A system of 3 masses attached by springs is oscillating

[Freya]

Homework Statement

A system has 3 identical masses each connected by springs with stiffness k, and also with the end masses attached to a wall by a spring. The system is oscillating vertically. Write down the equations of motion for each of the masses, with the displacements of each mass denoted ψa, ψb and ψc.
Consider the scenario where ψb=0 and also ψa=-ψc and find the angular frequency of the system.

Homework Equations

F=ma, F=-kx

The Attempt at a Solution

For the equations of motion I got;
for the first mass: m(d²ψa/dt²)=(-Fψa/a)+(f/a)(ψb-ψa)
2nd: m(d²ψb/dt²)=(-Fψb/a)+(f/a)(ψc-ψb)
3rd: m(d²ψc/dt²)=(-Fψc/a)+(f/a)(ψc-ψb)

I have denoted a, myself, as the distance between the masses. I have also denoted F as the force exerted along the spring when the masses are displaced from equilibrium. Then by using small angle formulae (assuming the vertical displacement causes an angle between the spring and the horizontal to be <<1)

For the second part of this, I literally just plugged in the fact that ψb=0 and ended up with
ω=√(2F/ma)

Thank you for any assistance.


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[OldEngr63]

This is a problem regarding the transverse oscillations of a string of block? A figure would really help.

What is your function (I guess it is a function) cap psi?

 

[BvU]

Something like this, but then vertically ? (stolen from Russell, PennState -- my apologies)

Here is the picture (explicit (c) makes me refer to it instead of steal it...)

 

[Freya]

Something like this, but then vertically ? (stolen from Russell, PennState -- my apologies)

Here is the picture (explicit (c) makes me refer to it instead of steal it...)

Yes that's right. I've spoken to course mates who have similar to me, I've changed it slightly now, but if you could let me know if the general gist is right, would be greatly appreciated.

 

[BvU]

I re-read your post #1 and expect a "No that's not right" ! The masses move up and down, not left-right !

And now I have to re-read some more to find out what a, F and f are. And why the four lengths of the four springs disappear from the equations

[edit] you need to express x in $\Psi_a$, a , etc.