Spiral motion & angular acceleration (Question)

Fjolvar

If the elasticity, plasticity, and density is known.. Does anyone know what motion/force equations I would have to reference?

haruspex

In your case, I'm sure you don't want to enter the plastic phase as that implies damage. In the elastic phase it behaves like a spring: force is proportional to deformation.
The issues will be the shear force on the bolt and the pressure of the bolt on the edges of the slots. I'm not a trained mechanical engineer, so don't really know how to go about analysing all of that.
Why not go with my suggestion of making the bolt work against a spring, so you can arrange that almost all of the energy has been absorbed before the bolt hits the stop?

Fjolvar

Unfortunately I cannot implement any design changes, since the product is already in production. I'm only assigned to determine the force of the locking bolt against the plastic hard stop of the gear. This requires using equations that will take into account change of inertia, since the spiral is directed inwards towards the center of the wheel or axle. Do you happen to know any equations to recommend?

haruspex

As I said, I'm not a qualified engineer, but my judgment is that a calculation sufficiently accurate will be a much more time consuming and difficult task than simply measuring it. You would need to model the detailed shape of the bolt and the grooves it contacts, plug in the properties of the materials, think about the different failure modes, the uncertainties in the variables...
If you have to go down that path:
- the bolt is a cylinder which sits in a groove at each end (yes?)
- there is some slack around the bolt and between the vertical plates carrying the grooves
- at impact, the longitudinal section through the bolt will look like a rectangle ABCD at some small angle to the horizontal; A is in space in the 1st groove, B contacts the upper side of the 1st groove, C is in space, D contacts the lower side of the 2nd groove; point E on BC contacts the upper lip of the 2nd groove; point F on DA contacts the lower lip of the 1st groove; there is a horizontal gap between the two plates
- there is a shear force on the bolt section between the plates
- there is a bending moment about E and about F
- there are compressive forces on the bolt and grooves at all contact points
Each of these should be analysed wrt elastic limit.
In practice, the bolt corners will be or will become a little rounded. This increases its ability to deal with compressive forces at B and D, but increases the slack, allowing the skew angle to increase; not sure whether that makes the shear worse, but if the angle gets steep enough it will create a significant force prising the plates apart. One possible failure mode is that the bolt escapes a groove entirely.