2.1 Terminology | unit 2 helical bevel worm and worm wheel

TOM2

Unit - 2

Helical, Bevel, Worm and Worm Wheel

2.1 Terminology

Helix angle:- It is a constant angle made by the helices with the axis of rotation.

Axial pitch:- It is the distance, parallel to the axis, between similar faces of adjacent teeth. It is the same as circular pitch and is therefore denoted by

The actual pitch may also be defined as the circular pitch in the plane of rotation or the diametral plane.

Normal pitch:- it is the distance between similar faces of adjacent teeth along a helix on the pitch cylinders normal to the teeth. It is denoted by

The normal pitch may also be defined as the circular pitch in the normal plane which is a plane perpendicular to the teeth. Mathematically, normal pitch,

2.2 Geometrical relationships

Though the proportions for helical gears are not standardized, yet the following are recommended by American gear manufacturers association.

Pressure angle in the plane of rotation, ϕ =15° to 25°

Helix angle α = 20° to 45°

Addendum=0.8m

Dedendum= 1m

Minimum total depth= 1.8m

Minimum clearance=0.2m

Thickness of tooth=1.5708m

2.3 Tooth forces & torque transmitted

The force exerted by a helical gear on its matting gear acts normal to the contacting surfaces if friction is neglected.

However, a normal force in the case of helical gears has three components.

Apart from tangential and radial components that are present in spur gears, a third component parallel to the axis of the shaft of the gear also exist.

This is known as the actual is the thrust force component.

The figure shows the normal force and its component acting on a helical gear.

The gear shown is the driving gear and the forces are exerted on it by the driving gear.

Let =total normal force

=tangential force

=axial force

=radial force

=normal force in the plane of

=Pressure angle

=Normal pressure angle

=Helix angle

Then

2.4 Efficiency

A pair of spiral gears 1 and 2 in the mesh is shown in the figure.

Let the gear 1 be the driver and the gear 2 the driven.

The forces acting on each of a pair of teeth in contact are shown in the figure.

The forces are assumed to act at the center of the width of each teeth and in-plane tangential to the pitch cylinders.

force applied tangentially on the driver,

=resisting force acting tangentially on the driven.

axial or end thrust on the driver

=axial or end thrust on the driven

= normal reaction at the point of contact,

= Angle of friction

R= resultant reaction at the point of contact

From triangle OPQ,

Work input to the driver

From triangle OST,

The work output of the driven

The efficiency of spiral gears,

………… 1

t pitch circle diameter of gear 1,

And pitch circle diameter of Gear 2

……………. 2

We know that ……………. 3

Multiplying equation 2 and 3 we get

Substituting this value in equation 1 we get

…….. 4

………… 5

Since the angles are constants, therefore the efficiency will be maximum, when. ) Is maximum.

Since , therefore

Similarly,

Similarly, in the equation we get

2.5 Virtual number of teeth for helical gears

The virtual or equivalent number of teeth for helical gear may be defined as the number of teeth that can be generated on the surface of a cylinder having a radius equal to the radius of curvature at a point at the tip of the minor axis of an ellipse obtained by taking a section of the gear in the normal plane.

The pitch cylinder of the helical gear is cut by plane A-A which is normal to the tooth elements as shown in the figure.

When a cylinder is cut by a plane, which is inclined to its base, the cut section is an ellipse.

We will apply this rule of engineering drawing to the pitch cylinder.

The intersection of the plane A-A, which is inclined to the base of the cylinder and the pitch cylinder produces an ellipse.

This ellipse is shown by the dotted line.

The semi-major and semi-minor axis of the ellipse is

respectively.

It can be proved from analytical geometry that the radius of curvature r' at point B is given by,

Where a and b are semi-major and semi-minor axis respectively.

Substituting the values of a and b in the expression for r',

In the design of helical gears, an imaginary spur gear is considered in-plane A-A with the center at O' having a pitch Circle radius of r' and module m.

It is called a formative or virtual spur gear. This formative gear is shown in the figure.

The pitch circle diameter d' of the virtual gear is given by

The number of teeth z' on this imaginary spur gear is called the virtual number of teeth. It is given by

Where, T= actual number of teeth on a helical gear, and

=Helix angle

2.6 Terminology of bevel gears

Pitch cone:- It is a cone containing the pitch elements of the teeth.

Cone center:- it is the apex of the pitch cone. It may be defined as that point where the axis of two mating gears intersects each other.

Pitch angle:- It is the angle made by the pitch line with the axis of the shaft. It is denoted by

Cone distance:- It is the length of the pitch on the element. It is also called 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 center. It is denoted by Mathematically, addendum angle,

a= Addendum and OP=Cone distance

6. Dedendum angle:- It is the angle subtended by the dedendum of the tooth at the cone center. It is denoted by mathematically, dedendum angle,

d= Dedendum and OP=Cone distance

7. Face angle:- It is the angle subtended by the face of the tooth at the cone center. It is denoted by . 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 center. It is denoted by . It is equal to the pitch angle minus the dedendum angle.

9. Back 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 It is also called a back cone radius.

11. Backing:- It is the distance of the pitch. 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 from the cone center, parallel to the axis of the gear. It is denoted by

13. Mounting height:- It is the distance of the back of the Boss from the cone center. It is denoted by

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,

Pitch circle diameter

a= addendum and

=Pitch angle

16. Inside or dedendum cone diameter:-the inside of the dedendum cone diameter is given by

=Inside diameter, and

d= dedendum

2.7 Geometrical relationships

The proportions for the bevel gears may be taken as follows:-

Addendum a=1m

Dedendum d =1.2m

Clearance = 0.2m

Working depth =2m

The thickness of tooth= 1.5708m

Where m is the module.

2.8 Tooth forces & torque transmitted

Consider a bevel gear and pinion in the mesh as shown in the figure.

The normal force on the tooth is perpendicular to the tooth profile and thus makes an angle equal to the pressure angle to the Pitch Circle.

Thus, a normal force can be resolved into two components, one is the tangential component and the other is the radial component.

The tangential component produces the bearing reactions while the radial component produces end thrust in the shafts.

The magnitude of the tangential and radial components is as follows:

These forces are considered to act at the mean radius.

From the geometry of the figure, we find that

Now the radial force acting at the mean radius may be further resolved into 2 components

in the axial and radial directions as shown in the figure.

Therefore, the axial force acting on the pinion shaft,

And the radial force acting on the pinion shaft,

A little consideration will show that the axial force on the pinion shaft is equal to the radial force on the gear shaft, but their directions are opposite.

Similarly, the radial force on the pinion shaft is equal to the axial force on the gear shaft but act in opposite directions.

2.9 Terminology of worm gear

Axial pitch:- It is also known as linear pitch of a worm. It is the distance measured actually from a point on one thread to the corresponding point on the adjacent thread on the worm, as shown in the figure. It may be noted that the axial pitch of a worm is equal to the circular pitch of the mating worm gear when the shafts are at right angles.

2. Lead:- It is the linear distance through which a point on a thread moves ahead in one revolution of the worm. For single start thread lead is equal to the axial pitch, but for multiple start threads, lead is equal to the product of pitch and number of stars. Mathematically,

= axial pitch; n=number of starts.

3. Lead angle:- It is the angle between the tangent to the thread helix on the pitch cylinder and the plane normal to the axis of the worm. It is denoted by .

m =module, and

=pitch circle diameter of the worm.

4. Tooth pressure angle:- It is measured in a plane containing the axis of the warm and is equal to one half of the thread profile angle as shown in the figure.

5. Normal pitch:- It is the distance measured along the normal to the threads between two corresponding points on two adjacent threads of the worm. Mathematically,

Normal pitch,

6. Helix angle:- It is the angle between the tangent to the thread helix on the pitch cylinder and the axis of the worm. It is denoted by.

7. Velocity ratio:- It is the ratio of the speed of worm in r.p.m. to the speed of the worm gear in the r.p.m. mathematically velocity ratio.

Let. l= lead of the worm, and

= pitch circle diameter of the worm gear.

2.10 Geometrical relationships

Proportions for the worm

Sr No.	Particulars	Single and double threaded worms	Triple and quadruple threaded worms
1	Normal-pressure angle	14.5°	20°
2	Pitch circle diameter for worms integral with the shaft
3	Pitch circle diameter for worm’s board to fit over the shaft	2.4
4	Maximum bore for shaft
5	Hub diameter
6	Face length
7	Depth of tooth
8	Addendum

Proportions for worm gear

Sr No.	Particulars	Single and double threads	Triple and quadruple threads
1	Normal-pressure angle		20
2	Outside diameter
3	Throat diameter
4	Face width
5	The radius of gear face
6	The radius of gear rim

2.11 Tooth forces & torque transmitted

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

Forces on a worm gear are equal in magnitude to that of the worm, but opposite in direction to those shown in fig

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

Tangential force on the worm

= Axial force or thrust on the worm gear.

The tangential force on the worm produces a twisting moment of magnitude x and bands the worm in the horizontal plane.

2. The axial force of thrust on the worm

=tangential force on the worm gear.

The axial force on the worm tends to move the worm axially, includes an axial load on the bearings and bends the worm in a vertical plane with a bending moment of magnitude x .

3. Radial or separating force on the worm

Radial or separating force on the worm gear.

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

Numerical :

1. The center distance between two matching spiral gears is 260 mm and the angle between the shafts is 65°. The normal circular pitch is 14mm and the gear ratio is 2.5. The driven gear has a helix angle of 35°. Find the

Number of teeth on each wheel

Exact center distance

Efficiency assuming the friction angle to be 5.5°

G =2.5 C =260mm

Let the gear with a smaller number of teeth be the driver.

Take

Then

=262.4mm

=0.883

2. Two start worm rotating at 800 rpm drives a 26-tooth worm gear. The worm has a pitch diameter of 54 mm and a pitch of 18 mm. If the coefficient of friction is 0.06 find the

The helix angle of the worm

Speed of gear

Center distance

Lead angle for maximum efficiency

Efficiency

Maximum efficiency

Solution

(i) Unwrap one thread of the worm,

Helix angle

(ii) The pitch of wheel =Axial pitch of the worm

=18mm

(iii)

(iv) For maximum efficiency

=43.29

(v)

=0.77

(vi)

3. A drive is made up of two spiral gear wheels of the same hand, the same diameter and of the normal pitch of 14mm. The center distance between the ages of the shafts is approximately 150mm. The speed ratio is 1.6 and the angle between the shafts is 75°. Assuming of friction angle of 6° determine the

The spiral angle of each wheel

Number of teeth on each wheel

The efficiency of the drive

Maximum efficiency

Solution

(i) Let be the spiral angle of the wheel 1.

Then the spiral angle of the wheel 2,

Now velocity ratio, VR=

(ii) The center distance

say 20

=152.4mm

(iii) Efficiency

(iv) Maximum efficiency

= 0.872

4. A pair of spiral gears are required to connect two shafts 175 mm apart, the shaft angle being 70°. The velocity ratio is to be 1.5 to 1, the faster wheel having 80 teeth, and a pitch circle diameter of 100mm. Find the spiral angles for each wheel. If the torque on the faster wheel is 75N-m. Find the axial thrust on each shaft, neglecting friction.

Solution:- Given L=175mm=0.175m;