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Helical Gears Technical Information
HELICAL GEARS
In contrast to the spur gear, the helical gear has teeth that are twisted in an axial direction along a helical path. In the axial direction, it appears as though there are a number of staggered spur gears, but in the plane of rotation, it resembles the spur gear.


in the design above there are a number of different features relative to the spur gear, two of the most important being as follows:
- Tooth strength is improved because of the elongated helical wraparound tooth base support.
- Contact ratio is increased due to the axial tooth overlap. Helical gears thus tend to have greater load carrying capacity than spur gears of the same size. Spur gears, on the other hand, have a somewhat higher efficiency.
Helical gears are used in two forms:
- Parallel shaft applications, which is the largest usage.
- Crossed-helicals (also called spiral or screw gears) for connecting skew shafts, usually at right angles.
Generation Of The Helical Tooth
The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear. However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in three dimensions to show changing axial features.


Referring to the figure above, there is a base cylinder from which a taut plane is unwrapped, analogous to the unwinding taut string of the spur gear illustration below.


On the plane there is a straight line AB, which when wrapped on the base cylinder has a helical trace AoBo. As the taut plane is unwrapped, any point on the line AB can be visualized as tracing an involute from the base cylinder. Thus, there is an infinite series of involutes generated by AB, all alike, but displaced in phase along a helix on the base cylinder.
Again, a concept analogous to the spur gear tooth development is to imagine the taut plane being wound from one base cylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugate helical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, a complete involute helicoid tooth is formed.
Fundamentals Of Helical Teeth
In the plane of rotation, the helical gear tooth is involute and all of the relationships governing spur gears apply to the helical. However, the axial twist of the teeth introduces a helix angle. Since the helix angle varies from the base of the tooth to the outside radius, the helix angle β is defined as the angle between the tangent to the helicoidal tooth at the intersection of the pitch cylinder and the tooth profile, and an element of the pitch cylinder. See figure below


The direction of the helical twist is designated as either left or right. The direction is defined by the right-hand rule.
For helical gears, there are two related pitches – one in the plane of rotation and the other in a plane normal to the tooth. In addition, there is an axial pitch. Referring to the figure below, the two circular pitches are defined and related as follows:




The normal circular pitch is less than the transverse radial pitch, pt, in the plane of rotation; the ratio between the two being equal to the cosine of the helix angle. Consistent with this, the normal module is less than the transverse (radial) module.
The axial pitch of a helical gear, px, is the distance between corresponding points of adjacent teeth measured parallel to the gear's axis – see the figure below. Axial pitch is related to circular pitch by the expressions:






A helical gear such as shown in the figure above is a cylindrical gear in which the teeth flank are helicoid. The helix angle in standard pitch circle cylinder is β, and the displacement of one rotation is the lead, L.
The tooth profile of a helical gear is an involute curve from an axial view, or in the plane perpendicular to the axis. The helical gear has two kinds of tooth profiles – one is based on a normal system, the other is based on an axial system.
Circular pitch measured perpendicular to teeth is called normal circular pitch, pn. and pn divided by π is then a normal module, mn.


The tooth profile of a helical gear with applied normal module, mn, and normal pressure angle αn belongs to a normal system.
In the axial view, the circular pitch on the standard pitch circle is called the radial circular pitch, pt. And pt divided by π is the radial module, mt.


Equivalent Spur Gear
The true involute pitch and involute geometry of a helical gear is in the plane of rotation. However, in the normal plane, looking at one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch. However, the shape of the tooth corresponds to a spur gear of a larger number of teeth, the exact value depending on the magnitude of the helix angle.


The geometric basis of deriving the number of teeth in this equivalent tooth form spur gear is given in the illustration above. The result of the transposed geometry is an equivalent number of teeth, given as:


This equivalent number is also called a virtual number because this spur gear is imaginary. The value of this number is used in determining helical tooth strength.
Helical Gear Pressure Angle


Although, strictly speaking, pressure angle exists only for a gear pair, a nominal pressure angle can be considered for an individual gear. For the helical gear there is a normal pressure, αn , angle as well as the usual pressure angle in the plane of rotation, α. The illustration above shows their relationship, which is expressed as:


Importance Of Normal Plane Geometry
Because of the nature of tooth generation with a rack-type hob, a single tool can generate helical gears at all helix angles as well as spur gears. However, this means the normal pitch is the common denominator, and usually is taken as a standard value. Since the true involute features are in the transverse plane, they will differ from the standard normal values. Hence, there is a real need for relating parameters in the two reference planes.
Helical Tooth Proportions
These follow the same standards as those for spur gears. Addendum, dedendum, whole depth and clearance are the same regardless of whether measured in the plane of rotation or the normal plane. Pressure angle and pitch are usually specified as standard values in the normal plane, but there are times when they are specified as standard in the transverse plane.
Parallel Shaft Helical Gear Meshes
Fundamental information for the design of gear meshes is as follows:
- Helix angle – Both gears of a meshed pair must have the same helix angle. However, the helix direction must be opposite; i.e., a left-hand mates with a right-hand helix.
- Pitch diameter – This is given by the same expression as for spur gears, but if the normal module is involved it is a function of the helix angle. The expressions are:


- Center distance – Utilizing the equation above, the center distance of a helical gear mesh is:


Note that for standard parameters in the normal plane, the center distance will not be a standard value compared to standard spur gears. Further, by manipulating the helix angle, β, the center distance can be adjusted over a wide range of values. Conversely, it is possible:
- to compensate for significant center distance changes (or errors) without changing the speed ratio between parallel geared shafts; and
- to alter the speed ratio between parallel geared shafts, without changing the center distance, by manipulating the helix angle along with the numbers of teeth
Helical Gear Contact Ratio
The contact ratio of helical gears is enhanced by the axial overlap of the teeth. Thus, the contact ratio is the sum of the transverse contact ratio, calculated in the same manner as for spur gears, and a term involving the axial pitch.


Details of contact ratio of helical gearing are given later in a general coverage of the subject; see SECTION 11.1.
Design Considerations
Involute Interference
Helical gears cut with standard normal pressure angles can have considerably higher pressure angles in the plane of rotation (see the equation below) depending on the helix angle.


Therefore, the minimum number of teeth without undercutting can be significantly reduced, and helical gears having very low numbers of teeth without undercutting are feasible.
Normal Vs. Radial Module (Pitch)
In the normal system, helical gears can be cut by the same gear hob if module mn and pressure angle αn are constant, no matter what the value of helix angle β.
It is not that simple in the radial system. The gear hob design must be altered in accordance with the changing of helix angle β, even when the module mt and the pressure angle αt are the same.
Obviously, the manufacturing of helical gears is easier with the normal system than with the radial system in the plane perpendicular to the axis.
Helical Gear Calculations
Normal System Helical Gear
In the normal system, the calculation of a profile shifted helical gear, the working pitch diameter dw and working pressure angle αwt in the axial system is done per the equations below. That is because meshing of the helical gears in the axial direction is just like spur gears and the calculation is similar.




The table above shows the calculation of profile shifted helical gears in the normal system. If normal coefficients of profile shift xn1, xn2 are zero, they become standard gears.
If center distance, ax, is given, the normal coefficient of profile shift xn1 and xn2 can be calculated from the table below. These are the inverse equations from items 4 to 10 of the table above.


The transformation from a normal system to a radial system is accomplished by the following equations:


Sunderland Double Helical Gear


A representative application of radial system is a double helical gear, or herringbone gear, made with the Sunderland machine. The radial pressure angle, αt, and helix angle, β, are specified as 20° and 22.5°, respectively. The only differences from the radial system equations in the table above, are those for addendum and whole depth. The table below presents equations for a Sunderland gear.


Helical Rack
Viewed in the normal direction, the meshing of a helical rack and gear is the same as a spur gear and rack. In the table below Calculationof a Helical Rack in the Normal System, the calculation examples for a mated helical rack with normal module and normal pressure angle standard values. Similarly, in the Calculation of a Helical Rack in the Radial System table below the examples for a helical rack in the radial system (i.e., perpendicular to gear axis).




The formulas of a standard helical rack are similar to those of the table Calculation of a Helical Rack in the Normal System, (above) with only the normal coefficient of profile shift xn = 0. To mesh a helical gear to a helical rack, they must have the same helix angle but with opposite hands. The displacement of the helical rack, l, for one rotation of the mating gear is the product of the radial pitch, pt, and number of teeth.


According to the equations of the table above (the Calculation of a Helical Rack in the Radial System), let radial pitch pt = 8 mm and displacement l = 160 mm. The radial pitch and the displacement could be modified into integers, if the helix angle were chosen properly. In the axial system, the linear displacement of the helical rack, l, for one turn of the helical gear equals the integral multiple of radial pitch.

