Meck. Mack. Theory Vol. 29, No. I.pp. 25--42,1994 0094-I14X194 $6.00+ 0.00 Printed ia Great Bnutin. All rights ~ CoP~Sht ~ 1993 Pergamon Preu Lul GEOMETRY FOR TROCHOIDAL-TYPE MACHINES WITH CONJUGATE ENVELOPES J. B. SHUNG and G. R. PENNOCK School of Mechanical Engineering, Purdue University. West lafayette, IN 47907-1288, U.S.A. (Recewed 9 No~ember 1992~recewed for publication 14 December 1992) Abstrsct--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 ! conjugate envelope. Finally. the paper includes a detailed discussion of the characteristics and the relationships of the different types of trochoid and conjugate envelope. I. INTRODUCTION Trochoidal-type machines belong to the category of planetary rotation machines and offer significant advantages over types of machinery; for example, simplicity and reliability; the possibility of higher speeds; and a wide variety of applications, namely, engines, pumps, compressors and blowers. There are two major components; i.e. a rotor (or piston) and a chamber (or cylinder). If one component is referred to as the trochoid then the other is referred to as the envelope. For example, in the trochoidal-type pump, the rotor is a peritrochoid and the chamber is an outer envelope, whereas in the Wankel engine, the rotor is an inner envelope and the chamber is the peritrochoid. The inner and outer conjugate envelopes are defined as the limiting cases of the envelope for which there is no interference on the bottom and on the top of the trochoid lobe, respectively, during the entire motion [Wydra, 1986]. The conjugate envelope provides the maximum compression ratio, the lowest contact stresses, and the best geometry for sealing. The earliest trochoidal-type machinery with a conjugate envelope is believed to be the rotary steam engine invented by Cooley [Yamamoto, 198 I]. The rotor was a peritrochoid and the chamber was an outer conjugate envelope. Later, Wallinder and Skoog invented a rotary engine where the chamber was a hypotrochoid and the rotor was an inner conjugate envelope [Yamamoto, 1981]. The first gerotor (abbreviation for generated rotor) pump was invented by Hill, the profile of the rotor was a peritrochoid generated by a circular arc instead of a point [Beard et al., 1987]. Wankel invented the first rotary internal combustion engine where the chamber was a peritrochoid and the rotor was an inner conjugate envelope [Yamamoto, 1981]. Colbourne [1974] defined eight types of conjugate envelope for each trochoid where the number of envelope lobes was either one more, or one less, than the number of trochoid lobes. He generated the eight types of envelope, i.e. two type ! inner, two type ! outer, two type 2 inner and two type 2 outer envelopes, by a numerical method. In spite of the important advantages of a conjugate envelope, the existing literature only contains closed-form equations of the geometry for two types of conjugate envelope [Ansdale and Lockley, 1969]. The authors demonstrated the value of the existing closed-form equations in the design of a Wankel rotary engine. Then Robinson and Lyon [1976] were able to modify the equations by introducing a constant which accounts for the space that is required in the sealing design. The closed-form equations again proved to be useful in a recent design of a trochoidal-type gas compressor [Hoffmann, 1985]. Since the existing closed-form equations have proved to be important in the analysis and design of planetary rotation machines, the focus of this paper is (i) to derive closed-form parametric equations for nine types of conjugate envelope; and (ii) to study the geometric characteristics and 25