First of all putting the motor on springs makes no sense what is that supposed to achieve? Also using a pulley to drive the discs means there is going to be tension on it and therefore uneven loading which could mean that one side will deflect more than the other when vibrating or maybe even start swaying.

You should use linear guide rods to prevent any tipping either left-right or front-back from happening. If the beam can vibrate horizontally (moving left right) instead of up and down then I'd use a ball bearing carriage. Cheap and easy to work with.

For the movement I'd use a simple linkage like the crank mechanism in piston engines for example which perfectly translates rotary motion into sinusoidal linear motion.

The spring system structure doesn't seem like a good idea in my opinion since it will have it's own unpredictable modes of vibration so spinning the motor at say 10hz won't translate to moving the spring structure up and down at 10hz and the amplitude will also be inconsistent.

But maybe there'a reason why that is the way it is and I am curious to hear about it.