Natural frequencies and mode shapes for equal mass spring system

[acpower89]

Homework Statement

Obtain the expressions for the natural frequencies and mode shapes for the case where m1=m2=m.

Homework Equations

The Attempt at a Solution

I apologise for not making an attempt, this is just unlike anything I've seen.

I'd appreciate if anyone could demystify this problem.

 

[haruspex]

Do the usual free body analysis and write the force/acceleration equations for each of the two masses.

 

[haruspex]

Consider mass m₁. If at some time t it is displaced vertically downward wrt m₂ by an amount x₁, what are the forces in the springs? What is ythe net vertical force on it? What is its displacement relative to the ground? What is its acceleration? What differential equation does that give you?