EP1546560B1 - Closed system rotary machine - Google Patents

Abstract

A closed system rotary machine has an inner shaped element (1) and an outer shaped element (2) which define therebetween cavities or capsules having a variable volume (V1, V9). The contact points which define the capsules (C1, C9) are disposed along lines of action (CA1, CA2, CA3) which are concurrent at junction points BN and BM, where the cavities begin and end respectively. The contacts (C2) at points located on the tangent (T) common to both pitch circles (6, 7) are osculating elements with a shared centre of curvature which is situated at the rolling point (R) of the pitch circles (6, 7). The invention can be used to ensure that the capsules form and disappear very gradually and to facilitate the distribution of the capsules when they are forming and disappearing in order to increase the leak paths.

Description

The present invention relates to a rotary machine with capsulism.

"Capsulism machine" means a machine in which two profiled members have annular profiles which mesh with each other by defining between them chambers - or capsules - with variable volume.

The invention is more particularly concerned with machines in which one of the profiles is internal to the other, one being m-lobed and the other (m-1) -lobed, where the integer m is greater or equal to 2.

A "m-lobed" profile is called an annular profile defined by a pattern forming a lobe dome and a lobe hollow, this pattern repeating itself around the center of a pitch circle associated with this profile.

A (m-1) -lobed profile is an annular profile defined by a pattern forming a lobe dome and a lobe hollow, this pattern repeating itself (m-1) times around the center of a pitch circle associated with this profile. .

The profiles cooperate with each other by a kind of meshing in which their respective primary circles roll over each other at a running point which is fixed relative to a connecting member with respect to which the two profiled members, each along an axis passing through the center of its original circle.

Capsulism machines may for example be hydraulic motors, hydraulic pumps, compressors or expansion machines.

EP-A-0870926 discloses a capsulism machine of the type called "gerotor", that is to say in which the inner profile member is (m-1) -lobed. The geometry of this machine is classic in itself. The document relates more particularly to the realization of a determined game between the profiles.

EP-539273-B1 discloses various machines with capsulism, in particular machines with two lobes on the inner profile and three on the outer profile, and conversely machines with three lobes on the inner profile and two lobes only on the outer profile.

US-A-1 892 217 discloses the Sparrow pump. Instead of having cylindrical profiles, this gerotor type machine has helical profiled members with a total helix angle of several turns. The capsules are formed at one axial end of the profiled members and are transported without volume variation to the other end, where they disappear. Two remarkable results are obtained: The distribution is simplified to the extreme since it suffices that the capsules open freely on the intake at one end and the discharge at the other end. And on the other hand, the flow is strictly constant.

Numerous documents such as US-A-6 106 250, DE 42 04 186 A1, EP 0 094 379 B1, DE 44 25 429 A1, EP 0 799 966 A2, describe machines with a Wankel geometry, it is that is to say with a generally triangular rotor with curved faces effecting a planetary movement in a bi-lobed stator.

WO 93/08402 discloses improvements to the Moineau pump.

In the prior art the profiles are often conjugated only in an approximate manner. Elastic sealing members are provided to compensate for the approximations of the conjugation. For example, in the Moineau pump (US-A-1 892 217), the inner lining of the outer profiled member is resilient. In most Wankel type machines, retractable segments are provided at the ends of the triangular rotor and sometimes also at the vertices of the lobes of the outer profiled member. Even in the best known machines, the leakage paths between successive capsules are relatively short and there are problems to switch a capsule from the intake to the discharge.

The object of the present invention is to seek an optimization with regard to the quality of the contacts between the profiles, the switching between the suction and the delivery by the distribution, and the progressivity of the birth and the disappearance of each capsule.

It has more particularly been found according to the invention a family of geometries, and associated determination methods, by which the profiles are in contact with the osculator at the stages of birth and the disappearance of a capsule. By osculator contact means a point of contact where the curvatures of the two profiles are continuous, equal and in the same direction. At the birth of a capsule, the osculating contact splits into two contacts between which the capsule is formed. At the disappearance of a capsule, two distinct contacts come closer and closer to becoming a single osculator contact, then simple.

• deux organes profilés, intérieur et respectivement extérieur, qui portent un profil annulaire intérieur et respectivement un profil annulaire extérieur,
• un organe de liaison lié de façon rotative avec chacun des deux organes profilés selon un axe de rotation respectif,

et dans laquelle :

• les profils sont l'un m-lobé, l'autre (m-1)-lobé, et sont définis autour de l'axe de rotation de leur organe profilé respectif par m et respectivement (m-1) motif(s) comprenant un arc de dôme de lobe et un arc de creux de lobe,
• chaque profil est l'enveloppe de l'autre lors de rotations relatives des organes profilés autour de leur axe de rotation respectif avec engrènement de leurs profils qui définissent entre eux les contours de capsules, et roulement sans glissement entre deux cercles primitifs centrés sur les axes de rotation respectifs,

est caractérisé en ce que dans les positions relatives des organes profilés pour lesquelles un point de contact entre les profils se trouve sur la tangente aux deux cercles primitifs en leur point de roulement mutuel, les organes profilés présentent audit point de contact des courbures continues égales et de même sens ayant pour centre commun ledit point de roulement.According to the invention, the capsulism machine comprising:

• two profiled members, inside and outside, which carry an inner annular profile and respectively an outer annular profile,
• a connecting member rotatably connected to each of the two profiled members according to a respective axis of rotation,

and wherein:

• the profiles are one m-lobed, the other (m-1) -lobed, and are defined around the axis of rotation of their respective profiled member by m and respectively (m-1) pattern (s) comprising a lobe dome arch and a lobe hollow arch,
• each profile is the envelope of the other during relative rotations of the profiled members around their respective axis of rotation with meshing of their profiles which define between them the contours of capsules, and rolling without sliding between two primitive circles centered on the respective axes of rotation,

is characterized in that in the relative positions of the profiled members for which a point of contact between the profiles is on the tangent to the two primitive circles in their mutual point of rotation, the profiled members have at said point of contact equal continuous curvatures and of the same meaning having as common center said running point.

• des points M d'un premier des deux arcs du profil m-lobé étant définis par deux fonctions ρ(δ) et σ(δ) reliant les paramètres ρ et σ au paramètre δ considéré comme une coordonnée sur l'arc et qui sont :
  • ρ : mesurée le long de la normale à l'arc au point M, la distance entre le point M et le milieu N entre les deux points d'intersection P et D, proximal et respectivement distal, de ladite normale avec le cercle primitif de centre O du profil m-lobé, et de rayon supposé égal à 1, le point d'intersection proximal P étant situé entre le point M de l'arc donné et le point d'intersection distal D,
  • δ : demi-distance angulaire entre D et P par rapport au centre O, mesurée dans le sens direct
  • σ : angle polaire du point d'intersection proximal P par rapport à O, diminué de δ,
  • les fonctions ρ(δ) et σ(δ) ayant un domaine de définition allant de δ=0 à δ=π,
• deux arcs du motif du profil (m-1)-lobé sont un arc conjugué proximal et un arc conjugué distal définis ci-après dans un repère cartésien ayant pour origine le centre O du cercle primitif associé au profil m-lobé :
  • a) arc conjugué proximal : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
    $$y_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)-\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
  • b) arc conjugué distal : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
    $$y_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)+\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    

Preferably, the capsulism machine is characterized in that

• points M of a first of two arcs of the m-lobed profile being defined by two functions ρ (δ) and σ (δ) connecting the parameters ρ and σ to the parameter δ considered as a coordinate on the arc and which are:
  • ρ: measured along the arc normal at the point M, the distance between the point M and the middle N between the two points of intersection P and D, proximal and respectively distal, of said normal with the pitch circle of center O of the m-lobed profile, and of radius assumed equal to 1, the proximal intersection point P being situated between the point M of the given arc and the distal intersection point D,
  • δ: half-angular distance between D and P with respect to the center O, measured in the forward direction
  • σ: polar angle of the proximal intersection point P with respect to O, minus δ,
  • the functions ρ (δ) and σ (δ) having a domain of definition ranging from δ = 0 to δ = π,
• two arcs of the profile pattern (m-1) -lobed are a proximal conjugate arc and a distal conjugate arc defined below in a Cartesian coordinate system originating from the center O of the pitch circle associated with the m-lobed profile:
  • a) proximal conjugate arc: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
    $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)-\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
  • b) Distal Conjugate Bow: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    
    $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)+\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
    

If one refers to the mathematical complexity associated with the design of the machines with capsulism, the solution proposed according to the invention is remarkably simple.

We can choose a first arc of one of the profiles and a primitive circle for this profile, then we mathematically define this arc in the very particular parameterization that has been imagined according to the invention, by establishing the two functions ρ (δ) and σ (δ). This arc, initially chosen, is called a "given arc".

Then, by applying the formulas according to the invention, the proximal conjugate arc and the distal conjugate arc are obtained directly by their Cartesian coordinates originating from the center O of the pitch circle associated with the given arc. The conjugate profile of the given arc is obtained by concatenation of the proximal conjugate arc and the distal conjugate arch. The concatenation means that the two arcs, taken each in its whole extent corresponding to a variation of δ over the interval [0, π], are connected end to end by the points where δ = 0. The formulas automatically realize that the two arcs, proximal and distal, have not only the same tangent but also the curvature at their point of connection and this curvature is also the same as that at a corresponding end of the given arc. The normal to the conjugate profile at the point of connection is tangent to the respective primitive circles of the selected arc and the conjugate profile at the point of rotation of these circles one over the other. Since the radius of the pitch circle of the given arc has been chosen arbitrarily equal to 1, the radius of the pitch circle of the conjugate profile is equal to (m-1) / m. The primitive circle of the conjugate profile is thus determined. The complete conjugate profile is then obtained concatenating (m-1) times the pattern consisting of the proximal conjugate arc and the distal conjugate arc according to (m-2) rotations of angle 2π / (m-1) around the center O 'of the pitch circle of the profile conjugate.

For the second arc of the m-lobed profile, or arc complementary to the given arc, two cases are possible according to the geometry chosen for the given arc. According to the invention, these two cases can be distinguished from the value of the derivative ρ 'of the function ρ with respect to its variable δ at the points 0 and π.

$$-\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{<}\mathrm{\rho ʹ}\left(\pi \right)<\mathrm{0}$$

le profil m-lobé est alors intérieur au profil (m-1)-lobé, et
le motif m-lobé est complété par un arc complémentaire proximal défini par ses coordonnées dans ledit repère cartésien : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{mcos}\left(\mathrm{\delta }\right)\cos \left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

$$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)-\mathrm{mcos}\left(\mathrm{\delta }\right)\sin \left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

In a first case, the derivative ρ 'with respect to δ for δ = 0 and δ = π satisfies the following strict inequalities: $$\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{>}\mathrm{\rho \text{'}}\left(\mathrm{0}\right)\mathrm{>}\mathrm{0}$$

$$-\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{<}\mathrm{\rho \text{'}}\left(\pi \right)<\mathrm{0}$$

the m-lobed profile is then inside the profile (m-1) -lobed, and
the m-lobed pattern is completed by a proximal complementary arc defined by its coordinates in said Cartesian coordinate system: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

$${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)-\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

A first class of machines according to the invention is thus produced in which the inner profile has one lobe more than the outer profile.

For this first class of machines, the two conjugate arcs, proximal and respectively distal, defined by the formulas according to the invention, are placed radially outside the given arc, and the complementary arc of the given arc completes the m-lobed profile within the conjugate (m-1) -lobed profile.

$$\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{>}\mathrm{\rho ʹ}\left(\pi \right)>\mathrm{0}$$

In a second case, the derivative ρ 'with respect to δ for δ = 0 and δ = π satisfies the following strict inequalities: $$-\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{<}\mathrm{\rho \text{'}}\left(0\right)<\mathrm{0}$$

$$\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{>}\mathrm{\rho \text{'}}\left(\pi \right)>\mathrm{0}$$

$$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)+\mathrm{mcos}\left(\mathrm{\delta }\right)\sin \left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

The m-lobed profile is exterior to the profile (m-1) -lobed; and
the m-lobed pattern is completed by a distal complementary arc defined by the next set of Cartesian coordinates around the center O: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

$${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)+\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$

We then have a second class of machines in which the conjugated (m-1) -lobed profile is automatically defined as lying radially inside the m-lobed profile to which the given arc belongs.

The preceding formulas, whether relating to the first or the second class of machines, do not require that the given arc has an axis of symmetry.

If the given arc does not have an axis of symmetry, machines are made in which the processes of growth and decay of the capsules are not symmetrical to each other.

Other features and advantages of the invention will emerge from the description below, relating to non-limiting examples.

• la figure 1 est une vue de face des organes profilés, montrant certaines particularités géométriques d'une machine de la première classe selon l'invention ;
• les figures 2A à 2F sont des vues analogues à la figure 1, mais à plus petite échelle, et montrant six états successifs de la machine de la figure 1 ;
• la figure 3 est une vue analogue à la figure 1 mais relative à une machine de seconde classe ;
• les figures 4A à 4F sont des vues analogues à la figure 3, mais à plus petite échelle, et montrant six états successifs de la machine ;
• la figure 5 est une construction géométrique illustrant la détermination des paramètres des profils selon l'invention ;
• les figures 6A, 6B et 6C montrent le détail, à grande échelle, du passage des profils par l'osculation, dans l'exemple de la figure 1, la figure 6B étant celle relative à l'osculation, les figures 6A et 6C étant décalées d'une rotation de trois degrés du profil intérieur dans un sens et dans l'autre ;
• les figures 7A et 7B montrent, en deux états différents, une machine de la première classe selon l'invention avec profil intérieur bi-lobé ;
• les figures 8A et 8B montrent, en deux états différents, une machine de la première classe selon l'invention avec profil intérieur tri-lobé ;
• les figures 9A et 9B montrent, en deux états différents, une machine de la première classe selon l'invention avec profil intérieur octo-lobé ;
• les figures 10A à 10I représentent neuf géométries différentes pour une machine de la première classe selon l'invention, à profil intérieur quadri-lobé ;
• les figures 11A, 11B et 11C représentent trois géométries différentes pour une machine de la première classe selon l'invention à profil intérieur penta-lobé ;
• la figure 12 est une vue de la machine de la figure 11B à échelle agrandie, avec schématisation de certains moyens de distribution ;
• la figure 12A est une vue de détail montrant une variante pour la distribution dans la réalisation de la figure 12 ;
• la figure 13 est une vue analogue à la figure 12 mais relative à la machine de la figure 1 ;
• la figure 14 est une vue schématique en perspective d'une machine dont les organes profilés sont hélicoides avec des profils successifs selon la figure 1;
• la figure 15 est une demie-vue en coupe axiale, schématique, d'une machine selon l'invention ;
• la figure 16 est une vue en coupe axiale partielle d'une machine selon l'invention, à cylindrée variable ;
• les figures 17A et 17B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur mono-lobé ;
• les figures 18A et 18B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur di-lobé ;
• les figures 19A et 19B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur tri-lobé ;
• les figures 20A et 20B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur quadri-lobé ;
• les figures 21A et 21B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur penta-lobé ;
• les figures 22A et 22B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur hepta-lobé ;
• les figures 23A et 23B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur tri-lobé dans une géométrie différente de celle des figures 19A et 19B ;
• les figures 24A et 24B sont analogues aux figures 23A et 23B respectivement, mais dans une géométrie encore différente ;
• les figures 25A et 25B sont analogues aux figures 23A et 23B respectivement, mais dans une géométrie encore différente ;
• les figures 26A et 26B représentent dans deux états différents une machine de la deuxième classe selon l'invention, à profil intérieur di-lobé mais dans une géométrie différente de celle des figures 18A et 18B, plus particulièrement appropriée pour la réalisation d'un compresseur ;
• les figures 27A et 27B sont analogues aux figures 26A et 26B, mais avec des profils asymétriques ;
• les figures 28A à 28F représentent très schématiquement dans six états différents un premier mode de réalisation d'une machine gigogne selon l'invention, avec un organe profilé intermédiaire di-lobé monté entre deux profils tri-lobés ; et
• les figures 29A à 29F représentent très schématiquement dans six états différents un deuxième mode de réalisation d'une machine gigogne selon l'invention, avec un organe profilé intermédiaire tri-lobé monté entre deux profils di-lobés ;

In the accompanying drawings:

• Figure 1 is a front view of the profiled members, showing certain geometric features of a machine of the first class according to the invention;
• Figures 2A to 2F are views similar to Figure 1, but on a smaller scale, and showing six successive states of the machine of Figure 1;
• Figure 3 is a view similar to Figure 1 but relating to a machine second class;
• Figures 4A to 4F are views similar to Figure 3, but on a smaller scale, and showing six successive states of the machine;
• FIG. 5 is a geometric construction illustrating the determination of the parameters of the profiles according to the invention;
• FIGS. 6A, 6B and 6C show the large-scale detail of the passage of the profiles by the oscillation, in the example of FIG. 1, FIG. 6B being that relating to the oscillation, FIGS. 6A and 6C being offset by a three-degree rotation of the inner profile in one direction and the other;
• FIGS. 7A and 7B show, in two different states, a machine of the first class according to the invention with a bi-lobed inner profile;
• FIGS. 8A and 8B show, in two different states, a machine of the first class according to the invention with tri-lobed inner profile;
• Figures 9A and 9B show, in two different states, a machine of the first class according to the invention with octo-lobed inner profile;
• FIGS. 10A to 10I represent nine different geometries for a machine of the first class according to the invention, with an inner four-lobed profile;
• FIGS. 11A, 11B and 11C show three different geometries for a machine of the first class according to the invention with penta-lobed inner profile;
• Figure 12 is a view of the machine of Figure 11B on an enlarged scale, with schematization of certain distribution means;
• Fig. 12A is a detail view showing an alternative for dispensing in the embodiment of Fig. 12;
• Figure 13 is a view similar to Figure 12 but relating to the machine of Figure 1;
• FIG. 14 is a schematic perspective view of a machine whose members profiles are helicoid with successive profiles according to Figure 1;
• Figure 15 is a half-view in axial section, schematic, of a machine according to the invention;
• Figure 16 is a partial axial sectional view of a machine according to the invention, variable displacement;
• FIGS. 17A and 17B show, in two different states, a machine of the second class according to the invention, with an inner lobed profile;
• FIGS. 18A and 18B show, in two different states, a machine of the second class according to the invention, with a di-lobed inner profile;
• Figures 19A and 19B show in two different states a machine of the second class according to the invention, tri-lobed inner profile;
• FIGS. 20A and 20B show, in two different states, a machine of the second class according to the invention, with an inner four-lobed profile;
• FIGS. 21A and 21B show, in two different states, a machine of the second class according to the invention, with a penta-lobed inner profile;
• FIGS. 22A and 22B show, in two different states, a machine of the second class according to the invention, with a hepta-lobed inner profile;
• FIGS. 23A and 23B show, in two different states, a machine of the second class according to the invention, having a tri-lobed inner profile in a geometry different from that of FIGS. 19A and 19B;
• Figures 24A and 24B are similar to Figures 23A and 23B respectively, but in a still different geometry;
• Figures 25A and 25B are similar to Figures 23A and 23B respectively, but in a still different geometry;
• FIGS. 26A and 26B show, in two different states, a machine of the second class according to the invention, with a di-lobed inner profile but in a geometry different from that of FIGS. 18A and 18B, more particularly suitable for producing a compressor ;
• Figs. 27A and 27B are similar to Figs. 26A and 26B, but with asymmetric profiles;
• FIGS. 28A to 28F show very diagrammatically in six different states a first embodiment of a nesting machine according to the invention, with a di-lobed intermediate profile member mounted between two tri-lobed profiles; and
• 29A to 29F very schematically show in six different states a second embodiment of a nesting machine according to the invention, with a tri-lobed intermediate profile member mounted between two di-lobed profiles;

In the example shown in FIG. 1, the machine comprises an inner profiled member 1 and an outer profiled member 2 which surrounds the inner profiled member 1.

The inner profiled member 1 has on its outer periphery a lobed profile 3 and the outer profiled member 2 has on its inner periphery a lobed profile 4 which surrounds the lobed profile 3 of the inner profiled member 1.

One of the profiles has one lobe more than the other. In the example of Figure 1, which corresponds to what is called in the context of the invention a machine of the first class, it is the inner profile 3 which has one lobe more than the outer profile 4. it is said that the inner profile 3 is m-lobed and that the outer profile 4 is (m-1) -lobed.

In the example of Figure 1, m = 6, so that the inner profile 3 is hexa-lobed and the profile 4 of the outer profiled member 2 is penta-lobed.

Each profile 3, 4 has rotational symmetry around the origin of the primitive circle associated with it and the order of this symmetry is the number of its lobes.

Thus, the profile 3 of the inner member 1 has a symmetry of order 6 around a center O, and the profile 4 of the outer profiled member 2 has a symmetry of order 5 around a center O ' .

There is between the centers O and O 'a distance 1 / m along an Ox axis.

Each lobe is defined by a respective pattern, the profile 3 or 4 being defined by plotting m times or respectively (m-1) times its respective pattern by rotation of 2π / m or respectively 2π / (m-1) around the center of symmetry O or respectively O '.

Each of the profiles 3, 4, has a pitch circle 6, 7, of center O and respectively O '. The rays of the primitive circles are proportional to the number of lobes of the profile with which they are respectively associated, so that they are tangent to each other at a point R located on the Ox axis.

Each pattern consists of a "lobe dome" and a "lobe hollow". A "lobe dome" is a protruding part, so a part radially away from the center when it comes to the inner profile and a part radially close to the center when it comes to the outer profile. Conversely, a "lobe hollow" is a generally concave part, so close to the center when it comes to the inner profile and away from the center when it comes to external profile. The "peak of lobe" is the culmination of a lobe dome and "lobe bottom" the deepest point of a lobe hollow.

In the example shown, the profiles have mirror symmetry with respect to rays passing through the lobes and lobes, but this symmetry is not essential to the meaning of the invention, as will be seen later. .

The m-lobed profiled member 1 is articulated to a connecting member, not shown in Figure 1, along an axis of rotation coinciding with the center O. Similarly, the member section (m-1) -lobé 2 is articulated to the connecting member along an axis of rotation coinciding with the center O 'of its original circle.

In operation, the two profiled members perform relative to the connecting member a rotation about their respective axis of rotation O, O ', so that the two primitive circles 6, 7 roll one on the other at point R which remains motionless with respect to the connecting member. Therefore, the Ox mark, Oy is stationary relative to the connecting member, as are the centers O and O '. Furthermore, the description made up to now also implies that the m-lobed profiled member 1 executes (m-1) / m revolution when the profiled member (m-1) -lobed 2 performs a complete revolution.

During this combined movement of the two profiled members 1 and 2, each lobe dome of each profile 3 or 4 is in contact with the other profile. In a region to the right of FIG. 1 and more precisely radially beyond a common tangent T to the two primitive circles 6 and 7 at their mutual running point R, each lobe dome of one of the profiles forms a single contact with a lobe dome of the other profile. Such a single contact C ₁ is shown in particular. On the other side of the common tangent T, each lobe dome of one of the profiles is in contact with a lobe recess of the other profile. Thus, contacts C ₃ , C ₅ , C ₇ and C _{9 are seen} between a dome of the m-lobed profile and a hollow of the (m-1) -lobed profile, which alternate with contacts C ₄ , C ₆ and C _8. between a dome of the profile (m-1) -lobed and a hollow of the m-lobed profile.

Action curves are called the trajectories of the contact points with respect to the connecting member represented by the reference Oxy. In the region to the right of the common tangent T, there is a single AC ₁ action curve whose ends are points B _N and B _M located on the tangent T. On the other side of the tangent T, there are two AC ₂ and AC ₃ action curves that correspond to the trajectory of the contact points formed by the domes of the m-lobed profile 3, and respectively by the contact points. formed by the domes of the (m-1) -lobed profile 4. The ends of the two AC ₂ and CA ₃ action curves are also constituted by the points B _N and B _M , which will be called bifurcation points of the curves. Action.

In the particular situation shown in FIG. 1, one of the points of contact, denoted by C ₂ , coincides with the bifurcation point B _N. This point of contact marks the boundary between a hollow and a dome on a slope of the pattern of each of the two profiles. In another situation, shown in FIG. 2C, a point of contact coincides with the bifurcation point B _M and marks the limit between a hollow and a dome on the other side of the pattern of each of the two profiles.

According to an important feature of the present invention, the profiles, determined in a manner to be described later, define an osculator contact between the two profiles when the contact point is made in B _N or B _M. This means that the profiles have at their point of contact located in B _N or B _M not only a common tangent, but also have continuous curvatures, equal and in the same direction.

In addition, the center of curvature common to the two profiles in their oscillation coincides with the rolling point R, so that their radius of curvature is equal to the distance between R and B _N , or respectively B _M. This oscillation ensures between the two profiles a contact which is of excellent quality.

When the profiled member 1 rotates about its center O in the direction indicated by the arrow F, the contact such as C _1, the following CA action curve ₁ until it comes to coincide with the branch point B _N to form the aforementioned oscillation. From there, the contact splits into two distinct contacts each following one of the two AC ₂ and AC ₃ action curves. Then these two distinct contacts come again to merge into an osculating contact at the point of bifurcation B _M.

Capsules - or chambers - are defined between the two profiles 3 and 4 and between the contact points successive. In the situation shown in Figure 1, a capsule is being born at the point of contact C ₂ . During the rotation of the inner profiled member 1 and the corresponding rotation of the outer profiled member 2, the capsule being born at the bifurcation point B _N will successively form the capsules V ₁ , V ₂ , ... , V ₉ . The capsules V ₁ to V ₄ are in the volume growth phase whereas the capsules V ₅ to V ₉ are in a phase of volume decrease. The growth phase extends over almost a complete turn, the decay phase also, so that the complete cycle extends over a little less than two turns. If the machine is a hydraulic motor, the hydraulic fluid is at high pressure in the capsules V ₁ to V ₄ in the growth phase, and at low pressure in the capsules V ₅ to V ₉ in the decay phase. The capsules in the growth phase and subjected to pressure alternate with the capsules in the decay phase and which are not subjected to pressure. If the hydraulic machine operates as a pump, the same alternation is observed except that it is the capsules in the decay phase that are subjected to pressure and the capsules in the growth phase that are in the process of admission of the fluid to be pumped.

This results in two consequences. On the one hand the radial load on the bearings of the machine is low. On the other hand, there is in each contact point self-lubrication due to leaks between high pressure and low pressure. This self-lubrication should in particular promote the start of the machine, without gluing effect.

Moreover, the osculative contact at the birth and the death of the capsules at bifurcations B _N and B _M, respectively, has the consequence that each capsule is born and dies on a relatively large contact surface and secondly with a very slow growth of its volume. These two circumstances favor the creation of appropriately sized orifices to start feeding and to terminate the delivery of each capsule, at birth and respectively at death, as will be seen later.

Figures 2A to 2F show six successive angular positions of the two profiled members 1 and 2 of the machine of Figure 1, from the situation shown in Figure 1, which is also that of Figure 2A. The situation shown in FIG. 2F corresponds to the passage of capsule V4 by its maximum volume. These views make it possible in particular to follow the evolution of the capsule which is formed at the point B _N in FIG. 2A. We also see how the capsule V ₉ of Figure 2A dies at the point of branching B _M in Figure 2C.

The example of Figure 3 will be described only for its differences from that of Figure 1.

The m-lobed profile 13 is now external to the profile (m-1) -lobed 14, and belongs to a profiled member 11 which is external and surrounds the profiled member 12 bearing the profile (m-1) -lobed 14.

This time, there are two CB ₂ and CB ₃ action curves radially beyond the R rolling point and a single CB ₁ action curve on the other side of the tangent T. The action curves are concurrent at bifurcation points B _N and B _M situated on the common tangent T as before, except that the bifurcation B _N corresponding to the birth of the capsules is now located upstream relative to the direction of rotation F taken as an example, with respect to the bifurcation B _M corresponding to the death of the capsules. Beyond the point B _M , the capsules V ₂ , V ₃ and V ₄ are all growing then the capsules V ₅ , V ₆ and V ₇ are decreasing while it is being born by osculation at point B _N in the situation represented a new capsule growing. There is therefore alternation of capsules growing and decreasing only radially beyond the tangent T. The contact points are less numerous than in the first class machine of Figures 1 and 2A to 2F.

FIGS. 4A to 4F represent six successive states of the machine of FIG. 3, from the situation shown in Figure 3, which is also that of Figure 4A.

In the situation shown in FIG. 4F, the capsule V ₄ has reached a position where it is symmetrical with respect to the axis Ox so that the direction of variation of its volume is changing. This is why it is to this figure that has also been shown the inlet ports 8 and discharge 9 made through a flange which, moreover, laterally closes the capsules. The capsule V ₄ communicates neither with the light 8 nor with the light 9. The capsules in the growth phase communicate with the light 8 which extends to the point of rear contact C ₄ of the capsule V ₄ . The capsules in decay phase communicate with the discharge light 9 which starts from the point of contact before C ₅ of the capsule V ₄ . The flange (s) in which (or) are defined the lights 8, 9, are integral with the connecting member symbolized by the Oxy mark.

We will now describe with reference to Figure 5, the particular parameterization for implementing the geometric profile definitions of the invention.

We consider in the Euclidean plane the circle of center O and radius 1, intended to constitute the primitive circle of the m-lobed profile. The arc M ₀ M _π is chosen arbitrarily, which in the example of FIG. 5 is represented identical to the dome of a lobe of the profile 3, including as regards its distance and its orientation with respect to the center O, and a ray from this center. By the expression "one chooses arbitrarily", one does not mean that any bow can agree, and one will give further necessary conditions that must verify this choice. Outside of the types of arc to be excluded, one can also choose the shape and the dimension of the arc, as well as its position with respect to the center O according to desiderata on the sought geometry, taking into account for example different examples of geometry shown and described later. We call "arc given" the arc M ₀ M _π , and we call M any point of the given arc. One of the characteristics that the given bow must present is that its normals N ₀ and N _π at the ends M ₀ and M _π are tangents at two different points of the pitch circle 6.

We call P and D the two intersections of the normal to the arc in M with the pitch circle 6, the point P being situated between M and D. We also call N the middle of the segment PD. We call 2δ the angle DOP, measured between 0 and 2π in the forward direction, so that δ is between 0 and π. We call σ the polar angle of P decreased by δ, which is also the polar angle of D increased by δ. We observe that for δ <π / 2, σ is the polar angle of N and that for δ> π / 2, σ is the polar angle of the symmetrical point of N with respect to the origin 0.

Finally, ρ is the distance MN counted positively.

The quantities (δ, σ, ρ) are defined univocally by the point M. Conversely, the point M is univocally defined by these quantities: we build the half-line of origin O and of polar angle σ, then the points P and D by taking the angles ± δ from this half-line. The point N is the middle of the PD segment and we build M by carrying the length MN = ρ on the line PD on the P side.

We choose the given arc as a differentiable arc on which the angle δ is a coordinate between 0 and π. This means that when the point M traverses this arc, the angle δ associated with it takes once and only one each value between 0 and π. We are therefore interested in arcs whose normal sweeps regularly (from a tangent N ₀ to a tangent N _π ) the primitive circle, when we go from the origin to the end. These arcs form two classes according to the relative direction of the course and the scan, and these two classes are associated with the two aforementioned classes of conjugated profiles and consequently of machines.

By choosing δ as a parameter along the arc, the arc is characterized by the two functions ρ (δ) and σ (δ). These two functions are not independent; they are linked by the following relation between their derivatives ρ '(δ) and σ' (δ) with respect to δ: $$\mathrm{\sigma \text{'}}\left(\mathrm{\delta }\right)\cos \left(\mathrm{\delta }\right)\mathrm{=}\mathrm{\rho \text{'}}\left(\mathrm{\delta }\right)$$

cette intégration étant faite de τ = δ₀ à τ = δ, et où τ est une variable d'intégration muette et où l'arbitraire sur la constante d'intégration δ₀ correspond à une rotation arbitraire de l'arc autour de l'origine O.The addition of a constant to the function σ (δ) corresponds to an overall rotation of the arc around the origin O. Because in the problems of conjugation, we are interested in arcs defined to such an rotation, it is natural to characterize the arcs by the function ρ (δ), the function σ (δ) being deduced by the quadrature: $$\mathrm{\sigma \text{'}}\left(\mathrm{\delta }\right)\mathrm{=}{\displaystyle \underset{{\mathrm{<figure-callout\; id="0"\; label="\delta "\; filenames="imgf0015.png,imgf0016.png"\; state="\{\{state\}\}">\delta </figure-callout>}}_{\mathrm{0}}}{\overset{\mathrm{\delta }}{\mathrm{\int }}}}\mathrm{}\frac{\mathrm{\rho \text{'}}\mathrm{\left(}\mathrm{\tau }\mathrm{\right)}d\mathrm{\tau }}{\cos \left(\mathrm{\tau }\right)}$$

this integration being made from τ = δ ₀ to τ = δ, and where τ is a dummy integration variable and where the arbitrary on the integration constant δ ₀ corresponds to an arbitrary rotation of the arc around the origin O.

$$y\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\sin \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\cos \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)$$

With these definitions, the Cartesian coordinates (x (δ), y (δ)) of an arc defined by the function ρ (δ) and a choice of the constant in σ (δ) are written: $$x\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\sin \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)$$

$$\mathrm{there}\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\sin \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\cos \left(\mathrm{\sigma }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)$$

• arc conjugué proximal : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
  $$y_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)-\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
• arc conjugué distal : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
  $$y_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)+\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
• arc complémentaire proximal : $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{mcos}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
  $$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)-\mathrm{mcos}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
• arc complémentaire distal $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{mcos}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
  $$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)+\mathrm{mcos}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  

Given an arc defined as above by the function ρ (δ) and an integer m≥2, we define its four associated arcs by the following expressions:

• proximal conjugated arc: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
  $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)-\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{-}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
• distal conjugate arch: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
  $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)+\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \left(\mathrm{\delta }\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m\sigma }\left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}\mathrm{m}$$
  
• proximal complementary arch: $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
  $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)-\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}\mathrm{-}\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
• distal complementary arch $${\mathit{x}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\mathrm{+}\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  
  $${\mathrm{there}}_{\mathit{VS}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\mathrm{\delta }\right)\mathrm{=}\left(-\left(2\sin \left(\mathrm{\delta }\right)+\mathrm{m\rho }\left(\mathrm{\delta }\right)\right)\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">cos</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)+\mathrm{MCOS}\left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\frac{2\mathrm{\delta }}{\mathrm{m}}+\mathrm{\sigma }\left(\mathrm{\delta }\right)\right)\right)\mathrm{/}\mathrm{m}$$
  

From a given arc defined by the function ρ (δ) and associated arcs, we define a pair of conjugated profiles.

As has been indicated above, there are two classes of such profiles, which correspond to the two relative directions of scanning of the circle by the normal to the given arc, when one traverses this arc.

These two classes are very simply characterized by the sign of derivatives ρ '(0) and ρ' (π).

The profiles are generated, for one, by the concatenation (ie put end to end maintaining the relative orientation) of the given arc and one of the complementary arcs: it is the completed profile; for the other, by the concatenation of the two conjugated arcs: it is the conjugate profile.

The given arc is first class when: ρ '(0)> 0 and ρ' (π) <0

The study of the regularity of the connections shows that one must have more precisely: $$1/\mathrm{m}>{\mathrm{\rho }}^{\text{'}}\left(0\right)>0\mathrm{and}-1/\mathrm{m}<{\mathrm{\rho }}^{\text{'}}\left(\mathrm{\pi }\right)<0$$

In this case, the completed profile is constituted by the concatenation of the given arc and the proximal complementary arc, repeated by rotations of 2π / m around the origin. The profile is of order m, that is to say, it is preserved by the rotation of 2π / m (around the origin) and that it presents m lobes or teeth. This is the profile shown partially in Figure 5.

The conjugate profile is constituted by the concatenation of the proximal conjugate arc and the distal conjugate arc, repeated by rotations of 2π / (m-1) around the center O 'of coordinates (1 / m, 0). The profile is of order (m-1), in the same sense as before. The ratio of rotation speeds is (m-1) / m.

The completed profile is internal to the conjugate profile.

The given arc is of second class when: ρ '(0) <0 and ρ' (π)> 0.

The study of the regularity of the connections shows that one must have more precisely: $$-1/\mathrm{m}<{\mathrm{\rho }}^{\text{'}}\left(0\right)<\mathrm{O\; and}1/\mathrm{m}>{\mathrm{\rho }}^{\text{'}}\left(\mathrm{\pi }\right)>0$$

In this case, the completed profile is constituted by the concatenation of the given arc and the distal complementary arc, repeated by rotations of 2π / m around the origin. The profile is of order m.

The conjugate profile is constituted, as for the first class, by the concatenation of the proximal conjugate arc and the distal conjugate arc, repeated by rotations of 2π / (m-1) around the center O 'of coordinates (1 / m, 0). The profile is of order (m-1). The ratio of rotation speeds is (m-1) / m.

The completed profile is external to the conjugate profile.

The inequalities relating to ρ '(0) and ρ' (π) are strict. This point controls the continuity of the curvature of the profiles at the joints between the arches.

These inequalities are necessary and sufficient for the regularity of the connections, but do not ensure the regularity of the arcs themselves, which must be studied elsewhere. In other words, any function ρ (δ) does not necessarily lead to a pair of regular conjugated profiles.

Here are some indications on the regularity at the interior points of the associated arcs.

et cette condition exprime simplement que le quotient par cos(δ) de la dérivée du carré du rayon vecteur garde un signe constant.It can be shown that the only singularities that can appear on the arcs associated with a given regular arc are of the dovetail type: two cusps framing a self-intersection. The condition for this not to happen is simply that the velocity vector (vector derived from the current point on the arc with respect to the parameter) does not cancel on the interval] 0, π [. These four velocities (corresponding to the four arcs of which the two profiles are formed) are expressions depending on δ, on ρ (δ) and on the derivative ρ '(δ). The non-cancellation of these expressions is therefore a constraint on the function ρ (δ). This constraint must be approached from the angle of verification, since it does not know how to solve the systems of nonlinear differential inequalities. For the given arc, the condition on the amplitude of the speed is written: $$\mathrm{V}\left(\mathrm{\delta }\right)=\left(\mathrm{\rho }\left(\mathrm{\delta }\right){\mathrm{\rho }}^{\text{'}}\left(\mathrm{\delta }\right)\right)/\cos \left(\mathrm{\delta }\right)-\mathrm{<figure-callout\; id="0"\; label="sin\; \delta \; \ne "\; filenames="imgf0015.png,imgf0016.png"\; state="\{\{state\}\}">sin\; </figure-callout>}\left(\mathrm{\delta }\right)\ne 0$$

and this condition simply expresses that the quotient per cos (δ) of the derivative of the square of the vector ray keeps a constant sign.

• pour l'arc complémentaire proximal : $${\mathrm{V}}_{\mathit{CpP}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\mathrm{2}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{mcos}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{2}\mathrm{m\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$
  
• pour l'arc complémentaire distal : $${\mathrm{V}}_{\mathit{CpD}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\mathrm{2}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{mcos}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{2}\mathrm{m\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$
  
• pour les arcs conjugués : $${\mathrm{V}}_{\mathit{CjP}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$
  
  $${\mathrm{V}}_{\mathit{CjD}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$
  

The corresponding expressions for the associated arcs are less simple. Here they are :

• for the proximal complementary arch: $${\mathrm{V}}_{\mathit{CPP}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\mathrm{2}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho \text{'}}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{MCOS}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{2}\mathrm{m\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$
  
• for the distal complementary arch: $${\mathrm{V}}_{\mathit{CpD}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\mathrm{2}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho \text{'}}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{MCOS}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{2}\mathrm{m\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0002.png,imgf0006.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$
  
• for the conjugated arches: $${\mathrm{V}}_{\mathit{CJP}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho \text{'}}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$
  
  $${\mathrm{V}}_{\mathit{CJD}}\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho \text{'}}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$
  

An interesting family of pairs of profiles of the first class is obtained from arcs of shortened epicycloids. These are typical solutions, rather than an example.

$$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{n}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\phi }\right)\right)\mathrm{+}\left(\mathrm{\delta }\mathrm{/}\mathrm{n}\right)$$

These arcs depend on three parameters: n is the order of the epicycloid, which can be chosen real (positive and not too small), φ is an angular parameter between 0 and π / 2, which describes the shortening (or eccentricity); finally, ρ ₀ is the parameter of parallelism, ie a parameter characterizing the distance to the basic epicycloid. The calculation of ρ (δ) and σ (δ) gives: $$\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{not}\right){\left(\mathrm{1}\mathrm{/}\cos {\left(\mathrm{\phi }\right)}^{\mathrm{2}}\mathrm{-}\cos {\left(\mathrm{\delta }\right)}^{\mathrm{2}}\right)}^{\mathrm{1}\mathrm{/}\mathrm{2}}\mathrm{+}\left(\mathrm{1}\mathrm{/}\mathrm{not}\right)\sin \left(\mathrm{\delta }\right)\mathrm{<figure-callout\; id="0"\; label="+\; \rho "\; filenames="imgf0015.png,imgf0016.png"\; state="\{\{state\}\}">+\; </figure-callout>}{\mathrm{\rho }}_{}{\mathrm{}}_{\mathrm{0}}$$

$$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{not}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\phi }\right)\right)\mathrm{+}\left(\mathrm{\delta }\mathrm{/}\mathrm{not}\right)$$

The best oscillation of the profiles is found for n close to 2m-2; ρ ₀ should not go too far from 0; small φ correspond to fine teeth and when φ tends to n / 2, the profiles round off and expand without limit; the reasonable values of φ are to π / 3 or π / 4.

$$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{n}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\varphi }\right)\right)-\left(\mathrm{\delta }\mathrm{/}\mathrm{n}\right)$$

A family of example profiles of the second class is similarly provided by: $$\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{not}\right){\left(\mathrm{1}\mathrm{/}\cos {\left(\mathrm{\phi }\right)}^{\mathrm{2}}\mathrm{-}\cos {\left(\mathrm{\delta }\right)}^{\mathrm{2}}\right)}^{\mathrm{1}\mathrm{/}\mathrm{2}}-\left(\mathrm{1}\mathrm{/}\mathrm{not}\right)\sin \left(\mathrm{\delta }\right)-{\mathrm{<figure-callout\; id="0"\; label="\rho "\; filenames="imgf0015.png,imgf0016.png"\; state="\{\{state\}\}">\rho </figure-callout>}}_{\mathrm{0}}$$

$$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{not}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\phi }\right)\right)-\left(\mathrm{\delta }\mathrm{/}\mathrm{not}\right)$$

The variability of the parameters (before stumbling on a singularity) is greater than in the previous case, in particular with regard to ρ ₀ .

In short, the given arc must have the following property: when it is traversed from its origin to its end, its normal "regularly scans" the primitive circle, and in particular, the normals at the origin and at the end of the arc are tangent to the primitive. The possible arcs are divided into two disjoined classes: those whose normal sweeps the primitive circle "in the opposite direction" of the current point M and those whose normal sweeps it "in the same direction" as the current point M.

To these two possibilities correspond the two classes of solutions already discussed with regard to the problem of maximum internal conjugation. The first class consists of pairs of profiles such that the inner profile has one more lobe than the outer profile; the second, conversely, is such that the inner profile has a lobe less than the outer profile. These two classes have very different morphologies and properties as previously described.

In the general case, the formulas obtained for arcs are invertible, in that one can construct the family of four arcs that define the two profiles, from any one of them. This does not mean that they play completely symmetrical roles: in fact, of the two arcs that make up each profile, one of them comes into contact with the two arcs of the other profile, and the other with only one of them. them. This is the maximal conjugation, from which it follows that the action curves are formed of three arcs concurring at two points of bifurcation B _M and B _N. The passage of the contact by these "triple points" occurs at the connection between the two arcs which constitute each of the two profiles.

• le contact entre l'arc donné et son conjugué proximal est la courbe d'action proximale, d'équation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
  $$y\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• le contact entre l'arc donné et son conjugué distal est la courbe d'action distale, d'équation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)+\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
  $$y\left(\mathrm{\delta }\right)\mathrm{=}-\cos \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)+\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• le contact entre le complémentaire proximal de l'arc donné et son conjugué proximal est la courbe d'action complémentaire proximale, d'équation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\mathrm{(}\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{)}$$
  
  $$y\left(\mathrm{\delta }\right)\mathrm{=}-\cos \left(\mathrm{\delta }\right)\left(\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• le contact entre le complémentaire distal de l'arc donné et son conjugué distal est la courbe d'action complémentaire distale, d'équation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\mathrm{(}\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)-\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{)}$$
  
  $$y\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\left(\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)-\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  

Ces quatre arcs sont concourants aux points δ = 0 et δ = π. Les courbes d'action proximale et complémentaire distale passent radialement au-delà du point de roulement R, et les deux autres de l'autre côté de l'origine O par rapport au point de roulement R. Seulement trois de ces quatre courbes d'action interviennent: la courbe d'action complémentaire distale est absente pour la première classe pour laquelle l'arc complémentaire distal n'intervient pas, et la courbe d'action complémentaire proximale est absente de la seconde classe pour laquelle l'arc complémentaire proximal n'intervient pas.The parameterization according to the invention made it possible to determine, for the machines according to the invention, simple mathematical expressions for the action curves, namely:

• the contact between the given arc and its proximal conjugate is the proximal action curve of equation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
  $$\mathrm{there}\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)\mathrm{-}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• the contact between the given arc and its distal conjugate is the distal action curve of equation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)+\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
  $$\mathrm{there}\left(\mathrm{\delta }\right)\mathrm{=}-\cos \left(\mathrm{\delta }\right)\left(\sin \left(\mathrm{\delta }\right)+\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• the contact between the proximal complement of the given arc and its proximal conjugate is the proximal complementary action curve of equation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\mathrm{(}\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{)}$$
  
  $$\mathrm{there}\left(\mathrm{\delta }\right)\mathrm{=}-\cos \left(\mathrm{\delta }\right)\left(\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)\mathrm{+}\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  
• the contact between the distal complement of the given arc and its distal conjugate is the distal complementary action curve of equation: $$x\left(\mathrm{\delta }\right)\mathrm{=}\mathrm{1}\mathrm{-}\sin \left(\mathrm{\delta }\right)\mathrm{(}\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)-\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{)}$$
  
  $$\mathrm{there}\left(\mathrm{\delta }\right)\mathrm{=}\cos \left(\mathrm{\delta }\right)\left(\left(\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\mathrm{/}\mathrm{m}\right)\sin \left(\mathrm{\delta }\right)-\mathrm{\rho }\left(\mathrm{\delta }\right)\right)$$
  

These four arcs are concurrent at the points δ = 0 and δ = π. The proximal and distal complementary action curves pass radially beyond the running point R, and the two others on the other side of the origin O with respect to the running point R. Only three of these four action curves are involved: the distal complementary action curve is absent for the first class for which the complementary arc distal does not intervene, and the proximal complementary action curve is absent from the second class for which the proximal complementary arc is not involved.

FIGS. 7A, 7B, 8A, 8B, 9A, 9B show different embodiments of machines of the first class. It appears that when the number of lobes is small, for example equal to 2 or 3, the lobar troughs are simply less prominent regions, whose profile can even be convex with respect to the inner profiled member.

In the very particular case where the (m-1) -lobed profile has only one lobe (FIGS. 7A and 7B), the lobe vertex and the lobe hollow are diametrically opposed, if the profile is symmetrical.

FIGS. 10A to 10I represent nine variants of geometries for a quadrilobelled inner profile in a tri-lobed outer profiled member.

FIGS. 11A to 11C show three examples of a first-class machine with penta-lobed inner rotor.

The embodiment of FIG. 11B is characterized by the fact that the two osculating contacts take place simultaneously, on either side of a capsule V ₁ whose volume is then maximal.

By comparison, the embodiment of FIG. 11A is analogous to that of FIG. 1, in the sense that a capsule V ₂ whose rear edge has passed the bifurcation point B _M and has therefore disappeared behind it. V capsule _1, has not yet reached its front edge the other bifurcation point B _N which will be born before her future new capsule V ₃ is therefore indicated by a dash.

In contrast, in the embodiment of FIG. 11C, the same capsule V ₂ covers both of the two bifurcation points B _N , B _M so that it is still followed by a capsule V ₁ dying and already preceded by a nascent capsule V ₃ .

A mode of distribution for a machine, in particular hydraulic, of the first class will now be described with reference to FIG.

In FIG. 12, the case of the machine of FIG. 11B is considered. It is considered that against each radial face of the profiled members 1 and 2 a flange laterally closing the capsules with the exception of the lights that will be described. These flanges are integral in rotation of the outer profile 2. In the flange located on the side of the observer in Figure 12, lights 16 in the form of drops or commas whose angular tip coincides with the connection of the two constituent arcs of the profile outside, on the rear slope of the lobes, were formed through the flange (the flange itself not being shown).

From their point coinciding with the connection of the constituent arcs of the profile 4, the lights extend generally towards the axes O and O '. These lights 16, depending on whether they are covered or not by the profiled member m-lobed, selectively communicate the capsules with the admission. In the other flange, located at the axial end which is hidden for the observer of FIG. 12, lights 17 are made which are symmetrical with the lights 16 with respect to rays passing through the lobes of the profile (m -1) -lobed 4, and whose angular tip coincides with the connection of the two constituent arches of the profile (m-1) lobed 4 on the front slope of each lobe. The lights 17 communicate with the hydraulic discharge of the machine.

Thanks to the peculiarity of the represented geometry, according to which the capsule V ₁ is adjacent on the one hand to a dying capsule at the point B _M and on the other hand to a nascent capsule at the point B _N , the capsule V ₁ is not isolated only for a short time while its volume is maximum and is therefore not changing. At the moment before, the dying capsule still communicated with the light of delivery 17 neighbor while the capsule V ₁ communicated with the intake port 16. At the instant following the new capsule will communicate with the corresponding intake port 16, while the capsule V1 will communicate with the discharge light 17.

FIG. 12A shows that, in place of or in addition to the lights 16 and 17, it is also possible to provide, in the profiled (m-1) -lobed member, intake and discharge channels 19 which open through the respective slopes; lobes of the outer profile 4, substantially to the connections between the two constituent arcs of the profile 4 so as to be closed when the profiles are in contact osculator, then to be gradually released by the capsule formed between the two contacts resulting from the disintegration of the osculator contact, in the case of the birth of a capsule for admission, or to be gradually closed with regard to the discharge, in the case of the death of a capsule.

In the example shown in Figure 13, the machine has a geometry corresponding to that of Figure 1, apart from the number of lobes. The situation is also that shown in FIG. 11A, but when the profiled members 1 and 2 are in a different angle around their respective axes.

The situation shown in Figure 13 corresponds substantially to that of Figure 2A. By observing FIG. 2D, it will be understood that the capsule V ₄ whose rear edge has already passed the bifurcation point B _M and therefore already communicates with the delivery light of a distribution according to FIG. 12 has not yet reached the point B _N and always communicate with the intake light of such a distribution, which is also necessary since the volume of the capsule V ₄ is still growing. It is therefore the communication with the light of repression that must be suppressed. This is why it is provided in Figure 13 a mask 21 secured to the housing (the connecting member) and which extends over a certain angular distance forward relative to the direction of rotation defined by the arrow F, from the bifurcation point B _M , to obscure the discharge light in this area.

For reasons quite symmetrical, a mask 22 is provided to obscure the intake ports over a certain angular zone from the B _N bifurcation point back relative to the direction of rotation.

In the situation shown in FIG. 11C, the capsule V ₂ undergoes volume variations between the moment when its front edge comes to cover the bifurcation point B _N and until its rear edge no longer covers the other point of Bifurcation B _M.

In this angular range, the capsule V _{2 would} no longer communicate with any of the lights of a distribution such as that of Figure 12. To overcome this difficulty, additional communications, controlled for example by a cam when a capsule such as V ₂ passes in this zone, are in necessary principles, where other analogous solutions.

FIG. 14 represents a particularly preferred embodiment for a machine having a profile according to FIG. 1. The distribution principle is the same as in FIG. 12, and in each plane perpendicular to the axes the profiles 3 and 4 are those of FIG. 1. However, from one plane to another, each profile 3 or 4 is offset angularly by a determined pitch about its respective axis so as to give all the profiled members a helical shape. The angular offset between the profiles of the two ends is such that in the situation shown, where the capsule V ₅ on the intake side reaches the bifurcation point B _N , this capsule itself having a spiral shape has just left its rear edge. other oscillation at the other bifurcation point B _M. The situation which was obtained by a profile in a single plane in the case of FIGS. 11B and 12 is thus restored by means of helicity, namely that one and the same cavity is adjacent to a nascent cavity by its front edge and to a cavity. dying by its back edge. This cavity V ₅ is therefore not isolated only at a short moment when the speed of instantaneous variation of its volume is equal to zero. In Figure 14, there is shown in solid lines the vertices of the profile 3 of the inner profiled member and in phantom with crosses some of the vertices of the lobes of the profile of the outer profiled member 4. The centers O and O ' profiles of the successive planes are aligned along parallel axes of rotation which are also parallel to a straight line R _R on which the _R points of alignment are aligned.

Figure 15 shows schematically an embodiment of a first class machine according to the invention. The inner profiled member 1 is secured to a drive shaft 23 which is a motor in the case of a pump and receiver in the case of a hydraulic motor. The shaft 23 is supported in rotation, on either side of the profiled member 1, by two bearings 24 in a fixed housing 25 which constitutes the connecting member according to the invention. The outer profiled member 2 is supported in rotation by peripheral bearings 26 installed between the outer peripheral wall of the profiled member 2 and a peripheral ring 27 forming part of the housing 25. The geometric axis of the shaft 23 corresponds to the center O whereas the geometric axis, not shown, of the bearings 26 corresponds to the center O '. In the zone where the profiles 3 and 4 are formed, the profiled members 1 and 2 are installed between two flanges 28, 29 through which the inlet and discharge ports 16 and respectively 17 are formed.

The profiled members 1 and 2 have planar and coplanar end faces on which the corresponding flat end faces of the flanges 28 and 29 are sealingly and slidingly abutted so as to close the capsules except for the communications selectively set by lights 16 and 17.

There is between each flange 28 or 29 and a corresponding end wall 31 or 32 of the casing, a respective axial abutment 33, 34. The flanges 28, 29 are connected in rotation with the outer profiled member 2 while being free in translation with respect to it thanks to grooves 36. The internal space between the end wall 31 of the casing on the one hand and the flange 28 and the corresponding face of the profiled member 1 on the other hand is arranged in a chamber subjected to the inlet pressure . Similarly, a chamber subjected to the discharge pressure is formed between the other end wall 32 of the housing on the other hand and the other flange 29 and the other end face of the inner profiled member 1 on the other hand. These two chambers are closed by dynamic sealing devices 38, 39, 41, 42 which prevent the hydraulic fluid from accessing the bearings 24 and 26, and prevent the two chambers from communicating with each other between the two. outer profiled member 2 and the crown 27 of the housing.

In operation, that of the two chambers which is subjected to the high pressure (the admission in the case of a motor and the discharge in the case of a pump) compresses the axial stack constituted by the two flanges and the two bodies sections 1 and 2 mounted sandwiched between them, in axial bearing against the axial stop of the opposite chamber. The area exposed to pressure to provide this axial pressing force is chosen so that the axial thrust is appropriate for sealing between the flanges and the profiled members, but without being excessive.

In addition, if the profiled members are helical as described with reference to FIG. 14, the axial thrust thus created must be sufficient to balance the tendency of the profiled members to "unscrew" with respect to one another under the action of the forces of work exerted between the profiles 3 and 4.

For example, if with the embodiment shown in Figure 15 the axial thrust retained is too strong, it can be radially outwardly beyond the axial abutments 33 and 34, so between each flange and the end wall. 31 corresponding to the housing, the sealing devices 41 and 42 shown acting in contact with the shaft 23. Furthermore, the shaft 23 must be mounted with a certain freedom of axial sliding for allow the axial floating of the profiled member 1 between the flanges 31 and 32. The outer profiled member 2 is free to rotate so that its drive results from its cooperation with the profiled member 1 and the working fluid.

In the example shown in Figure 16, the machine is variable displacement. For this, the profiled members 1 and 2 are axially sliding relative to each other. In the example shown, the profiled member 2 is axially fixed while bearing against the casing 25 by means of an axial abutment 53 and a flange 51. The profiled member 1 is axially sliding relative to the casing by means of an actuator 49 which is only schematically shown, acting on the member 1 by means of an axial stop 54 and a flange 52. The flange 51 rests in a sealed manner against a flat end face of the outer profiled member 2 and has as a radially inner edge a profiled face 47 which is exactly complementary to the profile 3 of the profiled member 1. Thus, the flange 51 is in sealing contact with the profile 3 all around the profiled member 1, to slide axially relative to the profiled member 1 while being rotated by the profiled member 1.

Similarly, the flange 52 is sealingly supported against a flat end face of the profiled member 1 and has on its outer periphery a shaped face 48 which is exactly complementary to the profile 4 of the profiled member 2 so that to support it sealingly, axially sliding, and ensuring the rotational drive of the flange 52 with the profiled member 2. The distribution is provided by channels 18, 19 according to the embodiment of Figure 12A.

FIGS. 17A to 22B show various embodiments, each in two operating states, for machines of the second class, with lobe numbers ranging from 1 for the inner profiled member and 2 for the outer profiled member (FIGS. 17A and 17B), to 7 for the inner profiled member and 8 for the outer profiled member (Figures 22A and 22B).

In comparison with the embodiment of FIGS. 19A and 19B in the case where the inner profiled member is tri-lobed and the outer profiled member is quadrilobed, FIGS. 23A to 25B represent three other possible geometries which illustrate the great variety of feasible geometries for second class machines.

In the case of second class machines, there are two action curves on the side of the running point and only one on the opposite side. The outer curves are simple arcs. The inner curve may have a loop whose double point is the running point; it's not a singularity of the profiles. At the moment the contact passes through the rolling point, the relative movement of the two profiles is a rolling without sliding. In the limit cases for which the action curve has a cusp point at the running point, the speed of the contact point is canceled.

The description of the capsular cycle is somewhat complicated by the possible occurrence of the phenomenon of "capsular scission" briefly described below. In all cases, a capsule is born at the passage of the slopes before the lobes of the external profile by the osculating contact, at the intersection B _N of the action curves located upstream of the axis O _x bearing the point R. It passes by its maximum after a rotation of a little more than half a turn. The capsule is then on the opposite side to the running point relative to the pivots. The closure of the capsule is symmetrical with its opening, and the "life time" of the capsule is a little over one turn.

The phenomenon of capsular scission is likely to occur for capsules in the vicinity of their birth or death, that is to say when two lobes are strongly engaged one in the other side of the running point. The volumes of the capsules concerned are small. The course is as follows: at a point inside a capsule being closed, both profiles come into an exceptional osculator contact, and the capsule is cut into two sub-capsules. The new osculator contact disintegrates into two simple contacts between which a new capsule is born. Each of these two contacts joins the corresponding edge of one of the two subcapsules being closed and these disappear (usually at different times), one in a normal way to the passage by the confluence of the curves of action, and the other exceptionally through an oscillation that disappears on the spot. At this point the new capsule coalesces with another new capsule that was born normally at the bifurcation of the action curves.

This somewhat delicate phenomenon of capsular scission takes place in the case where the profiles become tangent to the external action curve on the side of the running point, but outside the Ox axis.

• les capsules sont entièrement vidées; on peut donc par un simple clapet supprimer le reflux vers la basse pression ;
• la courbure relative des surfaces en "contact" (en général, ces machines ne sont pas auto-entraînées et on ne va pas jusqu'au contact) est bornée ; les fuites se font donc à travers un passage qui est non seulement aussi étroit que le permet la précision de fabrication, mais qui de plus reste étroit sur une certaine longueur.

Figures 26A and 26B show a geometry particularly well suited to the realization of a compressor. It is a second class machine, with a di-lobed inner profiled member and a tri-lobed outer profiled member. A machine of this type and more generally a machine according to the invention has for the realization of a compressor the following advantageous features, which are both in the direction of the limitation of leakage:

• the capsules are completely emptied; it is therefore possible by a single valve to remove the reflux to the low pressure;
• the relative curvature of the "contact" surfaces (in general, these machines are not self-driven and there is no contact) is bounded; the leaks are therefore through a passage which is not only as narrow as the precision of manufacture allows, but which moreover remains narrow for a certain length.

It is desired to raise the maximum of obstacles between the low pressure side and the high pressure side of the compressor. It is therefore natural to focus on the second class of conjugated profiles: in fact, during the growth phase, the consecutive capsules remain at the intake pressure, and during the volume decay phase, the compression is progressive. It is only at the end of compression that the capsule being closed is adjacent to two capsules at low pressure: along the external action curve with a nascent capsule and along the internal action curve with a growing capsule. In both cases the surfaces in contact have their concavities in the same direction and the relative curvature is small (it vanishes at the end of repression). We will choose a profile that like that of Figures 26A and 26B does not give rise to capsular split.

Helicoid execution is possible and we find the good contact qualities of the right case.

In the case of a compressor, it may be preferable to keep the external profile fixed (which then becomes that of the casing) and to give the rotor a planetary movement, the connecting member is then rotated relative to the casing around the axis. 0 of the outer profiled member.

The case of the compressor is also one in which the properties of the fluid change between intake and delivery; moreover, the parameters to be optimized are not the same at intake (limitation of pressure drop) and at discharge (limitation of leaks). For these reasons, it may be preferable to use asymmetric profiles. An example is given in Figures 27A and 27B.

In the example shown in FIGS. 28A to 28F, an intermediate profiled member 62 comprises a first profile 64 of order m-1 on its radially inner face, and a second profile 74 of order (m-1) on its radially face. exterior. Both profiles have the same primitive circle centered in O '. Each of the profiles (m-1) -lobed 64, 74, cooperates with an m-lobed profile 63, 73 of a profiled member 61 which is shown fixed in this example. The two profiles 63, 73 also have a common pitch circle, which is centered at O. The profiles 63 and 64 form a machine of the first class according to the invention and the profiles 73 and 74, a machine of the second class according to the invention. 'invention.

In the example shown in FIGS. 29A to 29F, the difference is that the intermediate profiled member 82 carries two m-lobed profiles cooperating with two (m-1) -lobed profiles belonging to the profiled member 81.

Such a geometry could make it possible to manufacture an internal combustion engine in which, for example, the inner machine would be used for intake and compression, while the outer machine would serve for expansion and exhaust.

Of course, the invention is not limited to the examples described and shown.

In the examples described, and more particularly in that of FIG. 15, the inner profiled member is rotated and the outer profiled member rotates thanks to the moment of rotation transmitted to the points of contact between the inner profiled member and the inner profile member. external shaped member which is free to rotate in the housing. In addition, in motor operation, the pressure of the hydraulic fluid tends to make the cavities subjected to this pressure evolve in the direction of the enlargement of their volume, which contributes to urging the external profiled member in the direction of rotation. desired. But it is also possible to provide an external drive, for example by gearing, which forces the two profiled members to rotate in a gear ratio corresponding to the ratio of the number of their lobes. It is also possible to drive the external profiled member and leave the internal profiled member free. It is also possible to fix one of the two profiled members to the casing and to drive the other profiled member in a planetary movement by rotating the center of the pitch circle of the other profiled member around the center of the pitch circle of the fixed profiled member. . In this configuration, said other profiled member can be left to position freely around its own axis or on the contrary determine, for example by a gear, its angular position as a function of the angular position of the connecting member around the center of the fixed profiled member.

The invention is compatible with the Sparrow principle according to which, as described in US Pat. No. 1,892,217, the helical shape of the two profiled members extends over a sufficient number of helical steps so that no cavity simultaneously leads to two axial ends of the machine. Thanks to the precision and quality of the geometry according to the invention, it is possible to limit the total angular offset between the profiles at the two ends of the machine to a value barely greater than the lifetime of the capsule in each plane. perpendicular to the axes ..

The pitch of the helix is not necessarily the same all along the machine, and the profile can still be varied along the axes of the machine. This allows for example to achieve a compressor or a relaxation machine in which the volume of the capsules being transferred varies gradually.

Claims (34)

1. A displacement machine comprising:

   - two profiled members (1, 2; 12, 11), inner and outer respectively, that have an annular inner profile and an annular outer profile respectively (3, 4; 14, 13),

   - a connecting member (25) rotatably connected to each of the profiled members (1, 2; 12, 11) along a respective axis of rotation (0, O'; O', O),

   and in which:

   - one of the profiles (3; 13) is m-lobed and the other (4; 14) is (m-1)-lobed, and they are defined around the axis of rotation of their respective profiled member by m and (m-1) respectively, pattern(s) comprising a lobe dome arc and a lobe hollow arc,

   - each profile is the envelope of the other during relative rotations of the profiled members around their respective axis of rotation with meshing of their profiles, which define the chamber contours between them, and slideless rolling between two pitch circles centred on the respective axes of rotation,

   characterised in that the relative positions of the profiled members (1, 2; 12, 11) for which a contact point (C₂) between the profiles is located on the tangent (T) to the two pitch circles (6, 7) at their mutual rolling point (R), the profiled members (1, 2; 12, 11) have at said contact point equal continuous curvatures which are in the same direction and which have said rolling point (R) as their common centre.
2. A machine according to claim 1, characterised in that:

   - the points M on a given arc that is one of the two arcs of the m-lobed profile being defined by two functions ρ(δ) and σ(δ) connecting parameters ρ, δ and σ, which are:

   ρ: measured along the normal to the arc at point M, the distance between point M and the middle N between the two intersection points P and D, proximal and distal respectively, of the said normal with the pitch circle with centre O of the m-lobed profile, and with a radius assumed equal to 1, the proximal intersection point P being located between point M on the given arc and the distal intersection point D,

   δ: angular half-distance between D and P relative to the centre O, measured clockwise,

   σ: polar angle of the proximal intersection point P relative to O, minus δ,

   the functions ρ(δ) and σ(δ) having a domain of between δ=0 and δ=π,

   - two arcs of the pattern of the (m-1)-lobed profile are a proximal conjugate arc and a distal conjugate arc defined below in a Cartesian reference system with their origin at the centre O of the pitch circle associated with the m-lobed profile:

   a) proximal conjugate arc: $${\mathrm{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\delta \right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\delta \right)\mathrm{-}\mathrm{m}\begin{array}{}\end{array}\rho \left(\delta \right)\right)\sin \left(\frac{\delta \mathrm{-}\mathrm{m}\begin{array}{}\end{array}\sigma \left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(m\mathrm{-}\mathrm{1}\right)\cos \left(\delta \right)\cos \left(\frac{\delta \mathrm{-}\mathit{m\sigma }\left(\delta \right)}{m\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}m$$

   

   $${\mathrm{y}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{P}}\left(\delta \right)\mathrm{=}\left(\left(\sin \left(\delta \right)\mathrm{-}\mathrm{m}\begin{array}{}\end{array}\rho \left(\delta \right)\right)\cos \left(\frac{\delta \mathrm{-}\mathrm{m}\begin{array}{}\end{array}\sigma \left(\delta \right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)-\left(m\mathrm{-}\mathrm{1}\right)\cos \left(\delta \right)\sin \left(\frac{\delta \mathrm{-}\mathit{m\sigma }\left(\delta \right)}{m\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}m$$

   

   b) distal conjugate arc: $${\mathrm{x}}_{\mathit{C}\begin{array}{}\end{array}\mathit{j}\begin{array}{}\end{array}\mathit{D}}\left(\delta \right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\left(\sin \left(\delta \right)+\mathrm{m}\begin{array}{}\end{array}\rho \left(\delta \right)\right)\sin \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m}\begin{array}{}\end{array}\sigma \left(\mathrm{\delta }\right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)\mathrm{+}\left(m\mathrm{-}\mathrm{1}\right)\cos \left(\delta \right)\cos \left(\frac{\mathit{\delta }\mathit{+}\mathit{m\sigma }\left(\delta \right)}{m\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}m$$

   

   $${\mathrm{y}}_{\mathit{C}\begin{array}{}\end{array}\mathit{jD}}\left(\delta \right)\mathrm{=}\left(-\left(\sin \left(\delta \right)+\mathrm{m}\begin{array}{}\end{array}\rho \left(\delta \right)\right)\cos \left(\frac{\mathrm{\delta }\mathrm{+}\mathrm{m}\begin{array}{}\end{array}\sigma \left(\delta \right)}{\mathrm{m}\mathrm{-}\mathrm{1}}\right)+\left(m\mathrm{-}\mathrm{1}\right)\cos \left(\delta \right)\sin \left(\frac{\mathit{\delta }\mathit{+}\mathit{m\sigma }\left(\delta \right)}{m\mathrm{-}\mathrm{1}}\right)\right)\mathrm{/}m$$

   
3. A machine according to claim 2, characterised in that the derivative ρ' relative to δ where δ = 0 and δ = π satisfies the following strict inequalities: $$1/\mathrm{m}>{\mathrm{\rho }}^{\prime }\left(0\right)>0$$

   

   $$-1/\mathrm{m}<{\mathrm{\rho }}^{\prime }\left(\mathrm{\pi }\right)<0$$

   

   
   in that the m-lobed profile is inside the (m-1)-lobed profile, and
   in that the m-lobed profile is complemented by a proximal complementary arc defined by its coordinates in the said Cartesian reference system: $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\delta \right)\mathrm{=}\left(\left(2\sin \left(\delta \right)\mathrm{-}\mathit{m\rho }\left(\delta \right)\right)\sin \left(\frac{2\delta }{m}\mathrm{-}\sigma \left(\delta \right)\right)\mathrm{+}m\begin{array}{}\end{array}\cos \left(\delta \right)\cos \left(\frac{2\delta }{m}\mathrm{-}\sigma \left(\delta \right)\right)\right)\mathrm{/}m$$

   

   $$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}\mathit{P}}\left(\delta \right)\mathrm{=}\left(\left(2\sin \left(\delta \right)\mathrm{-}\mathit{m\rho }\left(\delta \right)\right)\cos \left(\frac{2\delta }{m}\mathrm{-}\sigma \left(\delta \right)\right)-m\begin{array}{}\end{array}\cos \left(\delta \right)\sin \left(\frac{2\delta }{m}\mathrm{-}\sigma \left(\delta \right)\right)\right)\mathrm{/}m$$

   
4. A machine according to claim 3, characterised by the fulfilment of the following conditions over the entire interval ]0,π[ of variation of the coordinate δ: $$\left(\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\right)\mathrm{/}\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{-}\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{\ne }\mathrm{0}$$

   

   $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)-\mathrm{2}\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{mcos}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)-\left(\mathrm{2}\mathrm{m\rho }\left(\mathrm{\delta }\right)+\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{m}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$

   

   $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$

   

   $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$

   
5. A machine according to claim 3 or 4, characterised in that the functions ρ(δ) and σ(δ) are: $$\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{n}\right){\left(\mathrm{1}\mathrm{/}\cos {\left(\mathrm{\phi }\right)}^{\mathrm{2}}\mathrm{-}\cos {\left(\mathrm{\delta }\right)}^{\mathrm{2}}\right)}^{\mathrm{1}\mathrm{/}\mathrm{2}}\mathrm{+}\left(\mathrm{1}\mathrm{/}\mathrm{n}\right)\sin \left(\mathrm{\delta }\right)\mathrm{+}{\mathrm{\rho }}_{\mathrm{0}}$$

   

   $$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{-}\mathrm{1}\mathrm{/}\mathrm{n}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\phi }\right)\right)+\left(\mathrm{\delta }\mathrm{/}\mathrm{n}\right)$$

   

   
   that define the given arc as a curve parallel to a shortened epicycloid, and where:

   n is a real number that is the order of the epicycloid,

   ϕ is an angular parameter of between 0 and π/2, which describes the shortening,

   ρ₀ is a parameter characterising the distance to the base epicycloid.
6. A machine according to claim 5, characterised in that n is taken as close to 2m-2.
7. A machine according to any one of claims 3 to 6, characterised in that it comprises:

   - two flanges (28, 29) between which the profiled members (1, 2) are installed, and which are connected for common rotation with one of the profiled members;

   - inlet ports (16) through a first (28) of the flanges near a side of each of the lobe domes of the profile of the profiled member with which the flanges (28, 29) are connected for common rotation therewith;

   - discharge ports (17) through a second of the flanges near another side of each of the said lobe domes.
8. A machine according to claim 7, characterised in that it comprises means (21, 22) for selectively closing at least some of the ports in at least an angular area close to an intersection between a common tangent (T) of the pitch circles (6, 7) and on the other hand action curves (CA₁, CA₂, CA₃) defined by the trajectories of the contact points between profiles.
9. A machine according to claim 7, characterised in that there is an angular displacement between a profile of the profiled members (1,2) on the side of one of the flanges and a profile of the profiled members on the side of the other flange, such that each chamber (V₅ - FIG. 15) passing through its maximum volume ceases to communicate with a port (16) through one of the flanges approximately at the moment when it starts to communicate with a port (17) through the other flange.
10. A machine according to any one of claims 3 to 9, characterised in that it comprises in the profiled outer member, distribution channels (18, 19) opening on the one hand into the profile (4) at the connection of the arcs and communicating on one side of the lobe domes with the intake and on the other side of the lobe domes with the discharge.
11. A machine according to claim 2, characterised in that the derivative ρ' relative to δ where δ = 0 and δ = π satisfies the following strict inequalities: $$-\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{<}\mathrm{\rho ʹ}\left(0\right)<\mathrm{0}$$

    

    $$\mathrm{1}\mathrm{/}\mathrm{m}\mathrm{>}\mathrm{\rho ʹ}\left(\pi \right)>\mathrm{0}$$

    

    
    in that the m-lobed profile is outside the (m-1)-lobed profile;
    and
    in that the m-lobed pattern is complemented by a distal complementary arc defined by its coordinates in the said Cartesian reference system with centre O: $${\mathit{x}}_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\delta \right)\mathrm{=}\left(\left(2\sin \left(\delta \right)+\mathit{m\rho }\left(\delta \right)\right)\sin \left(\frac{2\delta }{m}+\sigma \left(\delta \right)\right)\mathrm{+}m\begin{array}{}\end{array}\cos \left(\delta \right)\cos \left(\frac{2\delta }{m}+\sigma \left(\delta \right)\right)\right)\mathrm{/}m$$

    

    $$y_{\mathit{C}\begin{array}{}\end{array}p\begin{array}{}\end{array}D}\left(\delta \right)\mathrm{=}\left(-\left(2\sin \left(\delta \right)+\mathit{m\rho }\left(\delta \right)\right)\cos \left(\frac{2\delta }{m}+\sigma \left(\delta \right)\right)\mathrm{+}m\begin{array}{}\end{array}\cos \left(\delta \right)\sin \left(\frac{2\delta }{m}+\sigma \left(\delta \right)\right)\right)\mathrm{/}m$$

    
12. A machine according to claim 11, characterised by the fulfilment of the following conditions over the entire interval ]0,π[ of variation of the coordinate δ: $$\left(\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\right)\mathrm{/}\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{-}\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{\ne }\mathrm{0}$$

    

    $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\mathrm{2}\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\mathrm{mcos}\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{2}\mathrm{m\rho }\left(\mathrm{\delta }\right)-\left({\mathrm{m}}^{\mathrm{2}}\mathrm{-}\mathrm{4}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}{\mathrm{m}}^{\mathrm{2}}\mathrm{\ne }\mathrm{0}$$

    

    $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)\mathrm{-}\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{-}\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\mathrm{+}\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$

    

    $$\left(\mathrm{m\rho }\left(\mathrm{\delta }\right)+\sin \left(\mathrm{\delta }\right)\right)\mathrm{\rho ʹ}\left(\mathrm{\delta }\right)\mathrm{/}\left(\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\cos \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)+\left(\mathrm{\rho }\mathrm{\left(}\mathrm{\delta }\mathrm{\right)}-\left(\mathrm{m}\mathrm{-}\mathrm{2}\right)\sin \mathrm{\left(}\mathrm{\delta }\mathrm{\right)}\right)\mathrm{/}\left(\mathrm{m}\mathrm{-}\mathrm{1}\right)\mathrm{\ne }\mathrm{0}$$

    
13. A machine according to claim 11 or 12, characterised in that the profiles only pass through a single point of tangency with the outermost trajectory (CB₃) followed by the contact points.
14. A machine according to any one of claims 11 to 13, characterised in that the functions ρ(δ) and σ(δ) are: $$\mathrm{\rho }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{n}\right){\left(\mathrm{1}\mathrm{/}\cos {\left(\mathrm{\varphi }\right)}^{\mathrm{2}}\mathrm{-}\cos {\left(\mathrm{\delta }\right)}^{\mathrm{2}}\right)}^{\mathrm{1}\mathrm{/}\mathrm{2}}-\left(\mathrm{1}\mathrm{/}\mathrm{n}\right)\sin \left(\mathrm{\delta }\right)-{\mathrm{\rho }}_{\mathrm{0}}$$

    

    $$\mathrm{\sigma }\left(\mathrm{\delta }\right)\mathrm{=}\left(\mathrm{1}\mathrm{+}\mathrm{1}\mathrm{/}\mathrm{n}\right)\arccos \left(\cos \left(\mathrm{\delta }\right)\cos \left(\mathrm{\varphi }\right)\right)-\left(\mathrm{\delta }\mathrm{/}\mathrm{n}\right)$$

    

    
    that define the given arc as a curve parallel to a shortened epicycloid and where:

    n is a real number that is the order of the epicycloid,

    ϕ is an angular parameter of between 0 and π/2, which describes the shortening,

    ρ₀ is a parameter characterising the distance to the base epicycloid.
15. A machine according to any one of claims 1 to 14, characterised in that each lobe is symmetrical relative to an axial plane passing through the lobe vertex.
16. A machine according to any one of claim 1 to 14, characterised in that each lobe is dissymmetrical relative to an axial plane passing through the lobe vertex.
17. A machine according to any one of claims 1 to 16, characterised in that the connecting member is firmly attached to a housing (25), and in that one of the profiled members is at least indirectly connected to a drive shaft (23) for common rotation therewith.
18. A machine according to claim 17, characterised in that the other profiled member rotates freely around its axis of rotation.
19. A machine according to any one of claims 1 to 18, characterised in that the profiles are each subject to evolution along the axis of rotation of their respective profiled member, the points of tangency of the pitch circles being aligned on a straight line parallel to the two axes of rotation.
20. A machine according to claim 19, characterised in that the profiles are subject to evolution by angular displacement of a constant profile around the axis of rotation.
21. A machine according to claim 20, characterised in that the profiles evolution is in a constant pitch helix fashion.
22. A machine according to any one of claims 1 to 18 or 21, characterised in that the profiles are constant along their respective axis of rotation, have a constant angular displacement pitch, finite or infinite, along their respective axis of rotation, in that the profiled members are axially movable relative to each other, and in that the machine comprises at each end a flange (51, 52) complementary to a respective one of the profiles and leak-tightly resting against an end surface of the profiled member holding the other profile.
23. A machine according to any one of claims 19 to 21, characterised in that the angular displacement of the profiles from one end surface of the profiled members to the other is hardly greater than the lifetime angle of each chamber relative to the respective profiled member.
24. A machine according to any one of claims 1 to 23, characterised in that the profiled members are mounted between two flanges (28, 29) closing the chambers at their axial ends, and in that the machine comprises pressing means to press the flanges axially against the profiled members.
25. A machine according to claim 24, characterised in that each flange (28, 29) is connected for common rotation with one of the profiled members.
26. A machine according to claim 24 or 25, characterised in that the pressing means are means of subjecting at least part of the outer surface of a first of the flanges to the high pressure of the working fluid to push the first flange against the profiled members and thus push the profiled members against the second flange.
27. A machine according to claim 26, characterised in that the machine comprises distribution means that comprise at least one port (16, 17) formed in the first flange (28, 29) for the high-pressure working fluid.
28. A machine according to claim 27, characterised in that the distribution means comprise at least one port formed in the second flange for the low-pressure fluid.
29. A machine according to claim 27 or 28, characterised in that the ports (28, 29) are connected for common rotation with the outer profiled member (2).
30. A machine according to any one of claims 1 to 26, characterised in that it comprises distribution means that comprise ports connected for common rotation with one of the profiled members, preferably the (m-1)-lobed profiled member (2), and that are selectively revealed and hidden by the other profiled member (1).
31. A machine according to claim 5 or 30, characterised in that the ports have tips coinciding with the connection point of the arcs forming the profile with which the ports (16, 17) are integral, on the profile side where the chambers appear for the inlet ports and on the profile side where the chambers disappear for the discharge ports.
32. A machine according to any one of claims 1 to 30, characterised in that one of the profiled members (61, 81) has two m-lobed profiles, one on a radially inner annular surface and the other on a radially outer annular surface, which have the same pitch circle and each cooperate with an (m-1)-lobed profile, and in that the (m-1)-lobed profiles have the same pitch circle and are held by the other profiled member.
33. A machine according to claim 32, characterised in that the two m-lobed profiles (83, 93) are facing away from each other and are radially between the two (m-1)-lobed profiles (84, 94).
34. A machine according to claim 32, characterised in that the two m-lobed profiles (63, 73) are facing towards each other and are radially on either side of the two (m-1)-lobed profiles (64, 74).

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