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Formative or equal no. Of teeth for bevel gears? Tredgold’s Approximation?

Introduction:

The involute teeth for a spur gear may be generated by the edge of a plane as it rolls on a base cylinder.

Tredgold’s Approximation:

• A true section of the resulting involute lies on the surface of a sphere. But it is not possible to represent on a plane surface the exact profile of a bevel gear tooth lying on the surface of a sphere.

• Therefore, it is important to approximate the bevel gear tooth profiles as accurately as possible.

• The approximation (known as Tredgold’s approximation) is based upon the fact that a cone tangent to the sphere at the pitch point will closely approximate the surface of the sphere for a short distance either side of the pitch point, as shown in Fig. (a).

• The cone (known as back cone) may be developed as a plane surface and spur gear teeth corresponding to the pitch and pressure angle of the bevel gear and the radius of the developed cone can be drawn.

• This procedure is shown in Fig. (b).

Let θ_P = Pitch angle or half of the cone angle,
R = Pitch circle radius of the bevel pinion or gear, and
R_B = Back cone distance or equivalent pitch circle radius of spur pinion or gear.
Now from Fig.(b), we find that
 

R_B = R sec θ_P

We know that the equivalent (or formative) number of teeth,

where T = Actual number of teeth on the gear

Notes :

1. The action of bevel gears will be same as that of equivalent spur gears.
2. Since the equivalent number of teeth is always greater than the actual number of teeth, therefore a given pair of bevel gears will have a larger contact ratio. Thus, they will run more smoothly than a pair of spur gears with the same number of teeth.

Determination of Pitch Angle for Bevel Gears

Introduction:

The bevel gears are used for transmitting power at a constant velocity ratio between two shafts whose axes intersect at a certain angle.

Pitch Angle for Bevel Gears

Consider a pair of bevel gears in mesh,

Let θ_P1 = Pitch angle for the pinion,
θ_P2 = Pitch angle for the gear,
θ_S = Angle between the two shaft axes,
D_P = Pitch diameter of the pinion,
D_G = Pitch diameter of the gear, and

Terms used in bevel gears

Introduction:

Bevel Gears include Pitch cone, Pitch angle., Addendum angle etc.

Terms used in bevel gears:

Terms used in bevel gears

1. Pitch cone. It is a cone containing the pitch elements of the teeth.
2. Cone centre. It is the apex of the pitch cone. It may be defined as that point where the axes of two mating gears intersect each other.
3. Pitch angle. It is the angle made by the pitch line with the axis of the shaft. It is denoted by ‘θ_P’.
4. Cone distance. It is the length of the pitch cone element. It is also called as a pitch cone radius. It is denoted by ‘OP’. Mathematically, cone distance or pitch cone radius,

5. Addendum angle. It is the angle subtended by the addendum of the tooth at the cone centre. It is denoted by ‘α’ Mathematically, addendum angle,

where a = Addendum, and OP = Cone distance.

6. Dedendum angle. It is the angle subtended by the dedendum of the tooth at the cone centre. It is denoted by ‘β’. Mathematically, dedendum angle,

where d = Dedendum, and OP = Cone distance.

7. Face angle. It is the angle subtended by the face of the tooth at the cone centre. It is denoted by ‘φ’. The face angle is equal to the pitch angle plus addendum angle.
8. Root angle. It is the angle subtended by the root of the tooth at the cone centre. It is denoted by ‘θR’. It is equal to the pitch angle minus dedendum angle.
9. Back (or normal) cone. It is an imaginary cone, perpendicular to the pitch cone at the end of the tooth.
10. Back cone distance. It is the length of the back cone. It is denoted by ‘RB’. It is also called back cone radius.
11. Backing. It is the distance of the pitch point (P) from the back of the boss, parallel to the pitch point of the gear. It is denoted by ‘B’.
12. Crown height. It is the distance of the crown point (C) from the cone centre (O), parallel to the axis of the gear. It is denoted by ‘HC’.
13. Mounting height. It is the distance of the back of the boss from the cone centre. It is denoted by ‘HM’.
14. Pitch diameter. It is the diameter of the largest pitch circle.
15. Outside or addendum cone diameter. It is the maximum diameter of the teeth of the gear. It is equal to the diameter of the blank from which the gear can be cut. Mathematically, outside diameter,

DO = DP 2 a cos θ_P

Classification of Bevel Gears

Introduction:

The bevel gears may be classified depending upon the angles between the shafts and the pitch surfaces.

Classification:

The bevel gears may be classified into the following types, depending upon the angles between the shafts and the pitch surfaces.

1. Mitre gears.

When equal bevel gears (having equal teeth and equal pitch angles) connect two shafts whose axes intersect at right angle, as shown in Fig. (a), then they are known as mitre gears.
 

2. Angular bevel gears.

When the bevel gears connect two shafts whose axes intersect at an angle other than a right angle, then they are known as angular bevel gears.

3. Crown bevel gears.

When the bevel gears connect two shafts whose axes intersect at an angle greater than a right angle and one of the bevel gears has a pitch angle of 90º, then it is known as a crown gear. The crown gear corresponds to a rack in spur gearing, as shown in Fig.b).

4. Internal bevel gears.

When the teeth on the bevel gear are cut on the inside of the pitch cone, then they are known as internal bevel gears.

Note : The bevel gears may have straight or spiral teeth. It may be assumed, unless otherwise stated, that the bevel gear has straight teeth and the axes of the shafts intersect at right angle.

Introduction of bevel Gears

Introduction:

The bevel gears are used for transmitting power at a constant velocity ratio between two shafts whose axe n intersect at a certain angle.

Bevel Gears:

The pitch surfaces for the bevel gear are frustums of cones.

• The elements of the cones, as shown in Fig. (a), intersect at the point of intersection of the axis of rotation.

• Since the radii of both the gears are proportional to their distances from the apex, therefore the cones may roll together without sliding.

• In Fig (b), the elements of both cones do not intersect at the point of shaft intersection. Consequently, there may be pure rolling at only one point of contact and there must be tangential sliding at all other points of contact.

• Therefore, these cones, cannot be used as pitch surfaces because it is impossible to have positive driving and sliding in the same direction at the same time.

The elements of bevel gear pitch cones and shaft axes must intersect at the same point.

Design of Worm Gearing

Introduction:

In designing a worm and worm gear, the quantities like the power transmitted, speed, velocity ratio and the centre distance between the shafts are usually given and the quantities such as lead angle, lead and number of threads on the worm are to be determined.

Design of Worm Gearing

In order to determine the satisfactory combination of lead angle, lead and centre distance, the following method may be used:

the centre distance,  

The centre distance may be expressed in terms of the axial lead (l), lead angle (λ) and velocity
ratio (V.R.), as follows :

Since the velocity ratio (V.R.) is usually given, therefore the equation (i) contains three variables i.e. x, l_N and λ. The right hand side of the above expression may be calculated for various values of velocity ratios and the curves are plotted.

The lowest point on each of the curves gives the lead angle which corresponds to the minimum value of x / l_N. This minimum value represents the minimum centre distance that can be used with a given lead or inversely the maximum lead that can be used with a given centre distance

Worm gear design curves

Note : The lowest point on the curve may be determined mathematically by differentiating the equation (i) with respect to λ and equating to zero, i.e.

Forces Acting on Worm Gears

Introduction:

The radial or separating force tends to force the worm and worm gear out of mesh. This force also bends the worm in the vertical plane.

Forces Acting on Worm Gears

• When the worm gearing is transmitting power, the forces acting on the worm are similar to those on a power screw.

• Fig shows the forces acting on the worm. It may be noted that the forces on a worm gear are equal in magnitude to that of worm, but opposite in direction to those shown in Fig.

The various forces acting on the worm may be determined as follows :

1. Tangential force on the worm

= Axial force or thrust on the worm gear

The tangential force (WT) on the worm produces a twisting moment of magnitude (WT × DW / 2) and bends the worm in the horizontal plane.

2. Axial force or thrust on the worm,

W_A = W_T / tan λ = Tangential force on the worm gear

The axial force on the worm tends to move the worm axially, induces an axial load on the bearings and bends the worm in a vertical plane with a bending moment of magnitude (WA × DW / 2).

3. Radial or separating force on the worm,

WR = WA . tan φ = Radial or separating force on the worm gear

The radial or separating force tends to force the worm and worm gear out of mesh. This force also bends the worm in the vertical plane.

thermal rating of worm gearing:

Introduction:

In the worm gearing, the heat generated due to the work lost in friction must be dissipated in order to avoid over heating of the drive and lubricating oil.

Calculations of thermal rating of worm gearing:

The quantity of heat generated (Q_g) is given by

Q_g = Power lost in friction in watts = P (1 – η)                 ...(i)

where

P = Power transmitted in watts, and
η = Efficiency of the worm gearing.

The heat generated must be dissipated through the lubricating oil to the gear box housing and then to the atmosphere. The heat dissipating capacity depends upon the following factors :

1. Area of the housing (A),

2. Temperature difference between the housing surface and surrounding air (t₂ – t₁), and

3. Conductivity of the material (K).

Mathematically, the heat dissipating capacity,

Qd = A (t₂ – t₁) K                  ...(ii)

From equations (i) and (ii), we can find the temperature difference(t₂ – t₁). The average value of K may be taken as 378 W/m²/°C.

Notes :

1. The maximum temperature (t₂ – t₁) should not exceed 27 to 38°C.

2. The maximum temperature of the lubricant should not exceed 60°C.

3. According to AGMA recommendations, the limiting input power of a plain worm gear unit from the standpoint of heat dissipation, for worm gear speeds upto 2000 r.p.m., may be checked from the following relation, i.e.

where

P = Permissible input power in kW,
x = Centre distance in metres, and
V.R. = Velocity ratio or transmission ratio.