Lagranguan / Coupled Oscillator

[Trenthan]

Homework Statement

WIthin the framework of an idealised model, let a square plate be a rigid object with side "w" and mass "M", whose corners are supported by massless springs, all with a spring constant "k". The string are confined so they stretch and compress vertically with upperturbed length L.

Look for solutions proportional to exp(-iwt). Show that three modes of oscillation exist with angular frequencies of

Homework Equations

The Attempt at a Solution

Ok I've tried this several ways but my working attached is the one i believe most to correct, however it differs from the answer the most!

Ive assumed 3 general coordinates, one for the COM (center of mass) translation vertically "z", and two for rotation about the x-z "theta", and y-z "thi", with both angles being the angular rotation of the plate with respect to the axis of the plate in its equilibrium position (x,y,z)_equilibrium = (0,0,0)

My working attached go through how i figure it out, however I am stuck with determining the angular frequency due to the "Mg" which has no "z" component. I am unable to move any further from this point if any1 can give me a hint.

page 1

page 2

2nd I've also worked through this, which is basically identical except i ignored the gravitational PE on the Center of mass, and also the left velocity of each spring. SInce i had no "Mg" to deal with i was able to continue to get 3 modes
My Lagranguan in this case is

This results nicely. i get w1, but w2=w3= ((2*k)/M)^1/2 which doesn't match!

If any1 can point out what I've done wrong or anything i have overlooked or looked too deeply when forming my Lagranguan thanks heeps

 

[Jhero]

!

Hello there,

I have reviewed your work and here are some suggestions and corrections:

1. In your first attempt, you have correctly identified the three general coordinates as z, theta and phi. However, the Lagrangian you have written is incorrect. The kinetic energy term should include the rotational kinetic energy of the plate as well, which is given by (1/2)Iw^2, where I is the moment of inertia of the plate and w is the angular velocity. Also, the potential energy term should include the gravitational potential energy of the plate, which is given by Mgz, where M is the mass of the plate and g is the acceleration due to gravity. So the correct Lagrangian would be:

L = (1/2)M((dz/dt)^2 + (Ldtheta/dt)^2 + (Lsin(theta)dphi/dt)^2) + (1/2)I(theta-dot)^2 + (1/2)I(sin(theta)phi-dot)^2 + Mgz - (1/2)k(dz/dt)^2 - (1/2)k(Ldtheta/dt)^2 - (1/2)k(Lsin(theta)dphi/dt)^2

2. In your second attempt, you have correctly included the rotational kinetic energy term in the Lagrangian. However, the potential energy term is incorrect. The potential energy term should include the gravitational potential energy of the plate, which is given by Mgz, where M is the mass of the plate and g is the acceleration due to gravity. So the correct Lagrangian would be:

L = (1/2)I(theta-dot)^2 + (1/2)I(sin(theta)phi-dot)^2 + Mgz - (1/2)k(Ldtheta/dt)^2 - (1/2)k(Lsin(theta)dphi/dt)^2

3. In both your attempts, you have not taken into account the fact that the springs are confined to stretch and compress vertically with unperturbed length L. This means that the springs will experience a restoring force only in the vertical direction, and not in the horizontal directions. This will affect the potential energy terms in your Lagrangian.

4. Once you have the correct Lagrangian, you can proceed with solving the equations of motion using the Euler-Lagrange equations. This will give you the three