Document Sample

Engineering Letters, 15:1, EL_15_1_4 ______________________________________________________________________________________ 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)

DOCUMENT INFO

Shared By:

Categories:

Tags:
Gears, Google Gears

Stats:

views: | 114 |

posted: | 10/29/2011 |

language: | English |

pages: | 5 |

Description:
Gears, formerly known as Google Gears, is a software developed by Google, allows users offline access, but is still in beta stage. The software side through the local SQLite database so the information can scratch up. So the page is achieved through the buffer, not made ??from the actual network. Moreover, Web-related programs through the Gears will be periodic temporary local data and information on the network to do synchronization. If the network is temporarily unavailable, the synchronization process will be delayed until the network is restored. So, Gears of Web-related applications are not immediate. Gears is a free and open source software to BSD license.

OTHER DOCS BY bestt571

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.