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Spur Gears Technical Information

This introduction sets the tone for a technical discussion on gear fundamentals while accommodating both beginners and experienced users. Here's a breakdown of its intent and structure:

1. Purpose:

  • Provide a broad introduction to the fundamentals of gears.
  • Serve as both an educational guide for beginners and a design reference for experts.

2. Audience:

  • Geared toward designers and engineers, regardless of prior exposure to gear systems.
  • Acknowledges that not everyone has in-depth knowledge of the subject.

3. Structure:

  • First-time readers: Encouraged to follow the material sequentially for a logical and comprehensive understanding.
  • Experienced users: Can use the document selectively, accessing specific sections as needed for reference.

4. Importance:

  • Highlights the necessity of understanding gear basics for proper application and system design.
  • Serves as a starting point and resource for further exploration of advanced gear concepts.

This structured approach ensures the material is versatile and useful for a wide audience, fostering both foundational learning and practical application.


The Basic Geometry Of Spur Gears

The basic geometry and nomenclature of a spur gear mesh is shown in above.
The essential features of a gear mesh are:

  1. Center distance.
  2. The pitch circle diameters (or pitch diameters).  

    Pitch Circle:
    an imaginary circle passing through the teeth of a gearwheel, concentric with the gearwheel, and having a radius that would enable it to be in contact with a similar circle around a mating gearwheel (source, thefreedictionary.com).
  3. Size of teeth (or module).
  4. Number of teeth.
  5. Pressure angle of the contacting involutes

    Pressure Angle:
    is the angle between the toothface and the gear wheel tangent. It is more precisely the angle at a pitch point between the line of pressure (which is normal to the tooth surface) and the plane tangent to the pitch surface. The pressure angle gives the direction normal to the tooth profile. The pressure angle is equal to the profile angle at the standard pitch circle and can be termed the "standard" pressure angle at that point. Standard values are 14.5 and 20 degrees (source, thefreedictionary.com).

The Law Of Gearing

Gears must have consistent angular velocities or proportionality of position transmission. Accurate placement is required for precision devices. High-speed and/or high-power gear trains must also be transmitted at constant angular velocities to avoid significant dynamic issues.

Constant velocity (or constant ratio) motion transmission is characterized as "conjugate action" of the gear teeth profiles. A geometric relationship can be constructed for the shape of the tooth profiles to give conjugate action, which is stated as the Law of Gearing as follows:

"A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point." Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves.


The Involute Curve

  • 1. Prevalence of the Involute Profile:

    Although various curves meet the Law of Gearing, involute teeth dominate modern gear systems (except for clock gears).

2. Key Advantages of the Involute Profile:

  1. Conjugate Action Independence: The constant velocity ratio is unaffected by slight changes in center distance, enhancing operational flexibility and manufacturing tolerances.
  2. Straight-Sided Basic Rack:
    • Simplifies the design and manufacturing of cutting tools.
    • Ensures high accuracy in gear tooth production.
  3. Universal Tooling: A single cutter can be used to generate gear teeth across all numbers in the same pitch, reducing complexity in tooling and manufacturing.

3. Geometric Definition:

  • Involute Curve Generation: The involute is formed as the trace of a point on the end of a taut string unwinding from a cylinder (called the base circle). The base circle is fundamental but invisible in the actual gear.
Generation of an involuteby taut stringGeneration of an involuteby taut string
  • Visualization of Mating Teeth: Imagine a taut string simultaneously unwinding from one base circle and winding onto another:

    • The traced curves on the rotating planes of each base circle form conjugate involutes.

    • At all points of contact, the common normal is the common tangent, passing through a fixed pitch point on the line-of-centers.
Generation and Action of Gear TeethGeneration and Action of Gear Teeth

4. Accommodating Motion Reversal:

  1. A second string wound in the opposite direction produces oppositely curved involutes.
  2. When properly spaced, these oppositely curved involutes form a gear tooth pair, enabling motion reversal and bidirectional operation.

5. Result:

  1. Involute Gear Tooth Formation: Properly spaced pairs of involutes create gear teeth with optimal conjugate action, as illustrated in the figure above.
  2. This explanation emphasizes the efficiency, simplicity, and versatility of the involute profile, solidifying its role as the standard for gear systems.

Pitch Circles

Defintion of pitch pointDefintion of pitch point

In the figure above, the tangent to the two base circles represents the line of contact, or line-of-action in gear vernacular. The pitch point, P, is defined as the point at which this line crosses the line of centers. This sets the size of the pitch circles, also known as pitch diameters. The ratio of the pitch diameters gives the velocity ratio:

Velocity ratio of gear 2 to gear 1 is:

Velocity ratio of 2:1Velocity ratio of 2:1

Pitch And Module
The size, or spacing, of the teeth along the pitch circle is crucial for determining the gear shape. There are two fundamental forms of this, which is known as pitch.

  1. Circular Pitch: A naturally occurring linear measure along the tooth spacing's pitch circle is called "circular pitch." It is the linear distance between corresponding points of neighboring teeth (measured along the pitch circle arc), as illustrated in image below.
Definition of Circular PitchDefinition of Circular Pitch

It is calculated by dividing the number of teeth by the pitch-circle circumference:

Circular PitchCircular Pitch
  1. Module:  The American inch unit, diametral pitch, is replaced by the amount module m in metric gearing. The pitch diameter length per tooth is the module. So:

    equation for module
  2. Relation of Pitches: From the geometry that defines the two pitches, it can be shown that module and circular pitch are related by the expression:

    Relation of Pitches

    This relationship is simple to remember and permits an easy transformation from one to the other.

    Diametral pitch iis widely used in England and America to represent the tooth size. The relation between diametral pitch and module is as follows:

    Diametral Pitch relationship to module

Module Sizes And Standards

Module m represents the size of involute gear tooth. The unit of module is mm. Module is converted to circular pitch p, by the factor π.

Module m repreents the size of involute gear toothModule m repreents the size of involute gear tooth
Standard Values of ModuleStandard Values of Module

The table above is extracted from JIS B 1701-1973 which defines the tooth profile and dimensions of involute gears. It divides the standard module into three series.

Comparative Size of Various Rack TeethComparative Size of Various Rack Teeth

The illustration above shows the comparative size of various rack teeth. 

When a specific desired spacing is required, such as to obtain an integral feed in a mechanism, the circular pitch, p, is also employed to denote tooth size. In this instance, a circular pitch with an integer or unique fractional value is selected.

When designing position control systems, this is frequently the option chosen. The driving of printing plates to supply a certain feed is another specific use. The majority of involute gear teeth have a standard pressure angle of α = 20° and a standard entire depth. The tooth profile of a full depth standard rack tooth and mating gear is displayed in drawing on the right. Its dedendum is ≥ 1.25m and its addendum is = 1m.

A tooth is referred to as "stub" if its depth is less than its full depth, and "high" if it is deeper than its whole depth.

The addendum and dedendum of the most commonly used stub tooth are 0.8 and 1 meters, respectively. Although the contact ratio is decreased, stub teeth are stronger than full depth gears. However, a high depth tooth weakens the tooth while increasing the contact ratio.

In the standard involute gear, pitch p times the number of teeth becomes the length of pitch circle:

Length of pitch circleLength of pitch circle
The tooth profile and dimension of standard rack The tooth profile and dimension of standard rack

DETAILS OF INVOLUTE GEARING

Generation and Action of Gear TeethGeneration and Action of Gear Teeth
Definition of pitch circle and pitch pointDefinition of pitch circle and pitch point

The pressure angle is defined as the angle between the line-of-action (common tangent to the base circles in the illustrations above) and a perpendicular to the line-of-centers. 

Definition of Pressure AngleDefinition of Pressure Angle

See the above illustration. From the geometry of these illustrations, it is obvious that the pressure angle varies (slightly) as the center distance of a gear pair is altered. The base circle is related to the pressure angle and pitch diameter by the equation:

The base circle is related to the pressure angle and pitch diameter by this equationThe base circle is related to the pressure angle and pitch diameter by this equation

where d and α are the standard values, or alternately:

where d' and α' are the exact operating values.

According to the fundamental formula, the base circle gets smaller as the pressure angle increases. As a result, the base circles of 14.5° pressure angle gears are substantially closer to the tooth roots than those of 20° gears for conventional gears. It is because of this that 14.5° gears have more undercutting issues than 20° gears. 

Proper Meshing And Contact Ratio

The illustration above shows a pair of standard gears meshing together. The contact point of the two involutes, as shown, slides along the common tangent of the two base circles as rotation occurs. The common tangent is called the line-of-contact, or line-of-action.

 

A pair of gears can only mesh correctly if the pitches and the pressure angles are the same. Pitch comparison can be module (m), circular (p), or base Pb. That the pressure angles must be identical becomes obvious from the following equation for base pitch:

Thus, if the pressure angles are different, the base pitches cannot be identical. The length of the line-of-action is shown as ab in the illustration below.


Contact Ratio

When one set of teeth stops making contact, the next set must have already engaged in order to provide smooth, continuous tooth movement. It is desirable to have as much overlap as possible. Having as much overlap as feasible is good. The contact ratio is a measure of this overlap. This represents the line-of-action length divided by the base pitch. The illustration above shows the geometry.

The length-of-action is determined from the intersection of the line-of-action and the outside radii. For the simple case of a pair of spur gears, the ratio of the length-of-action to the base pitch is determined from:

It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1, calculated for all tolerances at their worst-case values.

A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such a high contact ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or specially designed nonstandard external spur gears.

Geometry ofm Contact RatioGeometry ofm Contact Ratio

The Involute Function

The illustration above shows an element of involute curve. The definition of involute curve is the curve traced by a point on a straight line which rolls without slipping on the circle.

The circle is called the base circle of the involutes. Two opposite hand involute curves meeting at a cusp form a gear tooth curve. We see above, that the length of the base circle arc ac equals the length of straight line bc.

The length of base circle arc ac equals the length of straight line bc.The length of base circle arc ac equals the length of straight line bc.

The θ in the below illustration can be expressed as inv α + α, then the formula below will become:

Function of α, or inv α, is known as involute function. Involute function is very important in gear design. Involute function values can be obtained from appropriate tables. With the center of the base circle O at the origin of a coordinate system, the involute curve can be expressed by values of x and y as follows:

The Involute CurveThe Involute Curve

SPUR GEAR CALCULATIONS

Standard Spur Gear

The meshing of standard gearsThe meshing of standard gears

The meshing of standard spur gears is depicted in the above illustration. Pitch circles of two gears make contact and roll with one another when standard spur gears mesh.

The table below contains the calculating formulas of standard spur gears.

The calculation of teeth numberThe calculation of teeth number

All calculated values in the Calculation of Standard Spur Gears table are based upon given module (m) and number of teeth (z1 and z2). If instead module (m), center distance (a) and speed ratio (i ) are given, then the number of teeth, z1 and z2, would be calculated with the formulas as shown in the table Calculation of Teeth Number below.

The Calculation of Teeth NumberThe Calculation of Teeth Number

Keep in mind, that using the calculations from the table above, the number of teeth will most likely not be an integer value. The designer is required to select a set of integer tooth counts that are as near to the theoretical values as feasible. Both the center distance and the gear ratio will probably be slightly altered as a result. In the event that the center distancebe inviolable, it will then be necessary to resort to profile shifting.

The Generating Of A Spur Gear

Involute gears can be readily generated by rack type cutters. The hob is in effect a rack cutter. Gear generation is also accomplished with gear type cutters using a shaper or planer machine.

The generating of a standard spur gearThe generating of a standard spur gear

The creation of an involute gear tooth profile is shown in the illustration above. It demonstrates how a spur gear is created when a rack cutter's pitch line rolls on a pitch circle.

Undercutting

Geometry of Contact RatioGeometry of Contact Ratio
Example of undercut standard design gearExample of undercut standard design gear

It is evident from the illustration above that the line-of-contact's maximum length is constrained by the common tangent's length. In addition to being worthless, any tooth addendum that extends past the tangent points (T and T') obstructs the mated tooth's root fillet area. As a result, the illustration Example of Undercut Standard Design Gear  displays the typical undercut tooth. The undercut eliminates some of the beneficial involute next to the base circle in addition to weakening the tooth with a wasp-like waist.

The condition for no undercutting in a standard spur gearThe condition for no undercutting in a standard spur gear

This indicates that the minimum number of teeth free of undercutting decreases with increasing pressure angle. For 14.5° the value of zc is 32, and for 20° it is 18. Thus, 20° pressure angle gears with low numbers of teeth have the advantage of much less undercutting and, therefore, are both stronger and smoother acting.

For more information, including, camparisons of enlarged and undercut standard pinions, calculations of internal gears, the fundamentals of helical gears, and the elements of metric gear technology, please click here.

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