Design of Spur Gears

In this tutorial, we see the design process of a simple spur gear, starting from the very basics of the process to the different design considerations needed to endure a string and efficient design. In short, you will learn to define all design parameters for a spur gear and ensure the design is strong and safe for industrial use.

Contents:

1. What are Spur Gears?
2. What Forces act on Gears?
3. Beam Strength of Gear Tooth
4. Effective Load on Gear Tooth
5. Estimating Module based on Beam Strength
6. Wear Strength of Gear Tooth
7. Estimating Module based on Wear Strength
8. Design Sketch of Spur Gears

What are Spur Gears?

Gears are used to transmit torque in a variety of applications. They help in cases where there is a requirement of changing the input speed, modify the torque or simply change the direction of the input gear. Spur gears are among the most common types of gears in the application.

• Spur Gears have their tooth axis in the direction of the shaft axis. This means that their teeth are straight compared to the other types like helical gear.
• Due to this characteristic of their teeth, the spur gear is expected to work only on parallel shafts for power transmission.
• Spur gears are simple and compact in design and this makes them easy to handle, design, and install, even in restricted space conditions.
• Their simple meshing allows for high precision when it comes to increasing or decreasing shaft velocity or simply maintain a constant velocity.

What Forces act on Gears?

In gears, power is transmitted using a force exerted by the tooth of the driving gear on the meshing tooth of the driven gear. The following figure shows the tooth of the driving pinion exerting a force PN on the tooth of the driven gear.

• Following the fundamental law of gearing, the resultant force acts along the pressure line of the meshing gears. This resultant force can be divided into two components, the radial, and the tangential component.
• The tangential component P_t is the useful load because it helps determine the magnitude of the torque and the power which is being transmitted.
• The radial component P_r is a separating force generated and is always directed towards the center of the gear.
• The torque transmitted by the gear is given by the following equation
  Mt=\(\frac{60 × 10^6 × (kW)}{2πN}\)
  Where kW is the power being transmitted in kW and N is the speed in RPM.
• The tangential component acts at the pitch circle radius and is given by the following equation
  Pt × \(\frac{d}{2}\) = M_t
• For the radial component, we use the relation depicted in the diagram and get the following equation
  P_r=P_t tan α

Beam Strength of Gear Tooth

Wilfred Lewis was the first to analyze the bending forces on a gear tooth and published his findings in the year 1892. The Lewis equation is one of the most fundamental equations used in gear design.
The following figure shows the beam strength analysis

• Lewis considered the gear tooth as a cantilever beam with the tangential component acting as the load at the end of the beam. This force was considered to be the cause for bending stresses induced.
• The Lewis analysis ignores the effect of the radial component and the resultant compressive stresses induced in the tooth. Also, at any time, only one gear is assumed to be in mesh.
• The diagram shows the application of the tangential component on the gear tooth and the weakest section of the tooth where the stresses are maximum.
  For the section
  M_b = P_t × h
  I = \(\frac{bt^3}{12}\)
  y = \(\frac{t}{2}\)
  Where Mb is the bending moment, I is the second moment of inertia and y is the location of the centroidal axis.
• Using the standard bending equation
  σ_b = \(\frac{M_b y}{I}=\frac{(P_t × h)\frac{t}{2}}{\frac{bt^3}{12}}\)
  Rearranging the terms
  P_t = \(bσ_b (\frac{t^2}{6h})\)
  Multiplying the numerator and denominator by m and arranging the terms
  P_t = \(mbσ_b (\frac{t^2}{6mh})\)
  P_t= mbσ_bY
  Where Y is the Lewis Form Factor.
• When the stress reaches the permissible magnitude of bending stresses, the corresponding force (P_t) is called the beam strength.
• Therefore, the beam strength (S_b) is the maximum value of the tangential force that the tooth can transmit without bending failure.
  S_b= mbσ_bY
  This is the Lewis equation.
• To avoid the breakage of the gear tooth due to bending, the beam strength should be more than the effective force between the meshing teeth.

Effective Load on Gear Tooth

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.
• When gears rotate, it is necessary to account for the dynamic forces acting on the teeth. These forces are a result of inaccuracies in profile, errors in tooth spacing, misalignment, elastic materials, or simply the inertia developed during rotation. The dynamic forces can be calculated in the initial and the end stages of gear design.
• In the initial stages, the velocity factor can be used to calculate the dynamic load. The velocity factors are calculated by the empirical relations listed in the table below
  

  Cv=\(\frac{3}{3+v}\)	When v < 10 m/s
  Cv=\(\frac{6}{6+v}\)	When v < 20 m/s
  Cv=\(\frac{5.6}{5.6+\sqrt{v}}\)	When v > 20 m/s

  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_tP_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.

Estimating Module based on Beam Strength

The module of the gear is an important parameter required for the design of gears. This value must be calculated from the available parameters provided for the design. We saw the relation between the beam strength and the effective load and can use this relation to calculate the module of the gear.
S_b > P_eff
We add the factor of safety to the above equation for a safe design
S_b > P_eff × (fs)
We saw the relation between the tangential and the effective force. The tangential component can be written as
P_t = \(\frac{2M_t}{d}=\frac{2M_t}{mZ}=\frac{2}{mZ}(\frac{60 × 10^6 × (kW)}{2πN})\)
And the effective load is given as
P_eff = \(\frac{C_s P_t}{C_v} = \frac{C_s}{C_v} × \frac{60 × 10^6}{π} (\frac{(kW)}{mZN})\)
From Lewis Equation
S_b = mbσ_bY = m²b/m σ_bY
Placing the relations in the inequality, we can solve for the module.

Wear Strength of Gear Tooth

The failure of the gear tooth due to pitting occurs due to contact stresses between two meshing teeth exceeding the surface endurance strength of the material. Pitting is characterized by small pits on the surface of the gear tooth.

• To avoid this type of failure, the proportions of the gear tooth and surface properties should be selected in a way so that the wear strength of the gear tooth is more than the effective load between the meshing teeth.
• The wear strength is the maximum value of the tangential force that the tooth can transmit without pitting failure. Like the beam strength of the gear tooth, the equation of wear strength is given by the following equation.
  S_w = Qbd_pK
• Q is ratio factor given by \(\frac{2z_g}{zg+zp}\) where z is the number of gear teeth, b is the face width, d_p is the pinion diameter and K is the corresponding contact stress given by 0.16 \((\frac{BHN}{100})^2\) with BHN as the Brinell Hardness Number of the material.

Estimating Module based on Wear Strength

The module of the gear is an important parameter required for the design of gears. This value can also be calculated from the available parameters provided for the design. We saw the relation between the wear strength and the effective load and can use this relation to calculate the module of the gear.
S_w > P_eff

• We add the factor of safety to the above equation for a safe design
  S_w > P_eff × (fs)
• We saw the relation between the tangential and the effective force. The tangential component can be written as
  P_t = \(\frac{2M_t}{d}=\frac{2M_t}{mZ}=\frac{2}{mZ}(\frac{60 × 10^6 × (kW)}{2πN})\)
  And the effective load is given as
  P_eff = \(\frac{C_s P_t}{C_v} = \frac{C_s}{C_v} × \frac{60 × 10^6}{π} (\frac{(kW)}{mZN})\)
• From Lewis Equation
  S_w = Qbd_p K=m² \(\frac{b}{m}\) Q(mz_p)K
• Placing the relations in the inequality, we can solve for the module.

Design Sketch of Spur Gears

The design of gears is complete when all the parameters required for the manufacture of the gear have been calculated. Spur gears are manufactured primarily by casting or forging.
The following figure gives specifications of a spur gear manufactured by casting.

The following figure gives specifications of a spur gear manufactured by forging.