forced and damped oscillations

Question 1

What is the equation for forced damped SHM

Answer 1

x’’ + 2kx’ + ω^2x = F(t)

Question 2

For undamped unforced motion what are the equations and solutions

k = 0, F(t) = 0

Answer 2

x’‘+ω^2x =. 0

this has solutions

x = Acos(ωt)+Bsin(ωt)
or
x = C cos(ωt − φ).

Question 3

For damped unforced motion what are the equations and solutions

k > 0, F(t) = 0

Answer 3

x’’ + 2kx’ + ω^2x =0

solving the ODE we get solution
p^2 + 2kp + ω^2 = 0
if we solve this quadratic we get -k+-root(ω^2-k^2)

this gives us three cases

k < ω0 - light damping - complex roots

x = e^−kt(A cos ωdt + B sin ωdt)

where ωd = root(ω0^2-k^2)

k > ω0: heavy damping
both roots for p are real and negative
x = Ae^(−k+q)t +
Be^(-k-q)t

k = ω0: critical damping. The roots for p are equal

x = (A + Bt)e^−kt.

Question 4

Undamped, forced motion: k = 0, F(t) =/= 0

Answer 4

F(t) = γ sin ωt, so that
F(t) = γ sin ωt,
x ̈ + ω0^2x = γ sin ωt.

gives us two cases

Forcing frequency ω different from natural frequency ω0
x = A sin ω0t + B cos ω0t +γsin(ωt)/(ω0^2-ω^2)

or the same

x = Asinωt+Bcosωt - (γ tcosωt)/2ω