Design of Helical Gears

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


  1. What are Helical Gears?
  2. Virtual Number of Teeth
  3. What Forces act on the Helical Gear?
  4. Effective Load on Helical Gear Tooth
  5. Beam Strength Criterion
  6. Wear Strength Criterion
  7. What are Herringbone Gears?
  8. What are Crossed Helical Gears?

What are Helical Gears?

Helical Gears have their teeth cut in the form of a helix of a pitch cylinder. That is, the teeth are cut at an angle from the axis of the shaft. This angle is called the helix angle and is represented by Ψ. The helix angle governs the design of helical gear and differentiates it from a spur gear.

  • In helical gears, the contact between meshing teeth begins with a point on the leading edge of the tooth and gradually extends along the diagonal line across the tooth. There is a gradual pick-up of load by the tooth, resulting in smooth engagement and quiet operation even at high speeds.
  • Parallel helical gears operate on two parallel shafts. In this case, the magnitude of the helix angle is the same for the pinion and the gear, however, the hand of the helix is opposite. A right-hand pinion meshes with a left-hand gear and vice versa.
  • Crossed helical gears are mounted on shafts with crossed axes. Crossed helical gears are not recommended for applications that must transmit large torque or power. They are nevertheless frequently used in light-load applications, such as distributor and speedometer drive of automobiles.
  • The helix angle may typically range from about 10 to 45°.
  • The module of the helical gear, also called the transverse module, are a function of the helix angle and normal module. The two modules are related by the following relation
    mn=m cos Ψ.
  • The pitch circle diameter of the helical gear is given as πd=pz. This can be written as

Virtual Number of Teeth

The diagram below shows the formative gear creation.

Formative Gear

The pitch cylinder of the helical gear is cut by the plane A–A, which is normal to the tooth elements. The intersection of the plane A–A and the pitch cylinder (extended) produces an ellipse. This ellipse is shown by a dotted line.
The semi-major and semi-minor axes of this ellipse are \(\frac{d}{2 cos⁡ψ}\) and d/2. The radius of curvature can be given as r = a2/b where a and b are the semi-major and semi-minor axes.
r = \(\frac{d}{2(cosψ)^2}\)
The number of virtual teeth is given by the following equation
z= \(\frac{2πr}{p_n} = \frac{2π ×\frac{d}{2(cosψ)^2}}{π m_n} =\frac{d}{m_n(cosψ)^2}\)
Substituting the value of the pitch circle diameter,
z=\(\frac{z}{(cos Ψ)^3}\)
Where z is the actual number of teeth and z’ is the virtual number of teeth.

What Forces act on the Helical Gear?

The following figure shows the forces acting on the helical gear.

Forces acting on a helical gear

The resultant force P acting on the tooth of a helical gear is resolved into three components, Pt, Pr, and Pa.
Pt = tangential component
Pr = radial component
Pa = axial or thrust component
The normal pressure angle is in the plane ABC, shown to be shaded, while helix angle Y is in the lower plane BCD.
From triangle ABC
Pr=P sin αn
From triangle BDC
Pa=BC sin Ψ =P cos αn sin Ψ
Pt=BC cos Ψ =P cos αn cos Ψ
From the above equations
Pa=Pttan Ψ
Pr=\(P_t \frac{tan \,α_n}{cos⁡ψ}\)
The tangential component is calculated from the transmitted torque relation.

  • The direction of the tangential component for a driving gear is opposite to the direction of rotation. The direction of the tangential component for a driven gear is the same as the direction of rotation.
  • The radial component on the pinion acts towards the center of the pinion. The radial component on the gear acts towards the center of the gear.
  • The direction of the thrust component for the driven gear will be opposite to that for the driving gear. It is the axial force that is created when transmitting power by gears.

Effective Load on Helical 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 Cs 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
    Peff= \(\frac{C_s P_t}{C_v}\)

  • In the end stages, the Buckingham equation is used to calculate the effective load with the following equation
    Peff= CsPt+Pd
    Pd= \(\frac{21v(Ceb(cos⁡ψ)^2+P_t)cos⁡ψ}{21v+\sqrt{Ceb(cos⁡ψ)^2+P_t}}\)
    Where C is the deformation factor and e is the sum of errors between meshing teeth.

Beam Strength Criterion

To determine beam strength, the helical gear is considered to be equivalent to a formative spur gear. The formative gear is an imaginary spur gear in a plane perpendicular to the tooth element. The pitch circle diameter of this gear is d’, the number of teeth is z’, and the module mn. The beam strength of spur gear is given by the following equation.
Sb = mbσbY
The following figure shows the equivalent to the beam strength for a helical gear.

Beam Strength of a Helical Gear

(Sb)n = \(\frac{mbσ_b Y}{cos⁡ψ}\)
(Sb)n= Sb cos Ψ
From the above equations,
Sb= mnbY

Beam strength (Sb) indicates the maximum value of tangential force that the tooth can transmit without bending failure. It should be always more than the effective force between the meshing teeth.

Wear Strength Criterion

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.
    Sw= Qbdp K
  • Referring to the previous figure, we can derive the actual relationship.
    (Sw)n=\(\frac{Qbd_p K}{(cos⁡ψ)^3}\)
  • The component of (Sw)n in the plane of rotation is denoted by Sw.
    (Sw)n= Sw cos Ψ
  • Substituting in the previous equation, we solve the wear strength of the helical gear.
    Sw=\(\frac{Qbd_p K}{(cos⁡ψ)^2}\)
  • The wear strength (Sw) indicates the maximum tangential force that the tooth can transmit without pitting failure. It should be always more than the effective force between the meshing teeth.

What are Herringbone Gears?


When a pair of helical gears are transmitting power, both the input and the output shafts are subjected to thrust loads. These forces cause reactions on the bearings. The thrust forces on input and output shafts can be eliminated by using herringbone or double helical gears.

  • There is a basic difference between herringbone and the double-helical gear. There is a groove between two helical gears in the case of double helical gear, while a gear without a groove is called herringbone gear.
  • Double helical gears are cut on a single gear blank, by a hob with a tool run-out groove between the hands of helices. Herringbone gear is cut by two cutters, which reciprocate 180° out of phase to avoid clashing.
  • They develop opposite thrust reactions and thus cancel out the thrust force within the gear itself. The net axial force that acts on the bearings is zero.
  • High pitch line velocities can be attained with herringbone and double helical gears while having a high-power transmitting capacity.
  • Balancing of thrust forces depends on the equal distribution of load on the gear. A high degree of precision is required to locate herringbone and double helical gears axially on the shaft. They must be aligned accurately to take exactly one-half of the load.
  • Herringbone and double helical gears are used in high-power applications such as ship drives and turbines.
  • The helix angle for a single helical gear is from 15° to 25°. Herringbone and double helical gears permit higher helix angles because there is no thrust force. The helix angle for herringbone and double helical gears is from 20° to 45°.
  • The design procedure and design equations for herringbone and double helical gears are the same as for single helical gear. In design, a herringbone or double helical gear is considered equal to two identical helical gears, each transmitting one-half power.

What are Crossed Helical Gears?

Helical gears, which are mounted on non-parallel shafts, are called crossed helical gears. In these gears, the axes of two shafts are neither parallel nor intersecting like worm gears. There is line contact between meshing teeth of parallel helical gears.

  • There is a point contact between the meshing teeth of crossed helical gears. Since the contact area of a point is very small, the contact pressure is high, and wear is comparatively rapid.
  • Crossed helical gears have very low load-carrying capacity. They are not recommended for high-power transmission. They are particularly useful in light-duty applications.
  • When the shaft angle is small, the opposite hand of the helix is used. On the other hand, the same hand of helix is used when the shaft angle is more. For a particular case, when the shaft angle is 90°, the gears must have the same hand of the helix and each helix angle is 45°.

Key Points to Remember

Here is the list of key points we need to remember about “Design of Helical Gears”.

  • Helical gears have teeth cut at an angle to the axis of the shaft and are more efficient in a quieter environment.
  • The helix angle is the angle made by the teeth to the shaft. It is usually around 10 to 45°.
  • The virtual number of teeth are the teeth on the formative spur gear and is an important aspect in helical gear design.
  • Three forces act on the contact point of helical gears in mesh, the tangential force, the radial force, and the axial force.
  • The effective load on a helical gear is a combination of tangential and dynamic loads.
  • Beam Strength and Wear strength criteria are applied in helical gear design with modifications for the formative spur gears.
  • Herringbone gears are used to minimize thrust loads and maximize power transfer capacity.
  • Crossed helical gears are used to transmit power between non-intersecting, non-parallel shafts.

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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