Geometry for trochoidal-type machines with conjugate envelopes

Abstract This paper presents unified and compact equations describing the geometry and the geometric properties of the different types of trochoid. For the first time, the double-generation theorem is expressed in an explicit manner and a complete classification of all trochoids is documented. Unified and compact equations describing the geometric properties of a conjugate envelope are also presented in the paper. A new type of conjugate envelope for a given trochoid is discovered. An important contribution to the existing literature on trochoidal-type machines is the derivation of closed-form parametric equations for nine types of conjugate envelope. These equations provide significant geometrical insight into the design and analysis of trochoidal-type machines. The paper also presents the necessary and sufficient conditions for a closed type 1 conjugate envelope. Finally, the paper includes a detailed discussion of the characteristics and the relationships of the different types of trochoid and conjugate envelope.