Strength of Worm Gear Teeth

Introduction:

In finding the tooth size and strength, it is safe to assume that the teeth of worm gear are always weaker than the threads of the worm.

Strength of Worm Gear Teeth

In worm gearing, two or more teeth are usually in contact, but due to uncertainty of load distribution among themselves it is assumed that the load is transmitted by one tooth only. We know that according to Lewis equation,

W_T = (σ_o . C_v) b. π m . y

where

W_T = Permissible tangential tooth load or beam strength of gear tooth,
σ_o = Allowable static stress,
C_v = Velocity factor,
b = Face width,
m = Module, and
y = Tooth form factor or Lewis factor.

Notes :

1. The velocity factor is given by

where v is the peripheral velocity of the worm gear in m/s.

2. The tooth form factor or Lewis factor (y) may be obtained in the similar manner as discussed in spur gears

3. The dynamic tooth load on the worm gear is given by

where W_T = Actual tangential load on the tooth.

The dynamic load need not to be calculated because it is not so severe due to the sliding action between the worm and worm gear.

4. The static tooth load or endurance strength of the tooth (WS) may also be obtained in the similar manner as discussed in spur gears, i.e.

W_S = σ_e.b π m.y

where σ_e = Flexural endurance limit. Its value may be taken as 84 MPa for cast iron and 168 MPa for phosphor bronze gears.

Efficiency Of Worm And Worm Gearing

Machine Design

Gear Drives

Introduction:

The efficiency of worm gearing may be defined as the ratio of work done by the worm gear to the work done by the worm.

Expression for the efficiency of worm gearing:

Mathematically, the efficiency of worm gearing is given by

The efficiency is maximum, when

In order to find the approximate value of the efficiency, assuming square threads, the following relation may be used :

 ...(Substituting in equation (i), φ = 0, for square threads)

where φ₁ = Angle of friction, such that tan φ₁ = μ.

The coefficient of friction varies with the speed, reaching a minimum value of 0.015 at a rubbing speed 

  between 100 and 165 m/min. For a speed below 10 m/min, take μ = 0.015. The following empirical relations may be used to find the value of μ, i.e.

Note :

If the efficiency of worm gearing is less than 50%, then the worm gearing is said to be self locking, i.e. it cannot be driven by applying a torque to the wheel. This property of self locking is desirable in some applications such as hoisting machinery.

Proportion Of Worm And Worm Gears

Machine Design

Gear Drives

Introduction:

Worms proportions include pitch circle diameter, face length etc.

Proportions for Worms

The following table shows the various porportions for worms in terms of the axial or circular pitch ( p_c ) in mm.

Notes:

1. The pitch circle diameter of the worm (DW) in terms of the centre distance between the shafts (x) may be taken as follows :

... (when x is in mm)

 

2. The pitch circle diameter of the worm (D_W ) may also be taken as D_W = 3 pc, where pc is the axial or circular pitch.

3. The face length (or length of the threaded portion) of the worm should be increased by 25 to 30 mm for the feed marks produced by the vibrating grinding wheel as it leaves the thread root.

Proportions for Worm Gear

The following table shows the various proportions for worm gears in terms of circular pitch ( p_c ) in mm.

Term Used In Worm Gearing

Machine Design

Gear Drives

Introduction:

The worm gears are widely used for transmitting power at high velocity ratios between non-intersecting shafts that are generally, but not necessarily, at right angles.

Terms used in Worm Gearing

1. Axial pitch.

It is also known as linear pitch of a worm.It is the distance measured axially (i.e. parallel to the axis of worm) from a point on one thread to the corresponding point on the adjacent thread on the worm.It may be noted that the axial pitch (p_a) of a worm is equal to the circular pitch ( p_c ) 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 threads, lead is equal to the axial pitch, but for multiple start threads, lead is equal to the product of axial pitch and number of starts.Mathematically,

Lead, l = p_a . n
where p_a = Axial pitch ; and 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 λ.If one complete turn of a worm thread be imagined to be unwound from the body of the worm, it will form an inclined plane whose base is equal to the pitch circumference of the worm and altitude equal to lead of the worm,

From the geometry of the figure, we find that

where m = Module, and
DW = Pitch circle diameter of worm.

The lead angle (λ) may vary from 9° to 45°. It has been shown by F.A. Halsey that a lead angle less than 9° results in rapid wear and the safe value of λ is 12½°.For a compact design, the lead angle may be determined by 

the following relation, i.e. 

where N_G is the speed of the worm gear and N_Wis the speed of the worm.

4. Tooth pressure angle. It is measured in a plane containing the axis of the worm and is equal to one-half the thread profile angle

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, p_N = p_a.cos λ

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
α_W,The worm helix angle is the complement of worm lead angle, i.e.

α_W λ = 90°

the helix angle on the worm is generally quite large and that on the worm gear is very small. Thus, it is usual to specify the lead angle (λ) on the worm and helix angle (α_G) on the worm gear. These two angles are equal for a 90° shaft angle.

7. Velocity ratio. It is the ratio of the speed of worm (N_W) in r.p.m. to the speed of the worm gear (N_G) in r.p.m. Mathematically, velocity ratio, 

Let l = Lead of the worm, and
D_G = Pitch circle diameter of the worm gear.

We know that linear velocity of the worm,  

and linear velocity of the worm gear,

Types Of Worm And Worm Gears

Machine Design

Gear Drives

Introduction:

Worm gear is used mostly where the power source operates at a high speed and output is at a slow speed with high torque. It is also used in some cars and trucks.

Types of Worms

The following are the two types of worms :

1. Cylindrical or straight worm, and
2. Cone or double enveloping worm.

Fig1. Types of worms

The cylindrical or straight worm, as shown in Fig.1 (a), is most commonly used.The shape of the thread is involute helicoid of pressure angle 14 ½° for single and double threaded worms and 20° for triple and quadruple threaded worms.The worm threads are cut by a straight sided milling cutter having its diameter not less than the outside diameter of worm or greater than 1.25 times the outside diameter of worm.The cone or double enveloping worm, as shown in Fig.1 (b), is used to some extent, but it requires extremely accurate alignment.

Types of Worm Gears

The following three types of worm gears are important from the subject point of view :

Fig 2. Types of worms gears

1. Straight face worm gear, as shown in Fig2 (a),

2. Hobbed straight face worm gear, as shown in Fig.2 (b), and

3. Concave face worm gear, as shown in Fig 2 (c).

The straight face worm gear is like a helical gear in which the straight teeth are cut with a form cutter.Since it has only point contact with the worm thread, therefore it is used for light service.The hobbed straight face worm gear is also used for light service but its teeth are cut with a hob, after which the outer surface is turned.The concave face worm gear is the accepted standard form and is used for all heavy service and general industrial uses.The teeth of this gear are cut with a hob of the same pitch diameter as the mating worm to increase the contact area.

Strength Of Helical Gear

Introduction:

In order to find the strength of helical gears, a modified Lewis equation is used.

Strength of Helical Gears

In helical gears, the contact between mating teeth is gradual, starting at one end and moving along the teeth so that at any instant the line of contact runs diagonally across the teeth.Therefore in order to find the strength of helical gears, a modified Lewis equation is used. It is given by

W_T = (σ_o × C_v) b.π m.y'

where W_T = Tangential tooth load,
σ_o = Allowable static stress,
C_v = Velocity factor,
b = Face width,
m = Module, and
y' = Tooth form factor or Lewis factor corresponding to the formative or virtual or equivalent number of teeth.

Notes :

1. The value of velocity factor (Cv) may be taken as follows :

2. The dynamic tooth load on the helical gears is given by

where v, b and C have usual meanings as discussed in spur gears.

3. The static tooth load or endurance strength of the tooth is given by

W_S = σ_e.b.π m.y'

4. The maximum or limiting wear tooth load for helical gears is given by

where D_P , b, Q and K have usual meanings as discussed in spur gears.

In this case,  

where φ_N = Normal pressure angle.

Equivalent Number Of Teeth, Proportions for Helical Gears

Machine DesignGear Drive

Introduction:

The formative or equivalent number of teeth for a 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.

Formative or Equivalent Number of Teeth for Helical Gears

Mathematically, formative or equivalent number of teeth on a helical gear,

TE = T / cos³ α

where

T = Actual number of teeth on a helical gear, and

α = Helix angle.

Proportions for Helical Gears

Though the proportions for helical gears are not standardised, yet the following are recommended by American Gear Manufacturer's Association (AGMA).

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

Helix angle, α = 20° to 45°

Addendum = 0.8 m (Maximum)

Dedendum = 1 m (Minimum)

Minimum total depth = 1.8 m

Minimum clearance = 0.2 m

Thickness of tooth = 1.5708 m

Face Width Of Helical Gears

Machine Design

Gear Drives

Introduction:

To have more than one pair of teeth in contact, the tooth displacement (i.e. the advancement of one end of tooth over the other end) or overlap should be atleast equal to the axial pitch.

Face Width

Overlap = p_c = b tan α                           ...(i)

The normal tooth load (W_N) has two components ; one is tangential component (W_T) and the other axial component (W_A). The axial or end thrust is given by

W_A = W_N sin α = W_T tan α  

From equation (i), we see that as the helix angle increases, then the tooth overlap increases. But at the same time, the end thrust as given by equation (ii), also increases, which is undesirable. It is usually recommended that the overlap should be 15 percent of the circular pitch.

Overlap = b tan α = 1.15 p_c

where b = Minimum face width, and
m = Module.

Notes :                                                                                                                                                                                        Fig: Face width of helical gear.

1. The maximum face width may be taken as 12.5 m to 20 m, where m is the module. In terms of pinion diameter (D_P), the face width should be 1.5 D_P to 2 D_P, although 2.5 D_P may be used.

2. In case of double helical or herringbone gears, the minimum face width is given by

The maximum face width ranges from 20 m to 30 m.

3. In single helical gears, the helix angle ranges from 20° to 35°, while for double helical gears, it may be made upto 45°.

Terms used in Helical Gears

Machine Design

Gear Drives

Introduction:

A helical gear has teeth in form of helix around the gear. Two such gears may be used to connect two parallel shafts in place of spur gears. The helixes may be right handed on one gear and left handed on the other.

Terms

Helical gear

The pitch surfaces are cylindrical as in spur gearing, but the teeth instead of being parallel to the axis, wind around the cylinders helically like screw threads.The teeth of helical gears with parallel axis have line contact, as in spur gearing.This provides gradual engagement and continuous contact of the engaging teeth. Hence helical gears give smooth drive with a high efficiency of transmission.The helical gears may be of single helical type or double helical type.In case of single helical gears there is some axial thrust between the teeth, which is a disadvantage.In order to eliminate this axial thrust, double helical gears herringbone gears) are used.It is equivalent to two single helical gears, in which equal and opposite thrusts are provided on each gear and the resulting axial thrust is zero.

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

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

3. 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 pN. 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,

pN = pc cos α

Note : If the gears are cut by standard hobs, then the pitch (or module) and the pressure angle of the hob will apply in the normal plane. On the other hand, if the gears are cut by the Fellows gear-shaper method, the pitch and pressure angle of the cutter will apply to the plane of rotation. The relation between the normal pressure angle (φN) in the normal plane and the pressure angle (φ) in the diametral plane (or plane of rotation) is given by

tan φN = tan φ × cos α

Design of Arms of Spur Gears

Machine Design

Gear Drive

Introduction:

The cross-section of the arms is calculated by assuming them as a cantilever beam fixed at the hub and loaded at the pitch circle.

Design Procedure:

It is assumed that the load is equally distributed to all the arms.The arms are designed for the stalling load.The stalling load is a load that will develop the maximum stress in the arms and in the teeth.This happens at zero velocity, when the drive just starts operating.The stalling load may be taken as the design tangential load divided by the velocity factor.

                  

D_G = Pitch circle diameter of the gear,
n = Number of arms, and
σ_b = Allowable bending stress for the material of the arms.         

Now, maximum bending moment on each arm,

and the section modulus of arms for elliptical cross-section,

where a₁ = Major axis, and b₁ = Minor axis.

The major axis is usually taken as twice the minor axis. Now, using the relation, σ_b = M / Z, we can calculate the dimensions a1 and b1 for the gear arm at the hub end.

Note :

The arms are usually tapered towards the rim about 1/16 per unit length of the arm (or radius of the gear).
∴ Major axis of the section at the rim end

Designing of the shafts and spur gear

MACHINE DRIVE

GEAR DRIVE

Introduction:

Designing of the shafts include calculation of the diameter of the shaft.

Design procedure:

In order to find the diameter of shaft for spur gears, the following procedure may be followed.

1. First of all, find the normal load (WN), acting between the tooth surfaces. It is given by

W_N = W_T / cos φ
where W_T = Tangential load,

 and φ = Pressure angle.
A thrust parallel and equal to WN will act at the gear centre.


2. The weight of the gear is given by

W_G = 0.001 18 T_G.b.m²(in N)
where T_G = No. of teeth on the gear,
b = Face width in mm, and
m = Module in mm.

3. Now the resultant load acting on the gear,
    


 Load acting on the gear

4. If the gear is overhung on the shaft, then bending moment on the shaft due to the resultant load
M = WR × x
where x = Overhang i.e. the distance between the centre of gear and the centre of bearing.

5. Since the shaft is under the combined effect of torsion and bending, therefore we shall determine the equivalent torque. We know that equivalent torque


where T = Twisting moment = W_T × D_G / 2

6. Now the diameter of the gear shaft (d ) is determined by using the following relation, i.e.


where τ = Shear stress for the material of the gear shaft.

Note : Proceeding in the similar way as discussed above, we may calculate the diameter of the pinion shaft.

Spur Gear Construction

Machine Design

Gear Drive

Introduction:

The gear construction may have different designs depending upon the size and its application.

Construction of spur gears

• When the dedendum circle diameter is slightly greater than the shaft diameter, then the pinion teeth are cut integral with the shaft as shown in Fig(a).

• If the pitch circle diameter of the pinion is less than or equal to 14.75 m 60 mm (where m is the module in mm), then the pinion is made solid with uniform thickness equal to the face width, as shown in Fig(b).

• Small gears upto 250 mm pitch circle diameter are built with a web, which joins the hub and the rim.

• The web thickness is generally equal to half the circular pitch or it may be taken as 1.6 m to 1.9 m, where m is the module.

• The web may be made solid as shown in Fig(c) or may have recesses in order to reduce its weight.
  

Gear with arms

The hub diameter is kept as 1.8 times the shaft diameter for steel gears, twice the shaft diameter for cast iron gears and 1.65 times the shaft diameter for forged steel gears used for light service. The length of the hub is kept as 1.25 times the shaft diameter for light service and should not be less than the face width of the gear.
The thickness of the gear rim should be as small as possible, but to facilitate casting and to avoid sharp changes of section, the minimum thickness of the rim is generally kept as half of the circular pitch (or it may be taken as 1.6 m to 1.9 m, where m is the module). The thickness of rim (t_R) may also be calculated by using the following relation, i.e.

where T = Number of teeth, and
n = Number of arms.

The rim should be provided with a circumferential rib of thickness equal to the rim thickness.

Design procedure for spur gear

Machine Design

Gear Drive

Introduction:

The values for service factor are for enclosed well lubricated gears. In case of non-enclosed and grease lubricated gears, the values given in the above table should be divided by 0.65.

Design Procedure:

In order to design spur gears, the following procedure may be followed :

1. First of all, the design tangential tooth load is obtained from the power transmitted and the pitch line velocity by using the following relation :

P = Power transmitted in watts,
*v = Pitch line velocity in m / s , = πD N/60
D = Pitch circle diameter in metres,

N = Speed in r.p.m., and
C_S = Service factor.

The following table shows the values of service factor for different types of loads :

2. Apply the Lewis equation as follows :

W_T = σ_w.b.p_c.y = σ_w.b.π m.y
= (σ_o.C_v) b.π m.y ...             (Q σ_w = σ_o.C_v)

Notes :

(i) The Lewis equation is applied only to the weaker of the two wheels (i.e. pinion or gear).
(ii) When both the pinion and the gear are made of the same material, then pinion is the weaker.
(iii) When the pinion and the gear are made of different materials, then the product of (σ_w × y) or (σ_o × y) is the deciding factor. The Lewis equation is used to that wheel for which (σ_w × y) or (σ_o × y) is less.

(iv) The product (σ_w × y) is called strength factor of the gear.
(v) The face width (b) may be taken as 3 p_c to 4 p_c (or 9.5 m to 12.5 m) for cut teeth and 2 p_c to 3 p_c (or 6.5 m to 9.5 m) for cast teeth.

3. Calculate the dynamic load (W_D) on the tooth by using Buckingham equation, i.e.

W_D = W_T W_I

In calculating the dynamic load (W_D), the value of tangential load (W_T) may be calculated by
neglecting the service factor (C_S) i.e.
W_T = P / v, where P is in watts and v in m / s.

4. Find the static tooth load (i.e. beam strength or the endurance strength of the tooth) by using the relation, W_S = σ_e.b.p_c.y = σ_e.b.π m.y
For safety against breakage, W_S should be greater than W_D.

5. Finally, find the wear tooth load by using the relation,  Ww = D_P.b.Q.K

The wear load (W_w) should not be less than the dynamic load (W_D).

Gear tooth failure

Machine Design

Gear Drive

Introduction:

There are different modes of failure of gear teeth and their possible remedies to avoid the failure.

Causes of Gear Tooth Failure

1. Bending failure.

• Every gear tooth acts as a cantilever.
• If the total repetitive dynamic load acting on the gear tooth is greater than the beam strength of the gear tooth, then the gear tooth will fail in bending, i.e. the gear tooth will break.
• In order to avoid such failure, the module and face width of the gear is adjusted so that the beam strength is greater than the dynamic load.

2. Pitting.

• It is the surface fatigue failure which occurs due to many repetition of Hertz contact stresses.
• The failure occurs when the surface contact stresses are higher than the endurance limit of the material.
• The failure starts with the formation of pits which continue to grow resulting in the rupture of the tooth surface. In order to avoid the pitting, the dynamic load between the gear tooth should be less than the wear strength of the gear tooth.

3. Scoring.

• The excessive heat is generated when there is an excessive surface pressure, high speed or supply of lubricant fails. It is a stick-slip phenomenon in which alternate shearing and welding takes place rapidly at high spots.
• This type of failure can be avoided by properly designing the parameters such as speed, pressure and proper flow of the lubricant, so that the temperature at the rubbing faces is within the permissible limits.

4. Abrasive wear.

• The foreign particles in the lubricants such as dirt, dust or burr enter between the tooth and damage the form of tooth.
• This type of failure can be avoided by providing filters for the lubricating oil or by using high viscosity lubricant oil which enables the formation of thicker oil film and hence permits easy passage of such particles without damaging the gear surface.

5. Corrosive wear.

• The corrosion of the tooth surfaces is mainly caused due to the presence of corrosive elements such as additives present in the lubricating oils.
• In order to avoid this type of wear, proper anti-corrosive additives should be used.