Potential in a double vertical spring-mass system

[Ffop]

Homework Statement

I want to find an expression for the eigenfrequencies of a vertical spring-mass-spring-mass system by using the potential energy V and kinetic energy T in matrixform. Both springs have the springconstants k, and both masses are m. The springs unstretched length are l.

I have chosen y_1 to represent the displacement from equilibrium of mass 1 (upper mass) and y_2 the displacement for equilibrium of mass 2 (lower mass). I have also chosen the positive y-direction to be towards the ceiling.

My skills in mechanics is a bit rusty and i need help to find an expression for the potential energy V, i think i can do the rest from there.

The Attempt at a Solution

My attempt at writing an expression for V is

V = mg(y_1 + y_2) + k/2(y_1)^2 + k/2(y_2 - y_1)^2

The y_2 - y_1 is there because when the upper mass is displaced downwards this will compress the lower string.

I do realize i need to have something about the equlibrium for the system in V, but I am not sure how.

Thanks for any help!

 

[haruspex]

That looks ok, provided you are giving g negative value.
I didn't understand you here:

i need to have something about the equilibrium for the system in V

 

[Herman Trivilino]

Homework Statement

I want to find an expression for the eigenfrequencies of a vertical spring-mass-spring-mass system by using the potential energy V and kinetic energy T in matrixform. Both springs have the springconstants k, and both masses are m. The springs unstretched length are l.

I have chosen y_1 to represent the displacement from equilibrium of mass 1 (upper mass) and y_2 the displacement for equilibrium of mass 2 (lower mass). I have also chosen the positive y-direction to be towards the ceiling.

If you do it this way the spring potential energy will be $\frac{1}{2}ky_1^2+\frac{1}{2}ky_2^2$.

But then the gravitational potential energy will not be $mg(y_1+y_2)$.

Best to make a drawing, I think, and define y to be the height above some point of your choosing. Note that the gravitational potential energy of the upper body must always be greater than that of the lower body, something your current expression does not accomplish. Then you can write expressions for the potential energies in terms of y and l.

 

[haruspex]

If you do it this way the spring potential energy will be $\frac{1}{2}ky_1^2+\frac{1}{2}ky_2^2$.

But then the gravitational potential energy will not be $mg(y_1+y_2)$.

I believe Ffop is defining y2 as m2's displacement from its position when the whole system is at equilibrium.

 

[Ffop]

I believe Ffop is defining y2 as m2's displacement from its position when the whole system is at equilibrium.

This is exactly what i mean. I think i will take a closer look at the potential gravitational energy term i suppose. Still not sure how a quadratic term will arise from the gravitational potential (for matrix form).

Thank you all for the answers anyway.

 

[haruspex]

This is exactly what i mean. I think i will take a closer look at the potential gravitational energy term i suppose. Still not sure how a quadratic term will arise from the gravitational potential (for matrix form).

or, have a linear term arise from the matrix form, maybe by making the vector (y₁, y₂, 1)?