artificial insemination

First of all, tomatoes have "perfect" flowers. That is, they have both male and female organs, and are self-fertilizing.

I suspect your plants are being stressed by environmental conditions. Have you been running high temps? That will do it. High heat will cause the plants to either not set flowers, or, if they do, the flowers abort. You'll know if that's the case as the night tempertures go down to the high 60s and low 70s. Of a sudden you'll be covered up with flowers.

Another possibility is humidity. If the humidity is too high the pollen clumps, and cannot drop properly. Result: Fertilization does not take place, and the flowers abort. I would guess this to be the real culprit, being as you have been shaking the plants, and they still don't pollinate. If the pollen was not clumped, shaking usually does the job. 

Inconsistent watering can also lead to problems. But I suspect, given that other folks are experiencing the same problem, that this is not the cause.

Hand pollinating tomatoes is an exercise in futility. Modern tomatoes (we'll define that in a minute) do not have exposed stiles. That's why they don't cross-pollinate readily; the pollen is not available to pollinators. So you can rub flowers together from now until doomesday to no avail.

"Modern" means "more sophisticated." And that translates to mean they developed later, evolving from more primitive ones that do have extruded stiles. That includes potato leaf varieties, current types, and, maybe, double-blossomed beefsteaks.  In other words, about 90+% of the tomatoes we grow are modern types.