Damped Oscillator equation. Finding Energy and general solution for coupled oscillato
Hi! I can do (a) easily enough but (b) has me totally stumped altogether! I can't seem to figure out how they get it in that form! Also (c) I'm having the same problem. I have two exams tomorrow and this is the first one so im juggling between the two subjects going through the past papers so if anyone could help with a solution for this it would be very helpful! Thanks
1. (a) The damped oscillator equation
m d2y/dt2 + dy/dt + ky = 0 has a solution of the form
y(t) = Ae−α t cos(wt − ϕ ).
Determine α and ϕ.
(b) Show that the energy of the system in (a) given by E = 1/2mẋ^2 + 1/2kx^2
satisfies dE/dt = −mvẋ
(c) Two coupled oscillators are described by the equations
mẍ1 = −kx1 + k(x2 − x1)
mẍ2 = −k(x2 − x1) − kx2.
Construct the general solution (normal modes) for this system.
Re: Damped Oscillator equation. Finding Energy and general solution for coupled oscil
Quote:
Originally Posted by
paddy543211 
Hi! I can do (a) easily enough but (b) has me totally stumped altogether! I can't seem to figure out how they get it in that form! Also (c) I'm having the same problem. I have two exams tomorrow and this is the first one so im juggling between the two subjects going through the past papers so if anyone could help with a solution for this it would be very helpful! Thanks
1. (a) The damped oscillator equation
m d2y/dt2 + dy/dt + ky = 0 has a solution of the form
y(t) = Ae−αt cos(wt − ϕ ).
Determine α and ϕ.
(b) Show that the energy of the system in (a) given by E = 1/2mẋ^2 + 1/2kx^2
satisfies dE/dt = −mvẋ
(c) Two coupled oscillators are described by the equations
mẍ1 = −kx1 + k(x2 − x1)
mẍ2 = −k(x2 − x1) − kx2.
Construct the general solution (normal modes) for this system.
(b) What is x relative to y? What is v relative to y?
(c) Try a trial solution of the form
=B_{1}e^{i\omega t})
=B_{2}e^{i\omega t},)
and solve for the 
's.
