This paper addresses the fundamental issue on the conjugation for any gearing systems after tooth modification. It presents a rigorous theory on compensated conjugation for gear transmission error (TE) balance. The basic idea is that conjugation impaired by the loading condition can be compensated by modifying the transmission function. Thus, conjugation holds true after tooth modification. Because the modification is based on the universal concept of transmission rather than the tooth geometry, the proposed tooth modification method is universal rather than limited to involute or even planar gearing. A theorem about the continuity of motion and conjugate geometries is presented and proved for any desirable modification. The proposed theory is consistent with the standard manufacturing process for tooth modification. Tooth geometries and cutter geometries can be obtained after the theoretical TE function is designed. The proposed method is highlighted and demonstrated with an involute gear design, in which a convenient and practical method with a direct rack-cutter modification is presented and rigorously analyzed based on kinematics and differential geometry. Examples are presented to show the effectiveness of the methodology.

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