Planar kinematic geometry - cyclic motions

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The motion determined by fixed and moving centrodes where both centrodes are circles or one centrode is a straight line and the second is a circle is called cyclic motion. Trajectories of moving points and envelopes of moving curves generated during cyclic motion are called cycloids. According to the shape of centrode, the cyclic motions can be classified as follows:

• Cycloidal motion - the fixed centrode is a straight line, the moving centrode is a circle.

• Involute motion - the fixed centrode is a circle, the moving centrode is a straight line.

• Epicycloidal motion - both centrodes are circles, the moving centrode is rolling by its external circumference along the external circumference of the fixed centrode.

Hypocycloidal motion - both centrodes are circles, the moving centrode is rolling by its external circumference along the internal circumference of the fixed centrode.

• Pericycloidal motion - both centrodes are circles, the moving centrode is rolling by its internal circumference along the external circumference of the fixed centrode.

The following animations are presented as an example of each cyclic motion mentioned above: trajectory of moving points, envelope of moving circle and envelope of moving straight lines. Tangent lines of trajectories and points of contact between moving circle of straight line and its envelope are drawn in animations, too.

Individual figures are distinguished by the following colours:

 - fixed centrode - moving centrode - operating centre of instantaneous rotation (point of contact between fixed and moving centrodes); normal line of trajectory of moving point; normal line of trajectory of moving circle, normal line of moving circle, normal line of envelope of moving circle; normal line of moving straight line, normal line of envelope of moving straight line - moving point, trajectory of moving point, tangent line of trajectory of moving point; a branch of envelope of moving circle, point of contact between moving circle and its envelope; moving straight line, envelope of moving straight line - moving point, trajectory of moving point, tangent line of trajectory of moving point; a branch of envelope of moving circle, point of contact between moving circle and its envelope; moving straight line, envelope of moving straight line - moving point, trajectory of moving point, tangent line of trajectory of moving point; moving circle, centre of moving circle; moving straight line, envelope of moving straight line