Applying envelope theory and deviation function to tooth profile design

In this work, a mathematical model of an elastic conjugate element is presented, using envelope theory and a deviation function. A basic deformation curve in a two-dimensional system is defined, and determined by the maximum deformation of the originally generated curve. The set of these points of maximum deformation in the moving coordinate frame determines the deviation of the originally generated curve. The degree of deviation is described by a deviation function. A deviation function is chosen to reshape the originally generated curve. The reshaped curve is called a basic deformation curve. If the basic deformation curve is known, an envelope associated with them can be determined. The results are applied to rotary gear pump design. The reshaped curve is superior to the originally generated curve. The reshaped curve has lower deformation, von-Mises stress, and greater bending strength than the originally generated curve. This investigation indicates that envelope theory and deviation function can avoid singular points of conjugate elements, using an example.