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Generation of Wildhaber-Novikov gears

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					                           Engineering Letters, 15:1, EL_15_1_4
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      Generation of Crowned Parabolic Novikov gears
               Somer M. Nacy, Member, IAENG, Mohammad Q. Abdullah, and Mohammed N.Mohammed

      Abstract - The Wildhaber-Novikov gear is one of the            where u i is a variable parameter that determines the
circular arc gears, which has the large contact area between         location of the current point in the normal section and
the convex and concave profiled mating teeth. In (June 28,
                                                                      ai is the parabolic coefficient.
1999), a new geometry of W-N gear with parabolic profile in
normal section has been developed. This paper studies the
generation of rack-cutters for parabolic crowned profiles
with its generation in order to select the requirements of W-
N gears.

Index Terms- crowned parabolic profile, generation of
gears, Novikov gears


                      I. INTRODUCTION
   Circular arc helical gears were proposed by Wildhaber
and Novikov. However, there is a significant difference
between the ideas proposed by the previously mentioned
inventors. Wildhaber’s idea [1] is based on generation of
the pinion and gear by the same imaginary rack-cutter
that provides conjugate gear tooth surfaces that are in line
contact at every instant. Novikov [2] proposed the
application of two mismatched imaginary rack-cutters
that provide conjugated gear tooth surfaces that are in                     Fig. 1- Parabolic profile of rack-cutter in
point contact at every instant. Point contact of Novikov                    normal section.
gears has been achieved by application of two
mismatched rack-cutters for generation of the pinion and
gear, respectively.
                                                                        II. DERIVATION OF PINION TOOTH SURFACE
  There are two versions of Novikov gears (with circular-
arc profile), the first having one zone of meshing, and the           A. Pinion Rack-Cutter Surface ∑ c
other having two zones of meshing. The design of gears
with two zones of meshing was an attempt to reduce high                The derivation of rack-cutter surface ∑ c is based on
bending stresses caused by point contact.                            the following procedure:
  The proposed new version of helical gears is based on              1. The normal profile of ∑ c is a parabola and is
the following ideas [3]:                                             represented in coordinate system S a , Fig. 2-b, by
1. The bearing contact is localized and the contact
                                                                     equations that are similar to (1):
                                                                                [                 ]
stresses are reduced because of the tangency of concave-                                              T
convex tooth surfaces of the mating gears.                           ra (u c ) = u c        2
                                                                                       ac u c 0   1                         (2)
2. The normal section of each rack-cutter is a parabola as           where ac is the parabolic coefficient; u c is the variable
shown in Fig.1. A current point of the parabola is
                                                                     parameter.
determined in an auxiliary coordinate system S i by the
                                                                     2. The normal profile is represented in Sb by matrix
equations
                                                                     equation
xi = u i       y i = ai u i2                                  (1)    rb (u c ) = M ba r a (u c )                            (3)
  Manuscript received January 27, 2007.                              M ba indicates the 4×4 matrix used for the coordinate
  S. M. Nacy is with Al-Khawarizmi College of Engineering,           transformation from coordinate system S a to Sb [4]:
University of Baghdad, Baghdad, Iraq.
Phone 009647901387055                                                3. Consider that rack-cutter surface ∑ c is formed in
e-mail: smnacy@elearningiq.net
  M. Q. Abdullah is with the College of Engineering, University of
                                                                      S c while coordinate system Sb with the normal profile
Baghdad, Baghdad, Iraq.                                              performs a translational motion in the direction a-a of the
  M. N. Mohammed is with Al-Khawarizmi College of Engineering,       skew teeth of the rack-cutter, Fig. 3. Surface ∑ c is
University of Baghdad, Baghdad, Iraq.



                                        (Advance online publication: 15 August 2007)
                           Engineering Letters, 15:1, EL_15_1_4
______________________________________________________________________________________
                                                            components; k a is the unit vector of axis z a . The
                                                            transverse section of rack-cutter ∑ c is shown in Figs. 4-a
                                                            and b.




                                                                     Fig. 4- Rack-cutter transverse profiles.
                                                                (a) Mating profiles. (b) Pinion rack-cutter profile.
                                                                           (c) Gear rack-cutter profile.

                                                              B. Determination Of Pinion Tooth Surface ∑ 1
                                                              The determination of ∑ 1 is based on the following
                                                            considerations:
                                                            1. Movable coordinate systems S c and S1 , Fig. 5, are
                                                            rigidly connected to the pinion rack-cutter and the pinion,
                                                            respectively. The fixed coordinate system S m is rigidly
Fig. 3- For derivation of pinion rack-cutter surface ∑ c    connected to the cutting machine.
                                                            2. The rack-cutter and the pinion perform related motions,
                                                            as shown in Fig. 5, where s c = rp1 ψ 1 is the displacement
determined in coordinate system S c in two-parameter
                                                            of the rack-cutter in its translational motion, and ψ 1 is the
form          by     the        following matrix equation
rc (u c , θ c ) = M cb (θ c ) r b (u c )                    angle of rotation of the pinion.
                                                            3. A family of rack-cutter surfaces is generated in
(4)
                                                            coordinate system S1 and is determined by the matrix
4. The normal N c to rack-cutter surface ∑ c is
                                                            equation
determined by matrix equation, [4]
                                                            r1 (u c , θ c ,ψ 1 ) = M 1c (ψ 1 ) rc (u c , θ c )
 N c (u c ) = Lcb Lba N a (u c )                      (5)
                                                            (8)
Here
                                                            Here
                     ∂r                                      M 1c (ψ 1 ) =M 1m M mc
 N a (u c ) = k a × a                                 (6)                                                              (9)
                    ∂u c
                                                            The pinion tooth surface ∑ 1 is generated as the
and the unit normal to the surface is
                                                            envelope of the family of surface r1 (u c ,θ c ,ψ 1 ) .Surface
                N         N c (u c )
n c (u c ) = c =                                      (7)   ∑ 1 is determined by
                Nc       1+ 4a2 u2
                         c   c
                                                            f1 p (u c , θ c ,ψ 1 ) = 0                               (10)
where Lcb indicates the 3×3 matrix that is the sub-matrix
                                                            simultaneous consideration of vector function
of M cb and is used for the transformation of vector        r1 (u c , θ c ,ψ 1 ) and the so-called equation of meshing .


                                   (Advance online publication: 15 August 2007)
                           Engineering Letters, 15:1, EL_15_1_4
______________________________________________________________________________________
4. To derive the equation of meshing (10), apply the
theorem of [5] and [4] to obtain,
                                                                        ∑ c . The normal profile of ∑ t is a parabola represented
                                                                        in S e , referring to Fig. 2-c,

                                                                                    [
                                                                        re (u t ) = u t at ut2 0        1 ] T
                                                                                                                                      (17)
                                                                        which is similar to (2). Use coordinate systems S k , Fig.
                                                                        2-c, and S t that are similar to Sb and S c , Fig. 3, to
                                                                        represent         surface         St        by   matrix   equation,
                                                                        rt (ut ,θ t ) = M tk (θ t ) M ke re (ut )                     (18)
                                                                        The normal to the surface ∑ t is determined by equations
                                                                        similar to (5) to (7). The difference in the representation
                                                                        of ∑ t is the change in the subscript c to t.
                                                                           B. Determination Of Gear Tooth Surface ∑ 2
                                                                          The generation of ∑ 2 by rack-cutter surface ∑ t is
                                                                        represented schematically in Fig. 6. The rack-cutter and
                                                                        the gear perform related translational and rotational
                                                                        motions designated as st = rp2 ψ 2 and ψ 2 .
                                                                        The gear tooth is represented
                                                                        r2 = r2 (ut ,θ t ,ψ 2 )
     Fig. 5- Generation of pinion by rack-cutter ∑ c                    (19)
                                                                         f 2t (ut ,θ t ,ψ 2 ) = 0
Oc O1 = −rp1 i + rp1ψ 1 j                                       (11)    (20)
                                                                        Equation (20) represents in S 2 the family of rack-cutter
v c = ω (1) rp1 j     where ω (1) = ω k                         (12)
                                                                        surfaces ∑ t determined as,
                                   (1)
v1 = ω    (1)
                × rc + Oc O1 × ω                                (13)    r2 (ut ,θ t ,ψ 2 ) = M 2t (ψ 2 ) rt (ut ,θ t )                (21)
   The relative velocity is                                             Here
v c1 = v c − v1 = −ω [(rp1ψ 1 − yc ) i + xc ]                    (14)   M 2t (ψ 2 ) = M 2 m (ψ 2 ) M mt (ψ 2 )                        (22)
Thus,            the       equation          of       meshing      is
N c • v c1 = 0                                                  (15)
That yields
 f1c (u c ,θ c ,ψ 1 ) = (rp1ψ 1 − yc ) N xc + xc N yc = 0       (16)
where ( xc , yc , z c ) are the coordinates of a current point of
∑ c ; (N c ) is the normal to the surface ∑ c ; ω is the
angular velocity; v c and v1 are the velocities of the rack-
cutter c and pinion respectively; v c1 represent the
relative velocity (sliding velocity) between the rack-
cutter and pinion, Fig. 5.
      Equations (8) and (16) represent the pinion tooth
surface by three related parameters. Taking into account
that the equations above are linear with respect to θ c ,
hence θ c may be eliminated and represent the pinion
tooth surface by vector function r1 (u c ,ψ 1 ) .

    III. DERIVATION OF GEAR TOOTH SURFACE

  A. Gear Rack-Cutter Surface ∑ t
  The derivation of rack-cutter surface ∑ t is based on                         Fig. 6- Generation of gear by rack-cutter ∑ t
the procedure similar to that applied for derivation of

                                          (Advance online publication: 15 August 2007)
                           Engineering Letters, 15:1, EL_15_1_4
______________________________________________________________________________________


The derivation of the equation of meshing (20) may be                 Also to find the slope of fillet curve at any point, the
accomplished              similarly       to      that     of (16),   circle equation is:-
 f 2t (ut ,θ t ,ψ 2 ) = (rp2 ψ 2 − yt ) N xt + xt N xt = 0    (23)    ( y − h) 2 + ( x − k ) 2 = r f2
Equations (21) and (23) represent the gear tooth surface                                                                         (25)
                                                                      y 2 − 2 h y + h 2 + x 2 − 2 k x + k 2 = r f2
by three related parameters. The linear parameter θ t can
                                                                      differentiating (25) with respect to y as follows:
be eliminated and the gear tooth surface represented in
                                                                                 dx     dx                     dx     dx
two-parameter form by vector function r2 (ut ,ψ 2 ) .                  y−h+ x −k            =0 ⇒ h = y+ x −k                  (26)
                                                                                 dy     dy                     dy     dy
                                                                      by substituting (13) , (24) and (26) in (25), thus obtaining
   IV. MATHEMATICAL SIMULATION OF RACK
                                                                      a non-linear equation which is solved numerically using
                 FILLET
                                                                      Secant Method to get the coordinates of smoothing point
  A fillet part is a circular arc which has coordinates x             (A).
and y, k and h –center coordinates and a radius rf.                   For the crowned parabolic profile, the fillet curve can be
  This arc lies between points A and B, where point A                 represented                        as               follows:
represent the mating point, which satisfy smoothing                         ⎡ xb ⎤ ⎡ r f sin (θ f ) + xof ⎤
contact, between the circular-arc or parabolic curve and              rb = ⎢ yb ⎥ = ⎢r f cos(θ f ) + yof ⎥
                                                                            ⎢ ⎥ ⎢                         ⎥                   (27)
                                                                         f
the fillet curve, point B represent the meeting point,                      ⎢ zb ⎥ ⎢                      ⎥
which satisfy smoothing contact, between the fillet curve                   ⎣ ⎦ ⎣              0          ⎦
and the horizontal straight line, as shown in Fig. 7.                 Here, r f is the fillet radius; ( xof , yof ) are the arc center
Therefore, to find the x and y-coordinates of points A and            coordinates; θ f is the variable parameter.
B, angles θ A and θ B which satisfy smoothing contact
                                                                        Using coordinates systems similar to Sb and S c , Fig. 3,
must be found.
  To satisfy smoothing contact at point B, angle θ B may              to represent surface S c f , the subscript f means the fillet

equal to 90 o because the radius of fillet may be                     surface, thus
perpendicular on the tangent, which is the horizontal                 rc f (θ f ,θ c ) = M c f b f (θ c ) rb f (θ f )            (28)
straight line at this point, also to satisfy smoothing                The unit normal to the surface can be found as
contact at point A the fillet radius may be perpendicular                    ∂rc f ∂rc f                     Nc f
on the tangent at this point, or in other words the slope of          Nc f =      ×           and     nc f =                     (29)
circular-arc or parabolic curve must be equal to the slope                   ∂θ f ∂θ c                       Nc f
of the fillet curve at this point, [6].                                 The representation of the pinion tooth fillet surface by
   Thus, to find the slope of parabolic curve at any point,           equations similar to (8) and (16), and also the
using (3), to get,                                                    representation of the gear tooth fillet surface by equations
 dx dx du cos(α n ) − 2ac u c sin(α n )
     =      *     =                                     (24)          similar to (21) and (23). Then the generation of pinion
 dy du dy sin(α n ) + 2ac u c cos(α n )                               and gear for parabolic profiles with fillet radius can be
                                                                      obtained as shown in Fig. 8.




                                                                                 Fig. 8- Generation of parabolic tooth with fillet
                                                                                                     radius.
               Fig. 7- Fillet part of rack-cutter.




                                        (Advance online publication: 15 August 2007)
                           Engineering Letters, 15:1, EL_15_1_4
______________________________________________________________________________________
                                                                            M ij , L ij               Matrices of coordinate transformation from

                                                                            coordinate system S i to S j .
                           V. CONCLUSIONS
                                                                            n i( j ) , N i( j )       Unit normal and normal to surface    ∑i   in coordinate
  The developed approach of design and generation of the
crowned parabolic Novikov gear drives has successfully                      system      Sj.
been applied. The conjugation of gear tooth surfaces with                   ri                       Position vector of a point in coordinate system   Si .
profile crowning is achieved by applying two rack-cutters
with crowned profile in normal section.                                     rf                    Fillet radius .

                                                                            rpi                   Radius of cylinder of pinion (i=1) or for gear (i=2).
                           REFERENCES
[1]
1926.
        Wildhaber, E., "Helical Gearing", U.S. Patent No. 1,601,750,        si                    Displacement of rack-cutter for pinion (i=1) or for gear
[2] Novikov, M.L., U.S.S.R., Patent No. 109,750, 1956.                      (i=2) .
[3] Litvin, F.L., Feny, P., and Sergei A. L., "Computerized Generation
and Simulation of Meshing of a New Type of Novikov-Wildhaber                Si (Oi,xi,yi,zi) Coordinate system (i=c,t,p,g,1,2,m,a,b,f,fs,cf,r,k,e)
Helical Gears", NASA/CR—2000-209415, 2000.
[4] Litvin, F.L., "Gear Geometry and Applied Theory", Prentice-Hall,
                                                                            αn                     Pressure angle in Normal section.
Englewood Cliffs, NJ, 1994.                                                 β                Helix angle .

                                                                             ∑i
[5] Litvin, F.L, "Theory of Gearing", NASA RP-1212 (AVSCOM 88-
C-035), 1989.                                                                                      Surfaces (i=c,t,p,g,1,2).

                                                                            φi
[6]     Mohammed Qasim Abdullah, "Computer Aided Graphics of
Cycloidal Gear Tooth Profile", University of Baghdad, Fifth                                        Angle of rotation of the pinion (i=1) or the gear (i=2) in
Engineering Conference, 2003.
                                                                            the process of generation for circular-arc profile .

                            Nomenclature                                    ψi                      Angle of rotation of profiled-crowned pinion (i=1), the

ai               Parabolic coefficients of profiles of pinion rack cutter   double crowned- profile (i=p) or for gear (i=2) in the process of
(i=c) and gear rack cutter (i=t).                                           generation for circular-arc profile .
 f ij          Equation of meshing between tooth surface (i) and rack-      (ui ,θ i )              Parameters of surface ∑ i .
cutter (j).
                                                                            θ A ,θ B               Angles which satisfy smoothing contact curves .
li             Parameter of location of point tangency Q for pinion
(i=c) or gear (i=t).                                                        ρi                     Profile radii (i=p,g,c,t,1,2).




                                            (Advance online publication: 15 August 2007)

				
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