After revolving the inside part of the tire, I created a sketch normal to the tire's surface. Then I drew up the FIRST logo. (It's just an imitation of the logo; I guess you could trace a sketch on top of the real logo image to make it exact. All those dimensions are there just because I wanted it to be fully constrained.)
Then I made a rectangular pattern of the logo, inside the same sketch, making sure that the width of the pattern did not extend past pi * diameter of tire surface.
Then I embossed the sketch, choosing a raised emboss and "Wrap to Face" with the tire surface.
You could also emboss one instance of the logo and circular pattern the Emboss as a feature.