Gears

Introduction

Gears transmit motion and power by direct contact between toothed members without any intermediate link. At the point of contact the motion between the two bodies is a combination of rolling and sliding along the common tangent. When sliding is negligible and rolling predominates the members behave as ideal gears; where sliding is significant, they behave more like rollers. No relative motion is possible along the common normal without either breaking contact or causing one member to penetrate the other.

Point P may be considered on either gear 1 or gear 2 depending on the diagram convention.

Symbols in the figures carry their usual mechanical meanings (pitch circle radii, centres, pressure angle, points of contact, etc.).

Classification of Gears Based on the Arrangement of Shafts

1. Parallel shafts

A. Spur gears

Spur gears have straight teeth parallel to the axis of rotation. Because the teeth are not inclined, spur gears do not produce axial thrust due to tooth load and are simple to manufacture and design.

B. Spur rack and pinion

A spur rack is a special case of a spur gear of effectively infinite diameter so that its pitch surface is a straight line. Rack-and-pinion pairs convert rotary motion to translatory motion, or vice versa. Racks are commonly used to drive slides and saddles on machine tools (for example the carriage on a lathe).

C. Helical gears (helical spur gears)

In helical gears the teeth are inclined to the axis of rotation and form a helix. Two mating gears must have the same helix angle but opposite hands (one right-hand and one left-hand) for correct meshing. Helical gears engage gradually over several teeth, giving smoother operation and higher load capacity than spur gears, but they generate axial thrust components.

D. Double-helical and herringbone gears

A double-helical gear is equivalent to two helical gears fixed side by side with opposite hands (right hand and left hand helix). This cancels axial thrust. When the two helices meet at a central apex with no gap, the gear is known as a herringbone gear.

2. Intersecting shafts

When shafts intersect, the geometry of meshing is equivalent to the rolling of two cones (assuming no slipping). Bevel gears connect intersecting shafts.

A. Straight bevel gears

Straight bevel gears have teeth that are straight and radial to the point of intersection of the shaft axes; teeth are tapered in cross-section. Bevel gears that connect two shafts at right angles and have equal sizes are called mitre gears.

B. Spiral bevel gears

When bevel gear teeth are curved or inclined to the face of the bevel, they are termed spiral bevel or helical bevel gears. Spiral bevels give gradual load application, reduced impact stresses and quieter running; they are commonly used in automotive differentials.

• Zero bevel gears: Spiral bevel gears with curved teeth but with a zero degree spiral angle are called zero bevel gears.

3. Skew shafts (non-parallel, non-intersecting)

• Pure rolling motion cannot give uniform rotary motion between skew shafts; some sliding is inevitable.
• If two hyperboloids (generating surfaces) touch each other along their entire length while rolling, sliding parallel to the line of contact must occur.
• If the two hyperboloids rotate on their respective axes, motion between them is a combination of rolling and sliding.
• The angle between two shafts equals the sum of the angles of generation of the two hyperboloids: q = y1 + y2.

q = y₁ + y₂

A. Crossed helical gears

• Crossed-helical (or spiral) gears are used for light loads and for connecting non-parallel, non-intersecting shafts. Applications include feed drives in machine tools, camshafts and some oil pumps in I.C. engines.

B. Worm gears

• Worm gearing is a special case of spiral gearing where the smaller member (the worm) is a screw and the larger member (the wheel) often has a concave or hollow profile.
• Sliding velocity at tooth contact in worm gears is higher compared with most other gear types.
• In a non-throated worm gear the contact is concentrated at a point; in a single-throated worm gear there is area contact between worm and wheel.
• Worms can be single-, double- or multi-start threads; higher starts change the effective lead and speed ratio.

C. Hypoid gear

• A hypoid pinion is larger and stronger than a comparable spiral bevel pinion and gives quieter, smoother action.
• Hypoid pairs have continuous pitch-line contact and typically have a larger number of teeth in contact than straight-tooth bevel gears.

Gear Terminology

Pitch circle - an imaginary circle such that pure rolling of two pitch circles gives a motion identical to the actual gear motion at the nominal centre distance.

• Pitch point: the point where the pitch circles of two mating gears touch.
• Pressure angle (φ): the angle between the common normal at the point of contact and the tangent at the pitch point.

The commonly used standard values of pressure angle are 14.5º, 20º and 25º.

• Module (m): the ratio of the pitch circle diameter (in mm) to the number of teeth, m = D(mm)/T.

m = D_(mm) / T

• Addendum circle: the circle through the tip of the tooth and concentric with the pitch circle. Addendum is the radial distance from pitch circle to addendum circle and is commonly equal to 1·m.

NOTE: Clearance = 0·157 m

• Dedendum circle: the circle through the bottom of the tooth space and concentric with the pitch circle. Dedendum is the radial distance from pitch circle to dedendum circle and is commonly equal to 1·157 m.
• Circular pitch (C): distance along the pitch circle between corresponding points on adjacent teeth.

D = pitch circle diameter
T = number of teeth

• Diametral pitch: ratio of number of teeth to the pitch circle diameter (diameter in mm must be used consistently).

• Relation between circular pitch (C) and diametral pitch (p_d):

C × p_d = π

• Tooth thickness: thickness of a tooth measured along the pitch circle.
• Tooth space: space between consecutive teeth measured along the pitch circle.
• Backlash: difference between tooth space and tooth thickness; provided to avoid jamming due to thermal expansion and manufacturing tolerances.
• Face: part of the tooth profile above the pitch surface.
• Flank: part of the tooth profile below the pitch surface.
• Profile: curvature formed by face and flank.
• Path of contact (POC): path travelled by the point of contact from the start of engagement to the end of engagement. POC = path of approach + path of recess.
• Arc of contact (AOC): the arc on the pitch circles swept by the point of contact from start to end of engagement.
• Gear ratio (G): G = T / t, where T = number of teeth on the gear (wheel) and t = number of teeth on the pinion (small gear).
• Velocity ratio (VR): VR = 1 / G (so the ratios of angular velocities are inversely proportional to the tooth counts).
• Angle of action: angle turned by the gear from beginning to end of engagement = angle of approach + angle of recess.
• Contact ratio: contact ratio = angle of action / pitch angle = arc of contact / circular pitch.

Law of Gearing

• The law of gearing states the condition that gear tooth profiles must satisfy to maintain a constant angular velocity ratio between two mating gears.

Let ω₁ = angular velocity of gear 1 (clockwise) and ω₂ = angular velocity of gear 2 (anti-clockwise).

Condition:

• If the angular velocities of two gears are to remain constant, the common normal at the point of contact of the two teeth must always pass through a fixed point P on the line of centres; this point divides the line of centres in the inverse ratio of the angular velocities.
• Equivalently, for a constant angular velocity ratio the common normal at the point of contact must pass through the pitch point.

Velocity of sliding

When curved tooth surfaces are in contact there will be sliding along the common tangent between the surfaces. The velocity of sliding at a contact point is given by the sum of the angular velocities multiplied by the distance from the pitch point to the contact point:

Velocity of sliding = (ω₁ + ω₂) × PC

where PC is the distance between the pitch point and the instantaneous point of contact measured along the common normal or tangent as per the diagram.

Type of Tooth Profiles

(i) Cycloidal profile teeth

A cycloid is the locus of a point on the circumference of a circle that rolls without slipping on another circle. For cycloidal gear teeth:

• The faces of the teeth are epicycloids and the flanks are hypocycloids.
• An epicycloid is the locus of a point on a circle rolling without slipping on the outside of another circle; a hypocycloid is the locus of a point on a circle rolling without slipping on the inside of another circle.
• A property of the hypocycloid is that at any instant the line joining the generating point to the contact point is normal to the hypocycloid.
• For correct meshing, the diameter of the generating circle for the face of one tooth must equal the diameter of the generating circle for the flank of the mating tooth (the pitch circle being common).
• For cycloidal teeth the pressure angle varies: it is maximum at the beginning of engagement, falls to zero at the pitch point and then increases in the opposite direction; therefore speed ratio is sensitive to centre-distance variation.

(ii) Involute profile

An involute is the curve generated by a point on a tangent that rolls without slipping on a base circle. The involute tooth profile is generated from a base circle; the radius of the base circle is fixed for a given gear and does not change with centre distance. The involute form is the most widely used tooth profile in modern gearing.

• For a pair of involute gears the velocity ratio is inversely proportional to the pitch circle diameters (and accordingly to the base circle diameters).

Path of contact for involute gears

Notation: r = pitch circle radius of pinion; R = pitch circle radius of wheel; r_a = addendum circle radius of pinion; R_a = addendum circle radius of wheel.

• Arc of contact (AOC): the arc on the pitch circles traversed by the contact point during meshing.
• Contact ratio (ε): ε = Arc of contact / Circular pitch.
• The addendum and dedendum must be chosen to avoid interference; diagrams and formulae (see figures) give maximum allowable values for addendum to prevent interference.

• Contact or mesh conditions, minimum tooth numbers to avoid interference and maximum addendum values are derived from geometry of base circles, pitch radii and pressure angle.

• Formulas and graphical constructions (see figures) give the maximum addendum of the wheel for given gear ratio, pressure angle and addendum coefficient a_w.

Example results and formulae for maximum tooth counts and addendum values are commonly tabulated for design reference.

• Point of contact for involute gears lies on a line tangent to the base circles.
• Base circle diameter = pitch circle diameter × cos φ (where φ is the pressure angle).
• For equal number of teeth on pinion and wheel the gear ratio G = 1.

• To avoid interference between a rack and a pinion, the addendum of the rack may be adjusted so that the path of contact limits coincide with the edge conditions shown in the diagrams.

Let the adopted value of the addendum coefficient for the rack be a_r (so addendum = a_rm).

• To avoid interference it is necessary that GE ≥ a_r m (notations as per figure).
• For 20º pressure angle typical minimum teeth values are given in design tables (for example t_min = 13 for certain rack-pinion conditions).

Condition for Interchangeable Gears

Gears are interchangeable if they have:

• the same module,
• the same pressure angle,
• the same addendum and dedendum, and
• the same tooth thickness.

Undercutting

• If the addendum of a mating gear is larger than the limiting value, interference occurs between the addendum of one gear and the dedendum of the mating gear; the gears may lock.
• When a cutting rack (with the interfering teeth) is used to cut the mating gear, the portion of material that would interfere is removed; the resulting gear is said to be undercut and the process is undercutting.

Comparison of Cycloidal and Involute Profiles

Cycloidal teeth

(a) Pressure angle varies from a maximum at the start of engagement to zero at the pitch point and back to a maximum at the end; this can lead to less smooth running.
(b) Tooth profile involves two curves (epicycloid and hypocycloid) which complicates manufacture.
(c) These are generally costlier because of manufacturing complexity.
(d) Exact centre distance is required to maintain a constant velocity ratio.
(e) Interference phenomenon is generally absent.
(f) Teeth have wider flanks and are thus stronger for the same pitch.
(g) A convex flank normally contacts a concave face, which can reduce wear.

Involute teeth

(a) Pressure angle is constant throughout engagement, which gives smoother running.
(b) Tooth profile involves a single involute curve and is simpler to manufacture.
(c) Involute teeth are cheaper to produce.
(d) Small variations in centre distance do not affect velocity ratio appreciably.
(e) Interference can occur if the minimum number of teeth condition is not satisfied.
(f) Teeth typically have radial flanks and are weaker than cycloidal flanks for the same pitch.
(g) Convex-to-convex tooth contact may give more wear compared with cycloidal form.

Helical and Spiral Gears

Helical and spiral gears have teeth inclined to the axis; the teeth can be right-hand or left-hand. The relation between helix angles and shaft angle depends on whether the gears are of the same or opposite hand.

Let y₁ = helix angle for gear 1 and y₂ = helix angle for gear 2.

• For intersecting or skew shafts the shaft angle q = y₁ + y₂ (when helix hands are the same).
• For opposite hands q = y₁ - y₂ (depending on convention).
• For parallel shafts (case of helical gears joining parallel shafts) y₁ = y₂ and q = 0.

Terminology:

• Helix angle (y): angle between the tooth and the axis of rotation (also called spiral angle).
• Normal circular pitch (p_n): shortest distance measured along the normal to the helix between corresponding points on adjacent teeth. For two mating helical gears the normal circular pitch must be equal.

Relationships:

p_n = p cos y

p = π m (circular pitch for an equivalent spur gear)

p_n = π m_n

m_n = m cos y

• Centre distance: geometry depends on normal module and helix angles (see figure).
• Efficiency: helical gears have slipping components that affect efficiency; specific formulae and factors are given in design handbooks and depend on pressure angle φ, helix angle y and shaft angle θ.

In the figures φ = pressure angle, y = helix angle, θ = angle between shafts.

Worm and Worm Gear

• Worm gears provide large speed reductions in a compact form and are used for skew shafts, typically at right angles.
• Load transmitted through worm gearing is limited; lubrication and sliding considerations are important because of high sliding velocities.
• A worm may be single-start, double-start, etc.; the number of starts affects the lead and therefore the velocity ratio.

Terminology:

• Axial pitch (P_a): distance between corresponding points on adjacent teeth measured along the axis.
• Lead (L): axial distance a helix advances in one full turn.
• Lead angle (l): angle which the helix makes with a plane normal to the axis (lead angle is complement of helix angle in some conventions): y + l = 90º.

Velocity ratio (VR): equals angle turned by the gear divided by angle turned by the worm; for a worm with z threads and a wheel with Z teeth, VR = Z / z (see figure).

• Centre distance: geometry determined by pitch radii of worm and wheel (see figure).
• Efficiency: efficiency depends on friction between worm and wheel, lead angle, coefficient of friction and lubrication (see figure).

Bevel Gear

• Bevel gears are used to transmit motion between intersecting shafts.

Let γ_g, γ_p = pitch angles of wheel (gear) and pinion respectively, and r_g, r_p = pitch radii of wheel and pinion respectively. The geometry, tooth cutting and design relations are illustrated in the following figures.