# Theory

A spline can best be described as a method of transmitting torque. There are many different applications for splines in industry today. These applications vary from the simple to the complex. An example of a simple application is an automotive flanged axle. A simple spline on the inboard end of the shaft transmits torque from the differential to the wheel. An automotive automatic transmission output shaft is an example of a somewhat complex application. These output shafts will sometimes have five or six splines rolled on them. The most obvious spline on these shafts is referred to as the yoke spline. This yoke spline transmits torque from the transmission to the drive shaft. This section will discuss the theory behind rolling the above mentioned splines as well as others onto shafts.

Most rolled splines are called Involute Splines. This is because the form that makes up the sides of the teeth are involute curves. An involute curve is best described as a tightly held string that is unwound from an unmovable diameter. The path that the end of the string would make as the string is unwound is called an involute curve (fig. 3). The unmovable diameter is called the Base Diameter. The base diameter is a relation between the pitch diameter and the pressure angle (base diameter = pitch diameter x the cosine of the pressure angle). As you can see by this formula, each spline of a different pitch and pressure angle are going to have a different base diameter. By rolling a pair of toothed racks with a given pressure angle over a shaft, we get a natural involute curve.

Many rolled splines on part prints, process sheets, etc. are referred to as Fillet Root – Side Fit. A fillet root is a root form that is made up of fillet radii (fig. 4). Side fit refers to the fact that only the sides of the male and female splines come in contact with each other. The major and minor diameters will have clearance. There are some cases where a major diameter fit is called for. A major diameter fit spline uses the major diameter of the male and female splines for the fitting of these splines. Most of the time major diameter fits are used to control runout. In these cases the major diameter of the male spline is ground after rolling.

When a spline is rolled between two racks, no chips are made. All the forming is done by displacement of metal. A blank diameter must be determined so that the part will roll correctly. The engineer will first calculate the cross sectional area of the spline. The dividing line between where the cross sectional area of the addendum (fig. 5) of the tooth, and the cross sectional area of the dedendum of the space is equal, becomes the blank diameter. Once the engineer has determined the blank diameter he then generates much of the rack geometry from the blank diameter.

One of the most common mistakes in the spline rolling industry is changing the blank diameter. Each rack has a given distance from the centerline of one tooth to the centerline of the next tooth. This distance is called the linear pitch. Each blank diameter has a given circumference (Dia. x PI). The linear pitch corresponds to a diameter which is slightly smaller than the blank diameter which is called the theoretical roll diameter. The theoretical roll diameter is the imaginary diameter that the first teeth of the rack would make. Therefore, if the blank diameter is changed so is the theoretical roll diameter. If the number of teeth in the spline, times the linear pitch of the rack, does not match the circumference of the theoretical roll diameter the proper number of teeth will not fit on that diameter. This condition creates spacing error which creates bad parts and wears out racks.

As you can see changing the blank diameter creates conditions which the rack is not designed around. Sometimes, due to major diameter growth, or other part changes, the blank diameter must be adjusted slightly. In most cases a slight change will not effect things too much. However, any time a change is made in the blank diameter, the rack supplier should be notified so that they can determine if a linear pitch change is necessary.

Many of the splines in use today have a pitch called out on part prints, process sheets, etc. that look like a fraction (ie. 24/48). This fraction is called the diametral pitch. The numerator of the fraction is the pitch of the spline. The denominator of the fraction represents the size of the addendum.

The pitch is best described as the number of teeth that will fit in PI (3.1415927″) inches of pitch diameter. In the example above (24/48), the pitch of the spline is 24. That means if the pitch diameter was one inch there would be 24 teeth around it (P.D. x PI = circumference or 1.000″ x 3.1415927). In other words a 1.000″ pitch diameter has PI inches of circumference. There is a good formula that describes the pitch:

Pitch = The number of teeth / pitch diameter

Pitch Diameter = The number of teeth / pitch

The denominator of the diametral pitch fraction tells us the size of the addendum of the spline. In our example above (24/48), 48 is the denominator. Therefore, the addendum of the spline is 1/48″ (.0208″) high. If we had a 16/32 spline, the pitch of the spline would be 16 (16 teeth on a 1.000″ pitch diameter), and the addendum would be 1/32″ (.031″). These type of splines are said to have stub teeth. In other words the addendum is shorter than a standard pitch addendum.

Some rolled splines have a straight pitch called for. In these cases the pitch is as described above. A 10 pitch spline would have ten teeth in a 1.000″ pitch diameter. The addendum is found by a standard formula:

Addendum = one / pitch

A metric version of pitch known as metric module is showing up more frequently on spline data. This module is the same a the pitch except that it is a relationship between the pitch diameter expressed in millimeters to the number of teeth.

Module = pitch diameter in mm’s / number of teeth

One very large misconception in the spline rolling industry is what happens to the spline when the verniers are adjusted. Many people think that they are changing the pitch diameter, this is not true. Remember the formula, pitch diameter = number of teeth / pitch. In order to change the pitch diameter you must change the number of teeth or the pitch. When making a size change with the verniers, we are doing neither of these. What we are doing is changing the thickness of the teeth. As we move the verniers we are changing the daylight opening between the racks. Because the rack teeth are in a V shape, the teeth of the spline get thicker or thinner. The minor and major diameters will change slightly. However, the change in tooth thickness is what the measurement over pins or, a NO GO gage will find.

Much of the time, a measurement over pins is taken to determine if the spline is to proper size. When checking a spline for measurement over pins, we are finding out the actual thickness of the teeth. There are a lot of calculations involved in converting the measurement over pins into tooth thickness. These formulas can be found in any good gearing book.

Most tooth thicknesses called for on process sheets, part prints, etc., are measured at the pitch diameter. The pin diameter used for checking splines is picked so that the pins touch the spline teeth on or as close to the pitch diameter as possible.

The pressure angle of a spline is the angle between where the pitch diameter meets the involute curve and a tangent point on the base diameter (fig. 4). There are three major pressure angles used in the spline rolling industry today. These angles are 30, 37.5 and 45 degrees.

The 30 degree pressure angle is most often used when the two members are allowed to slide. Another common application for 30 degrees is when one of the two members has a thin walled section on or about the spline area. The 30 degree pressure angle will normally roll a straighter lead with less fall-off on the ends of the spline. Tool life is prone to be shorter than the same spline of a higher pressure angle.

The 45 degree pressure angle is most often used for axles and other torque delivering members which are not subjected to bursting forces. The 45 degree pressure angle is the most economical in terms of tool life. However, it does not roll the nice straight leads that the 30 degree pressure angle splines do. These 45 degree splines are prone to larger o.d. fall-off and the lead charts will usually have a crowned look to them.

The 37.5 degree pressure angle is a compromise between the other two pressure angles. It is often used when the shaft material is harder than normal and the application will not allow a 45 degree spline.

The pressure angle of a spline cannot be checked. The involute form of a given spline can be checked. The involute curve is controlled by the base diameter which is a product of the pressure angle and the pitch diameter (B.D. = P.D. x cos. P.A.). The most common method of checking the involute form is with an involute checking machine. This machine uses a stylus which touches the spline tooth. The part is then rotated about a simulated base diameter. While the part is being rotated, the stylus follows the form of one side of the tooth. An electronic signal is sent to a recording unit and a chart of the form is recorded. If the involute form is correct, a straight line will be recorded on the chart paper. Another method of checking the involute form is with measurement over pins. A series of calculations would permit several different sized pins to be used to check the tooth thickness of the spline at several different diameters. The pin method of checking forms is time consuming and somewhat inaccurate.

There is a diameter called for on most splines called the Form Diameter. This diameter is sometimes referred to as the T.I.F. or S.A.P. T.I.F. which stands for “true involute form diameter”, S.A.P. stands for start of “active profile”. They all mean the same thing. The form diameter is the diameter where the fillet radius in the root stops and the involute curve begins. This diameter is used in conjunction with the last point of contact of the mating part plus some clearance. The minor diameter of the GO composite gage will be the same as the form diameter.

A standard known as the effective fit concept is widely used to control the fit between male and female splines. This standard uses the minimum actual and the maximum effective tooth thicknesses as outer controll limits for a rolled spline. The min. actual tooth thickness is the minimum thickness of any one tooth about the spline. The max. effective tooth thickness is the min. actual tooth thickness plus all the allowable form errors of a given spline. You will see that the minimum M.O.P. is usually called out as a reference dimension. This is because, if you convert the minimum M.O.P. into tooth thickness you will get the min. actual tooth thickness. The tooth thicknesses are what the effective fit concept looks at.

As mentioned above, the maximum effective tooth thickness is the minimum actual tooth thickness plus the allowable form errors. Each given spline has its own effective tooth thickness. Most of the errors that are created during the spline rolling process come in the form of spacing or index error.

Spacing error or index error, as it is sometimes referred to, is the maximum radial misslocation of the spline teeth. In other words, if we have a spline with 36 teeth we should have the centerline of a tooth every 10 degrees around the pitch diameter (360 degrees divided by 36 or the number of teeth). Spacing error is the maximum error of this radial location in opposite directions. If a given spline has a min. actual tooth thickness of 0.096″, and 0.003″ of spacing error it can be said to have a effective thickness of 0.099″ (assuming there is no detectable lead, involute, or other form error). If the same spline had 0.003″ of spacing and, 0.0015″ of plus condition on the involute forms, the effective thickness of this spline would be 0.1005″ (0.096″ + 0.003″ + 0.0015″ = 0.1005″, same assumptions).

Spacing error is one of the biggest headaches for fitting gages and running six sigma capabilities. In the spline rolling process there are many factors that create spacing error. Other form errors have only one or two causes, therefore, they are not so prone to problems. Spacing error has over one dozen possible causes.

When a manufacturer wants to find out if his spline rolling machines are capable or not, the study is based around the min. act. tooth thickness and the max. eff. tooth thickness. There is in reality a max. actual tooth thickness for a given class of fit. There is also a min. eff. tooth thickness for a given class of fit. These two are normally not found on part prints, etc.. The next logical question has to be, “what are the limits for a capability study ?”. The min. act. tooth thickness is used as the lower control limit and the max. eff. tooth thickness is used as the upper control limit. The “as rolled” min. act. and the max. eff. tooth thicknesses are run separately to those control limits. The max. eff. bell curve usually is centered in the top half of the tolerance band. The min. act. bell curve is usually centered in the lower half of the tolerance. When looking at the numbers for a given spline, the closer the min. act. tooth thickness is to the max. eff. tooth thickness the better the spline is.

When running a capability study with parts that are not qualified you are checking the whole process capability. In order to check the spline roller’s capability six sigma capable blanks must be used.

The gages available to check min. act. and max. eff. tooth thicknesses are usually electronic. These gages will come up with min. act. quite readily. However, the max. eff. gages are somewhat under fire around the industry. This is due to the the fact that one type only tells if the spline is in a go condition. It does not give an effective value. While the other type of gage will actually give an effective number. Some people question the accuracy of the gage that gives an effective value.

This writer does not want to get into the debate. However, if you want to know if a spline roller is cabable you need to know the effective values. Spacing error is the biggest spline rolling problem. The biggest part of most rolled spline effective thicknesses is spacing.