# Preliminary Results of Testing of Low-Tooth-Count Bevel Gears of a Novel Design, Part I

This paper deals with bevel gears having a low-tooth-count (further, LTC ­– gears, for simplicity).

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In a general sense, LTC bevel gears feature the start-of-active-profile cone angle, ϒl, smaller than the base cone angle, ϒb, of the gear.  For LTC bevel gears the inequality ϒlϒb is always observed.  In a narrower sense,  LTC bevel gears are viewed as those with 12 teeth and a fewer. Bevel gears with 12 teeth and a fewer are not thoroughly investigated yet, and the accuracy requirements for gearing of this particular type are covered neither by national, nor by international standards on gearing.1

Precision, that is, geometrically accurate (ideal) bevel gearing is the main focus of the paper in particular.

### Preamble

In order to be referred to as geometrically accurate, bevel gearing needs to meet the following three conditions all of which are of fundamental importance:

Condition of contact is the first condition to be fulfilled in geometrically accurate bevel gearing.  According to the condition of contact, at every point of contact, K , of tooth flanks of the gear, G, and the pinion,P, vector of the resultant relative motion,VΣ, of the teeth flanks G and P is always locate in a common tangent plane to the surfaces G and P  at K. Commonly, the condition of contact is specified by Shishkov’s equation of contact2, n • VΣ=0 , where n is the unit vector of a common perpendicular to the tooth flanks G and P  at contact point K. Other forms of representation of the condition of contact are known as well.

• Condition of conjugacy of tooth flanks of the gear, G, and the pinion, P,  is the second condition to be fulfilled in geometrically accurate bevel gearing. To fulfill the condition of conjugacy, a straight line along the common perpendicular, n, at every point of contact, K, of tooth flanks of the gear, G, and the pinion, P, must intersect the pitch line, Pln.Pitch line in bevel gearing is the axis of instant rotation of the pinion in relation to the gear (or of the gear in relation to the pinion when the pinion is considered stationary).  More in detail the condition of conjugacy in bevel gearing is discussed in [1].

• Equality of base pitches of the gear, Φb.g, and the pinion,Φb.p, to the operating base pitch,Φb.op, is the third condition to be fulfilled in geometrically accurate bevel gearing.  In other words, at every point of contact,K, of tooth flanks of the gear, G, and the pinion,P, the identity  Φb.p  Φb.p  Φb.op must be fulfilled.  More in detail this condition in bevel gearing is discussed in [1].

A bevel gear pair is capable of transmitting a smooth rotation from the driving shaft to the driven shaft if and only if all of the above-listed three conditions are fulfilled.  Bevel gears produced in the nowadays industry meet only the first condition and do not satisfy the second and the third conditions listed above.  Therefore, all of them are the approximate gears.

### Development of CAD model for geometrically accurate bevel gears

The development of a CAD model for geometrically accurate bevel gearing begins with the analytical description of the tooth flanks of the gear,G, and the pinion,P.  A belt-and-pulley model of a bevel gear pair is helpful to develop an analytical description for the tooth flanks of the gear and the pinion.

2.1. A belt-and-pulley model for bevel gearing. Consider a right-angle bevel gear pair as an example.  The tooth count of the gear is Ng=14, the tooth count of the pinion is Np=10 and the transverse pressure angle φ t.w =22.5° .

Generation of the tooth flank of the gear,G, is illustrated in Fig. 1.  (Tooth flank,P, of the mating pinion is illustrated in the similar manner.).

In Fig. 1, the gear and the pinion are rotated about their axes of rotation, Og and Op, accordingly.  The axes  Og and Op form a right angle with each other (that is, the shaft angle Σ=90°).  Point of intersection of the axes  Og and O is the common apex, Apa, of the gear pair.

The rotation of the gear, ωg, and the pinion, ωp, are synchronized with one another.

Base cones are associated with the gear and the pinion.  The base cone angle of the gear is designated as ϒg, and the base cone angle of the pinion is designated as ϒb.  The plane of action, PA, is tangent to the base cone of the gear and the pinion. lcg is the line of tangency of the plane of action,PA, with the base cone of the gear.  Similarly, lcp is the line of tangency of the plane of action,PA, with the base cone of the pinion.

The plane of action forms the transverse pressure angle, φt.w, with the plane through the common apex  Apa and that is perpendicular to the axial plane, that is, the plane through the axes Og and Op .  The pitch line,Pln, in a bevel gear pair is the line of intersection of the axial plane by the plane of action. The pitch line makes the gear pitch cone angle, ϒ, with the axis of rotation of the gear, Og. With the axis of rotation of the pinion, Op , the pitch line makes the pinion pitch cone angle,ϒ.

If the rotations of the gear, ωg, and the pinion, ωp, are viewed as vectors ωg and ωp along the corresponding axis of rotation, Og and Op, then the instant rotation,ωpl, of the pinion in relation the motionless gear is represented by the vector of instant rotation,ωpl.  Vector ωp is along the pitch line, Pln.

When the gears rotate, the plane of action, PA, is unwrapped from the base cone of the pinion, and is wrapped on the base cone of the gear.  Under such a scenario, the plane of action is viewed as a zero thickness film that is free for bending about the base cones and is absolutely rigid in all the rest of directions.  The plane of action is rotated,ωpa, about its axis,Opa. The axis Opa is through the common apex, Apa, and is perpendicular to the plane of action, PA. Rotation of the plane of action, ωpa, can also be represented in vector notation as the vector ωpa that is pointed along the axis Opa.

An elementary analysis of Figure 1 reveals that bevel gears of the proposed design are insensitive to the shaft angle alterations.

* As known, formulae and equations those derived for  bevel gears are also valid for gears with a large tooth count and not vice versa.
This equation was proposed by Prof. Shishkov as early as in 1948 (or even earlier) in his paper: Shishkov, V.A., “Elements of Kinematics of Generating and Conjugating in Gearing”, in: Theory and Calculation of Gears, Vol. 6, Leningrad: LONITOMASH, 1948.  In detail, this equation is also discussed in the monograph: Shishkov, V.A., Generation of Surfaces in Continuously Indexing Methods of Surface Machining, Moscow, Mashgiz, 1951, 152 pages.