left, H is on the right)

θ₁: angle between left most spring and the horizontal
θ₂ angle between the middle spring and the horizontal
θ₃ angle between the right most spring and the horizontal

mẍₐ = -Tsin(θ₁) + Tsin(θ₂)
mẍₕ = -Tsin(θ₂) -Tsin(θ₃)

sub in sin(θ₁) = xₐ/l , sin(θ₂) = (xₐ-xₕ)/l , sin(θ₃) = xₕ/l (small angle approx)
where x is the vertical displacement

then sub in the trial solution, set up a matrix and solve for the eigenvalues.

Question 8

derive the differential equation for an RLC circuit

Answer 8

Kirchoffs 2nd: sum of emf = sum of voltage in a closed loop

V = L* dI/dt = Lq(..)
V = IR = Rq(.)
V = q/C

Lq(..) + Rq(.) + q/C = 0
(= 0 if there’s no driving voltage)

Question 9

equation of oscillation for an RLC circuit and compare it to the mechanical oscillator equation

Answer 9

Lq(..) + Rq(.) + 1/C q = V₀ * e^(iωt) (<– driving voltage)
L: inductance
R: resistance
C: capacitance
q: charge

this gives
I = I₀ * e^(iωt)
I: Current

hence:
γ = R/L
ω₀² = 1/LC

Question 10

derive the phase difference between the current and voltage in an RLC circuit

Answer 10

assume the driving voltage is V = V₀ * e^(iωt)
hence the current must have the same frequency:
I = I₀ * e^(iωt)

Inductor:
V = L* dI/dt
= iωL* I
i = e^(iπ/2)
hence, the voltage has a phase difference of π/2 (the voltage is leading the current by a phase of π/2)

Capacitor:
V = q/C = 1/C * ∫I dt
= 1/iωC * I
= -i/ωC * I
-i = e^(-iπ/2)
hence the voltage lags behind the current by a phase of π/2

Resistor:
V= RI
hence they are in phase

Question 11

Impedance equation

Answer 11

Z² = R² + (ωL - 1/ωC)²

Question 12

what happens in an RLC circuit at the resonant frequency

Answer 12

since ω₀² =1/LC
ωL - 1/ωC = 0
(the impedance has no complex component)