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If it is smooth by itself and only becomes twitchy when you add the weight and inertia of the road wheels, then it sounds like a steering damper cylinder may help.
Hi KB. Interesting construction project!As others indicate, the vibration in the steering could be remedied by using a damper or changing springs or weight of the wheels.It is difficult to determine whether this fully follows the natural frequency.I saw this passing by and thought, maybe I can help somehow to draw up a small model. In order to map out the natural frequency in a simplified manner of the control device. And on the basis of this change parameters such as spring strength and or application of a damper.This might create some insight.As an example I have made a model in which a certain mass and corresponding radius get a corner rotation. This mass is connected to a spring (2nd model + damper) over a certain distance from the center of the mass.This model gives an outline of how the rotation of the mass would take place with the use of only a spring or with a damper.I find it interesting to look at this. For this I do need the masses and dimensions of the components of the steering system. And any spring constant of the mounted spring.Let me know if this seems like something to you!