ENHANCED BI-HELICAL TOOTHED WHEEL WITH VARIABLE HELIX ANGLE AND NON-ENCAPSULATING TOOTH PROFILE FOR HYDRAULIC GEAR

Abstract

Bi-helical toothed wheel (1) with non-encapsulating profile (4) for hydraulic gear apparatuses (2), configured to be bound to a support shaft (5) to form a driving or driven wheel of said hydraulic apparatus and comprising a plurality of teeth (6) extending with variable helix angle with a composite function in the longitudinal or axial direction of the tooth (6), wherein the teeth profile (4) keeps a shape continuity in each cross section thereof, wherein each tooth (6) in the longitudinal direction is divided into five zones: initial (A), proximal intermediate (B), central (C), distal intermediate (D) and terminal (E), where said initial (A), central (C) and terminal (E) zones have a variable helix angle () and said proximal intermediate (B) and distal intermediate (D) zones have a constant helix angle ().

Claims

1. A bi-helical toothed wheel for hydraulic gear apparatuses, configured to be bound to a support shaft to form a driving or driven wheel of said hydraulic apparatus and comprising a plurality of teeth extending with variable helix angle with a continuous function in the longitudinal or axial direction of the tooth, wherein each tooth has a central zone with variable helix angle wherein a helix transition from right-handed to left-handed occurs; wherein the profile of the teeth keeps a shape continuity in each cross section thereof, wherein each tooth comprises at least an initial and a terminal zone, both with variable helix angle, at the two opposite lateral ends of the toothed wheel, wherein in these initial and terminal zones the helix angle decreases when approaching the lateral end of the toothed wheel.

2. The bi-helical toothed wheel according to claim 1, wherein each tooth further comprises a proximal intermediate zone and a distal intermediate zone with constant helix angle which connect the initial and terminal zones to the central zone.

3. The bi-helical toothed wheel according to claim 1, wherein the initial and terminal zones and the central zone with variable helix angle are directly connected by inflection points.

4. The bi-helical toothed wheel according to claim 1, wherein the helix angle is null at the opposite lateral ends of the toothed wheel, that is the profile of each single tooth joins at a right angle the opposite lateral faces of the toothed wheel.

5. The bi-helical toothed wheel according to claim 4, wherein the axial component of the force exchanged by said wheel meshed during use with another identical wheel is null at the lateral ends.

6. The bi-helical toothed wheel according to claim 1, wherein the profile of the teeth is mirrored with respect to a centre plane passing through the transition point between right-handed and left-handed helix.

7. The bi-helical toothed wheel according to claim 1, wherein the helix contact ratio parameter is comprised between 0.6 and 1.

8. The bi-helical toothed wheel according to claim 1, wherein the profile of the teeth of the toothed wheel is a non-encapsulating profile.

9. The bi-helical toothed wheel according to claim 8, wherein the non-encapsulating profile is defined by two arcs of circumference or elliptical crest and bottom portions connected by an involute profile comprised between two truncation diameters.

10. The bi-helical toothed wheel according to claim 9, wherein the lower involute truncation diameter is selected equal to: itr=pp*p1, with p pitch diameter and the parameter p1 comprised between 9.7% and 9.9%.

11. The bi-helical toothed wheel according to claim 9, wherein the upper involute truncation diameter is selected equal to: etr=p+p*p2, with p pitch diameter and the parameter p2 comprised between 12.1% and 12.3%.

12. The bi-helical toothed wheel according to claim 1, wherein the top of each tooth has a cutting edge, defined by a limited thickness projecting with respect to the profile.

13. A hydraulic gear apparatus comprising a pair of toothed wheels according to claim 1.

14. An apparatus according to claim 13, wherein said apparatus is a volumetric pump or a hydraulic gear motor.

15. The bi-helical toothed wheel according to claim 7, wherein the helix contact ratio parameter is comprised between 0.6 and 0.8.

16. The bi-helical toothed wheel according to claim 7, wherein the helix contact ratio parameter is equal to 0.65.

17. The bi-helical toothed wheel according to claim 10, wherein the lower involute truncation diameter is selected equal to: itr=pp*p1, with p pitch diameter and the parameter p1 is equal to 9.8%.

18. The bi-helical toothed wheel according to claim 11, wherein the upper involute truncation diameter is selected equal to: etr=p+p*p2, with p pitch diameter and the parameter p2 is equal to 12.2%.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0049] FIG. 1 shows a perspective and schematic view of a herringbone toothed wheel manufactured according to the prior art;

[0050] FIG. 2 shows a perspective and schematic view of a bi-helical toothed wheel with helices spaced apart from each other, manufactured according to the prior art;

[0051] FIG. 3 shows a perspective view of a bi-helical toothed wheel with variable angle helices, manufactured according to the prior art;

[0052] FIG. 4 shows a plan view of a pair of bi-helical toothed wheels according to the present invention coupled to each other in a hydraulic gear apparatus, for example a volumetric pump;

[0053] FIG. 5 shows a perspective view of a bi-helical toothed wheel with teeth developed along variable angle helices, manufactured according to the present invention;

[0054] FIG. 6 shows a lateral schematic view of a section of the wheel according to the invention;

[0055] FIG. 7 shows a geometric construction of the involute segment of the gear tooth profile;

[0056] FIG. 8a shows an enlarged detail F of FIG. 6;

[0057] FIG. 8b shows a lateral schematic view of two toothed wheels according to the present invention meshed with each other, with identification of the pressure straight line;

[0058] FIG. 8c shows an enlarged detail G of FIG. 8a;

[0059] FIG. 8d shows a geometric schematization of the contact angle of the teeth on the involute profile in the gear composed of toothed wheels according to the present invention;

[0060] FIG. 9a shows a front schematic view of the section of two adjoining teeth during meshing;

[0061] FIG. 9b shows a lateral schematic view of the gear in FIG. 9a;

[0062] FIGS. 10a and 10b show respective geometric schematizations related to a general helix;

[0063] FIGS. 11a and 11b show the development of a turn of the helix for a known herringbone gear;

[0064] FIGS. 12a-12c show respective schematic views of the geometry of the helix development in the toothed wheel according to the invention;

[0065] FIGS. 13 and 14 show geometric schematizations of the helix development in the toothed wheel according to the present invention;

[0066] FIGS. 15a and 15b show respective geometric diagrams to describe the kinematic analysis of the contact point between the helices of the gears on the pitch diameter as the rotation speed thereof varies;

[0067] FIGS. 16a to 19c show respective geometric schematizations for the calculation of the equations of the motion of the contact point between the two helices in the axial direction in three different zones;

[0068] FIGS. 20a and 20b show schematic lateral views of the fluid trapped in the chambers created by the teeth of the gears according to the present invention;

[0069] FIGS. 21a to 24c show respective geometric schematizations for the analysis of the fluid volumes trapped in the chambers created by the teeth of the gears;

[0070] FIG. 25a shows a lateral schematic view of two toothed wheels according to the present invention meshed with each other, with identification of the pressure straight line;

[0071] FIG. 25b shows an enlarged H of FIG. 25a;

[0072] FIG. 26 shows a geometric diagram for the analysis of the forces acting on the teeth of the gears;

[0073] FIG. 27a shows the orientation of the forces acting along the development of the teeth of a toothed wheel in a first gear according to the prior art;

[0074] FIG. 27b shows the orientation of the forces acting along the development of the teeth of a toothed wheel in a second gear according to the prior art;

[0075] FIG. 27c shows the orientation of the forces acting along the development of the teeth of a toothed wheel according to the present invention;

[0076] FIG. 28a shows a perspective schematic view of a gear according to the prior art with identification of the orientation of the exchanged forces;

[0077] FIG. 28b shows typical chipping in the toothed wheel profile according to the prior art of FIG. 28a;

[0078] FIG. 29 shows a perspective schematic view of a gear according to the present invention with identification of the orientation of the exchanged forces;

[0079] FIG. 30a shows the development of the tooth in a toothed wheel according to the prior art identifying the tooth angle with respect to a plane transverse to the lateral end thereof;

[0080] FIG. 30b shows the development of the tooth in a toothed wheel according to the present invention identifying the tooth angle with respect to a plane transverse to the lateral end thereof.

DETAILED DESCRIPTION

[0081] With reference to FIG. 5, a toothed wheel manufactured in accordance with the present invention and of the type with a bi-helical profile is globally and schematically indicated with 1.

[0082] The toothed wheel 1 is particularly, but not exclusively, intended for hydraulic gear apparatuses and the following description will refer to this specific field of application to simplify the exposition thereof.

[0083] For a better understanding of all the aspects of the present invention cylindrical helix is defined as a curve described by an animated point of continuous circular motion and, simultaneously, of various motion having a direction which is perpendicular to the rotation plane.

[0084] Moreover, helix pitch is defined as the distance travelled by the helix generator point in a complete turn in the axial direction.

[0085] Moreover, non-encapsulating profile is defined as a rotor tooth profile which lets the pumped fluid flow between the meshed teeth of the rotors, without trapping it or compressing it or subjecting it to volume variations when sliding inside the pump. In order to obtain this effect, the non-encapsulating profile is able to perfectly match a corresponding profile, without defining cavities interposed between the two meshed profiles. A non-encapsulating profile can be made with heads and feet manufactured as arcs of circle and connected to each other by flanks with circle involute.

[0086] The invention aims at manufacturing a bi-helical toothed wheel which can be used with a wheel of the same type in a gear for a volumetric pump which uses rotors with opposite helix. Advantageously, according to the invention, the wheel 1 has a non-encapsulating profile and such a helix shape as to suppress the angular point in the centre of the traditional herringbone gears manufactured according to the prior art and to suppress at the ends of the rotors a fragile edge which is present in the rotors manufactured according to the prior art.

[0087] The issues related to the machining of rotors having such a profile by means of machine tools are thereby suppressed at source.

[0088] FIG. 4 shows the view from above of the toothed wheel 1 which, coupled to a corresponding toothed wheel 1, defines a gear 2 of the bi-helical non-encapsulating type of a hydraulic apparatus, for example a volumetric pump.

[0089] The toothed wheel 1 is conventionally bound or fitted onto a support shaft 5 to form a driving or driven wheel depending on the role it was assigned in the hydraulic apparatus.

[0090] In the exemplary embodiment described herein by way of non-limiting example, the toothed wheel 1 has a front profile 4 with seven teeth, but nothing prevents to use a different plurality of teeth.

[0091] Advantageously, according to the invention, the bi-helical development 3 of the toothed wheel 1 varies with a continuous function and an arcuate pattern along the axial direction of the tooth, while keeping the shape continuity of the cross section thereof, which coincides with the front profile 4.

[0092] In the present invention, continuous function means a function which is devoid of discontinuity points. This function can be a unique functionfor example a cosinusoidor a composite functionfor example three arcs of circle, possibly connected by rectilinear transitions.

[0093] In other words, the gear 2 has neither any cusp, but nor any acute angle in the central zone thereof. Each corresponding tooth 6 is continuous and devoid of undercuts. Moreover, each corresponding tooth 6 has no cusp in the central zone and this geometry is defined both in the central zone of the gear and at the ends thereof, where the external sections end with a helix angle which is equal to 0.

[0094] This peculiar helix development, which will be further detailed below, allows a pair of rotors to be obtained, in which the pitch and helix angle vary with mathematical regularity and, in particular, in the preferred profile illustrated in the enclosed figures, a transmission continuity with a contact ratio which is equal to 0.65 is ensured.

[0095] In essence, this means that: before two teeth 6 are abandoned other two teeth 6 simultaneously begin to mesh. The contact is continuous and reversible and, depending on the right-handed or left-handed rotation and on the arrangement of the helices, it moves from the centre of the rotor outwards or vice versa.

[0096] Moreover, it should be noted that the teeth profiles are conjugated over the whole length of the rotor, that is the tangents to the profiles in the contact point coincide and the common normal passes through the centre of instantaneous rotation.

[0097] Referring to FIG. 5, a rotor covered by the present invention is shown in detail, which comprises several segments in which the helix angle is variable or constant, in particular: [0098] segment A: with variable helix angle; [0099] segment B: with constant helix angle; [0100] segment C: with variable helix angle; [0101] segment D: with constant helix angle; [0102] segment E: with variable helix angle.

[0103] In essence, the longitudinal development of the tooth of the rotor can be divided into five zones: initial, proximal intermediate, central, distal intermediate and terminal, where the segment A corresponds to the initial zone, the segment B corresponds to the proximal intermediate zone, the segment C corresponds to the central zone, the segment D corresponds to the distal intermediate zone and the segment E corresponds to the final zone.

[0104] It is noted that the division into five segments or zones is not essential for implementing the present invention, being it possible for the initial and distal segments to be directly connected to the central segment by inflection points, without any transition with constant helix angle. In this case, the profile of the tooth crest developed on a flat surface can be a cosinusoid, whereas in the preferred embodiment the terminal and distal segments can follow an arc of circleor other plane curveand the intermediate segments are rectilinear.

[0105] The lengths of the various segments A, B, C, D and E of the rotor are adjusted according to mechanical considerations and vary as the rotor band varies following geometric rules which will be described in detail below.

[0106] The pattern of the profile 4 of the single tooth 6 of the gear 2 is now described.

[0107] Referring to FIG. 6, the profile 4 of the teeth 6 of the gear 2 is formed by a segment with circle involute 8 comprised between two truncation diameters itr, etr which connects two arcs of circumference 7, 9, at the tooth crest and bottom.

[0108] Referring to FIG. 7, the involute segment is characterized by the following formulas:

[00001]$\{\begin{array}{c}\stackrel{\_}{\mathrm{OP}}\cos =R\left(\cos +\mathrm{sen}\right)\\ \stackrel{\_}{\mathrm{OP}}\mathrm{sen}=R\left(\mathrm{sen}-\cos \right)\end{array}$${\stackrel{\_}{\mathrm{OP}}}^2=R^2\left(1+{}^2\right)$$\tan =\frac{\mathrm{sen}+\cos }{\cos -\mathrm{sen}}$$\stackrel{\_}{\mathrm{OP}}=R\sqrt{1+{}^2}=R\left(\cos +\mathrm{sen}\right)\cos +R\left(\mathrm{sen}-\cos \right)\mathrm{sen}$$\{\begin{array}{c}R\left(\sqrt{1+{}^2}\right)\cos =\mathrm{Rx}\\ R\left(\sqrt{1+{}^2}\right)\sin =\mathrm{Ry}\end{array}$

[0109] The involute parameters used in the preferred embodiment of the invention are identified below, with reference to the aforementioned FIG. 6.

[0110] The lower involute truncation diameter is selected equal to:

[00002]$\mathrm{itr}=p-p*p1$ [0111] with p1 comprised between 9.7% and 9.9%, preferably equal to 9.8%.

[0112] The upper involute truncation diameter is selected equal to:

[00003]$\mathrm{etr}=p+p*p2$ [0113] with p2 comprised between 12.1% and 12.3%, preferably equal to 12.2%.

[0114] In both cases, p is the pitch diameter of the wheels 1.

[0115] With regard to the crest and bottom diameters, referring to FIGS. 8a, 8b and 8c, the following relations apply:

[00004]$\mathrm{ext}=p+p*p3$ [0116] with p3 comprised between 19.5% and 20.5%, preferably equal to 20%;

[00005]$\mathrm{int}=p-p*p4$ [0117] with p4 comprised between 19.5% and 20.5%, preferably equal to 20%.

[0118] Referring in particular to FIG. 8a, the top of each tooth 6 has a cutting edge 10, defined by a limited thickness projecting with respect to the profile 4, in order to easily run in the pump body and have narrower size tolerances on the external diameter of the gears.

[0119] In FIG. 8b the meshing between the two toothed wheels is illustrated, identifying with p the pitch diameters, with Cd the deferent circumferences and with Rp the pressure straight line. In the detail G of FIG. 8c the pressure angle is depicted, that is the angle between the pressure straight line and the tangent to the pitch diameters p.

[0120] Preferably, the parameters of the profiles 4 according to the present invention are such that the pressure angle is comprised between 28 and 32, preferably equal to 30.

[0121] From the above-listed design parameters, referring in particular to FIG. 8d, it results that the contact angle for two teeth engaging on the involute segment is equal to about 29.

[0122] With regard to the helix contact ratio, the profile 4 of the rotors 1 is, as already said, such as to avoid the fluid encapsulation between the groove bottom and crest of two meshing teeth. However, it results therefrom that the circumferential contact ratio Rc, defined as the ratio between the involute contact angle and the circumferential pitch subtended angle, is lower than 1; in this specific case equal to about 0.5.

[0123] This substantially means that the teeth which are in contact and transmit the motion detach before the following ones mesh, involving the need to manufacture the gear in the helical shape thereof.

[0124] Referring now in particular to FIGS. 9a and 9b, two adjoining teeth 6 in perpendicular section to the axis of rotation of the rotors 5 are indicated with I and II. The same teeth 6 in perpendicular section to the axis of rotation, at the distance L, are then indicated with I and II.

[0125] In order that the length of the contact line between the teeth 6 keeps constant along the axial direction during the mutual rotation of the rotors it is necessary that I and II are at a distance Lf respectively rotated by: 2.Math./(n teeth)

[0126] In this case it occurs that the helix contact ratio Re is equal to 1.

[0127] This is an ideal operating condition, since the hertzian contact stresses always discharge on the same surface.

[0128] However, based on the design specifications of the pumpwhich in the preferred case is designed to work at pressures of 250 bar and at speeds of 3600 rpmand on the calculations performed on the strength of the gears, it is necessary not to have helix angles which are higher than 45 and hence a helix contact ratio Re was established, which is equal to 0.65 and anyway not lower than 0.6, in particular comprised between 0.6 and 0.8.

[0129] The calculation of the coordinates of the helix development in the three-dimensional space is now described, which allows to determine the inter-tooth space of the rotor by means of a 3D software.

[0130] With reference to FIGS. 10a and 10b, some geometric definitions are recalled beforehand.

[0131] A helix is a curve in the three-dimensional space, depicted by a constant angle line wound around a cylinder.

[0132] The development of a turn of the helix is a straight-line segment corresponding to the hypotenuse of the right triangle having the pitch P and the length of the helix circumference .Math.d as catheti. The slope is determined by the angle comprised between the hypotenuse and the cathetus corresponding to the helix circumference. Hence obtaining: tan ()=P/(.Math.d).

[0133] Referring to FIGS. 11a and 11b, it is known that the development of a turn of the helix for a traditional version of a herringbone gear is a straight-line segment having a slope a. Considering a half gear, a helix contact ratio which is equal to 1 is obtained.

[0134] A similar construction is suggested, in FIGS. 12a and 12b, for the helix development suggested by the present invention. Obviously, in this case the hypotenuse is replaced by a curved line with a variable slope a. FIG. 12c illustrates the pattern of this curve in space.

[0135] For the sake of simplicity only half a gear with helix contact ratio which is equal to 1 is considered hereafter, this in order to simplify formulas and explanations. The right triangle in which the helix development is depicted is used as a base for the following description.

[0136] In the following schematizations, in the depiction of the helix developing triangle, in order to obtain a contact ratio 1, the respective variables p and (.Math.dp) for the horizontal and vertical catheti are replaced by new variables P/n teeth and .Math.dp/n teeth, with P helix pitch and dp pitch diameter used for the calculation of the average helix angle.

[0137] With these new variables, it goes from the graph of FIGS. 11a and 11b to the graph of FIGS. 12a-12c.

[0138] Given the band length, which determines the pump displacement, it occurs that the helix angle deriving from the prefixed geometric construction does not exceed the set design parameter.

[0139] In the construction according to the invention, all the sections which are perpendicular to the axis of rotation are equal to each other.

[0140] The choice of using, for manufacturing the helix, two identical arcs of circle for the development thereof in the rotor central and final part is mainly linked to making all the mathematical relationships deriving therefrom simpler; however, any other curve can also be used; moreover, also the rectilinear segment can be replaced by a curve which connects the two ends. In this specific case, the helix is formed by a straight line tangent to two arcs of circumference having the same radius comprised in an angle which is equal to the design one.

[0141] A variant providing the replacement of the rectilinear segment by a curved segment can be defined for example by a single cosinusoid, which will define in this case both the initial and terminal zones and the central zone, keeping the helix angle always variable along the whole tooth extension.

[0142] Referring to FIG. 13, the helix angle is obtained with the following relationship:

[00006]$\frac{F}{4}\mathrm{sena}=\left[\left(\frac{2}{n\mathrm{teeth}}{R}_e\right)r_p\right]\frac{1}{2}$

[0143] In this specific case, R.sub.e=1; n teeth=7 are used

[00007]$=\mathrm{arc}\mathrm{sen}\left(\frac{r_{p}4}{7F}\right)$

[0144] Referring to FIGS. 15a and 15b, the kinematic analysis of the contact point between the helices of the gears on the pitch diameter upon variation of the rotation speed thereof is now described.

[0145] It is assumed that the contact is punctual and at the pitch diameter.

[0146] The helix development in the plane is depicted by dividing the graph into three distinct zones respectively: [0147] a first zone z1 with variable helix angle; [0148] a second zone z2 with constant helix angle; [0149] a third zone z3 with variable helix angle.

[0150] Replacing on the x-axis and y-axis:

[0151] X.fwdarw.arc of circumference travelled by the generator point of the helix [C]

[0152] Y.fwdarw.space travelled in the axial direction by the generator point of the helix [F]

[0153] When rotating, the two gears keep in contact in a point up to travel the distance F which is equal to the band length.

[0154] The equations of the motion of the contact point between the two helices in the axial direction are written as a composite function of three clearly distinct: zones z1, z2 and z3.

[0155] For the first zone z1, referring to FIGS. 16a and 16b, the following relationships apply:

[00008]$\frac{C}{2}=\left[\left(\frac{2}{n\mathrm{teeth}}{R}_e\right)r_p\right]\frac{1}{2}\mathrm{with}\mathrm{Re}=1.fwdarw.\frac{C}{2}=\left[\left(\frac{2}{7}\right)r_p\right]\frac{1}{2}.fwdarw.\frac{C}{2}=\left[\left(\frac{}{7}\right)r_p\right]$

[0156] Indicating with

[00009]$k=\left(\frac{}{7}\right)$$\frac{C}{2}=\left[{\mathrm{kr}}_p\right]$$C1=1r_p$

[0157] It can be written:

[00010]$\{\begin{array}{cc}1.& F=\left[k{r}_p\mathrm{sen}\right]\\ 2.& C=\left[k{r}_p-k{r}_p\cos \right]\end{array}\mathrm{with}.fwdarw.0<<$$C=r_p$ [0158] by replacing in 2:

[00011]$r_p=\left[k{r}_p-k{r}_p\cos \right]$$r_p=k{r}_p\left(1-\cos \right)$$\begin{array}{cc}=k\left(1-\cos \right).fwdarw.=k-k\cos & 3.1\end{array}$$\cos =\frac{k-}{k}$$=\arccos \frac{k-}{k}$ [0159] by replacing in 1

[00012]$\begin{array}{cc}F1=k{r}_{p}sen\left[\arccos \left(\frac{k-}{k}\right)\right]& 3.2\end{array}$

[0160] Function which is valid for 0<< with max.fwdarw.=

[0161] Recalling 3.1:

[00013]$=k-k\cos \mathrm{replacing}=v$$\max =k-k\cos$$\begin{array}{cc}\max =k\left(1-\cos \right)& 3.3\end{array}$ [0162] the time law of the motion of the contact point between the two meshing helices is now obtained by replacing in 3.2:

[00014]$=t$$\begin{array}{cc}F1\left(t\right)=k{r}_{p}sen\left[\arccos \left(\frac{k-t}{k}\right)\right]& 3.4\end{array}$

[0163] Another important relationship which can be obtained from 3.3 by replacing max=*tmax with

[00015]$t\max =k\left(1-\cos \right)$$\begin{array}{cc}t\max =\frac{k\left(1-\cos \right)}{}& 3.5\end{array}$

[0164] Referring to FIG. 17, by deriving 3.4, 3.6 is obtained, which is the speed at which the point translates parallel to the rotor axis

[00016]$\begin{array}{cc}V1\left(t\right)=\frac{}{t}\left\{k{r}_{p}sen\left[\arccos \left(\frac{k-t}{k}\right)\right]\right\}=\frac{r_p\left(k-t\right)}{\sqrt{\frac{t\left(2k-t\right)}{k^2}}}& 3.6\end{array}$

[0165] By deriving 3.6, 3.7 is obtained, which is the equation of acceleration undergone by the fluid along the axial travel thereof towards the centre or the ends of the gear depending on the rotation direction.

[00017]$\begin{array}{cc}a1\left(t\right)=\frac{}{t}\left\{\frac{r_p\left(k-t\right)}{\sqrt{\frac{t\left(2k-t\right)}{k^2}}}\right\}=-\frac{r_p{}^2}{\sqrt{\frac{t\left(2k-t\right)}{k^2}}}-\frac{r_p\left(k-t\right)\left(\frac{\left(2k-t\right)}{k^2}-\frac{{}^{2}t}{k^2}\right)}{2\sqrt{{\left(\frac{t\left(2k-t\right)}{k^2}\right)}^3}}& 3.7\end{array}$

[0166] With regard to zone z2, referring to FIGS. 18a-18c, it can be written:

[00018]$\{\begin{array}{cc}1.& \tan =\frac{C}{F}\\ 2.& C=r_p\end{array}$

[0167] From 1

[00019]$F=\frac{C}{\tan }\mathrm{by}\mathrm{replacing}2$$F=\frac{r_p}{\tan }\mathrm{con}=t$$\begin{array}{cc}F2\left(t\right)=\frac{r_P}{\tan }t& 3.7\end{array}$${}_2={}_t-2{}_{\max }\mathrm{con}{}_t=\frac{2}{7}=2k$${}_2=2k-2k\left(1-\cos \right)$${}_2=2k\left[1-\left(1-\cos \right)\right]$$\begin{array}{cc}{}_2=2k\cos & 3.8\end{array}$$\begin{array}{cc}F2=\frac{r_P}{\tan }2k\cos & 3.9\end{array}$

[0168] From 3.8:

[00020]$t\max =2k\cos$$\begin{array}{cc}t\max =\frac{2k\cos }{}& 4.\end{array}$

[0169] By deriving 3.7:

[00021]$\begin{array}{cc}V2\left(t\right)=\frac{}{t}\left[\frac{r_p}{\tan }t\right]=\frac{r_p}{\tan }& 4.1\end{array}$

[0170] With regard to zone 3, referring to FIGS. 19a-19c, the same relationships as zone 1 apply, in particular it can be written:

[00022]$\frac{C}{2}=\left[\left(\frac{2}{n\mathrm{teeth}}{R}_e\right)r_p\right]\frac{1}{2}\mathrm{with}\mathrm{Re}=1.fwdarw.\frac{C}{2}=\left[\left(\frac{2}{7}\right)r_p\right]\frac{1}{2}.fwdarw.\frac{C}{2}=\left[\left(\frac{}{7}\right)r_p\right];$$k=\left(\frac{}{7}\right);$$\frac{C}{2}=\left[k{r}_p\right];$$C3=3r_p;$$\{\begin{array}{cc}1.& F=k{r}_p\mathrm{sen}-{\mathrm{kr}}_p\mathrm{sen}\\ 2.& C=k{r}_p\cos -{\mathrm{kr}}_p\cos \end{array}\mathrm{with}.fwdarw.0<<$

[0171] From 2.

[00023]$r_p=k{r}_p\cos -{\mathrm{kr}}_p\cos .fwdarw.r_p=k\cos -k\cos .fwdarw.\cos =\frac{+k\cos }{k}.fwdarw.$$\begin{array}{cc}=\mathrm{ar}\cos \left(\frac{+k\cos }{k}\right)& 4.2\end{array}$

[0172] By replacing in 1.

[00024]$F=k{r}_p\mathrm{sen}-k{r}_p\mathrm{sen}[\mathrm{ar}\cos \left(\frac{+k\cos }{k}\right)\mathrm{with}\mathrm{with}=t$$\begin{array}{cc}F3\left(t\right)={\mathrm{kr}}_p\mathrm{sen}-{\mathrm{kr}}_p\mathrm{sen}.Math.\mathrm{ar}\cos \left(\frac{t+k\cos }{k}\right).Math.& 4.3\end{array}$

[0173] By deriving 4.3 4.4 is obtained

[00025]$\begin{array}{cc}V3\left(i\right)=\frac{}{t}\left\{k{r}_p\mathrm{sen}-{\mathrm{kr}}_{p}sen\left[\mathrm{ar}\cos \left(\frac{t+k\cos }{k}\right)\right]\right\}=\frac{r_p\left(k\cos +\right)}{k\sqrt{1-\frac{{\left(t+k\cos \right)}^2}{k^2}}}& 4.4\end{array}$

[0174] Completing with the equation of acceleration 4.5:

[00026]$\begin{array}{cc}3\left(t\right)=\frac{}{t}\left\{\frac{r_p\left(k\cos +t\right)}{\sqrt{1-\frac{{\left(t+k\cos \right)}^2}{k^2}}}\right\}=\frac{r_p}{k\sqrt{1-\frac{{\left(t+k\cos \right)}^2}{k^2}}}+\frac{r_p{{}^2\left(k\cos +t\right)}^2}{k^3\sqrt{1-{\left(\frac{{\left(k\cos +t\right)}^2}{k^2}\right)}^3}}& 4.5\end{array}$

[0175] The fluid dynamics of the pump is now described to prove the efficiency of the solution described for attenuating vibrations.

[0176] The above geometric considerations point out how a development of the modified helix angle allows vibrations on the fluid to be reduced with respect to the prior art.

[0177] The fluid volumes trapped in the chambers created by the teeth of the gears are brought from the suction zone to the delivery one. Once arrived in the high-pressure zone, these volumes are ejected, being subjected to the compression they undergo because of the interpenetration of the solid tooth with the fluid itself. Being the teeth with helix, what is occurring is some sort of helical extrusion of the fluid.

[0178] Referring to FIG. 20, in order to explain the phenomenon, a section can be considered, which moves in the axial direction before being radially ejected from the pump. The section S moves in the axial direction with the same speed as the contact point of the helices of the two gears on the pitch diameter. Always assuming that the contact is punctual and only on the pitch diameter, (theoretical assumptions to explain the phenomenon), consideration is that the contact point on the helices of the gears continuously moves along the axial direction over an indefinite time: this occurs since at the end of the contact on the helix of a pair of meshed teeth the following one immediately conjugates.

[0179] Referring to FIGS. 21a-c which show in a graph the functions of the fluid section displacement in the axial direction with the related speed and to FIG. 22, how the described solution improves the management of the vibrations induced on the pumped fluid is shown.

[0180] Considering a tank in which the liquid is pumped through a constant-section pipe. The pump flow rate formula can be written:

[0181] With

[00027]$\stackrel{.}{Q}=\frac{m^3}{s}\mathrm{and}=\frac{\mathrm{kg}}{m^3}$

the maximum flow rate

[00028]$Q=\frac{kg}{s}$

is obtained

[0182] From the above-obtained relationships, the speed of the section S being considered is not constant and thus the flow rate is variable over time.

[0183] In order to understand the phenomenon linked to the vibrations induced on the fluid by the pump, consideration can be to have an infinitesimal mass of liquid m1 in the tank shown in FIG. 22, hit by a mass m0 exiting the pump, the mass m1 will change the speed thereof and will be able to do it in two ways: subjected to a small force for a long period of time or subjected to a great force for a short period of time.

[0184] Considering the second Newton's law in its simplest form:

[00029]$F=ma$ [0185] by multiplying both members by t, Ft=mat.fwdarw.Ft=mV.fwdarw.I=p is obtained [0186] with: I=impulse and p=momentum.

[0187] With reference to the two theorems of physics, i.e. the impulse theorem (which applies in the case of a single body) where the momentum variation is equal to the impulse of the force acting on a body and the momentum conservation principle where the total momentum of a system comprised of two particles only subjected to the mutual iteration thereof remains constant, it can be written:

[00030]$p=p_0+p_1=m_0{V}_0+m_1{V}_1$

[0188] Obtaining in the following instant t:

[00031]$p^{}=p_0^{}+p_1^{}=m_0{V}_0^{}+m_1{V}_1^{}$

[0189] Obtaining, according to the momentum conservation principle:

[00032]$p^{}=p=\mathrm{cost}$

[0190] These considerations lead to the following observation: an interaction between two particles produces a momentum exchange, so that the momentum lost by one of the two particles is equal to the momentum gained by the other particle.

[00033]$p=p_0+p_1=\mathrm{cost}$$p^{}=p_0^{}+p_1^{}=\mathrm{cost}$

[0191] By doing a member-to-member subtraction:

[00034]$p_0=-p_1$

[0192] Knowing that:

[00035]$\frac{dp}{dt}=\frac{d\left(\mathrm{mV}\right)}{dt}=m\frac{\mathrm{dV}}{dt}=ma=F$$F_0=-F_1$

[0193] When two particles interact, the force acting on a particle is equal and opposite to the force acting on the other one (which coincides then with the third principle of dynamics).

[0194] From FIGS. 21a-c and from the above formulas it is noted that the single particle is subjected to an abrupt acceleration by colliding with a following one having the same acceleration, with respect to the state of the art where instead the single particle is subjected to an abrupt acceleration by colliding with a following one having a low and constant speed.

[0195] Referring to FIG. 23, the contact line which is created between the rotating gears is now described, assuming the punctual contact on the pitch diameter, and considering that the contiguity occurs on a line which is created for the whole involute arc diagonally with respect to the gear band.

[0196] The calculation of the contact line length for a helical gear with constant helix is now described.

[0197] Indicating with: [0198] Rp: length of the line described by the contact on the involute of the rotating gears on the pressure straight line (direction along which pressure forces are exchanged). [0199] Rs: contact start radius on the involute [0200] Rf: contact end radius on the involute [0201] .sub.s: helix angle calculated for Rs [0202] .sub.ev: contact angle on the involute.

[00036]$\begin{array}{cc}{F}_p=\frac{R_s{}_{ev}}{\mathrm{tg}{}_s}& 4.6\end{array}$$\begin{array}{cc}\mathrm{LC}=\sqrt{R_p^2+F_p^2}& 4.7\end{array}$

[0203] By analysing the contact line for a gear with constant helix with helix contact ratio which is equal to 1 it occurs that it always keeps the same total length, indeed, when a pair of teeth gradually unmeshes the line shortens by the same amount which meshes on the pair of following teeth, as shown in FIGS. 24a-c.

[0204] The above description applies for a gear with helix contact ratio which is equal to 1, the case in which this value becomes lower than 1 is now analysed.

[0205] The contact line falls proportionally to the offset of the tooth following the meshed tooth at the opposed end of the gear band.

[0206] The value of Re selected for the present invention: Re=0.65 is considered.

[0207] This means that the gear front section rotates along the band by an angle which is equal to:

[00037]$\mathrm{Re}=0.65=\frac{13}{20}$

[0208] The offset of the tooth which is adjacent to the opposed end of the band will be thus equal to:

[00038]$\mathrm{Sf}=\frac{20}{20}-\frac{13}{20}=\frac{7}{20}$$=\frac{360}{7}\frac{7}{20}=18$

[0209] How the contact line length changes with respect to the case of Re=1 is now analysed

[00039]$F_p=\frac{R_s\left({}_{ev}-\right)}{\mathrm{tg}{}_s}$

[0210] It must be .sub.ev>0 to have a meshing continuity and always a meshed tooth.

[0211] For the present invention .sub.ev=29 was selected, so it is .sub.ev=11

[0212] The above equations can be used to see an important benefit brought by the following invention with respect to the prior art.

[0213] By recalling the equation 3.4 of the segment with variable helix angle and the equation 3.7 for the segment with constant helix angle:

[00040]$\begin{array}{cc}F1\left(\right)=k{r}_p\mathrm{sen}\left[\arccos \left(\frac{k-t}{k}\right)\right]& 3.4\end{array}$$\begin{array}{cc}F2\left(r\right)=\frac{r_p}{\tan }t& 3.7\end{array}$

[0214] By replacing the value of 11 in the equations 3.4 and 3.7 respectively it is obtained:

[00041]$\begin{array}{cc}F1\left(t\right)=5.85\mathrm{mm}.fwdarw.\mathrm{Variable}\mathrm{helix}\mathrm{segment}& 3.4\end{array}$$\begin{array}{cc}F2\left(t\right)=3.05\mathrm{mm}.fwdarw.\mathrm{Constant}\mathrm{helix}\mathrm{segment}& 3.7\end{array}$

[0215] As it is noted from the obtained numerical values, in the segment with variable helix angle the contact line has a length which is higher than in the segment with constant helix angle.

[0216] Otherwise it can be written that for the segment with variable helix angle the relationship applies:

[00042]$\begin{array}{cc}\max =k\left(1-\cos \right)& 3.3\end{array}$

[0217] 3.4 and 3.7 are rewritten as:

[00043]$\begin{array}{cc}F1\left(t\right)=k{r}_p\mathrm{sen}\left[\arccos \left(\frac{k-\max }{k}\right)\right]& 3.4\end{array}$$F1=k{r}_{p}sen\left[\arccos \left(\frac{k-k\left(1-\cos \right)}{k}\right)\right]\mathrm{by}\mathrm{simplifying}.fwdarw.F1=k{r}_p\mathrm{sen}\left[\arccos \left(\frac{k-k-k\cos )}{k}\right)\right]$$F1=k{r}_p\mathrm{sen}\left[\arccos \left(\frac{k-k-k\cos )}{k}\right)\right].fwdarw.F1=k{r}_{}sen\left[\arccos \left(\cos \right)\right]$$F1=k{r}_p\mathrm{sen}$$\begin{array}{cc}F2=\frac{r_P}{\tan }\max & \left(3.7\right)\end{array}$$F2=\frac{r_p}{\tan }k\left(1-\cos \right)$ [0218] thus:

[00044]$F1>F2.fwdarw.4.8k{r}_p\mathrm{sen}>\frac{r_p}{\tan }k\left(1-\cos \right)$

[0219] Formula 4.8 proves that, for a gear with a certain helix angle and with the selected geometric construction, the segment with variable helix angle has a longer contact line.

[0220] This allows the distribution of the efforts to be improved when the contact line takes the minimum length value thereof (since the helix contact ratio is lower than 1) with respect to the prior art.

[0221] The forces acting on the gear teeth are now analysed.

[0222] Starting with the formula to determine the power transferred to the fluid, from a hydraulic point of view it can be written:

[00045]$\begin{array}{cc}P=\frac{P\stackrel{.}{Q}}{{}_t}.& A\end{array}$ [0223] from a mechanical point of view:

[00046]$\begin{array}{cc}P=C.& B\end{array}$

[0224] Note from A. the power supplied to the fluid, the twisting torque transmitted to the shaft can be obtained with B:

[00047]$C_T=\frac{P\stackrel{}{Q}}{{}_t}$

[0225] The torque transmitted from the driving wheel to the driven wheel will be equal to

[00048]$C_r=\frac{C_T}{2}$

[0226] Referring to FIGS. 25a and 25b, the total force F directed according to the pressure angle t is schematically seen. In this view only a component of the total force F is depicted since the three-dimensional projection thereof sloping perpendicular to the helix angle is not visible.

[00049]$\begin{array}{cc}{C}_r=F_t\frac{D_p}{2}.fwdarw.F_t=\frac{2C_r}{D_p}& 4.9\end{array}$

[0227] Referring to FIG. 26, a complete three-dimensional depiction of the forces acting on the tooth of a toothed wheel with helical teeth can be seen. The force F acts in the direction which is normal to the contact point.

[0228] Indicating with: [0229] F: total force [0230] Ft: tangential component (force component which is useful for power transmission) [0231] Fr: radial component [0232] Fa: axial component

[0233] Between the total force and the components thereof the following relationships are obtained:

[00050]$\begin{array}{cc}{F}_t=F\cos {}_n\cos & 5.\end{array}$$\begin{array}{cc}{F}_r=F\mathrm{sen}{}_n& 5.1\end{array}$$\begin{array}{cc}{F}_a=F\cos {}_{n}sen& 5.2\end{array}$

[0234] Ft can be usually obtained directly from design data from 4.9 and the other forces can be obtained by the following expressions:

[00051]$\begin{array}{cc}{F}_r=F_t\tan {}_t=F_t\frac{\tan {}_n}{\cos }& 5.3\end{array}$$\begin{array}{cc}{F}_a=F_t\tan & 5.4\end{array}$$\begin{array}{cc}F=\frac{F_t}{\cos {}_n\cos }& 5.5\end{array}$

[0235] Referring to FIGS. 27a-c, the direction of the total force in the various points of the gear is evident, in the case of traditional herringbone rotors, prior art rotors with variable helix angle in three zones and the rotors according to the present invention respectively.

[0236] Referring to FIGS. 28a and 28b, it is evident how the direction of the total force has an axial component in the contact point at the ends of the gear, assuming a punctual contact on the pitch diameter, it is evident how this zone is subjected to cyclical stresses and efforts which in certain conditions of use can lead to the chipping ch of this edge.

[0237] At point a there are thus all the components of the total F, whose module is equal to:

[00052]$F=\frac{F_t}{\cos {}_n\cos }$

[0238] Referring to FIG. 29, it is instead noted how this edge b is subjected to a total force F which, besides having a lower module:

[00053]$F=\frac{F_t}{\cos {}_{n}1}$

has not even the axial component: F.sub.a=F.sub.t tan 0=0.

[0239] Moreover, it is evident how the section of the tooth 6 at the ends has a right-angle shape, whereas it is acute in the prior art, as also shown in comparative FIGS. 30a and 30b.

[0240] Furthermore, recalling the kinematic equations of the speed (at the ends) on the contact point of the helices of the gears on the pitch diameter (always using the punctual contact assumption).

[00054]$V1\left(t\right)=\frac{}{t}\left\{k{r}_{p}sen\left[\arccos \left(\frac{k-t}{k}\right)\right]\right\}=\frac{r_p\left(k-t\right)}{k\sqrt{\frac{t\left(2k-t\right)}{k^2}}}$$\underset{n.fwdarw.}{\lim }V1\left(t\right)=\frac{r_p\left(k-t\right)}{k\sqrt{\frac{t\left(2k-t\right)}{k^2}}}=$

[0241] Tending to infinity represents a paradox since dealing with a speed, but this is due to the fact that, in that segment, the gear has a helix development with angle being equal to 0. In the initial segment the rotor can be therefore considered in all respects with straight teeth and the contact in that zone will no longer appear to be punctual but will be distributed on a line whose length can be calculated and is moreover determined by the mechanical features of the material.

[0242] The method for manufacturing the above-described toothed wheel 1 comprises the following steps.

[0243] With reference to FIGS. 12a-12c,

Step 1:

[0244] Determining the equations of the rotations of all the profile sections in the axial direction calculated on the pitch diameter. A series of coordinates (x.sub.i; y.sub.i; z.sub.i) representing the helix path is obtained.

Step 2:

[0245] Making the rotor front profile slide to manufacture the solid model according to the helix path, shown in FIGS. 8b and 8c, with a 3D software.

Step 3:

[0246] Transferring the solid model to CAD-CAM

Step 4:

[0247] Possible thermal hardening treatment and creation of the inter-tooth space using the numerically controlled working station, for example a five-axis machine.

[0248] The invention brilliantly solves the technical problem and achieves several advantages, the first of which is given by the fact that it was allowed to manufacture gears with opposite helix with partially or totally variable helix angle, with non-encapsulating profile and such a shape as to suppress the cusp in the centre of the rotors.

[0249] Moreover, the accurate and continuous opposite slope of the teeth does not generate any axial force which can tend neither to displace the wheel which can thus be incorporated in gears which are devoid of axial compensation, nor to break terminal parts of the teeth.

[0250] In short, the invention allows to manufacture rotors with opposite helix, non-encapsulating profile and such a helix shape as to suppress the angular point in the centre of the rotors themselves and accordingly all the issues related to the machining thereof by means of machine tools.

[0251] Moreover, the invention allows to manufacture gears for hydraulic apparatuses with opposite helix with a partially or totally variable helix angle.

[0252] Obviously, in order to meet contingent and specific requirements, a person skilled in the art will be allowed to bring several modifications and variants to the above-described invention, all falling however within the scope of protection of the invention as defined by the following claims.