An Approach For Forming Spur Gear Tooth Profile

Due to high working speed requirement in industry of rotating components, gear design development become quite noticeable and rapid in the vicinity of engineering parameters which become to have a large effect on its performance. In this work the spur gear of straight tooth is chosen because of its wide usage in industry, also an equation for joining the involute profile and fillet curve is generated. Variation effect on the shape of tooth profile forming is studied. A suggested method for generating tooth profile is obtained and a computer program is written for drawing the involutes and trochoid curves. The results obtained from the new method are the true continuous involute and trochoid curves in comparison with those obtained from Mechanical Desk software which are approximated curves fitted results.


1.Introduction
Gear design is considered to be one of the most important and complicated fields of mechanical systems because of its wide usage and applications, so this makes their design be reviewed yearly.Norton,1998[2] developed the first equation for the fillet shape for the spur or helical gears, in which considered it as arcs of circles, while in actual fact it is a trochoid as will be considered in this work.

Construction of Involute Gear Tooth
The involute of a circle is a curve that can be generated by unwrapping a taut string from a cylinder that always tangents to the base circle and the centre of involute curvature is always at the point of tangency as shown in Fig.
(1), and to plot the involute, simply pressure angle φ values are assumed.A tangent to the involute is always normal to the string, which is the instantaneous radius of curvature of the involute curve [2], [9].The involute tooth profile can be generated using the following relations [10], [11] as shown in Fig.  8) pitch (diametral pitch) hob, and are easily obtainable.Gears that are cut using this hob are capable to match (mesh) each other.The distance (B+Rrt) on the hob tooth is equal to the dedendum of the generated gear tooth, Rrt is the hob tip radius with its center at point O, and B is the vertical distance from point O and the pitch lines as shown in Fig. (3).TP the hob tooth space, is equal to the tooth thickness of generated gear at gear pitch diameter.The work piece moves an angle (TP+TH)/rp, where rp is the gear pitch radius.(TP+TH) is the circular pitch of manufacturing gear.

The involute coordinates.
It was shown that the involute can be predicted if the base circle radius and the angle (θ) of involute are known, so the radius of involute curve can be found at any pressure angle φ by following equation [10] To find the coordinates with respect to the centre of tooth, the tooth thickness at any radius must be known.Let's start at the pitch diameter with pressure angle φ 1 , a pitch radius rp1, and a circular tooth thickness CTT1.The involute angle can be written asθ 1 = tan (φ 1 ) -(φ 1 ) - -----------(7) and from Fig. (4), it can be seen that the angle A is:  ( ) So to find X and Y coordinates of the involute at any radius R 2 , use following equations: The trochoid coordinates.
To find the trochoid coordinates on the desired X-Y system through the center of the tooth, they must be shifted through the angle (w) as in  ---------------

Discussion
the generated equation for the involute profile and the fillet curve is used to study the variation of tooth geometric on the shape of tooth profile.Its observed clearly from figure (8-16), how the involute and trochiod curve of tooth profile changes from case 1 to case 9. Case (4) is used for the comparison between the profile resulted from the present developed program with that of the Mechanical Disk Top Software [1].which shows that this approach gives exact true curve fit, also more accurate curve for trochiod, while from the software curve passes between the plotted point with interpolation method to deduce the profile, also trochiod plotted by an arc at imaginary centre point as in CAD QUST software [13] as shown in fig ( 17-18).conclusion.the constructed computer program using the developed equation for forming the involute and the trochiod of spur gear tooth in comparison with other works shown that present analysis gives a true profile especially at the portion of trochoid, it also showed that the program developed form exact profile because it follows the real path of the cutter that cut's the real profile during the manufacturing process.from that it can be conclude that construction of profile is more accurate than that formed by arc of imaginary centre.References.[ The involute is generated at the position of the base circle whose radius is given by the equation: rb = rp * cos (φ) = m * Z * cos (φ) * 0.5(4) and the root circle radius rf = m * (Z-2.5)* 0.5 ----------(5) Gear Tooth GenerationGenerating mean that the tool is cutting conjugate form, such a straight sided tooth is sometimes referred to as a rack.As the tool transverse and the work rotates, an involute is generated on the gear tooth flank and a [trochoid] in the root fillet, as shown in Fig.(2).Fig.(3) is a closer look at a hob tooth.This is a hob of pressure angle φ and diametral pitch capable of cutting a whole family of gears with the same pressure angle and diametral pitch.Such tools are standard as, for instance, a 20 o pressure angle and ( Fig.(2) when the hob traverses a distance (TH+TP) as in Fig.(3), the gear rotates through an angle (TH+TP)/rp; therefore the angle (w+v) between the center of the gear tooth and the center of tooth space is: ( ) Fig.(5) shows the trochoid generated by point O at its starting point and after the hob has moved a distance rp.β and the gear has rotated through an angle β the coordinate are: Xo = R O * sin (T-β)=R O * (sin(T) * cos (β) + cos (T) * sin (β))-----------------(17) Yo = R O * cos (T-β)=R O * (cos (T) * cos (β) + sin (T) * sin (β)-----------------(18)