PURPOSE: To obtain a compact apparatus by installing a shock absorber between a lever and the bracket of an upper cylinder and preventing the swing of the supporting lever of the lower cylinder under a front fork in the front-wheel suspension of a motor-cycle from interfere with an upper cylinder and reducing the amount of projection to the side of a chassis.

CONSTITUTION: In the front fork 10 of a front-wheel suspension, an upper cylinder 12 is axially supported in free steering onto a steering shaft 20 through a bracket. A lower cylinder 14 is installed, permitting being inserted and pulled-out from the upper cylinder 12, to form an air chamber 38. At the lower edge of the lower cylinder 14, the front edge of a swing lever 52 is supported by a pin through a bush link 76, and the rear edge of the lever 52 is supported by a pin onto the extension part of the bracket 50 on the lower-part rear surface of the upper cylinder 12. A shock absorber 80 is installed between the lever 52 and the bracket 86 of the upper cylinder 12. Therefore, the swing range of the lever 52 can be restricted only to the vicinity of the lower cylinder 14, and the inteference with the upper cylinder 12 can be obviated. Therefore, the apparatus itself can be made compact.