You can install our site as a web app on your iOS device by utilizing the Add to Home Screen feature in Safari. Please see this thread for more details on this.
Note: This feature may not be available in some browsers.
It is definitely not a wheel/tire issue. I have this same issue on my model s. I bought a brand new set of tires since mine were worn anyway and I road force balanced the wheel and tire. Your electric toothbrush description is spot on. I was describing it as a rumble strip type sensation but...