A variable speed transmission system, comprising: a rotary first transmission member (20 or 40 or 90 or 100) mounted on a rotation axis; a second transmission member (30 or 50 or 70 or 80 or 102 or 122 or 132) to be selectively coupled to and decoupled from said first transmission member; said second transmission member being wound around part of said first transmission member; said first transmission member including at least one gripping device (21 or 91 or 101 or 131) that is moveable radially with respect to said rotation axis and includes a plurality of laterally spaced and laterally displaceable gripping elements (26, 27 or 131 or 101); said second transmission member including a plurality of laterally spaced and laterally displaceable coupling elements (31 or 51 or 71 or 81) said transmission members being configured to be coupled together by said at least one gripping device and by said plurality of coupling elements on said second transmission member that are receivable in an interleaved manner between said gripping elements of said at least one gripping device as the first transmission member rotates, characterized in that: upon application of a lateral compressive force said gripping elements move toward each other and said interleaved coupling elements also move toward each other to decrease the spaces between all said interleaved elements; whereby the surfaces of each interleaved element presses on the surfaces of its adjacent elements and all the gripping and coupling elements clamp together to increase the friction force between said first and second transmission members according to a number of contact surfaces between the interleaved elements. a rotary first transmission member (20 or 40 or 90 or 100) mounted on a rotation axis; a second transmission member (30 or 50 or 70 or 80 or 102 or 122 or 132) to be selectively coupled to and decoupled from said first transmission member; said second transmission member being wound around part of said first transmission member; said first transmission member including at least one gripping device (21 or 91 or 101 or 131) that is moveable radially with respect to said rotation axis and includes a plurality of laterally spaced and laterally displaceable gripping elements (26, 27 or 131 or 101); said second transmission member including a plurality of laterally spaced and laterally displaceable coupling elements (31 or 51 or 71 or 81) said transmission members being configured to be coupled together by said at least one gripping device and by said plurality of coupling elements on said second transmission member that are receivable in an interleaved manner between said gripping elements of said at least one gripping device as the first transmission member rotates, characterized in that: upon application of a lateral compressive force said gripping elements move toward each other and said interleaved coupling elements also move toward each other to decrease the spaces between all said interleaved elements; whereby the surfaces of each interleaved element presses on the surfaces of its adjacent elements and all the gripping and coupling elements clamp together to increase the friction force between said first and second transmission members according to a number of contact surfaces between the interleaved elements. upon application of a lateral compressive force said gripping elements move toward each other and said interleaved coupling elements also move toward each other to decrease the spaces between all said interleaved elements; whereby the surfaces of each interleaved element presses on the surfaces of its adjacent elements and all the gripping and coupling elements clamp together to increase the friction force between said first and second transmission members according to a number of contact surfaces between the interleaved elements.