I was trying to come up with easy to build round bench dogs with a non damaging spring and after a bit of experimentation here is my method:
1. Cut your dowel to length and clamp in a vise with the top and front a little proud. I made my dogs four inches long.
2. Clamp a stop to the left of the blank so that a 00 size biscuit slot can be cut 1/2 inch from the bottom of the dog.
3. After the slot is cut turn the blank in the vise so the slot is facing upward.
4. Drill a 1/8 inch hole about 1/16th of an inch from each end of the slot. Drill the holes 1/4 inch deep and at a 45 degree angle, approximately perpendicular to the bottom of the slot.
5. Cut and insert a length of nylon string trimmer line into both holes so that it forms an arc that protrudes about 1/4 inch. I found that 0.065 line was too light and 0.13 line was too stiff, 0.080 or 0.095 provided the perfect friction.
6. Cut your flat on your dog so the "spring" is at right angles to the flat. I made my flat 1 inch deep and about 1/3 of the way into the dog. Some people say to put a 2 degree angle on the flat so that under heavy clamping pressure it still maintains full contact but this is your choice.