PSYCHOACOUSTIC GEAR TOOTH FLANK FORM MODIFICATION

Abstract

A method of producing a tooth flank surface on gear teeth by controlled removal of stock material from a work gear with a tool with the work gear and the tool being movable with respect to one another along and/or about a plurality of axes. The tool and work gear are engaged with one another and then moved relative to one another in a generating motion along and/or about the plurality of axes. Stock material is removed from the work gear to produce the tooth surface on the work gear. The generating motion along and/or about the plurality of axes comprises motion along and/or about at least one of the axes with the motion being defined by a function having a first level component and a second level component. The first level component defining a maximum flank form deviation amplitude for each tooth of the work gear, and the second level component defining a modification of the tooth surface of each tooth of the work gear.

Claims

1. A method of producing a tooth flank surface on gear teeth by controlled removal of stock material from a work gear with a tool, said work gear and said tool being movable with respect to one another along and/or about a plurality of axes, said method comprising: engaging said tool with said work gear, moving said tool and said work gear relative to one another in a generating motion along and/or about said plurality of axes, removing stock material from the work gear to produce the tooth surface on said work gear, wherein the generating motion along and/or about said plurality of axes comprises motion along and/or about at least one of the axes, said motion being defined by a function comprising a first level component and a second level component, said first level component defining a maximum flank form deviation amplitude for each tooth of the work gear, and said second level component defining a modification of the tooth surface of each tooth of the work gear.

2. The method of claim 1 wherein the first level component comprises at least one of a cosine function and a normal distribution.

3. The method of claim 1 wherein the second level component is determined along the path of contact of each tooth of the work gear.

4. The method of claim 3 wherein determination of the second level component includes tooth to tooth magnitudes obtained from the first level component.

5. A method of producing a tooth flank surface on gear teeth by controlled removal of stock material from a work gear with a tool, said method comprising: providing a gear producing machine, said machine having plurality of axes and comprising a work piece spindle rotatable about a work axis A and a tool spindle rotatable about a tool axis C, said work piece spindle and said tool spindle being movable with respect to one another along at least one of linear axes X, Y and Z and about a pivot axis B, providing a workpiece on said work piece spindle, providing a tool on said tool spindle, engaging said tool with said work gear, moving said tool and said work gear relative to one another in a generating motion comprising a generating roll along and/or about said plurality of axes, removing stock material from the work gear to produce the tooth surface on said work gear, wherein the generating motion along and/or about said plurality of axes comprises a modified motion along and/or about at least one of the axes, said modified motion being defined by a function comprising a first level component and a second level component, said first level component defining a maximum flank form deviation amplitude for each tooth of the work gear, and said second level component defining a modification of the tooth surface of each tooth of the work gear.

6. The method of claim 5 wherein said modified motion comprises motion along and/or about at least one of: angular motion about workpiece axis A, linear motion X in the direction of workpiece axis A, and linear motion Y in a direction parallel to pivot axis B.

7. The method of claim 6 wherein linear motion Y is in a vertical direction.

8. The method of claim 5 wherein the first level component comprises at least one of a cosine function and a normal distribution.

9. The method of claim 5 wherein the second level component is determined along the path of contact of each tooth of the work gear.

10. The method of claim 9 wherein determination of the second level component includes tooth to tooth magnitudes obtained from the first level component.

11. The method of claim 10 wherein the tooth to tooth magnitudes are applied to a tooth bound modification function comprising at least one of: a first order function, a sinusoidal function, and a third order function.

12. The method of claim 5 further comprising introducing a dwell section in the second level function, said dwell section being located at least at a center of roll position of the generating roll, wherein within said dwell section, the second level function has an amplitude of zero.

13. The method of claim 12 wherein the dwell comprises at least one of a toe dwell section and a heel dwell section, with the toe dwell section being located between the center of roll position and a toe end of a tooth and the heel dwell section being located between the center of roll position and a heel end of the tooth.

14. The method of claim 13 wherein each of the toe dwell section and the heel dwell section extend between 0° and 4° of the generating roll.

15. The method of claim 5 wherein said tool comprises a grinding wheel.

16. The method of claim 5 wherein the first level component comprises a normal distribution and the second level component comprises a sinusoidal function.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] FIG. 1 schematically illustrates a six-axis free-form bevel gear grinding machine.

[0020] FIG. 2 shows a three-dimensional view of a bevel gear tooth.

[0021] FIG. 3 illustrates modified material removal along the path of tooth contact showing a first order function, a sinusoidal function and a third order function.

[0022] FIG. 4 shows a coordinate measurement result of one tooth having a sinusoidal second level function.

[0023] FIG. 5 shows a normal distribution as a first level function.

[0024] FIG. 6 is a simplified two-dimensional depiction of a pressure angle change.

[0025] FIG. 7 shows a split sine function with different frequency and amplitude in the two sections.

[0026] FIG. 8 shows the second level function of FIG. 7 with toe and heel dwell sections around center of roll.

DETAILED DESCRIPTION OF THE INVENTION

[0027] The terms “invention,” “the invention,” and “the present invention” used in this specification are intended to refer broadly to all of the subject matter of this specification and any patent claims below. Statements containing these terms should not be understood to limit the subject matter described herein or to limit the meaning or scope of any patent claims below. Furthermore, this specification does not seek to describe or limit the subject matter covered by any claims in any particular part, paragraph, statement or drawing of the application. The subject matter should be understood by reference to the entire specification, all drawings and any claim below. The invention is capable of other constructions and of being practiced or being carried out in various ways. Also, it is understood that the phraseology and terminology used herein is for the purposes of description and should not be regarded as limiting.

[0028] The use of “including”, “having” and “comprising” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. The use of letters to identify elements of a method or process is simply for identification and is not meant to indicate that the elements should be performed in a particular order. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise and the term “and/or” includes any and all combinations of one or more of the associated listed items.

[0029] Although references may be made below to directions such as upper, lower, upward, downward, rearward, bottom, top, front, rear, etc., in describing the drawings, these references are made relative to the drawings (as normally viewed) for convenience. These directions are not intended to be taken literally or limit the present invention in any form. In addition, terms such as “first”, “second”, “third”, etc., are used to herein for purposes of description and are not intended to indicate or imply importance or significance unless explicitly stated.

[0030] The details of the invention will now be discussed with reference to the accompanying drawings which illustrate the invention by way of example only. In the drawings, similar features or components will be referred to by like reference numbers. The size and relative sizes of certain aspects or elements may be exaggerated for clarity or detailed explanation purposes. For a better understanding of the invention and ease of viewing, doors, casings, internal or external guarding, etc. may have been omitted from the drawings.

[0031] The inventive modifications are applicable to a workpiece gear manufactured by a generating method. If both members are generated, then the surface scattering can be applied to both members. In case of a Formate bevel gearset, which has a generated pinion and a non-generated ring gear, the modification can only be applied to the generated pinion.

[0032] The inventive modification is applied in at least two levels. The first and higher level controls the maximal flank form deviation magnitude for each individual tooth. The first level modification control is preferably defined by a cosine function or by a normal distribution. Also other mathematical functions (e.g. higher order functions), sine functions or random distributions may be applied to the first level modification.

[0033] The second and lower level controls the modification on the individual tooth surface itself. The second level may be defined as a first order, a third order and/or a sinusoidal function. Also, other higher order functions as well as cosine functions, normal distributions or random distributions may be utilized. The second level modifications on each single tooth are not conducted by common spiral angle and pressure angle corrections, but with roll position dependent functions, which are developed based on the center point of the flanks. Center point developed modifications will not show tooth thickness or indexing errors.

[0034] The inventive machining process modifies a single axis motion or multiple axes motions such as those available on a computer-controlled free-form bevel gear cutting or grinding machine (e.g. US 6,712,566) to superimpose the flank form (generated by the basic settings) with small modifications preferably in the single micron range. A principal design scheme of such a machine is shown in FIG. 1 and comprises a six-axis free-form bevel gear grinding machine with a monolithic column as the base structure. The linear axes of motion are X, Y and Z which are preferably mutually perpendicular with respect to one another. The rotational axes of motion are A, B and C. A is the work piece spindle rotation, B is the swing axis (i.e. pivot axis) which adjusts the correct angular inclination between the tool axis and the work piece axis and C is the tool spindle rotation. The modifications are determined such that the average flank form of all the teeth of a gear with flank form scattering will be identical to a gear without any modifications. Due to the roll position dependency of the single tooth corrections, the tooth indexing and tooth thickness will not vary from tooth to tooth, as long as the reference roll position is identical or close to the tooth center point.

[0035] Conventionally, it is known that a change to the work piece axis (A-axis) rotation angle alone is equivalent to a change of the ratio of roll. The ratio of roll modification causes a combined spiral angle and pressure angle change as shown in FIG. 2 which shows a three-dimensional view of a bevel gear tooth. A first order change of the A-axis rotation (depending on the distance of the actual roll position to the center roll position) removes lesser material (as required to machine the nominal flank surface) at the start roll position. The modification begins at the start roll position (heel-root in the drawing) and ends at the end roll position (toe-top in the drawing). There is no modification along the contact line which goes through the mean point of the flank. As the roll angle moves from start roll to center roll this amount becomes smaller and is zero along the contact line between tool and tooth flank at the center roll position. The contact lines between start roll and center roll are schematically shown in FIG. 2. As the roll progresses from center roll to end roll, proportionally more material than required for the nominal flank surface is removed, which is represented by the modified surface defined by the spiral angle modification line and pressure angle modification line of FIG. 2.

[0036] In the invention, one or more of the following machine motion modifications is preferably performed: [0037] A-axis angle modifications (angular movement about the workpiece axis A) [0038] Y-axis position modifications (linear movement in a direction parallel to the pivot axis B which is vertical in the machine configuration of FIG. 1) [0039] X-axis position modifications (linear movement in a direction along axis A).

[0040] The modification axes A, Y and/or X are labeled in the machine structure shown in FIG. 1.

[0041] FIG. 3 shows the modified material removal along the path of contact and perpendicular to the flank surface. In FIG. 3, a first order change of the A-axis is graphically represented. In addition to, or instead of, first order changes, one or more other more complex functions can be realized. For example, third order functions as well as sinusoidal functions may be applied and these functions are also shown in FIG. 3.

[0042] In a first step of the invention, the maximal modification amounts for each individual tooth are calculated in a first level of calculations. FIG. 5 shows a normal distribution as a first level function, which determines the maximal modification amount for each tooth of the pinion or gear which is subject to the modifications. At tooth number 1 the maximal modification amount has a large negative value. From tooth to tooth this amount becomes more positive until it reaches a large positive value at tooth number zm which is at the top of the normal distribution graph. Increasing number of teeth show a reduction of the modification, until a large negative value is reached at tooth number n+1, which is one tooth more than the last tooth and, therefore, is equal to the first tooth..

[0043] The maximal modification amount for one particular tooth is calculated as a cosine function and/or as a normal distribution. FIG. 5 shows symbolically how the tooth form changes from tooth to tooth, defined by the normal distribution. In FIG. 5 only the maximal change of each tooth is shown.

[0044] Cosine First Level Function type: cos [φ]:

[0045] The function should start at φ = -π at the first tooth and end at φ = +π at the last tooth plus 1. At the mean number of teeth (n+2)/2 (does not have to be integer) the argument φ should be zero.

[0046] From these boundary conditions:

[00001]$\text{φ}=-\text{π}\text{if}\text{zi}=1$

[00002]$\text{φ}=\text{+}\text{π}\text{if}\text{zi}=\text{n+1}$

[00003]$\text{φ}=0\text{if}\text{zi}=\text{zm}$

The cosine function becomes:

[00004]$\begin{array}{cc}\cos \left[\text{φ}\right]=\cos \left[\left(\text{n+2}\right)/\left(\text{-2}\right)\cdot \left(2\text{π}/\text{n}\right)+\text{zi}\cdot \left(2\text{π}/\text{n}\right)\right]& \text{­­­(1)}\end{array}$

where: [0047] φ... argument of cosine function [0048] zi... actual tooth [0049] zm... number of teeth at the middle of the function = (n+2)/2 [0050] n... number of teeth of target gear [0051] n+1... number of teeth at the end of the function

[0052] The amplitude of the cosine function in equation (1) changes from -1 to +1. In order to receive an amplitude with the desired correction amount Corr normal to the flank surface, in case of a modified A-axis rotation the cosine function has to be multiplied with the following term:

[00005]$\begin{array}{cc}\text{Corr}/\left(\cos \text{β}\cdot \cos \text{α}\cdot \text{RM}\cdot \text{sin}\text{γ}\right)& \text{­­­(2)}\end{array}$

where: [0053] Corr... correction amount perpendicular to flank surface (user input) [0054] β... spiral angle of member [0055] α... pressure angle of member [0056] RM... mean cone distance [0057] γ... root angle of member

[0058] The maximum A-axis modification amount for a respective tooth becomes:

[00006]$\begin{array}{cc}\begin{array}{l}\text{Δ}{\text{A}}_{\max }\left(\text{zi}\right)=\text{Corr}/\\ \left(\cos \text{β}\cdot \cos \text{α}\cdot \text{RM}\cdot \sin \text{γ}\right)\cdot \cos \left[\left(\text{n+2}\right)/\left(\text{-2}\right)\cdot \left(2\text{π}/\text{n}\right)+\text{zi}\cdot \left(2\text{π}/\text{n}\right)\right]\end{array}& \text{­­­(3)}\end{array}$

where: ΔA.sub.max(zi)... maximal A-axis modification amplitude for tooth zi

[0059] Normal Distribution as First Level Function type: e.sup.-η where: -η... argument of Euler function

[0060] For the present 13 tooth example the function should start at tooth number zi = 1 with ΔA.sub.max = 0.01463 as the threshold value. The threshold value is controlled by the chosen 0.1 multiplier in the exponent which delivers in the present example 1.463% of the Euler function magnitude. This results in a desirable cutoff before the function travels to an infinitely far away zero value. It should also end at tooth number zi = n+1 with ΔA.sub.max = 0.01463 as the threshold value.

[0061] The exponent -η is developed as -η = 0.1•(zi-n/2-1).sup.2 in order to fulfill the following boundary conditions: [0062] with zi = 1 and n = 13 or n/2 = 6.5, it becomes -η = -0.1.square-solid.(1-6.5-1).sup.2 = -4.225 [0063] also with zi = n+1 = 14 and n/2 = 6.5, it becomes -η = 0.1.square-solid.(14-6.5-1).sup.2 = -4.225 [0064] in case of zi = n/2 = 6.5, it becomes -η = 0With these definitions, the normal distribution becomes:

[00007]$\begin{array}{cc}{\text{e}}^{-0.1{\left(\text{zi}-\text{n}/2-1\right)}^2}\left(\text{e}\text{.g}\text{. for zi}=1.fwdarw.{\text{e}}^{-0.1{\left(6.5\right)}^2}=0.01463\right)& \text{­­­(4)}\end{array}$

[0065] In order to achieve a positive maximum of 1.0 and a minimum of (-1.0+2 • 0.02732) = -0.945 the Euler function is multiplied by 2 and shifted in ordinate direction by 1.0:

[00008]$\begin{array}{cc}2\cdot e^{-0.1{\left(zi-\frac{n}{2}-1\right)}^2}-1& \text{­­­(5)}\end{array}$

[0066] In order to receive an amplitude with the desired correction amount Corr normal to the flank surface, in case of a modified A-axis rotation the cosine function has to be multiplied with the following term from equation 2: Corr/(cosβ • cosα • RM • sinγ)

[0067] The maximum A-axis modification amount for a respective tooth becomes:

[00009]$\begin{array}{cc}\begin{array}{l}\text{Δ}{\text{A}}_{\max }\left(\text{zi}\right)=\text{Corr}/\\ \left(\cos \text{β}\cdot \cos \text{α}\cdot \text{RM}\cdot \sin \text{γ}\right)\cdot \left[2\cdot e^{-0.1{\left(\text{zi}-\text{n}/2-1\right)}^2}-1\right]\end{array}& \text{­­­(6)}\end{array}$

[0068] It is significant that both first level functions start at tooth number 1 and end at tooth number n+1. If the function would end at the last tooth, number n, then the last tooth and the first tooth would receive the same modification which is not ideal with respect to the scattering effect. The resulting function from equation 5 is graphically shown in FIG. 5. The function will reach the -1.0 magnitude in the positive and negative infinity. In order to design a useful normal distribution, the threshold at the desired starting and ending point of the function has to be defined.

[0069] The second level A-axis modifications are preferably determined along the path of contact of individual teeth and use the first level tooth to tooth magnitudes ΔA.sub.max applied to a tooth bound modification function. Three example functions are shown in FIG. 3.

[0070] The first order function boundary conditions are: [0071] qs ≤ qj ≤ qe [0072] if qj = qs => amplitude ΔA(zi,qj) = +1.0 [0073] if qj = qe => amplitude ΔA(zi,qj) = -1.0with these boundary conditions the second level first order function (FIG. 3) becomes:

[00010]$\begin{array}{cc}\text{Δ}\text{A}\left(\text{zi,qj}\right)=\text{Δ}{\text{A}}_{\max }\left(\text{zi}\right)\cdot 2\cdot \left(\text{qj}-{\text{q}}_0\right)/\left(\text{qs}-\text{qe}\right)& \text{­­­(7)}\end{array}$

where: ΔA(zi,qj)... roll position depending ordinate value of the second level functions

[0074] The sinusoidal function boundary conditions are: [0075] qs ≤ qj ≤ qe [0076] if qj = qs => amplitude ΔA(zi,qj) = 0.0 [0077] if qj = qe => amplitude ΔA(zi,qj) = 0.0 [0078] if qj = q.sub.0 => amplitude ΔA(zi,qj) = 0.0 [0079] maximal amplitude between qs and q.sub.0 => +1.0 [0080] maximal amplitude between q.sub.0 and qe => -1.0with these boundary conditions the second level sinusoidal function (FIG. 3) becomes:

[00011]$\begin{array}{cc}\text{Δ}\text{A}\left(\text{zi},\text{qj}\right)=\text{Δ}{\text{A}}_{\max }\left(\text{zi}\right)\cdot \sin \left[2\text{π}\cdot \left(\text{qj}-{\text{q}}_0\right)/\left(\text{qs}-\text{qe}\right)\right]& \text{­­­(8)}\end{array}$

[0081] The third order function boundary conditions are: [0082] qs ≤ qj ≤ qe [0083] if qj = qs => amplitude ΔA(zi,qj) = +1.0 [0084] if qj = qe => amplitude ΔA(zi,qj) = -1.0 [0085] if qj = q.sub.0 => amplitude ΔA(zi,qj) = 0.0with these boundary conditions the second level third order function (FIG. 3) becomes:

[00012]$\begin{array}{cc}\text{Δ}\text{A}\left(\text{zi},\text{qj}\right)=\text{Δ}{\text{A}}_{\max }\left(\text{zi}\right)\cdot \left[8\cdot {\left(\text{qj}-{\text{q}}_0\right)}^3/{\left(\text{qs}-\text{qe}\right)}^3\right]& \text{­­­(9)}\end{array}$

where: [0086] qs... start roll position [0087] q.sub.0... center roll position [0088] qe... end roll position

[0089] Instead of the combined spiral angle and pressure angle change by modifying the A-axis position as discussed above, a separate pressure angle change is also possible and can be applied solely or in addition to the A-axis change. The mechanism to create a pressure angle change requires a combined Y-axis position and A-axis rotational modification as shown in the two-dimensional drawing in FIG. 6 which shows a simplified representation in order to explain a pressure angle change. This change is achieved by small rotation of the work axis (A-axis) and a connected Y-axis move. The Y-axis move is calculated such that the tool profile follows the tooth slot center line, thereby achieving a pressure angle change.

[0090] Analogous to the sole A-axis modification described above, the first level function is either a cosine function (equations (10) and (11)) and/or a normal distribution (equations (12) and (13)). In case of a pressure angle modification two functions for the level 1 modifications have to be defined, one for the A-axis modification and a second one for the Y-axis modification: [0091] Cosine: [0092] Normal Distribution: where: [0093] Δα... angular correction amount to pressure angle (user input) [0094] ΔA*.sub.max(zi)... maximal A-axis modification amplitude for tooth zi [0095] ΔY.sub.max(zi)... maximal Y-axis modification amplitude for tooth zi

[0096] For the separate pressure angle modification, the second level function can be a first order function, a sinusoidal function and/or a third order function. Here, only the example of a preferred sinusoidal function is presented for the two addressed axes A and Y:

[00017]$\begin{array}{cc}\text{Δ}{\text{A}}^{\ast }\left(\text{zi},\text{qj}\right)=\text{Δ}{\text{A}}^{\ast }{}_{\max }\left(\text{zi}\right)\cdot \sin \left[2\text{π}\cdot \left(\text{qj}-{\text{q}}_0\right)/\left(\text{qs}-\text{qe}\right)\right]& \text{­­­(14)}\end{array}$

[00018]$\begin{array}{cc}\text{Δ}\text{Y}\left(\text{zi},\text{qj}\right)=\text{Δ}{\text{Y}}_{\max }\left(\text{zi}\right)\cdot \sin \left[2\text{π}\cdot \left(\text{qj}-{\text{q}}_0\right)/\left(\text{qs}-\text{qe}\right)\right]& \text{­­­(15)}\end{array}$

where: [0097] ΔA*(zi,qj)... roll position depending ordinate value of the second level functions - [0098] ΔY(zi,qj)... roll position depending ordinate value of the second level functions

[0099] An additional modification using the X-axis may be conducted as the only modification, or in combination with A-axis and/or Y-axis modifications. Analogous to the sole A-axis modifications discussed above, the first level function is either a cosine function (equation (16)) and/or a normal distribution (equation (17)). [0100] Cosine: [0101] Normal Distribution: where: [0102] ΔX... X-axis correction amount (user input) [0103] ΔX.sub.max(zi)... maximal X-axis modification amplitude for tooth zi

[0104] The second level function can be a first order function, a sinusoidal function and/or a third order function. Here, only the example of most preferred sinusoidal function is shown.

[00021]$\begin{array}{cc}\text{Δ}\text{X}\left(\text{zi},\text{qj}\right)\text{=Δ}{\text{X}}_{\max }\left(\text{zi}\right)\cdot \sin \left[2\text{π}\cdot \left(\text{qj}-{\text{q}}_0\right)/\left(\text{qs}-\text{qe}\right)\right]& \text{­­­(18)}\end{array}$

where:ΔX(zi,qj)... roll position depending ordinate value of the second level functions

[0105] A Y-axis modification may also be conducted as the only modification preferably in accordance with equation (15). Analogous to the sole X-axis modifications above, the first level function is either a cosine function (equation (16)) and/or a normal distribution (equation (17)). The second level function can be a first order function, a sinusoidal function and/or a third order function.

[0106] In order to provide second level tooth bound functions which can be optimized and adjusted regarding their amplitude and their wave length, the following equation (19) was developed. This equation only applies to the sinusoidal flank form modification: [0107] qm = (qs+qe)/2 [0108] qs ≤ qj < qm => f = f.sub.Toe (user input) ; Amp = A.sub.Toe (user input) [0109] qm ≤ qj < qe => f = f.sub.Heel (user input) ; Amp = A.sub.Heel (user input) where: [0110] qm.. mean roll position [0111] tToe... frequency toe section [0112] fHeel... frequency heel section [0113] f... actual frequency [0114] AToe... amplitude in toe section [0115] AHeel... Amplitude in heel section [0116] Amp... actual amplitude

[0117] A visualization of the control parameters for the wavelength and the amplitude is given in FIG. 7 which shows a diagram with a split sine function. The first half of the function begins at qs and ends at qm. This first half function has an amplitude of 0.6 and a frequency of 0.8 (extended wave length). The second half of the function begins at qm and ends at qe. This second half function has an amplitude of 1.3 and a frequency of 1.2 (reduced wave length). The graph of FIG. 7 is based on the definition that a standard sine function has an amplitude of 1.0 and a frequency of 1/(2π) equal a wave length of 2π.

[0118] Between toe and midface, the frequency factor f.sub.Toe is 0.8 (longer wavelength). Between midface and heel, the frequency factor f.sub.Heel is 1.2 (shorter wavelength). At this point the mean roll position is introduced. The center roll position is in case of hypoid pinions not at the geometrical center of the flanks. The mean roll position can assure a more centric second level modification function.

[0119] Preferably, a dwell is introduced at, and preferably adjacent to, the center of the second level function. The second level function has a zero amplitude at midface (center of the face width) at the center of roll position of the generating roll. In a coordinate measurement, the grid center point is used to determine the tooth to-tooth indexing error. In most practical cases, the grid center point will not match exactly the center of roll position but has a slightly different position. In order to avoid the introduction of tooth-to-tooth indexing errors, it is preferred that the second level function has a zero amplitude at the location of the measurement grid center point. This is preferably achieved with a toe dwell section and a heel dwell section. FIG. 8 shows the toe dwell section ΔToe between center of roll (midface) and toe and the heel dwell section ΔHeel between center of roll and the heel. When the machining takes place within the two dwell sections, the second level function is effectively turned off and no modification to the flank surface is machined (second level function has zero amplitude within the dwell sections). The preferred amounts of the toe and heel dwell sections are between 0° and 4° of roll.

[0120] Equation (19) can be applied to the sole A-axis modification (equation (8)), to the pressure angle modification (equation (14) and (15)) and to the X-axis modification (equation (18)).

[0121] FIG. 4 shows an example of a sinusoidal flank form modification of a generated bevel pinion represented on a symbolized three-dimensional tooth with a 9×5 surface point measurement grid.

[0122] The basic data of the bevel gear set of which the pinion represented in FIG. 4 is a member is as follows:

TABLE-US-00001 Method... Generated Pinion - Non-Generated Ring Gear Number of Teeth Pinion... 13 Number of Teeth Gear... 41 Transverse Module at Heel... 5.00 mm Spiral Angle Pinion... 45° Spiral Angle Gear... 29° Pressure Angle Pinion convex... 22° Pressure Angle Pinion Concave... 18° Pressure Angle Gear convex... 18° Pressure Angle Gear Concave... 22° Hypoid Offset... 25.4 mm Nominal Cutter Diameter... 127 mm (5 inch)

[0123] In the example, a grinding wheel is rotated about axis C (FIG. 1) and is moved relative to a workpiece so as to engage the tool with tooth surfaces of the workpiece (e.g. opposing tooth surfaces of a tooth slot). The grinding wheel and workpiece are moved relative to one another in a generating motion (i.e. roll) wherein the workpiece rolls relative to the grinding wheel (representing a tooth of the theoretical generating gear) to finish the profile and lead of the workpiece tooth surface. During the generating roll, the computer controlled (e.g. CNC) free form machine (e.g. FIG. 1) changes its axes positions to direct the grinding wheel and workpiece along the appropriate paths of motion relative to one another to perform the generating roll to produce the desired tooth surface modifications. In the example of FIG. 4, tooth surface modifications are introduced by a modification to the A-axis (workpiece axis in FIG. 1) defined by a normal distribution as a first level function and a sinusoidal function as a second level function. An example of such an A-axis modification being defined by equation (8).

[0124] The convention used in FIG. 4 is consistent with the standard output of coordinate measurements for gear metrology. The path of contact is drawn from heel-root to toe-top on the concave tooth flank and from the heel-top to the toe root on the convex tooth flank. The flat planes are the nominal flanks and the wobbled surfaces represent the modified surfaces. The sine function can be recognized along the path of contact. Along the contact line direction all modification values are equal, which leads to the three-dimensional modification function.

[0125] FIG. 4 shows that the modifications are zero in the flank center and at the entrance and exit points. The maximal flank form deviation amplitude is marked in FIG. 4 as the amplitude of the sine function. In spite of sizable sine function amplitude of 10 microns, the corner point deviations have desirable low amounts between zero and 3 microns. The tooth contact sweeps along the path of contact from entrance to exit. The instantaneous contact area is a line or a slim ellipse which is oriented in contact line direction. The active contact length in path of contact direction under light load is between the maximum and minimum point of the sine function (area with hash marks in the upper graphic labeled concave flank). The active contact area only covers about 50% of the tooth flank. The reason is the neighboring tooth pairs carry the load before and after these transfer points. In case of the sine function modification, the tooth mesh impact (marked in FIG. 4) happens in the area of zero slopes. Other functions, like the first order modifications show a slope in the impact area which worsens the impact condition. The sinusoidal tooth form variation changes the impact timing from tooth to tooth without worsening the impact conditions compared to the nominal flank form. This, in addition to the fact that the flank center point always stays unmodified, presents the optimal condition to achieve a psychoacoustic noise reduction without causing indexing errors or strength reducing edge contact.

[0126] Preferred results of the inventive manufacturing method are achieved when the first level tooth to tooth control of the maximum amount follows a normal distribution and when the second level control of the individual tooth modification follows a sinusoidal function. Sinusoidal second level modifications only cause very small tooth corner point deviations to a range of 5 microns, compared to twice this amount with the state of the art method. Furthermore, the second level modifications are driven by the roll position, relative to the center roll position or relative to any chosen roll position (for example the mean roll position). This means that at the center roll position (or the mean roll position) there will be the original flank surface present. Because the tooth spacing (or indexing) as well as the tooth thickness are measured at the tooth center point, the inventive machining method will not cause any indexing or tooth thickness errors.

[0127] A closer study of FIG. 4 makes it also evident that a sinusoidal function which is oriented in path of contact direction (= direction of contact movement), will reduce the risk of edge contact and compensate for small misalignments between pinion and gear.

[0128] Although the invention has been discussed with respect to modifications in the A, Y and/or X directions of motion, the invention is also applicable to the B and/or Z directions of motion. For grinding of bevel gears, the rotation of the tool about axis C is independent from all other motions and the generating method is not dependent on the C-axis rotational motion. Therefore, the present invention is not applicable to the C-axis motion.

[0129] While the inventive method has been discussed with respect to grinding, other generating processes such as cutting, skiving and grinding-from-solid may be utilized to produce the inventive modified flank forms. Furthermore, the inventive method is applicable to machining processes wherein both opposing tooth flank surfaces of a tooth slot are machined simultaneously or wherein only one tooth flank surface of a tooth slot is machined at a time.

[0130] While the invention has been described with reference to preferred embodiments it is to be understood that the invention is not limited to the particulars thereof. The present invention is intended to include modifications which would be apparent to those skilled in the art to which the subject matter pertains without deviating from the spirit and scope of the appended claims.