Design of Bevel Gears

In this tutorial, you will learn about the design process and working of a Bevel gear system. In short, you will learn about the various characteristics and parameters that are considered to design a bevel gear suitable for working under load.

Contents:

1. What are Bevel Gears?
2. Categories of Bevel Gears
3. Terminology of Bevel Gear
4. What Forces act on a Bevel Gear?
5. Effective Load on a Bevel Gear
6. Beam Strength Criterion
7. Wear Strength Criterion
8. Application of Bevel Gears

What are Bevel Gears?

Bevel gears are used to transmit power between two intersecting shafts. Bevel gears are cut on mating cones rather than the mating cylinders of spur or helical gears. Their axes are nonparallel and intersect at the apices of the mating cones. The angle between their axes can be any value and is often 90°.

• The elements of the teeth of the straight bevel gears are straight lines, which converge into a common apex point. The elements of the teeth of the spiral bevel gears are spiral curves, which also converge into a common apex point.
• Straight bevel gears are easy to design and manufacture and give reasonably good service when properly mounted on shafts. However, they create noise in high-speed conditions.
• Contact between the teeth of straight or spiral bevel gears has the same attributes as their analogous cylindrical counterparts, with the result that spiral bevels run quieter and smoother than straight bevels, and spirals can be smaller in diameter for the same load capacity. The most common pressure angle for bevels or spirals is φ = 20°.
• A maximum reduction of 10:1 is recommended for any bevel or spiral gear set. A 5:1 limit is recommended when used as a speed increaser.
• Bevel gears, straight, spiral, or hypoid, are not interchangeable and are always designed in pairs.

Categories of Bevel Gears

Bevel gears are classified based on pitch angle and the gear system arrangement. Some common bevel gear arrangements are listed below.

• When two identical bevel gears are mounted on shafts, which are intersecting at right angles, they are called ‘miter’ gears. The pinion and gear of miter gears rotate at the same speed.
• In a pair of bevel gears, when one of the gears has a pitch angle of 90° then that gear is called ‘crown’ gear. Such bevel gears are mounted on shafts, which are intersecting at an angle that is more than 90°. The crown gear is equivalent to the rack in spur gearing.
• When the teeth of bevel gear are cut on the inside of the pitch cone, it is called internal bevel gear. In this case, the pitch angle of the internal gear is more than 90° and the apex point is on the backside of the teeth on that gear.
• When two straight bevel gears are mounted on shafts, which are non-parallel and non-intersecting, they are called ‘skew’ bevel gears.
• Hypoid gears are like spiral bevel gears that are mounted on shafts, which are non-parallel and non-intersecting. Hypoid bevel gears are mainly used in automobile differentials to lower the center of gravity of the vehicle.
• Zerol gears are spiral bevel gears with zero spiral angles. These gears theoretically give more gradual contact and a slightly larger contact ratio.
• Face gears consist of a spur or helical pinion mating with a conjugate gear of disk form. The pinion of face gears is either a spur gear or a helical gear.

Terminology of Bevel Gear

A bevel gear is in the form of the frustum of a cone. The following figure highlights the important terms associated with designing a bevel gear.

• A Pitch cone is an imaginary cone, the surface of which contains the pitch lines of all teeth in the bevel gear.
• The apex of the pitch cone is called the cone center. It is denoted by O.
• Cone distance is the length of the pitch-cone element. It is also called the pitch-cone radius. It is denoted by Ao.
• The angle that the pitch line makes with the axis of the gear, is called the pitch angle. It is denoted by . The pitch angle is also called the center angle.
• Addendum angle is the angle subtended by the addendum at the cone center. It is denoted by .
• The Dedendum angle is the angle subtended by the dedendum at the cone center. It is denoted by .
• Face Angle It is the angle subtended by the face of the tooth at the cone center. Face angle = pitch angle + addendum angle.
• Root angle is the angle subtended by the root of the tooth at the cone center. Root angle = pitch angle – dedendum angle.
• The back cone is imaginary, and its elements are perpendicular to the elements of the pitch cone.

What Forces act on a Bevel Gear?

It is assumed that the resultant tooth force between two meshing teeth of a pair of bevel gears is concentrated at the midpoint along the face width of the tooth. As in helical gears, there are tangential, radial, and axial force components acting on a bevel or spiral gear.
The following figure shows the forces acting on a bevel gear tooth.

The resultant force P, shown by the dotted line, acts at the midpoint of the face width of the pinion. The three components are:
P_t = tangential or useful component (N)
P_r = radial component (N)
P_a = axial or thrust component (N)
P_s = separating component (N)
= pressure angle (degrees)
From the above figure, we can see that, for the separating force component

P_s = P_t tan tan
For the radial and axial component

P_r = P_s cos cos γ
P_a = P_s sin sin γ
Substituting the separating component
P_r = P_t tan tan cos cos
P_a= P_t tan tan sin sin
The above equation gives us the relation between two of the components in terms of the third.

We can calculate the tangential component using the transferred torque.
P_t = \(\frac{M_t}{r_m}\)
Where M_t is the transmitted torque and r_m is the radius of the pinion at the midpoint along the face width.

Effective Load on a Bevel Gear

The effective load acting on the tooth cannot be calculated without accounting for all the forces which act on the tooth. This includes the tangential forces and the dynamic forces which are introduced in the system during power transmission.

• The value of the tangential component depends upon the rated power and rated speed. In practical applications, the torque developed by the source of power varies during the work cycle. Similarly, the torque required by the driven machine also varies.
• In gear design, the maximum force (due to maximum torque) is the criterion. This is accounted for using a service factor. The service factor C_s is defined as the ratio of the maximum torque to the rated torque. The maximum tangential force is the product of the service factor and the tangential force from the rated torque.

The velocity factors are calculated by the empirical relations listed in the table below

Here v is the Pitch line Velocity given by
v = \(\frac{πdN}{60 × 10^3}\)
And the effective load is given by
P_eff = \(\frac{C_s P_t}{C_v}\)

In the end stages, the Buckingham equation is used to calculate the effective load with the following equation
P_eff = C_sP_t + P_d
Where,
P_d = \(\frac{21v(Ceb+P_t)}{21v + \sqrt{Ceb + P_t}}\)
Where C is the deformation factor and e is the sum of errors between meshing teeth. Bevel gears are made of steel and the deformation factor C is 11400 N/mm².

Beam Strength Criterion

The size of the cross-section of the tooth of a bevel gear varies along the face width. To determine the beam strength of the tooth of a bevel gear, it is considered to be equivalent to a formative spur gear in a plane perpendicular to the tooth element.

• Consider an elemental section of the tooth at a distance x from the apex O and having a width dx. Applying the Lewis equation to a formative spur gear at a distance x from the apex
  (S_b)= m_xb_xb Y
  Where
  (S_b) = beam strength of the elemental section (N)
  b_x = face width of elemental section (mm)
  m_x = module of the section (mm)
  Y = Lewis form factor based on virtual number of teeth
• Integrating the above equation in the limits of the tooth we get the following equation
  S_b = \(mbσ_b Y[1 – \frac{b}{A_o}]\)
  The additional term at the end is known as the bevel factor.
• The beam strength indicates the maximum value of the tangential force at the large end of the tooth that the tooth can transmit without bending failure. The beam strength should always be more than the effective force between the meshing teeth at the large end of the tooth.

Wear Strength Criterion

The contact between two meshing teeth of straight bevel gears is a line contact, which is like that of spur gears. To determine the wear strength, the bevel gear is equivalent to a formative spur gear in a plane that is perpendicular to the tooth at the large end.
S_w = Qbd_pK
where,
b = face width of gears (mm)
Q = ratio factor
d_p = pitch circle diameter of a formative pinion (mm)
K = material constant (N/mm²)
d_p‘= \(\frac{D_p}{cos }\)
Therefore, the wear strength equation becomes
S_w = \(\frac{QbD_pK}{cos }\)
In the case of bevel gears, either the pinion or the gear is generally overhanging. It is subjected to deflection under the action of tooth forces, and it has been found that to transmit the load, only three-quarters of the face width is effective. Modifying the equation to account for this effect
S_w = \(0.75 \frac{QbD_pK}{cos }\)

Application of Bevel Gears

Bevel gears find a diverse application in the industry, especially in marine applications, automobile industry, printing press factories, power tools, power plants, manufacturing plants, railways, etc.

• Bevel gears are used in the differentials of automobiles to transmit power to the two axles independently and thus allow them to spin at different speeds, especially when cornering.
• Bevel gears are used in power tools like drills as the main power transfer mechanism. They are used as a variable speed increaser and can also be used to change the direction of rotation.
• Industrial machines like planers utilize bevel gears to allow for easy disassembly and maintenance and to allow for some displacement in case of heavy loads.
• Systems using a rotorcraft drive, like helicopters, have requirements for applications at high speeds, high loads, and a large load cycle. Bevel gears find an application here to redirect the shaft from the horizontal engine to the vertical rotor.