EP0539273B1 - Volumetric machine with planetary movement and hypertrocoidal geometry - Google Patents

Description

The invention relates to a volumetric machine comprising a cylindrical capsulism essentially consisting of a cylindrical piston (male organ), a cylindrical capsule which surrounds it (female organ) and a third organ in rotoidal connection with the male organ. around the axis thereof, in rotoidal connection with the female member around the axis thereof, the shape of this third member imposing the parallelism of these two axes. In these machines, the cylinder defining the shape of the piston has an order of symmetry with respect to its axis equal to s _P , that of the capsule an order of symmetry equal to s _C ; s _P and s _C are chosen so that these values differ by one. In addition, the geometry of the piston and the capsule are chosen so that there is contact between these elements.
Many volumetric machines with planetary movement are known which conform to this description. We can mainly cite the machines which are described in the article PROJEKTIEREN DER ZYKLOIDENVERZAHNUNGEN HYDRAULISCHER VERDRAENGERMASHINEN published in MECHANISM AND MACHINE THEORY VOL 25 N ° 6 1990.
It will be observed that, in these machines with planetary movement, where the value of s _P is different from the unit, the axis of the cylinder defining the external shape of the piston must be confused with the axis of its rotoid connection with the third member . Likewise, in machines where the value of s _C is different from unity, the axis of the cylinder defining the interior shape of the capsule must be coincident with the axis of its rotoid connection with the third member. When s _P is equal to one, the axis of the cylinder defining the external shape of the piston can be chosen arbitrarily, provided that it is parallel to the axes of the third member. When s _C is equal to one, the axis of the cylinder defining the internal shape of the capsule can be chosen arbitrarily, provided that it is parallel to the axes of the third member.
The volumetric machines with planetary movement which are described in the article cited above are distinguished from the machines according to the invention by the geometry of the capsule and that of the piston. In fact, in these known machines, either the piston or the capsule has a director which is a shortened hypotrochoid or an epitrochoid, or a curve uniformly distant from a non-elongated hypotrochoid or an epitrochoid (that is to say ordinary or shortened). All these curves have only one or two shape parameters which can only be chosen between close limits. They do not make it possible to satisfy all the technological constraints, as is desired in modern machines.

• Les machines pour lesquelles D₁ est la directrice du piston et D₂ est la directrice de la capsule, celle-ci s'identifiant à l'enveloppe extérieure de D₁ dans le mouvement planétaire de D₁ relativement à D₂ pour lequel ${\text{R₁= s}}_{\text{P}}\text{E}$
  
  et ${\text{R₂=s}}_{\text{c}}{\text{E=(s}}_{\text{P}}\text{+1)E}$
  
  avec $\text{E =|O₁O₂|}$
  
  (famille I).
• Les machines pour lesquelles D₁ est la directrice du piston et D₂ est la directrice de la capsule, celle-ci s'identifiant à l'enveloppe extérieure de D₁ dans le mouvement planétaire de D₁ relativement à D₂ pour lequel ${\text{R₁= s}}_{\text{P}}\text{E}$
  
  et ${\text{R₂=s}}_{\text{c}}{\text{E=(s}}_{\text{P}}\text{-1)E}$
  
  avec $\text{E =|O₁O₂|}$
  
  et s_P>1 (famille II).
• Les machines pour lesquelles D₁ est la directrice de la capsule et D₂ est la directrice du piston, celle-ci s'identifiant à l'enveloppe intérieure de D₁ dans le mouvement planétaire de D₁ relativement à D₂ pour lequel ${\text{R₂= s}}_{\text{P}}\text{E}$
  
  et ${\text{R₁=s}}_{\text{c}}{\text{E=(s}}_{\text{P}}\text{-1)E}$
  
  avec $\text{E =|O₁O₂|}$
  
  et s_P>1 (famille III).
• Les machines pour lesquelles D₁ est la directrice de la capsule et D₂ est la directrice du piston, celle-ci s'identifiant à l'enveloppe intérieure de D₁ dans le mouvement planétaire de D₁ relativement à D₂ pour lequel ${\text{R₂= s}}_{\text{P}}\text{E}$
  
  et ${\text{R₁=s}}_{\text{c}}{\text{E=(s}}_{\text{P}}\text{+1)E}$
  
  avec $\text{E = |O₁O₂|}$
  
  (famille IV).

In contrast, the machines which are the subject of the invention have a geometry much richer in shape parameters and in certain cases have technological advantages which facilitate their realization. According to the invention, one of the organs, male or female, has a D₁ director which identifies with a curve uniformly distant from a closed hypertrochoid, presenting neither double point nor cusp, excluding hypertrochoids degenerated into hypotrochoids, epitrochoids or peritrochoids. It is clear that while remaining within the scope of the invention, the director D₁ can also be at zero distance from such a hypertrochoid and therefore identify with it. The definition of hypertrochoids is specified in patent FR-A-2,203,421.
The other member, male or female, of the machines which are the subject of the invention has a director D₂ which is the envelope of D₁ in a relative planetary movement defined by two circles C₁ and C₂ with respective centers and radii (O₁, R₁) and (O₂, R₂), these circles R₁ and R₂ being respectively integral with the directors D₁ and D₂ and rolling one on the other without sliding by internal contact, | O₁O₂ | specifying the center distance of the third organ. The machines according to the invention can be grouped into four families according to the nature of the member, the shape of which is defined by D₁ and according to the comparative values of the radii R₁ and R₂. A distinction should be made:

• The machines for which D₁ is the director of the piston and D₂ is the director of the capsule, the latter being identified with the outer envelope of D₁ in the planetary movement of D₁ relative to D₂ for which ${\text{R₁ = s}}_{\text{P}}\text{E}$
  
  and ${\text{R₂ = s}}_{\text{vs}}{\text{E = (s}}_{\text{P}}\text{+1) E}$
  
  with $\text{E = | WHERE |}$
  
  (family I).
• The machines for which D₁ is the director of the piston and D₂ is the director of the capsule, the latter being identified with the outer envelope of D₁ in the planetary movement of D₁ relative to D₂ for which ${\text{R₁ = s}}_{\text{P}}\text{E}$
  
  and ${\text{R₂ = s}}_{\text{vs}}{\text{E = (s}}_{\text{P}}\text{-1) E}$
  
  with $\text{E = | WHERE |}$
  
  and s _P > 1 (family II).
• The machines for which D₁ is the director of the capsule and D₂ is the director of the piston, the latter being identified with the inner envelope of D₁ in the planetary movement of D₁ relative to D₂ for which ${\text{R₂ = s}}_{\text{P}}\text{E}$
  
  and ${\text{R₁ = s}}_{\text{vs}}{\text{E = (s}}_{\text{P}}\text{-1) E}$
  
  with $\text{E = | WHERE |}$
  
  and s _P > 1 (family III).
• The machines for which D₁ is the director of the capsule and D₂ is the director of the piston, the latter being identified with the inner envelope of D₁ in the planetary movement of D₁ relative to D₂ for which ${\text{R₂ = s}}_{\text{P}}\text{E}$
  
  and ${\text{R₁ = s}}_{\text{vs}}{\text{E = (s}}_{\text{P}}\text{+1) E}$
  
  with $\text{E = | WHERE |}$
  
  (family IV).

It is clear that other machines according to the invention can be derived from machines belonging to one of the four preceding families. Indeed, one can use a director D₂ of which at least a part is identified with the envelope of D₁ in its movement relative to D₂ and of which at least a part is external to this envelope in the case of families I or II and is inside this envelope in the case of families III or IV, the different parts connecting to define a closed curve.

L'expression (1) fournit une relation entre γ et κ qui, introduite dans l'expression (2), permet la définition de Z₂ en fonction d'un seul paramètre cinématique γ ou κ. On observera qui si théoriquement, on a intérêt à rechercher l'ensemble {κ*} correspondant à une position particulière γ* de D₁, il est numériquement beaucoup plus facile de trouver l'ensemble des positions de D₁ définies par {γ**} pour lesquelles le contact s'établit en un point particulier de D₁ défini par κ**. On remarquera également que Z₂ correspond aux enveloppes intérieure et extérieure, qu'il convient de séparer ces deux enveloppes et d'utiliser l'une d'elles selon la famille de machines que l'on veut réaliser. Cette séparation peut par exemple se fonder sur la comparaison des rayons de courbure aux points de contact de D₁ et D₂.Knowing the parametric equation of the director D₁, which is written Z₁ (κ) in the complex plane O₁XY, (κ representing the kinematic parameter), we can quite easily obtain, in the complex plane O₂XY, the equation Z₂ of the envelopes of D₁ in the relative movement of D₁ with respect to D₂ defined by R₁ and R₂, by means of the two following relations, where γ represents the angle of rotation of D₁ compared to the third organ and or Z₃ is the conjugate complex number of the derivative of Z₁ with respect toκ: $$\begin{array}{c}\begin{array}{c}\begin{array}{}\text{(1)}\text{Re {Z₁Z₃ - R₁Z₃ expi (- γ)} = 0}\end{array}\\ \begin{array}{}\text{(2)}\text{Z₂ = (R₂ - R₁) expi {- γ R₁ / R₂} + Z₁ expi {γ (1-R₁ / R₂)}}\end{array}\end{array}\end{array}$$

Expression (1) provides a relationship between γ and κ which, introduced in expression (2), allows the definition of Z₂ as a function of a single kinematic parameter γ or κ. We will observe that if theoretically, it is in our interest to seek the set {κ *} corresponding to a particular position γ * of D₁, it is numerically much easier to find the set of positions of D₁ defined by {γ **} for which the contact is established at a particular point of D₁ defined by κ **. It will also be noted that Z₂ corresponds to the interior and exterior envelopes, that it is advisable to separate these two envelopes and to use one of them according to the family of machines that one wishes to produce. This separation can for example be based on the comparison of the radii of curvature at the contact points of D₁ and D₂.

• On peut immobiliser le troisième organe et rendre mobiles le piston et la capsule.
• On peut aussi immobiliser le piston et rendre mobiles la capsule et le troisième organe.
• On peut enfin, et c'est en principe la réalisation la plus simple, immobiliser la capsule et rendre mobiles le troisième organe et le piston.

Quels que soient les mouvements absolus retenus par les machines objet de l'invention, le mouvement planétaire relatif peut être réalisé par une transmission à rapport constant et notamment par un engrenage intérieur à axes parallèles, dont les roues E₁ et E₂ sont respectivement solidaires des piston et capsule et dont les rayons primitifs sont respectivement égaux à R₁ et R₂.
Lorsque l'on a recours à une transmission à rapport constant pour imposer le mouvement planétaire relatif, un jeu fonctionnel, ménagé entre la capsule et le piston, permet d'éviter le contact direct entre ces deux éléments et autorise un fonctionnement "à sec" de la machine.
Lorsque l'on accepte le contact direct entre le piston et la capsule, on peut, si la géométrie des surfaces en contact de ces deux éléments permet une conduite suffisante et si le fluide véhiculé dans la machine est suffisamment lubrifiant, supprimer la transmission à rapport constant, le mouvement planétaire relatif étant directement imposé par le contact piston-capsule. Il en résulte dans ce cas une grande simplicité de réalisation.The planetary movement of D₁ relative to D₂ can be achieved in the machines object of the invention in three different ways:

• The third member can be immobilized and the piston and the capsule made mobile.
• It is also possible to immobilize the piston and make the capsule and the third member mobile.
• Finally, and it is in principle the simplest embodiment, it is possible to immobilize the capsule and make the third member and the piston mobile.

Whatever the absolute movements retained by the machines which are the subject of the invention, the relative planetary movement can be achieved by a constant ratio transmission and in particular by an internal gear with parallel axes, the wheels E₁ and E₂ of which are respectively integral with the pistons and capsule and whose primitive radii are respectively equal to R₁ and R₂.
When a constant ratio transmission is used to impose the relative planetary movement, a functional clearance, formed between the capsule and the piston, makes it possible to avoid direct contact between these two elements and allows "dry" operation of the machine.
When accepting the direct contact between the piston and the capsule, it is possible, if the geometry of the surfaces in contact of these two elements allows sufficient conduct and if the fluid conveyed in the machine is sufficiently lubricating, to cancel the transmission to ratio constant, the relative planetary motion being directly imposed by the piston-capsule contact. In this case, this results in great simplicity of construction.

Whatever the mechanical organization of the machines which are the subject of the invention, these machines transform fluid energy into mechanical energy or vice versa.
Mechanical energy is exchanged with the outside by a tree. When the third organ is mobile, this tree identifies with it and in this case it is bent. When the third member is stationary, this shaft, of rectilinear shape, is distinct from it and it is integral with the capsule or the piston.
The fluid energy is introduced and extracted from the machine by a set of valves, lights and / or valves arranged in the capsule and / or the piston, according to the conventional techniques used in known volumetric machines and directly applicable by the skilled in the art. These fluid distribution devices can possibly be adjustable to allow a variation of the filling. Whether it is adjustable or not, the distribution of the fluid can be adapted to the nature thereof (incompressible or compressible fluid) and to the direction of energy transformation (fluid generating machine: compressor or pump and machine generator of mechanical energy: motor).
It will be observed that, in the particular case of machines where the third organ is immobile, when these machines belong to families I or II and when the director D₂ is not completely identified with the envelope of D₁ in the relative planetary movement, the part of the director D₂ which is external to this envelope can be deleted, the different parts of the director D₂ which identify with the envelope are then disjoint and the director no longer constitutes a closed curve. In these machines, a fixed casing which identifies with the third member surrounds the capsule to ensure sealing, the capsule and the piston ensuring the distribution of the fluid, by discovering and periodically closing in their absolute rotational movements, a light d at least one inlet and at least one exhaust light fixed in the machine.

dans laquelle Z₁ désigne l'affixe du point générateur de la directrice D₁, chaque point étant précisé par une valeur particulière du paramètre cinématique κ dont le domaine de variation est compris entre 0 et 2Sπ pour parcourir une seule fois la courbe, S est un nombre entier qui désigne l'ordre de symétrie de D₁ par rapport à l'origine du plan complexe et est choisi arbitrairement, expi représente la fonction exponentielle imaginaire, E et R_m sont deux longueurs choisies librement à condition que la courbe correspondante ne présente ni point double, ni point de rebroussement, ce qui limite indirectement la valeur du rapport E/R_m.A particularly interesting group of machines belonging to family I is the one whose director D₁ answers the following equation in the complex plane: $${\text{Z₁ = {(1 + S) / 2} E expi {κ (1 / S) -κ} + R}}_{\text{m}}\text{expi {κ (1 / S)} + {(1-S) / 2} E expi {κ (1 / S) + κ}}$$

in which Z₁ designates the affix of the generator point of the director D₁, each point being specified by a particular value of the kinematic parameter κ whose range of variation is between 0 and 2Sπ to traverse the curve once, S is a number integer which designates the order of symmetry of D₁ with respect to the origin of the complex plane and is chosen arbitrarily, expi represents the imaginary exponential function, E and R _m are two lengths chosen freely provided that the corresponding curve has neither point double, no cusp, which indirectly limits the value of the E / R _m ratio.

Figures 1 to 4 schematically represent a machine according to the invention. Figures 5 to 8 schematically show another machine according to the invention. These representations are the result of a computer numerical simulation.
Figures 9 and 10 show a compressor where the capsule is stationary and where the third member is a bent shaft.
Figures 11 and 12 show a machine where the third body, stationary, identifies with a housing surrounding the capsule, with which the piston and the capsule are in rotoid connections.
In the machines represented in FIGS. 9 to 12, the shape of the interior surface of the capsule and of the exterior surface of the piston correspond to the diagrams presented in FIGS. 1 to 4.

et R_m=45 mm. On distingue, sur ces deux figures la capsule (10) de directrice D₂ qui entoure le piston (11) de directrice D₁. On aperçoit nettement sur la figure 1, trois points de contact U₁,U₂,U₃ entre D₁ et D₂. La figure 3 représente la directrice D₁ (12), la figure 4 montre plusieurs positions de D₁ par rapport à la capsule, celle-ci n'étant pas figurée par souci de clarté.Figures 1 and 2 show, for two particular positions of the piston, a section, perpendicular to the axes, of a machine of family I characterized by s _P = 2, s _C = 3, E = 10 mm, R₁ = 20 mm, R₂ = 30 mm and whose director D₁ is in conformity to equation (3) with ${\text{S = s}}_{\text{P}}$

and R _m = 45 mm. We distinguish in these two figures the capsule (10) of director D₂ which surrounds the piston (11) of director D₁. We clearly see in Figure 1, three contact points U₁, U₂, U₃ between D₁ and D₂. Figure 3 shows the director D₁ (12), Figure 4 shows several positions of D₁ relative to the capsule, the latter not being shown for clarity.

On en déduit: $${\text{Re{Z₁Z₃} = -ER}}_{\text{m}}\text{sin(κ) -{(1-S²)/2}E²sin(2κ)}$$

et $${\text{Re{R₁Z₃expi(-γ)}={(1/S)-1}{(1+S)/2} R₁E sin{-κ(1/S)+κ-γ} + {(1/S)} R₁R}}_{\text{m}}\text{sin{-κ(1/S)-γ} + {(1/S)+1}{(1-S)/2} R₁E sin{-κ(1/S)-κ-γ}}$$

Si $$\begin{array}{c}\begin{array}{c}\begin{array}{}\text{(4)}\text{κ+κ(1/S)+γ = (2l+1)π avec l = 0,1,S=2}\end{array}\\ \text{sin{-κ-κ(1/S)-γ} = 0}\end{array}\end{array}$$

et, compte tenu de ce que R₁ = s_PE = SE , $${\text{Re{R₁Z₃expi(-γ)} = {(1/S)-1} {(1+S)/2} SE² sin{2κ+π} +{(1/S)} SER}}_{\text{m}}\text{sin{κ+π}}$$

ou $${\text{Re{R₁Z₃expi(-γ)} = {(1-S)/S} {(1+S)/2} SE² sin{2κ+π} +{(1/S)} SER}}_{\text{m}}\text{sin{κ+π}.}$$

The study of this machine gives the following results: $${\text{Z₃ = -i {{(1 / S) -1} {(1 + S) / 2} E expi {-κ (1 / S) + κ} + {(1 / S)} R}}_{\text{m}}\text{expi {-κ (1 / S)} + {(1 / S) +1} {(1-S) / 2} E expi {-κ (1 / S) -κ}}}$$

We can deduce: $${\text{Re {Z₁Z₃} = -ER}}_{\text{m}}\text{sin (κ) - {(1-S²) / 2} E²sin (2κ)}$$

and $${\text{Re {R₁Z₃expi (-γ)} = {(1 / S) -1} {(1 + S) / 2} R₁E sin {-κ (1 / S) + κ-γ} + {(1 / S)} R₁R}}_{\text{m}}\text{sin {-κ (1 / S) -γ} + {(1 / S) +1} {(1-S) / 2} R₁E sin {-κ (1 / S) -κ-γ}}$$

Yes $$\begin{array}{c}\begin{array}{c}\begin{array}{}\text{(4)}\text{κ + κ (1 / S) + γ = (2l + 1) π with l = 0.1, S = 2}\end{array}\\ \text{sin {-κ-κ (1 / S) -γ} = 0}\end{array}\end{array}$$

and, taking into account that R₁ = s _P E = SE, $${\text{Re {R₁Z₃expi (-γ)} = {(1 / S) -1} {(1 + S) / 2} SE² sin {2κ + π} + {(1 / S)} SER}}_{\text{m}}\text{sin {κ + π}}$$

or $${\text{Re {R₁Z₃expi (-γ)} = {(1-S) / S} {(1 + S) / 2} SE² sin {2κ + π} + {(1 / S)} SER}}_{\text{m}}\text{sin {κ + π}.}$$

: $${\text{Z₂ = E expi{-γ(S/S+1)} +{(1+S)/2} E expi{κ(1/S)-κ+γ-γ(S/S+1)} + R}}_{\text{m}}\text{expi{κ(1/S)+γ-γ(S/S+1)} +{(1-S)/2} E expi{κ(1/S)+κ+γ-γ(S/S+1)}}$$

En posant, pour simplifier l'écriture, $$\text{A={(1+S)/2} E,}$$

il vient : $${\text{Z₂={A expi(-κ)+ R}}_{\text{m}}\text{-A expi(+κ)} {expi{κ(1/S)+γ-γ(S/S+1)}}}$$

ou encore, compte tenu de (4) $${\text{Z₂={A expi(-κ)+ R}}_{\text{m}}\text{-A expi(+κ)} {expi{(1/S+1)(2l+1)π}}}$$

Le terme {Aexpi (-κ) + R_m - Aexpi (+κ)} de cette expression représente un segment de droite dirigé selon l'axe des ordonnées, passant par le point d'abscisse R_m et d'ordonnée 0. Sa longueur est égale à 4A, c'est-à-dire à (1+S)2E.
Le produit $${\text{{A expi(-κ)+ R}}_{\text{m}}\text{-A expi(+κ)} {expi{(1/S+1)(2l+1)π}}}$$

représente le même segment de droite tourné de $$\text{{-(1/S+1)(2l+1)π} avec l=0,1,S=2,}$$

c'est-à-dire de 60 degrés, 180 degrés et 300 degrés.Consequently, the relation (1) is well verified simultaneously with the relation (4).
The general expression of the relation (2) gives the expression of Z₂ which is written here taking into account (4) and what ${\text{R₂ = (s}}_{\text{P}}\text{+1) E = (S + 1) E}$

: $${\text{Z₂ = E expi {-γ (S / S + 1)} + {(1 + S) / 2} E expi {κ (1 / S) -κ + γ-γ (S / S + 1)} + R}}_{\text{m}}\text{expi {κ (1 / S) + γ-γ (S / S + 1)} + {(1-S) / 2} E expi {κ (1 / S) + κ + γ-γ (S / S + 1)}}$$

By posing, to simplify writing, $$\text{A = {(1 + S) / 2} E,}$$

he comes : $${\text{Z₂ = {A expi (-κ) + R}}_{\text{m}}\text{-A expi (+ κ)} {expi {κ (1 / S) + γ-γ (S / S + 1)}}}$$

or, taking into account (4) $${\text{Z₂ = {A expi (-κ) + R}}_{\text{m}}\text{-A expi (+ κ)} {expi {(1 / S + 1) (2l + 1) π}}}$$

The term {Aexpi (-κ) + R _m - Aexpi (+ κ)} of this expression represents a line segment directed along the ordinate axis, passing through the abscissa point R _m and ordinate 0. Sa length is equal to 4A, i.e. (1 + S) 2E.
The product $${\text{{A expi (-κ) + R}}_{\text{m}}\text{-A expi (+ κ)} {expi {(1 / S + 1) (2l + 1) π}}}$$

represents the same line segment turned from $$\text{{- (1 / S + 1) (2l + 1) π} with l = 0.1, S = 2,}$$

i.e. 60 degrees, 180 degrees, and 300 degrees.

From the previous result, obtained when the relation (1) is satisfied by the values of κ and γ compatible with the relation (4), it follows that D₂ has three line segments of length equal to (1 + S) 2E arranged at 2π / (S + 1) with respect to each other.
The connection of these three line segments is obtained for other relations between κ and γ satisfying relation (1). There correspond three arcs with variable curvature.

When relation (4) is verified, there exist, for all the angular positions of the piston defined by γ, three points of contact with the directrix defined by the three corresponding values of l and therefore of κ.
A value of κ and a value of γ verifying one of the determinations of the relation (4) define a point of contact located on one of the three line segments of D₂ and, for a particular value of γ, at each determination of the relation (4), corresponds a line segment of D₂. It follows that on the one hand the director of the capsule must identify with these three straight segments and can, outside of these segments, depart from the director D₂ provided that it is external to it. In this case, the contacts of the director of the capsule with the director D₁ of the piston are always made at three points and the relative planetary movement of the piston-capsule can be carried out directly by these contacts, without the need for recourse. to a gear materializing the wheels E₁ and E₂. This results in great ease of manufacture, since the number of constituent parts of the machine is reduced to the absolute minimum and that the machining of the capsule is extremely simple, since it is reduced to that of three planes. It will be observed that in this machine, there are permanently three working chambers into which the fluid can be introduced and from which it can escape.

Figures 5 to 8 respectively have the same meaning as that of Figures 1 to 4 {capsule (20), piston (21) and director D₁ (22) of the piston (21)}, but for a machine of family II with s _P = 3, s _C = 2 E = 10 mm, R₁ = 30 mm, R₂ = 20 mm and a directrix D₁ of the piston defined by the equation: $$\text{Z₁ = 15 expi (-2κ / 3) + 120 expi (+ κ / 3) - 5 expi (4κ / 3).}$$

The director D₂ of the corresponding capsule has a symmetry of order 2. The resolution of the relation (1) for all the relative positions piston-capsule shows that one has permanently three contacts between D₁ and its external envelope D₂. This leads to the existence of three working chambers for the fluid.

Referring to Figures 3 and 4 on the one hand, 7 and 8 on the other hand, we can still observe the following results:
FIG. 4 represents the planetary movement of a curve D₁ of order of symmetry equal to 2, represented in FIG. 3. The planetary movement is characterized by the rolling of a circumference C₁ of radius equal to 2E (with which is associated the director D₁) on a fixed circumference C₂ of radius equal to 3E. In Figure 4, we can observe the outer and inner envelopes integral with this fixed circumference C₂. These envelopes both have an order of symmetry equal to 3.

.
La machine correspondante appartient à la famille I.
Si on matérialise D₁ et son enveloppe intérieure D₂, avec R₁=2E et R₂=3E,
D₁ est la capsule
D₂ est le piston
s_C=2 s_P=3 , R₂ est bien égal à s_PE et R₁ à ${\text{s}}_{\text{C}}{\text{E=(s}}_{\text{P}}\text{-1)E}$

.
La machine correspondante appartient à la famille III.If we materialize D₁ and its outer envelope D₂, with R₁ = 2E and R₂ = 3E,
D₁ is the piston
D₂ is the capsule
s _P = 2 s _C = 3, R₁ is indeed equal to s _P E and R₂ to ${\text{s}}_{\text{VS}}{\text{E = (s}}_{\text{P}}\text{+1) E}$

.
The corresponding machine belongs to family I.
If we materialize D₁ and its inner envelope D₂, with R₁ = 2E and R₂ = 3E,
D₁ is the capsule
D₂ is the piston
s _C = 2 s _P = 3, R₂ is indeed equal to s _P E and R₁ to ${\text{s}}_{\text{VS}}{\text{E = (s}}_{\text{P}}\text{-1) E}$

.
The corresponding machine belongs to family III.

FIG. 8 represents the planetary movement of a curve D₁ of order of symmetry equal to 3, represented in FIG. 7. The planetary movement is characterized by the rolling of a circumference C₁ of radius equal to 3E (with which the director D₁ is associated) on a fixed circumference C₂ of equal radius to 2E. In Figure 8, we can distinguish the outer and inner envelopes integral with this fixed circumference C₂. These envelopes both have an order of symmetry equal to 2.

.
La machine correspondante appartient à la famille II.
Si on matérialise D₁ et son enveloppe intérieure D₂, avec R₁=3E et R₂=2E,
D₁ est la capsule
D₂ est le piston
s_C=3 s_P=2 , R₂ est bien égal à s_PE et R₁ à ${\text{s}}_{\text{C}}{\text{E=(s}}_{\text{P}}\text{+1)E}$

.
La machine correspondante appartient à la famille IV.If we materialize D₁ and its outer envelope D₂, with R₁ = 3E and R₂ = 2E,
D₁ is the piston
D₂ is the capsule
s _P = 3 s _C = 2, R₁ is indeed equal to s _P E and R₂ to ${\text{s}}_{\text{VS}}{\text{E = (s}}_{\text{P}}\text{-1) E}$

.
The corresponding machine belongs to family II.
If we materialize D₁ and its inner envelope D₂, with R₁ = 3E and R₂ = 2E,
D₁ is the capsule
D₂ is the piston
s _C = 3 s _P = 2, R₂ is indeed equal to s _P E and R₁ to ${\text{s}}_{\text{VS}}{\text{E = (s}}_{\text{P}}\text{+1) E}$

.
The corresponding machine belongs to family IV.

Figures 9 and 10 show a cross section and an axial section respectively, in a compressor where the compressed fluid is sufficiently lubricating to allow the piston-capsule couple to directly carry out the planetary movement.

In these sections, we distinguish the piston (11) of order of symmetry s _p = 2 and its director (12), the capsule (10) and its director of order of symmetry s _C = 3 consisting of three straight lines (13,14 and 15) as well as three arcs (16,17 and 18) outside the envelope of the piston between points A₁₃ B₁₃, A₁₄ B₁₄ and A₁₅ B₁₅. The third member, materialized by a bent shaft, (30) is in rotoid connection with the capsule (10) via the bearings (31 and 32) and is in rotoid connection with the piston (11) through the bearings (33 and 34). This bent shaft is driven by the pulley (35) wedged on it. The fluid is admitted into the compressor by the valves (41,42,43) located in the rear flange (101) of the capsule (10) and escapes therefrom by the valves (51,52,53) located in the part tubular (100) of the capsule (10). Controlled shutters such as (61), located in the front flange (102) of the capsule (10) allow maintenance at the intake pressure of one, two or three compressor working chambers. It is thus possible to regulate the flow in three steps and operate the compressor at zero flow without ceasing to drive it, thus avoiding the use of a clutch interposed between the bent shaft and the pulley or avoiding stopping the motor when it must continue to drive other machines.
Figure 11 is a machine which comprises a piston and a capsule, in rotoidal connection with a fixed casing; this view in the direction of the axes of the rotoid connections represents the machine without the flange located on the side of the drive.
Figure 12 is a section through the machine on a plane containing the axes of the two rotoid connections. In this section, the piston 11, the capsule 10 and the casing made up of a tubular part 130 and two flanges 230 and 330 are distinguished.
The piston 11 is, in the machine shown, in one piece with the shaft 111 whose bearings 112 and 113 materialize the rotoid connection of the piston 11 with the flanges 230 and 330 of the casing. The capsule 10 is in rotoidal connection by the plain bearing 110 with the tubular part 130 of the casing. The fluid is admitted into the machine through the light 140 connected in the flange 230 to the tube 340 and the exhaust is made through the light 150 connected to the tube 350 in the flange 330.

In the present description, the claimed shapes for the piston and the capsule as well as the planetary character of the movement are to be understood as nominal characteristics of the machines according to the invention.

Claims (8)

1. A volumetric machine comprising a cylindrical encapsulation essentially comprising a cylindrical piston (11) (male component) having with respect to its axis an order of symmetry expressed by a whole number s_p, a cylindrical capsule (10) which surrounds the said piston (female component) having with respect to its axis an order of symmetry expressed by a whole number sC and a third component rotatably connected to the male component about the axis of the said male component, rotatably connected to the female component about the axis of the said female component, the shape of this third component forcing these two axes to be parallel, the orders of symmetry s_p and s_C differing by one unit and the geometries of the piston (11) and the capsule (10) being defined so that these components are in contact CHARACTERISED IN THAT one of the male or female components has a directrix D₁ which becomes identical with a curve at a constant distance, the constant distance possibly being zero, from a closed hypertrochoid, not including the hypertrochoids transformed into hypotrochoids, peritrochoids and epitrochoids or into curves at a constant distance from these hypotrochoids, peritrochoids and epitrochoids, this hypertrochoid not having a double point nor cusp, the other component having a directrix D₂ which is the envelope of D₁ in a relating planetary movement described by two circles C₁ and C₂, with centres and radii (O₁, R₁) and (O₂, R₂) respectively, and each integral with the directrixes D₁ and D₂ and rolling on top of each other without slipping due to internal contact, |O₁O₂| determining exactly the centre distance of the third component.
2. A volumetric machine according to Claim 1, CHARACTERISED IN THAT D₁ (12) is the directrix of the piston (11), D₂ is the directrix of the capsule (10) which becomes identical with the outer envelope of D₁ in the planetary movement of D₁ with respect to D₂, defined by ${\text{R₁= S}}_{\text{p}}\text{E}$

   

   and ${\text{R₂ = S}}_{\text{c}}{\text{E = (S}}_{\text{p}}\text{+1) E}$

   

   

   where $\text{E = |O₁O₂|}$

   

   .
3. A volumetric machine according to Claim 1, CHARACTERISED IN THAT D₁ (22) is the directrix of the piston (21), D₂ is the directrix of the capsule (20) which becomes identical with the outer envelope of D₁ in the planetary movement of D₁ with respect to D₂, defined by ${\text{R₁ = S}}_{\text{p}}\text{E}$

   

   and ${\text{R₂ = S}}_{\text{c}}{\text{E = (S}}_{\text{p}}\text{-1) E}$

   

   

   where $\text{E = |O₁O₂|}$

   

   and S_p > 1.
4. A volumetric machine according to Claim 1 CHARACTERISED IN THAT D₁ (22) is the directrix of the capsule (21), D₂ is the directrix of the piston (20) which becomes identical with the inner envelope of D₁ in the planetary movement of D₁ with respect to D₂, defined by ${\text{R₂ = S}}_{\text{p}}\text{E}$

   

   and ${\text{R₁ = S}}_{\text{c}}{\text{E = (S}}_{\text{p}}\text{+1) E}$

   

   

   where $\text{E = |O₁O₂|}$

   

   .
5. A volumetric machine according to Claim 1, CHARACTERISED IN THAT the planetary movement is defined by ${\text{R₂ = S}}_{\text{p}}\text{E}$

   

   and ${\text{R₁ = S}}_{\text{c}}{\text{E = (S}}_{\text{p}}\text{-1) E}$

   

   where $\text{E = |O₁O₂|}$

   

   and S_p > 1.
6. A volumetric machine according to Claims 2 and 3, CHARACTERISED IN THAT at least one section of the directrix D₂ is outside the outer envelope of D₁ in its planetary movement with respect to D₂, and at least one other section of the directrix D₂ becomes identical with a section of this envelope, the different sections joining to describe a closed curve.
7. A volumetric machine according to Claims 4 and 5, CHARACTERISED IN THAT at least one section of the directrix D₂ is inside the inner envelope of D₁ in its planetary movement with respect to D₂, and at least one other section of the directrix D₂ becomes identical with a section of this envelope, the different sections joining to describe a closed curve.
8. A volumetric machine according to Claim 2, CHARACTERISED IN THAT the hypertrochoid corresponds, in the complex plane, to the equation: $${\text{Z₁ = { (1+S)/2} E expi {κ (1/S)-κ} + R}}_{\text{m}}\text{expi {κ (1/S)} + { (1-S)/2} E expi {κ (1/S) +κ},}$$

   

   in which, Z₁ denotes the affix of the generator point of the directrix D₁, each point being determined by a particular value of the kinematic parameter κ ranging between 0 and 2Sπ to pass through the curve once, S is a whole number which denotes the order of symmetry of the curve in relation to the origin of the complex plane and is chosen arbitrarily, expi is the imaginary exponential function, E and R_m are two lengths freely chosen provided that the corresponding curve does not show a double point nor a cusp, as this would indirectly restrict the value of the ratio E/R_m.

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