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−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.

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

$\displaystyle x_{1}(t)=B_{1}e^{i\omega t}$

$\displaystyle x_{2}(t)=B_{2}e^{i\omega t},$

and solve for the $\displaystyle \omega$'s.