coupled springs system...system of first order DE's

[Sarahisme]

hey all

 

damn this problem, i have spent hours at it and keep ending up in a big mess...

 

 

 

any ideas guys? :S

 

 

Sarah

[swansont]

Each spring will have a force of a general form F = -Kz. Write down the individual forces on each block, and recognize that the displacements of the two are coupled to each other, i.e. x and y will depend on each other through the middle spring. Remember that kx and nky are related to mg.

[Sarahisme]

Each spring will have a force of a general form F = -Kz. Write down the individual forces on each block, and recognize that the displacements of the two are coupled to each other, i.e. x and y will depend on each other through the middle spring. Remember that kx and nky are related to mg.

 

 

are you just talking about how to do the "show that the equations governing the motion are..."?

 

i can get this bit , writing down those equations, its the other stuff where everything goes horribly wrong

[swansont]

are you just talking about how to do the "show that the equations governing the motion are..."?

 

i can get this bit ' date=' writing down those equations, its the other stuff where everything goes horribly wrong [/quote']

 

 

OK. What are the normal modes?

[Sarahisme]

there are two normal modes:

 

(1) where the two masses are moving in the same way (in the same periodic motion)

 

(2) where the two masses are always moving in opposite directions (in a periodic way)

 

i think thats right?

 

i am guessing that i need to solve this problem using a system of first order linear equations.....?

[Sarahisme]

hmm the question seems to have disappeared, so here it is again :

 

[Sarahisme]

ok here goes, what do you think...:

 

i get the two eigen vectors to be [1,1] and [1,-1] (pretend that those brackets are vertical.

 

so i get the solution to be:

 

[math] x_1 = rsin(pt + \theta) + s sin(qt + \phi) [/math]

 

[math] x_2 = rsin(pt + \theta) - s sin(qt + \phi) [/math]

 

where [math] p = \sqrt{k} [/math] and [math] q = \sqrt{k+2nk} [/math]

 

 

and then choosing r = 1, s = 0 gives the 'moving together' type of normal mode oscillation and r = 0 , s = 1 gives the 'moving oppositely' type of normal mode oscillation.

 

is that the right sort of thing? l mean, does it answer the "under what conditions can this system be made to oscillate in a preciisely periodic manner?" bit of the question??

[swansont]

I think they want you to identify the actual frequencies of the normal modes. For the "precisely periodic" I think they are asking about relative phase of the oscillations.

[Sarahisme]

hmm ok let me have a go at that...

 

for the 'moving together' type of normal mode oscillation (mode 1)::

 

frequency = [math] \frac{p}{2 \pi} = \frac{\sqrt{k}}{2\pi} [/math]

 

for the 'moving oppositely' type of normal mode oscillation (mode 2)::

 

frequency = [math] \frac{q}{2 \pi} = \frac{\sqrt{k+2nk}}{2\pi} [/math]

 

so we can see that mode 1 is [math] \sqrt{1+2n} [/math] times faster than mode 2.

 

 

however i'm not sure about "relative phase of the oscillations."....??

[swansont]

You have the frequencies of the two normal modes, but you've already noted that they must be either in the same or opposite direction. That's zero phase, or 180 degrees out of phase. What happens if the relation is different? e.g. one lags the other by 90 degrees, so that one has zero displacement when the other has maximum displacement?

[Severian]

Sarahisme: where do you go to school? You seem to get a lot of homework.

[Sarahisme]

Sarahisme: where do you go to school? You seem to get a lot of homework.

 

a lot of it isnt homework, i just like to do extra work, i enjoy doing most of it too!

[swansont]

a lot of it isnt homework, i just like to do extra work, i enjoy doing most of it too!

 

 

Good for you!

[Sarahisme]

You have the frequencies of the two normal modes, but you've already noted that they must be either in the same or opposite direction. That's zero phase, or 180 degrees out of phase. What happens if the relation is different? e.g. one lags the other by 90 degrees, so that one has zero displacement when the other has maximum displacement?

 

do you mean that the system can be made to oscillat ein a precisely periodic manner if the ratio of the characteristic frequencies are a rational number?

 

hmmm i don't quite understand, do that want anything more than that or do they want actual numbers or something??? ahhh!

[swansont]

do you mean that the system can be made to oscillat ein a precisely periodic manner if the ratio of the characteristic frequencies are a rational number?

 

hmmm i don't quite understand' date=' do that want anything more than that or do they want actual numbers or something??? ahhh! [/quote']

 

It's not always clear what the point of a homework question is, unfortunately. "precisely periodic" means, to me, unchanging frequency, so my answer would be they have to either be in phase (at the lower frequency) or exactly out of phase (at the higher), otherwise you get beats as the oscillations move from one mass to the other

[Sarahisme]

yeah, i know what you mean. well i think i get the general idea of it anyway (the normal modes of oscillation stuff.)

 

thanks for all you help once again swansont