Abstract:
A geared hydraulic apparatus comprises a pair of gear wheels which mesh with each other with semi-encapsulation. Each gear wheel has a plurality of teeth with a profile which falls within a band of tolerance of ±15, more preferably ±20, and even more preferably ±30 with respect to the height of the tooth, with respect to a profile similar to a profile defined by a predetermined spline function passing through a plurality of nodal points having pre-established coordinates {X,Y} with origin on the rotational axis.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    The present invention refers to a gear wheel, of the type having a profile capable of meshing with semi-encapsulation in a geared hydraulic apparatus. 
         [0002]    Typical examples of geared hydraulic apparatus in which the gear wheels of the present invention, and to which specific reference shall be made hereinafter, find application are rotary positive displacement pumps, but the gear wheels of the present invention may also analogously be applied to hydraulic geared motors, which are thus deemed included within the scope of protection of the present invention. Rotary positive displacement pumps are generally made up of two gear wheels, in most cases of the straight cut gears, one of which, called driver, is connected to a driving shaft and rotates the other wheel, called driven. 
         [0003]    A disadvantage particularly revealed by the above-mentioned conventional gear pumps, generally with involute gear profile, lies in the fact that the pumped fluid is encapsulated, i.e. trapped, and compressed or however subjected to volume variations in the compartments enclosed between the gear profiles in the meshing area, thus leading to damaging and uncontrolled local stress peaks which are the source of direct operating noise. 
         [0004]    Besides the direct operating noise indicated above, there is also a known problem arising from the phenomenon of irregularity, or “ripple”, in the transfer of the fluid which entails an indirect operating noise, known as ripple noise, linked to the flow rate pulsation and therefore pressure pulsation in the user circuit. 
         [0005]    In other words, the oscillations in the fluid flow rate generate a pulsating wave which, via the fluid itself, is transmitted to the surrounding atmosphere and, in particular, to the walls of the pump, to the piping and to the delivery pipes. 
         [0006]    The induced noise may also reach unpredictable levels in the case where the aforesaid members resonate with the oscillation or ripple frequency. 
       PRIOR ART 
       [0007]    A series of studies and experiments have shown that such oscillations are intrinsically due to the configuration of the rotors or gear wheels of the abovementioned pumps which, in consecutive phases of the meshing thereof, produce a discontinuity in the variation of the volume which causes the transport of the fluid from intake to delivery. 
         [0008]    In other words, the ripple is due to the discontinuity in the variation of said volume with respect to time, or, rather, with respect to the reciprocal angular position of the rotors. 
         [0009]    The aforesaid phenomena are clearly and fully described in the articles by MORSELLI Mario Antonio, “ Mechanical and hydraulic noise in geared pumps”, Oleodinamica Pneumatica , January 2005, pp. 54-59, and February 2005, pp. 42-46, which also appeared in  Fluides  &amp;  Transmissions , No. 75, April 2005, pp. 34-37 and No. 77, May 2005, pp. 20-26. 
         [0010]    Some solutions which have addressed, with greater or lesser success, the problems illustrated above are known. 
         [0011]    Some of these solutions regard pumps with conventional teeth, having tooth side profiles, mostly, but not necessarily, that are involute, of the straight cut gear type or, more rarely, helical gear type, with clearance that is, with single contact of a tooth of one gear wheel with a corresponding tooth of the other gear wheel) or theoretically without clearance (that is, with double contact, where both the sides of the teeth are theoretically always in engagement, as in the pump from Bosch Rexroth AG known by the trade name SILENCE, or the pump from Casappa S.p.A. known by the trade name WHISPER). 
         [0012]    In these solutions, the fluid trapped between the teeth is, at least in part, “discharged”, that is, evacuated, via suitable outlets or pockets or ducts provided on the faces of the lateral abutment means, otherwise known as supports or bushes, of the gear wheels, that is, on the walls which face the flat lateral gear wheel ends, and which make it possible to discharge (or suction) the encapsulated volume of fluid towards the appropriate, respectively high or low pressure, port or gate. 
         [0013]    The provision of the pockets on the faces of the lateral abutment means, however, becomes much more complex when it is wished to produce helical gear wheels in order to reduce the problem of ripple noise. 
         [0014]    Moreover, the use of helical gear wheels in itself presents a series of additional problems, since in this case the volume of each fluid entrapment area also extends, like the teeth of the gear wheels, on a worm-like helical course over the entire width of the gear wheel, therefore representing a potential communication route or by-pass between the intake and delivery, if particular solutions are not adopted. In practice, either small helix angles of the gear wheels are used, or one is forced to use solutions that are very complex and expensive from the constructional point of view, such as that described in the document EP-0769104 of Brown David Hydraulics Ltd., in which the gear wheels have, for each of the cross-sections thereof, at least two teeth simultaneously in engagement. 
         [0015]    Such solutions, however, are very complex and substantially not very efficient, since they are developed on the basis of concepts that are closer to mathematical abstractions than to practical and technologically feasible possibilities; in practice, the geometry of said pockets is always a not entirely satisfactory compromise. 
         [0016]    In any case, all the known pump solutions, whether of the straight cut or helical gear type, single or double contact, which employ discharge pockets on the lateral abutment means have however a residual trapped volume which is subject to variations which cannot be discharged, and which therefore generate a certain residual noise, besides having a significant and damaging ripple. 
         [0017]    Other known solutions to the problems of direct and indirect noise mentioned above relate to pumps with teeth having a non-conventional profile, which may be defined as of the “continuous contact” type, which do not trap fluid between the head and bottom of the tooth. In practice, the gear wheels meshing with one another have profiles having a rounded form at the head of the tooth and a theoretical single point of contact which moves continuously from one side of the gear wheel to the other, so as not to generate any closed area of fluid entrapment during meshing, over the entire width of the gear wheels. 
         [0018]    This principle, theoretically stated in broad terms and quite generally in the documents U.S. Pat. No. 2,159,744, U.S. Pat. No. 3,164,099, U.S. Pat. No. 3,209,611 which has, however, never found any practical application, has been fully developed and described in the documents EP-A-1132618, EP-B-1371848, U.S. Pat. No. 6,769,891 of the same inventor and joint Applicant of the present application, as well as in the technical articles mentioned above, and has found a practical application in the pump known by the trade name Continuum® Settima Flow Mechanisms. 
         [0019]    The teeth types developed by the present inventor do not have a by-pass between intake and delivery of the pump, they have minimum pulsation of the fluid and a notable quietness of meshing. 
         [0020]    This last solution, although it has proved to be clearly superior from the point of view of quietness compared with conventional pumps, has however the drawback of a slightly lower displacement performance with respect to that of the known pump solutions in which there is fluid entrapment. 
         [0021]    The main reason lies in the low tooth height that can be produced with a profile designed according to the concept of “non-encapsulation”, and therefore a corresponding low efficient flow rate per unit of volume, considering the same number of teeth. In order to have efficient unitary flow rates, comparable with those of the pumps with encapsulation, the inventor, contrary to the traditional literature, identified an ideal range for this solution between 5 and 10 teeth, preferably 7 teeth, such number of teeth being low, but entailing greater volumetric losses due to the lower sealing between high pressure delivery and low pressure intake, since the teeth also function as labyrinth seals. 
         [0022]    All the problems discussed above are increased in the case of hydraulic apparatuses intended to operate with high pressure differentials, for example in the case of geared pumps for pressure differentials greater than a few tens of bars, and even more for pressures greater than 80-100 bars. 
         [0023]    International patent application WO 2008/111017 of the same applicants, whose contents are deemed integrally incorporated herein for reference, refers to an improved geared hydraulic apparatus, comprising a pair of meshing gear wheels, mutually rotating in a casing between an inlet side and an outlet side of a fluid having, in use, a flow substantially transverse with respect to the rotational axes of the gear wheels, the meshing gear wheels providing, in the mutual rotation thereof, progressive mutual configurations between the respective cooperating teeth, there being defined, in at least one of said progressive meshing configurations, in at least one cross-section of the gear wheels, at least one closed fluid entrapment area between respective teeth, said closed fluid entrapment area decreasing until it is substantially cancelled out at and around at least another, separate, progressive meshing configuration between the aforesaid respective cooperating teeth. 
         [0024]    In summary, the behaviour of the gear wheels of according to patent application WO 2008/111017 of the same applicants is such that an area for the entrapment or encapsulation of the fluid which gradually, during the rotational movement of the wheels, is reduced up to being substantially cancelled when the head of a tooth of a gear wheel touches the bottom of a tooth of another wheel, is formed between the teeth of the two wheels which mesh. Such behaviour, as regards with the present description, shall be referred to as “semi-encapsulation”. 
         [0025]    Experiments performed by the applicants on the various meshing solutions to be used in the aforementioned hydraulic apparatus reveal that there is a delimited field of teeth profiles which may be simultaneously efficient for reducing the noise of the pump and at the same time guarantee the possibility of a relatively easy construction, which may contribute to reducing the costs of producing the hydraulic apparatus and in particular positive displacement pumps which apply the principle of “semi-encapsulation”. 
         [0026]    Furthermore, this series of profiles identified specifically has the advantage of high reliability of use, which makes it particularly advantageous in case of use in the positive displacement pumps for high pressures. In this series of profiles, the proportion of teeth which is higher in the previously known solutions allows obtaining a considerably improved performance. 
       SUMMARY OF THE INVENTION 
       [0027]    With the aim of attaining the aforementioned objects, the invention has the object of a gear wheel with a plurality of teeth capable of meshing with the teeth of another corresponding gear wheel, the profile of each tooth of the gear wheel, in transverse section, being defined in the claims that follow. 
         [0028]    In particular, the profile of at least one tooth of one of the two rotors is defined by a spline function passing through a plurality of nodal points having pre-established coordinates, with a tolerance of ± 1/15, more preferably of ± 1/20, and even more preferably ± 1/30 of the height of the tooth of the gear wheel on the theoretical profile defined by the plurality of preferred nodal points. 
         [0029]    The nodal points are defined by a pair of values {X′, Y′} expressed in a system of cartesian coordinates having origin in the centre of the pitch circle of the gear wheel. 
         [0030]    Though it is clear from the description that follows, it is specified that the origin of the system of coordinates x,y is the line of the rotational axis of the wheel of a plane perpendicular to the axis itself, which is coincident with the centre of the pitch circle of the gear wheel itself. 
         [0031]    In the present description, the term “spline function” generally refers to any spline function which does not introduce errors, or a smoothing spline with a smoothing parameter sufficiently small not to introduce considerable errors with respect to the nodal points. 
         [0032]    In a preferred non-limiting embodiment of the present invention, the spline function used is a cubic natural spline function, i.e. an interpolation natural spline function of the third degree. 
         [0033]    Though the natural spline allows some theoretical advantages, the choice of the type of spline is not however binding, in that, depending on the case and for example on the format of the data required by the tooling machines, a man skilled in the art may find it more convenient to use different spline functions or even smoothing splines, also due to the fact that some of these spline functions are commonly available and used in CAD and CAD-CAM systems. 
         [0034]    The gear wheels are advantageously helical, and the face contact of the helical teeth is comprised between 0.4 and 1.2, preferably between 0.5 and 1.2, more preferably between 0.6 and 1.2, more preferably between 0.7 and 1.1, more preferably between 0.8 and 1.1, and even more preferably between 0.9 and 1. In a preferred non-limiting embodiment of the present invention, the face contact of the helical teeth is equal or close to one. 
         [0035]    Advantageously, a gear wheel according to the present invention has a ratio between dimensions of height and pitch circle comprised between 0.5 and 2, preferably between 0.6 and 1.8, more preferably between 0.65 and 1.5, and even more preferably between 0.7 and 1.25. In a preferred non-limiting embodiment of the present invention, the ratio between dimensions of height and pitch circle is close to one. 
         [0036]    The present invention also has the object of a geared hydraulic apparatus comprising a pair of meshing gear wheels having teeth profile of the previously described type. In particular, such hydraulic apparatus may be a hydraulic pump or a hydraulic motor. 
     
    
     
         [0037]    Further characteristics of the invention shall be clear from the description that follows of a preferred embodiment, with reference to the single attached FIGURE, purely provided by way of non-limiting example, which illustrates the profile of a tooth of a gear wheel according to the present invention, compared to the profile of a tooth of the prior art for a gear wheel without encapsulation. 
       
    
    
     DETAILED DESCRIPTION 
       [0038]    Though the description that follows was provided with reference to a pump, the same arguments and considerations may apply to analogous hydraulic motors. 
         [0039]    Now, with reference to the single FIGURE, a gear wheel  10  according to the present invention (illustrated only partly in the FIGURE) is intended for meshing with another corresponding gear wheel (not illustrated) for the use of a rotary positive displacement pump, preferably of the type for high operating pressures, where the pressure differential between intake and delivery are greater than a few tens of bars, more particularly greater than about 50 bars, and even more particularly greater than about 80-100 bar. 
         [0040]    The gear wheel  10  comprises a plurality of teeth  11  with a height H and a profile suitable for meshing with semi-encapsulation with the teeth of the other corresponding gear wheel. 
         [0041]    The profile of the teeth  11  is not describable as a succession of simple geometrical curves, but it may be defined by a cubic natural spline function (even though it is possible, according to the previously indicated terms, to use other spline functions or smoothing spline) passing through a plurality of nodal points  12  defined by a pair of values expressed in a system of cartesian coordinates having origin in the centre of the pitch circle  13  of the gear wheel  10 . 
         [0042]    In any case, the resulting profiles shall be conjugated, if not exactly from an analytical point of view, at least from a practical point of view, and i.e. the profiles must be capable of meshing correctly in the actual use in the hydraulic apparatus subject of the present invention. Regarding this, it is worth pointing out that even in the current art conventional “involute” gear wheels are not obtained according to the “pure” involute geometry, but with a few variations with respect thereto, leading to variously named profiles, such a “K” profile, tip relief, etc. 
         [0043]    Experimentations performed by the applicants led to identifying a series of teeth profiles especially suitable for providing gear wheels with seven, eight, nine or ten teeth each. The actual profile of the teeth  11  may fall within a band of tolerance whose width is of ± 1/15, more preferably ± 1/20, and even more preferably ± 1/30 of the height H of the tooth of the gear wheel. 
         [0044]    The single FIGURE shows a comparison between the profile of the tooth  11  of a gear wheel obtained according to the invention, and the profile of a tooth D of the prior art, drawn in a dash and dot line, designed according to the “non encapsulation” concept. It is immediately observable that the tooth  11  is considerably higher than the tooth D of the prior art, and it is thus understandable how a gear wheel with teeth  11  obtained according to the “semi-encapsulation” principle of the present invention leads to higher positive displacement performance or the gear wheels obtained according to the “non encapsulation” principle, at least due to the fact that a higher number of teeth may be employed considering the same capacity and overall dimension. 
         [0045]    Following are some examples regarding gear wheels of the present invention having a different number of teeth. 
       Example 1 
       [0046]    A gear wheel having a number of teeth equivalent to seven has a theoretical profile of the tooth defined by a cubic natural spline function (which may be replaced by another spline function or smoothing spline if required) passing through a plurality of nodal points defined by a pair of values {X′,Y′} expressed in a system of cartesian coordinates having origin in the centre O of the pitch circle P of the gear wheel. The coordinates of the nodal points are similar to the pairs of values {X,Y} of the list reproduced in the following table 1. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 X 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 −5.29 
                 10.99 
               
               
                   
                 −4.94 
                 11.21 
               
               
                   
                 −4.71 
                 11.37 
               
               
                   
                 −4.49 
                 11.54 
               
               
                   
                 −4.28 
                 11.74 
               
               
                   
                 −4.10 
                 11.98 
               
               
                   
                 −3.94 
                 12.24 
               
               
                   
                 −3.81 
                 12.53 
               
               
                   
                 −3.69 
                 12.86 
               
               
                   
                 −3.58 
                 13.25 
               
               
                   
                 −3.52 
                 13.62 
               
               
                   
                 −3.51 
                 13.84 
               
               
                   
                 −3.52 
                 14.06 
               
               
                   
                 −3.55 
                 14.35 
               
               
                   
                 −3.56 
                 14.61 
               
               
                   
                 −3.55 
                 14.78 
               
               
                   
                 −3.54 
                 14.95 
               
               
                   
                 −3.51 
                 15.12 
               
               
                   
                 −3.44 
                 15.46 
               
               
                   
                 −3.40 
                 15.63 
               
               
                   
                 −3.36 
                 15.79 
               
               
                   
                 −3.30 
                 16.04 
               
               
                   
                 −3.21 
                 16.38 
               
               
                   
                 −3.13 
                 16.62 
               
               
                   
                 −3.06 
                 16.79 
               
               
                   
                 −3.00 
                 16.94 
               
               
                   
                 −2.93 
                 17.09 
               
               
                   
                 −2.76 
                 17.41 
               
               
                   
                 −2.56 
                 17.71 
               
               
                   
                 −2.35 
                 18.01 
               
               
                   
                 −2.09 
                 18.28 
               
               
                   
                 −1.79 
                 18.51 
               
               
                   
                 −1.46 
                 18.70 
               
               
                   
                 −0.93 
                 18.92 
               
               
                   
                 −0.75 
                 18.98 
               
               
                   
                 −0.57 
                 19.03 
               
               
                   
                 −0.38 
                 19.06 
               
               
                   
                 −0.19 
                 19.07 
               
               
                   
                 0.00 
                 19.08 
               
               
                   
                   
               
             
          
         
       
     
       Example 2 
       [0047]    A gear wheel having a number of teeth equivalent to eight has a theoretical profile of the tooth defined by a cubic natural spline function (which may be replaced by another spline function or smoothing spline if required) passing through a plurality of nodal points defined by a pair of values {X′,Y′} expressed in a system of cartesian coordinates having origin in the centre O of the pitch circle P of the gear wheel. The coordinates of the nodal points are similar to the pairs of values {X,Y} of the list reproduced in the following table 2. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 X 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 0.00 
                 19.08 
               
               
                   
                 0.30 
                 19.06 
               
               
                   
                 0.61 
                 19.01 
               
               
                   
                 0.91 
                 18.93 
               
               
                   
                 1.20 
                 18.81 
               
               
                   
                 1.46 
                 18.64 
               
               
                   
                 1.70 
                 18.44 
               
               
                   
                 1.91 
                 18.23 
               
               
                   
                 2.11 
                 18.01 
               
               
                   
                 2.29 
                 17.77 
               
               
                   
                 2.42 
                 17.52 
               
               
                   
                 2.53 
                 17.26 
               
               
                   
                 2.60 
                 17.09 
               
               
                   
                 2.66 
                 16.92 
               
               
                   
                 2.73 
                 16.75 
               
               
                   
                 2.84 
                 16.50 
               
               
                   
                 2.90 
                 16.33 
               
               
                   
                 2.96 
                 16.15 
               
               
                   
                 3.00 
                 15.98 
               
               
                   
                 3.04 
                 15.80 
               
               
                   
                 3.07 
                 15.62 
               
               
                   
                 3.10 
                 15.44 
               
               
                   
                 3.13 
                 15.26 
               
               
                   
                 3.17 
                 14.99 
               
               
                   
                 3.19 
                 14.81 
               
               
                   
                 3.20 
                 14.63 
               
               
                   
                 3.20 
                 13.99 
               
               
                   
                 3.23 
                 13.76 
               
               
                   
                 3.29 
                 13.53 
               
               
                   
                 3.37 
                 13.29 
               
               
                   
                 3.45 
                 13.12 
               
               
                   
                 3.54 
                 12.94 
               
               
                   
                 3.70 
                 12.68 
               
               
                   
                 3.86 
                 12.45 
               
               
                   
                 4.05 
                 12.24 
               
               
                   
                 4.28 
                 12.06 
               
               
                   
                 4.66 
                 11.84 
               
               
                   
                 4.86 
                 11.72 
               
               
                   
                   
               
             
          
         
       
     
       Example 3 
       [0048]    A gear wheel having a number of teeth equivalent to nine has a theoretical profile of the tooth defined by a cubic natural spline function (which may be replaced by another spline function or smoothing spline if required) passing through a plurality of nodal points defined by a pair of values {X′,Y′} expressed in a system of cartesian coordinates having origin in the centre O of the pitch circle P of the gear wheel. The coordinates of the nodal points are similar to the pairs of values {X,Y} of the list reproduced in the following table 3. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 X 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 −4.47 
                 12.27 
               
               
                   
                 −4.34 
                 12.33 
               
               
                   
                 −4.09 
                 12.47 
               
               
                   
                 −3.85 
                 12.62 
               
               
                   
                 −3.64 
                 12.79 
               
               
                   
                 −3.45 
                 12.98 
               
               
                   
                 −3.19 
                 13.37 
               
               
                   
                 −3.03 
                 13.77 
               
               
                   
                 −2.98 
                 13.96 
               
               
                   
                 −2.95 
                 14.14 
               
               
                   
                 −2.91 
                 14.38 
               
               
                   
                 −2.89 
                 14.57 
               
               
                   
                 −2.89 
                 14.76 
               
               
                   
                 −2.88 
                 15.08 
               
               
                   
                 −2.86 
                 15.26 
               
               
                   
                 −2.85 
                 15.44 
               
               
                   
                 −2.83 
                 15.62 
               
               
                   
                 −2.80 
                 15.80 
               
               
                   
                 −2.77 
                 15.98 
               
               
                   
                 −2.73 
                 16.16 
               
               
                   
                 −2.68 
                 16.34 
               
               
                   
                 −2.62 
                 16.51 
               
               
                   
                 −2.55 
                 16.68 
               
               
                   
                 −2.48 
                 16.85 
               
               
                   
                 −2.41 
                 17.02 
               
               
                   
                 −2.34 
                 17.19 
               
               
                   
                 −2.28 
                 17.36 
               
               
                   
                 −2.21 
                 17.53 
               
               
                   
                 −2.13 
                 17.70 
               
               
                   
                 −1.97 
                 18.01 
               
               
                   
                 −1.82 
                 18.22 
               
               
                   
                 −1.64 
                 18.41 
               
               
                   
                 −1.44 
                 18.58 
               
               
                   
                 −1.22 
                 18.73 
               
               
                   
                 −1.00 
                 18.86 
               
               
                   
                 −0.77 
                 18.97 
               
               
                   
                 −0.52 
                 19.05 
               
               
                   
                 −0.26 
                 19.06 
               
               
                   
                 0.00 
                 19.08 
               
               
                   
                   
               
             
          
         
       
     
       Example 4 
       [0049]    A gear wheel having a number of teeth equivalent to ten has a theoretical profile of the tooth defined by a cubic natural spline function (which may be replaced by another spline function or smoothing spline if required) passing through a plurality of nodal points defined by a pair of values {X′,Y′} expressed in a system of cartesian coordinates having origin in the centre O of the pitch circle P of the gear wheel. The coordinates of the nodal points are similar to the pairs of values {X,Y} of the list reproduced in the following table 4. 
         [0000]    
       
         
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 X 
                 Y 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 −4.16 
                 12.80 
               
               
                   
                 −4.02 
                 12.86 
               
               
                   
                 −3.89 
                 12.92 
               
               
                   
                 −3.70 
                 13.03 
               
               
                   
                 −3.52 
                 13.15 
               
               
                   
                 −3.41 
                 13.24 
               
               
                   
                 −3.25 
                 13.38 
               
               
                   
                 −3.12 
                 13.53 
               
               
                   
                 −3.01 
                 13.68 
               
               
                   
                 −2.92 
                 13.83 
               
               
                   
                 −2.84 
                 14.03 
               
               
                   
                 −2.75 
                 14.33 
               
               
                   
                 −2.73 
                 14.44 
               
               
                   
                 −2.70 
                 14.65 
               
               
                   
                 −2.69 
                 14.75 
               
               
                   
                 −2.68 
                 14.96 
               
               
                   
                 −2.67 
                 15.19 
               
               
                   
                 −2.65 
                 15.37 
               
               
                   
                 −2.61 
                 15.63 
               
               
                   
                 −2.56 
                 15.89 
               
               
                   
                 −2.52 
                 16.15 
               
               
                   
                 −2.46 
                 16.41 
               
               
                   
                 −2.39 
                 16.66 
               
               
                   
                 −2.30 
                 16.92 
               
               
                   
                 −2.20 
                 17.16 
               
               
                   
                 −2.09 
                 17.40 
               
               
                   
                 −1.97 
                 17.64 
               
               
                   
                 −1.86 
                 17.88 
               
               
                   
                 −1.79 
                 18.02 
               
               
                   
                 −1.67 
                 18.22 
               
               
                   
                 −1.54 
                 18.41 
               
               
                   
                 −1.38 
                 18.57 
               
               
                   
                 −1.19 
                 18.72 
               
               
                   
                 −0.99 
                 18.83 
               
               
                   
                 −0.78 
                 18.93 
               
               
                   
                 −0.56 
                 19.00 
               
               
                   
                 −0.34 
                 19.06 
               
               
                   
                 −0.12 
                 19.07 
               
               
                   
                 0.00 
                 19.08 
               
               
                   
                   
               
             
          
         
       
     
         [0050]    Once the pitch between the meshing gear wheels of the positive displacement pump, or one of the characteristic circles of the wheels, for example the pitch circle or the head diameter are known or set, it is possible to obtain the values of coordinates {X′,Y′} starting from the pairs of values {X,Y} mentioned above by using simple conversion calculations. This allows obtaining values representing the points of the profiles of the teeth of the gear wheels which may be used in conjunction with a machine for cutting gear wheels of the known type, in particular for controlling the trajectory of a tool of a numerical control machine. 
         [0051]    The production (and design) tolerance of the gear wheels must be such to guarantee that the profile of the cut teeth is comprised in a band of tolerance of ± 1/15, more preferably ± 1/20, and even more preferably ± 1/30 of the height of the tooth of the gear wheel. 
         [0052]    Obviously, without prejudice to the principle of the invention, the construction details and the embodiments may widely vary with respect to what has been described and illustrated purely by way of example, without departing from the scope of protection of the present invention.