Involute gear

The involute is the trajectory of a point belonging to a straight line that rolls without sliding along a circle. This straight line is called the generating straight line, and the circle along which it rolls is called the main circle (Figure 1 a).

Picture 1

The involute has the following properties that are used in the theory of gear:

• 1. the shape of the involute is determined by the radius of the main circle;

  2. the normal to an involute at any point of it is tangent to the base circle. The point of tangency of the normal with the base circle is the center of curvature of the evolvent at the point in question;

  3. evolvent of the same basic circle are equidistant (equidistant from each other) curves.

The position of any point on the involute can be unambiguously characterized by the diameter of the circle on which it is located, as well as characteristic angles for the involute: a turning angle (denoted by ν), a profile angle (α), an involute angle – invα (Figure 1 b). Figure 1b shows these angles for a point Y arbitrarily chosen on the involute, therefore they have a corresponding index:

• • ν_Y is the angle of evolvent evolvent to point y;

  • α_Y – the angle of the profile at the point Y;

  • invα_Y is the involute angle at the point Y (on the circumference of diameter dY).

That is, the index shows on which circle the evolvent point under consideration is located; therefore, the characteristic circles use the indices given above.

For example: αa1 is the angle of the involute profile at a point lying on the circumference of the vertices of the first wheel;

invα is an involute angle at an evolvent point located on the pitch circumference of the wheel, etc.

Consider the properties of the involute. The first property has a rigorous mathematical proof, however, it is not given in this short course.

Since during the formation of an evolvent, the generating straight line rolls along the main circle without sliding, at the given moment it rotates around point N (N is the instantaneous center of velocity), describing an infinitely small arc of a circle that determines the curvature of the involute at a given point. Those. The segment NY is the radius of curvature of the involute at Y (NY = ρ_Y).

But the segment NY is exactly equal to the arc NY0 (this is the same arc only extended in a straight line). Thus, we have:

The larger the radius of the main circle, the greater the radius of curvature of the involute at any point (that is, the shape of the evolvent is actually determined by the radius of the main circle).

The second property is also easily visible. Since N is an instantaneous center of velocity, the velocity of the point Y is perpendicular to the radius of NY. But the speed of a point moving along a curved path is directed tangentially to this path — in this case, tangent to the involute at point Y.

The perpendicular to the tangent is the normal, so the straight line YN on the one hand is the normal to the involute at the point Y, on the other hand is the tangent to the main circle (as generating a line rolling along the main circle).

The fact that the point N is the center of curvature of the involute at the point Y is shown when considering the first property. We will write down some dependences that are used further in the study of the geometry of an involute link (obtained from consideration of Figure 1b):

The third property of the evolvent is evident from Figure 1 a. Indeed, if we take two points (A and B) on the generating line, then they will describe two absolutely identical evolvents, and, no matter how the generating line moves, the distance between these points does not change (A_iB_i = Const). Those. indeed, these are equidistant curves (equidistant from each other). But, most importantly, this distance A_iB_i is equal to the distance between these evolvents, measured along the arc of the main circle:

A sign that two curvilinear profiles are tangent (and not intersecting) is the presence of a common normal at the point of contact. In this regard, the contact of two involute profiles occurs on a common tangent to the main circles N₁N₂ (Figure 2), which will simultaneously be a common normal to these profiles at the point of their tangency at any moment in time (based on the second property of the involute).

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Figure 2

The locus of the contact points of the profiles, which they occupy in the process of working a pair of teeth, is called the line of engagement.

Thus, in an involute transmission, the line of engagement is the straight line N₁N₂ (common tangent to the main circles).

Figure 2a shows the engagement of two involute profiles at different points in time. In both positions, the straight line N₁N₂ is a general normal to these related profiles and passes through the pole of the link W (instantaneous center of relative rotation).

This, on the one hand, shows that evolvent profiles satisfy the basic law of gear, on the other hand, they ensure the constancy of the gear ratio, since the gear pole does not change its position in the process of the pair’s work (the O₂W/O₁W ratio remains constant).

With a change in the center distance, only the position of the engagement line will change, but the whole engagement pattern will remain the same, i.e. the basic linking law, the magnitude and constancy of the gear ratio will remain. This is a very important property of involute gear, since allows you to enter the transmission in different center distance, which is especially important when designing gear boxes, planetary and differential mechanisms.

Transmission is insensitive to inaccuracies of the axial distance, which reduces the requirements for assembly accuracy.

The angle between the line of engagement and the common tangent to the initial circles at the pole is called the angle of engagement. The angle of engagement, the angle of the profile on the initial circumference of the first wheel and the angle of the profile on the initial circumference of the second wheel are equal to each other (α_w1=α_w2=α_w), so they are all denoted the same – α_w (without a numerical index, see Figure 2 a).

The segment N₁N₂ is called the theoretical line of engagement. In this area, the normal operation of two unrestricted evolvent occurs.

In actual transmission, the evolvent is limited (“cut off”) to the circles of the vertices, so all the work of the pair takes place on the section of the P₁P₂ line of engagement between the circles of the vertices (Figure 2 b).

The P₁P₂ segment is called the working (active) part of the engagement line (sometimes called simply the “engagement working line”, or “active engagement line”). Figure 2b shows two positions of the same pair: at the beginning of engagement (the driven wheel tooth works with its top, the drive wheel tooth at the lower operating point of the P₁ profile), and at the end of engagement (the drive wheel tooth works at its top and the next moment out of engagement, the driven wheel tooth works with its lower operating point of profile P₂).

Note: here the term “lower” or “upper” point refers to the position of the points relative to the main circle, regardless of how these points are located one relative to another in space. Of the two considered points of the “lower” profile, there will be one that is located closer to the main circle.

As the radius of the main circle increases to infinity, the radius of curvature of the evolvent at any of its points also becomes infinitely large, i.e. the main circle and the evolvent turn into straight lines. Involute gear wheel turns into a rack with a straight tooth profile.

Thus, a rake with a straight tooth profile is a special case of an involute gear and has all its properties, i.e. can work with any involute wheel (with the same module) without violating the basic linking law. In this case, the rotational movement of the wheel is converted into a forward movement of the rail or the forward movement of the rail is converted into rotational movement of the wheel while observing the constancy of the gear ratio.

Because Tooth rake with a straight tooth profile on the one hand has simple shapes and it is easy to set the dimensions of its elements, on the other hand it is an involute gear, its parameters are the basis for the standardization of involute gear wheels. The standard gear rack is called the reference contour (Figure 3a).

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Figure 3

There are several standards for the original contours, taking into account the specifics of certain types of gear (fine modular, bevel, etc.).

In accordance with this standards, the original circuit has the following parameters:

α=20° is the angle of the source contour profile (the main parameter defining the series of evolvent used for gears in accordance with this standard, therefore, it is often said in design practice that a twenty-degree involute is used in our country);

h_a* = 1 – the ratio of the height of the head of the tooth;

c* = 0.25 – the coefficient of radial clearance (according to other standards, depending on the module and type of tool, s * may be equal to 0.2; 0.3; 0.35);

The coefficients shown are dimensionless values. The absolute value of any size is obtained by multiplying the corresponding coefficient by the modulus (For example: the height of the tooth head is h_a=h_a∙m; the size of the radial gap is c = c∙m, etc.).

Thus, the shape of the tooth remains constant, and the absolute dimensions are determined by the modulus (that is, the modulus is like a coefficient of proportionality).

The height of the tooth of the original contour is divided into the head and leg. This division is carried out by a dividing line. The paw line is the line on which the thickness of the tooth is equal to the width of the cavity (Figure 3 b).

The height of the toothfoil is somewhat larger than the head to ensure a radial gap between the tops of the teeth of one wheel and the circumference of the depressions of the other after the gear assembly.

Standard parameters of the original contour are “transferred” to the involute wheel through the pitch circle (at the pitch circle, the pitch is equal to the standard pitch of the original contour p = π ∙ m, the angle of the profile is equal to the angle of the profile of the original contour α = 20°).