Generalized coordinates of a couple harmonic oscillator

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  • #1
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Homework Statement



Suppose there is a square plate, of side a and mass M, whose
corners are supported by massless springs, with spring constants K, K, K, and k <= K
(the faulty one). The springs are confined so that they stretch and compress vertically,
with unperturbed length L. The density of the plate is uniform.

(it's a car so the square plate is the chassis and the springs are the suspension)

Explain why the system requires three generalised coordinates to be described
completely.

Homework Equations


-

The Attempt at a Solution


I'm just really having a hard time visualizing the actual motion of the plate on the springs.

I know there should be a generalized coordinate associated with the displacement of the spring, for each spring.. but I can't make the connection and reduce the number to 3 generalized coordinates, as specified.

I'm thinking that the springs on opposite corners are somehow related..i.e as one goes up the other goes down, but what happens to the 2 other springs in this case?

thanks in advance.
 

Answers and Replies

  • #2
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Try using angles ;))

How many angles do you need, to describe it if all of the springs were identical? ;)

Now what else changes when you replace one of the springs ;D
 
  • #3
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the angles taken from where? I'm trying to visualise the motion of the system, but I'm really struggling.

The angles taken from a plane parallel to the square plate.. so that the angle of deflection downward (at one spring) would equal the angle of deflection upward on the opposite side (for the opposite spring)

So you need 2 generalized coordinates if all the springs were the same.. but since they aren't you need on angle to describe the two opposite springs with the same spring constant, and then one more angle each for the remaining springs?

..is my reasoning correct?
 
  • #4
263
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No thats not quite correct.

Just to visualize, we put it in an xyz coordinate system

You need two angles, in the case when all the springs are the same. One of the angles, measure the tilting in the xz plane, and the other the tilting in the yz plane. With these two angles all other tilting combinations can be expressed. Now we dont need another coordinate, since in this case when every spring is the same, the center of the plate doesnt move vertically.

So if we replace one of the springs, then what other information do we need, other than the tilting ;)
 
  • #5
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I gotta run now, but I'll have a think about it and hopefully get back to you tomorrow.

Thanks for your time Thaakisfox, much appreciated.
 
  • #6
25
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bump: so the centre of the plate moves vertically if one of the springs is different?

and the generalized coordinates are 2 angles (measuring the tilt in each plane) and a displacement coordinate measuring the displacement of the centre of the plane from it's undisturbed position?

how do you know that the centre of the plane won't move if all the springs are identical?
(sorry if the questions seems silly or vague, but I've never dealt with a system configured as above)
 

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