Why is the spring strengthened in the middle?
It doesn’t seem to affect the spring’s buckling characteristics.
My speculation is that it’s to reduce spring noise. That strengthened region at the middle is where the spring will buckle outwards most, resting against the barely visible side rails on the inside of the case. Instead of just one wobbly contact point it now has three rigid ones as a “skate” to reduce the stick-slip noise when opening and retracting the tip. Is this right?
(The pen is a Mitsubishi Uni-Ball Power Tank, pretty much my favorite model.)
Right! But those are the same thing as number of coils is the spring length divided by a geometric constant. At free length there is no strain. At compression you reach max strain/torsion. Each coil turn, assuming all are equal, adds equally to the sum of restoring force. Looking at spring free length you’re just paying attention to the summed forces of the active coils.
The dead coils contribute negligibly because they would need to impinge the neighboring wire to deform. (Relegated to pure torsion) Which i think is basically what you were saying…