Also, unlike the cycloid, the shape of the involute tooth profile depends only on the size of the base circle, so involute gears will fit correctly at different center distances. The tangent to the involute is then obtained by drawing a line through P perpendicular to the normal. If two involute curves are in contact, the point of contact must be on the common normal and from the definition of the normal it follows that it must be tangent to both base circles.
If the base circles rotate about their fixed centers Op and Ow (see Fig. 4), the normal at any tangent point must be the common tangent (TpTo,) of the two base circles, i.e. all points of contact between the two involutes. they are on a common tangent, which is therefore a tangent path. The speed of the partial circle is the same for both involutes and is in the direction of the tangent of the partial circle at point P. The pressure between the two involutes is in the direction of the common normal (ie, the direction of action in the case of the involute).
The basic pitch (po) is the spacing of successive involutes, measured along the circumference of the basic circle, if the number of teeth = t. Circular pitch {p) is the measure of spacing along the circumference of a partial circle. The part of the involute that forms the gear tooth is between the addendum circle and the dedendum circle, thus defining the crest and root of the gear tooth. However, weakening of the gear tooth root will still occur as long as L < ro + A, i.e.
Now considering the following problem (figure given two complicated numbers, each generated from the same base circle (radius ro), find the distance of the two complicated numbers if one point on each complicated is given as shown in figure 2).
The involute function and the effect of correction on pressure angle and centre distance
The increased pressure engagement angle(s) and the required increase in center distance can be calculated if kp+kw is known. This can safely be done for so long. t —I)he correction for the gear can be. This means that if we denote the correction coefficient by which the addendum ine of the generator stand is tangent to the base circle by km (for the gear) and by Y km.
The relationship between kp+kw, the increased pressure angle 5tre and the required center distance extension (d) for p = IT Cm = P = 1) can be obtained by referring to Figures 14 and 15 and are given below. They can be calculated from the pressure angles r(re) at which the corrected gears will mesh without backlash. It can also be easily understood that the greater the number of teeth, the stronger the root will be. tooth, all other things being equal (Fig. 17), because the radius of curvature of the involute at point P is increasing as the base circle increases (Fig. 17B).
The relative force, however, will depend on the position of the point of application of maximum tooth pressure.). If the number of teeth on the wheel is much greater than on the rack, this may mean that the rack is much weaker if material of the same strength is used for both gears, and therefore the capacity of the gear combination is determined by the weakness. member (pinion). The conditions at the point of contact between two involute surfaces may be compared to those existing between two cylinders in contact having radii equal to the radius of curvature of the two involutes at the point of contact and which roll and slide at the same speed as the involute surfaces do at this point.
While the pressure is determined by the gear load, the relative radius of curvature and sliding speed can be affected by appropriate design. If the gear involute rotates at an angular velocity w1, the involute arc corresponding to a time interval dr can be obtained as follows (see Fig. 19). The involute arc of the gear will be R1.W1.dr, and the corresponding arc of the involute gear in contact in the same time interval is R2w2.dr, where w2 = the angular velocity of the gear.
If favorable contact conditions are required, the absolute value of the negative maximum should not be too far from the positive maximum and the speed change should not be too different for the rack and pinion steering. In the following example it will be shown how, by correction, the conditions for the Coating can be improved (Fig. 20), i.e., as the ratio. Summarizing the above discussions, it can be said that by using correction in an appropriately selected way, the following improvements can be achieved. a) eliminating or reducing the undercut in the pinion thus giving the tooth a stronger shape.
The increased pressure angle and increased center distance can be calculated as previously described in (5). From the initial analysis of the correction, it is clear that the correction method can be constructed for any pressure angle.
Introduction
This force (Fn) is in the direction of the line of action and the point of application is C4 on the tooth profile. If the point of intersection between the line of action and the axis of symmetry of the tooth is denoted by B, Fn can be considered to be acting at this time. By resolving it into two components in the direction of the axis of symmetry OB and in a direction perpendicular to this (LL) we get—. The tooth will be compressed by the Fn sinus and the Fn tose will cause bending and displacement.
The tooth thickness at the root can be found by drawing a parabola with vertex at B and tangent to the root radii of the tooth at points G and H. Point G can be found by trial considering that the tangent of the parabola at point G must be bisected by line LL, that is, GJ = JK.). As this happens, the actual contact area decreases all the time, thus causing a gradual increase in the surface tension. Note.—The effect of inertia and impact due to errors will be discussed in more detail later.
23 Fatigue cracks become more numerous and eventually complete destruction of the tooth surface occurs. The surface tension based on similarity will increase as the load increases and will decrease as the relative radius of curvature of the surfaces, at the point of contact, increases. 23, point C, will be the required point because the full load will be carried by one tooth pair from point CB to C4, but the relative radius of curvature is smallest at the point closer to the center of the pinion.
Influence of tooth length on distribution of tooth pressure along the tooth of the gear. The amount of deformation is dependent on the point of application of the load varying through a cycle. Deflection due to bending increases as the point of application approaches the tip of the gear tooth:).
The acceleration forces will also depend on the elasticity of the gears, etc., and dry will tend to level off with increasing flexibility of the gear teeth. If we consider that the above effects are important only when—. b) the errors in the tooth action are noticeable. Practice has also justified the use of the British Standard formulas, which, because of their simplicity and good agreement with the actual gear combination, can be safely used in all but extreme cases.
If we also remember that gears with finer pitch have more flexible teeth than gears with coarser pitch, a suitable balance can be found in the choice of the constant c and the diametral pitch P. The permissible horsepower is the smallest of the following four quantities :- gear strength Cbp .
