circle used to define the depth of involute profile control NOTE In the case of an external spline it is located near and above the minor diameter, and on an internal spline near and bel
General symbols
The general symbols used to designate the various spline terms and dimensions are given below
D Fe Form diameter, external spline mm
D Fe max Maximum form diameter, external spline mm
D Fi Form diameter, internal spline mm
D Fi min Minimum form diameter, internal spline mm
D Re Diameter of measuring ball or pin for external spline mm
D Ri Diameter of measuring ball or pin for internal spline mm
D ee Major diameter, external spline mm
D ee max Maximum major diameter, external spline mm
D ee min Minimum major diameter, external spline mm
D ei Major diameter, internal spline mm
D ei max Maximum major diameter, internal spline mm
D ei min Minimum major diameter, internal spline mm
D ie Minor diameter, external spline mm
D ie max Maximum minor diameter, external spline mm
D ie min Minimum minor diameter, external spline mm
D ii Minor diameter, internal spline mm
D ii max Maximum minor diameter, internal spline mm
D ii min Minimum minor diameter, internal spline mm
E Basic space width, circular mm
E max Maximum actual space width mm
E min Minimum actual space width mm
E v Effective space width, circular mm
E v max Maximum effective space width mm
E v min Minimum effective space width mm
F p Total cumulative pitch deviation àm
K e Approximation factor for external spline —
K i Approximation factor for internal spline —
M Re Measurement over two balls or pins, external splines mm
M Ri Measurement between two balls or pins, internal mm
S Basic tooth thickness, circular mm
S max Maximum actual tooth thickness mm
S min Minimum actual tooth thickness mm
S v Effective tooth thickness, circular mm
S v max Maximum effective tooth thickness mm
S v min Minimum effective tooth thickness mm
The measurement parameters for splines include the external spline measurement over k teeth in millimeters, the spline length, and form clearance Effective clearance, indicating looseness or interference, is represented as \(a_m\), with maximum and minimum values denoted as \(a_{m,\text{max}}\) and \(a_{m,\text{min}}\) Contact diameters are specified for external and internal splines as \(d_{ce}\) and \(d_{ci}\) respectively, in millimeters Fundamental deviations for external splines are marked as \(e_{sv}\), while form tooth height is \(h_s\) Involute angle \(\alpha\) is calculated with the formula \(\tan \alpha - \pi \cdot \alpha / 180^\circ\) The number of measured teeth is \(k\), and the module and circular pitch are given as \(m\) and \(p\) in millimeters, respectively, with the base pitch as \(p_b\) The total number of teeth is \(z\) Pressure angles vary, including \(\alpha\), \(\alpha_{Fe}\), \(\alpha_{Fi}\), \(\alpha_{ce}\), \(\alpha_{ci}\), \(\alpha_D\), \(\alpha_e\), and \(\alpha_i\), each corresponding to different contact or diameter points in degrees Deviation allowance is \(\lambda\), and the fillet radii of the basic rack are specified as \(\rho_{Fa}\) for external and \(\rho_{Fi}\) for internal splines in millimeters Fundamental deviations for external splines are represented by \(k; js; h; f; e; d\), indicating various tolerance measures.
Subscripts
Subsidiary subscripts such as "i" for minor or internal conditions, "e" for major or external conditions (typically in the last position), "b" to indicate the base, "c" at the contact point, and "d" for tolerance based on pitch diameter (D) are used to specify relative conditions or locations within the guidelines for gear and technical drawings These subscripts ensure precise communication of geometric tolerances and feature locations, adhering to industry standards for clarity and accuracy Proper use of these symbols enhances the understanding of manufacturing and inspection requirements in engineering designs.
E tolerance based on space width (E) or tooth thickness (S)
F pertaining to form diameter v effective
In electronic data processing (EDP), presenting symbols in their precise theoretical form can be challenging due to limitations of connected printing equipment To address this, alternative symbols are provided for EDP use, such as representing the base diameter (D_b) as "DB" when printing constraints prevent the accurate display of the original symbol These standardized alternatives ensure clarity and consistency in data representation within electronic systems.
Formulae for dimensions and tolerances for all fit classes
The formulae for dimensions and tolerances for all fit classes are given in Table 1
Table 1 — Formulae for dimensions and tolerances for all fit classes
Term Symbol Formula EDP representation
Fundamental deviation, external es v Resulting from fundamental deviation k, js, h, f, e and d ESV
30°, flat root D ei min m z ⋅ + ( 1,5) DEIMIN
30°, fillet root D ei min m z ⋅ + ( 1,8) DEIMIN
37,5°, fillet root D ei min m z ⋅ + ( 1,4) DEIMIN
45°, fillet root D ei min m z ⋅ + ( 1,2) DEIMIN
Maximum major diameter, internal D ei max D ei min + ( T + λ ) / tan α D (see Note 1) DEIMAX
30° flat root and fillet root D Fi min m z ⋅ + + ⋅ ( 1) 2 c F DFIMIN
37,5° fillet root D Fi min m z ⋅ + ( 0,9) 2 + ⋅ c F DFIMIN
45° fillet root D Fi min m z ⋅ + ( 0,8) 2 + ⋅ c F DFIMIN
Minimum minor diameter, internal D ii min D Fe max + ⋅ 2 c F (see Note 2) DIIMIN
Maximum minor diameter, internal: m u 0,75 D ii max D ii min + IT 10 DIIMAX
0,75 < m < 2 D ii max D ii min + IT 11 DIIMAX m W 2 D ii max D ii min + IT 12 DIIMAX
Minimum effective space width E v min 0,5 ⋅ π ⋅ m EVMIN
Maximum actual space width: class 4 E max E v min + ( T + λ ) (see Note 3) EMAX class 5 E max E v min + ( T + λ ) (see Note 3) EMAX class 6 E max E v min + ( T + λ ) (see Note 3) EMAX class 7 E max E v min + ( T + λ ) (see Note 3) EMAX
Minimum actual space width E min E v min + λ EMIN
Maximum effective space width E v max E v min + T v EVMAX
30°, flat root and fillet root D ee max m z ⋅ + + ( 1) es v /tan α D (see Note 4) DEEMAX
37,5°, fillet root D ee max m z ⋅ + ( 0,9) + es v /tan α D (see Note 4) DEEMAX
45°, fillet root D ee max m z ⋅ + ( 0,8) + es v /tan α D (see Note 4) DEEMAX
Minimum major diameter, external: m u 0,75 D ee min D eemax − IT 10 DEEMIN
0,75 < m < 2 D ee min D ee max − IT 11 DEEMIN m W 2 D ee min D ee max − IT 12 DEEMIN
Term Symbol Formula EDP representation
Maximum form diameter (see Note 5) D Fe max ( ) v 2
30°, flat root D ie max m z ⋅ − ( 1,5) + es v /tan α D DIEMAX
30°, fillet root D ie max m z ⋅ − ( 1,8) + es v /tan α D DIEMAX
37,5°, fillet root D ie max m z ⋅ − ( 1,4) + es v /tan α D DIEMAX
45°, fillet root D ie max m z ⋅ − ( 1,2) + es v /tan α D DIEMAX
Minimum minor diameter, external D ie min D ie max − ( T + λ )/tan α D (see Note 1) DIEMIN
Maximum effective tooth thickness S v max S es + v SVMAX
Minimum actual tooth thickness: class 4 S min S v max − ( T + λ ) (see Note 3) SMIN class 5 S min S v max − ( T + λ ) (see Note 3) SMIN class 6 S min S v max − ( T + λ ) (see Note 3) SMIN class 7 S min S v max − ( T + λ ) (see Note 3) SMIN
Maximum actual tooth thickness S max S v max − λ SMAX
Minimum effective tooth thickness S v min S v max − T v SVMIN
Total tolerance, space width or tooth thickness ( T + λ ) (see Note 6) TLAM
Maximum effective clearance c v max E v max − S v min CVMAX
Minimum effective clearance c v min E v min − S v max CVMIN
Form clearance c F see Note 5 CF
Form tooth height h s see Note 5 HS
Ball or pin diameter, internal spline D Ri see Note 7 DRI
Ball or pin diameter, external spline D Re see Note 7 DRE
Measurement between balls or pins M ri see Note 7 MRI
Measurement over balls or pins M re see Note 7 MRE
Change factor, internal K i see Note 7 KI
Change factor, external K e see Note 7 KE
NOTE 2 For all classes of fit, always take the D Fe max value corresponding to the H/h fit
NOTE 3 See Clause 8 and ISO 4156-2
NOTE 4 Take es v = 0 for fundamental deviation js and k
NOTE 5 For h s , see Figure 15 et Table 2
NOTE 7 See ISO 4156-3 concerning the choice of balls or pins
5 Concept of side fit splines
ISO 4156 specifies side fit involute splines with pressure angles of 30°, 37.5°, and 45°, enabling effective torque transmission solely through tooth flank contact These splines can operate in both clockwise and counterclockwise rotational directions Additionally, the major and minor diameters of the opposite tooth flanks must have sufficient clearance to ensure proper engagement and function.
Figure 2 — Side fit tooth flank contact
The nature of the involute profile divides the torque into two directions resulting in a centring force (see
Figure 3) This centring force enables side fit involute splines to be centralized by the tooth flanks a Centring force b Rotation force c Torque
The sizes of space width and tooth thickness (see Figure 4) are defined as the length of the arc at the theoretical pitch circle diameter
Figure 4 — Space width and tooth thickness
The major and minor diameters (see Figure 5) always have clearance and do not contact each other
To effectively machine internal spline spaces and external spline teeth, it is essential to adhere to a specific machining tolerance known as the actual machining tolerance Four classes of machining tolerance—classes 4, 5, 6, and 7—are available to meet various industrial requirements This machining tolerance, designated as T, applies to the internal spline space width and the external spline tooth thickness, ensuring precision and functionality in spline manufacturing.
The upper machining tolerance limit is referred to as maximum actual and the lower one is referred to as minimum actual
Form deviations in cylindrical fits, such as those between hubs and shafts, impact the maximum material condition and overall fit quality These deviations represent how much the actual geometry differs from an ideal cylinder Spline form deviations are more complex, affecting each flank of every tooth or space, and their cumulative effect is known as the effective deviation Proper understanding of these deviations is crucial for optimizing fit accuracy and ensuring reliable mechanical performance.
Form deviations in splines can be categorized into three types: profile deviation, index deviation, and helix deviation These deviations typically lead to a reduction in the effective space width of internal splines and an increase in the effective tooth thickness of external splines, ultimately decreasing the clearance between mating parts Detecting these deviations requires imagining a perfect mating spline that fits precisely without looseness or interference Key measurement points include the largest and smallest space widths, maximum and minimum actual tolerances, and the largest and smallest tooth thicknesses, which help assess the extent of form deviations and their impact on spline fit.
Profile deviation's positive material elements can lead to a smaller space or increased tooth thickness, impacting the fit with the mating part Specifically, internal splines and external splines may experience variations in space width and tooth thickness, both actual and effective, which influence the overall compatibility Understanding these deviations is crucial for ensuring proper engagement between the mating components and maintaining optimal performance.
Positive material elements of pitch deviation, such as increased tooth thickness or reduced space width, can adversely impact the fit between mating parts, including internal and external splines Specifically, deviations in actual and effective space widths and tooth thicknesses influence the precision of gear engagement, potentially leading to improper meshing with the mating component Understanding these deviations is crucial for ensuring optimal gear performance and accurate fitting in mechanical assemblies.
Helix deviation, as illustrated in Figure 10, leads to a reduction in space width or an increase in tooth thickness, both of which can impact the fit with a mating part Accurate assessment of internal and external splines, along with the actual and effective measurements of space width and tooth thickness, is essential for ensuring proper fit and function Proper alignment of these elements is critical for optimal performance of mating components.
The accumulated form deviations (see Figure 11) of each flank result in an effective size of space width or tooth thickness
`,,```,,,,````-`-`,,`,,`,`,,` - a Profile deviation b Pitch deviation c Helix deviation d Tooth e Maximum at Tooth 1 f Maximum at Tooth 4 g Maximum at Tooth 6 h Maximum at Tooth 5 i Accumulation = effective deviation λ j Theoretical maximum
Figure 11 — Influence of form deviations
The true effective size of a spline component with accumulated form deviations can only be accurately determined using an idealized, perfect mating spline that fits seamlessly without any looseness or interference This involves analyzing both internal and external splines with form deviations, where the internal spline's effective space width and the external spline's effective tooth thickness are key parameters Understanding these factors ensures precise measurement and optimal gear engagement, crucial for maintaining spline performance and longevity.
Figure 12 — True effective space width and tooth thickness
Spline parts are influenced not only by machining tolerances but also by form deviations, resulting in an effective tolerance (see Figure 13) For internal spline components, this leads to a minimum effective tolerance limit determined by the space width, while external spline parts are subject to a maximum effective tolerance limit based on tooth thickness (see Figure 14) Key parameters include the largest and smallest space widths, as well as maximum and minimum actual tolerances, which define the effective tolerances for space width and tooth thickness Understanding these tolerance limits is essential for ensuring proper fit and function of spline components.
Figure 13 — Actual and effective tolerances
This article discusses various measurements related to gear teeth and spacing, including internal space width (a), maximum actual space width (b), and maximum effective space width (c) It also covers minimum actual space width (d) and minimum effective space width (e), which are critical for ensuring proper gear engagement Additionally, the content highlights external tooth thickness (f), maximum effective tooth thickness (g), and the maximum and minimum actual tooth thicknesses (h and j), as well as the maximum and minimum effective tooth thicknesses (g and i) Understanding these measurements is essential for precise gear design, manufacturing, and maintenance to ensure optimal performance and durability.
Figure 14 — Graphical display of space width and tooth thickness theoretical tolerance zones
7 Basic rack profiles for spline
The basic rack is a fundamental component representing a section of the tooth surface on an involute spline with an infinitely large diameter, oriented perpendicular to the tooth surfaces It serves as the standard profile for defining tooth dimensions within involute spline systems, with a reference line crossing the profile to specify measurement standards The profile of the basic rack for standard pressure angle splines is illustrated in Figure 15 and detailed in Table 2, covering key features such as internal and external splines, major and minor tooth heights, form tooth height, and pitch line.
Table 2 — Dimensions of basic rack
Flat root Fillet root 37,5° 45° Major space height 0,75 m 0,9 m 0,7 m 0,6 m
The form diameter for internal splines (hub), generated from the basic rack, is consistently larger than the dimension listed in ISO 4156-2 tables, which reflects the maximum shaft diameter increased by the diametrical form clearance (2c F) Conversely, for external splines (shafts), the clearance (c F) is determined by generating from the basic rack (D Fe max) and considering the H/h fit, as noted in Note 2 to Table 1.
ISO 4156 specifies various fit classes—loose fit, zero fit, and press fit—to achieve desired clearances or interference between the space width and tooth thickness These fit types help determine the precise amount of clearance or interference needed for specific applications, ensuring optimal assembly and functionality.
This section of ISO 4156 defines the fundamental deviation parameters (k, js, h, f, e, and d) for the circular tooth thickness (S) at the pitch diameter of external splines, ensuring precise spline fit classes with either no looseness, maximum effective interference, or minimum clearance These standards facilitate consistent classification of spline fits and support the use of standardized composite GO gauges Additionally, Table 3 provides a graphical overview of the fundamental deviations and tolerance zones for the six spline fit classes, promoting clarity and uniformity in spline manufacturing and inspection.
Table 3 — Graphical representation of fundamental deviations for spline fit classes
To achieve the required maximum effective interference or minimum effective looseness, adjust the maximum effective tooth thickness from the zero line using the fundamental deviation es v, as detailed in Tables 3 and 5, while maintaining the specified machining tolerance T and deviation allowance λ The spline dimensions provided in the ISO 4156-2 tables are applicable to class H/h.
Table 4 — Effective interference and effective looseness of spline fit classes
Spline fit class Minimum effective looseness
H/d es v = fundamental deviation d c v min = − es v H/e es v = fundamental deviation e c v min = − es v H/f es v = fundamental deviation f c v min = − es v H/h es v = fundamental deviation h = zero c v min = − es v = zero
H/js es v = fundamental deviation js c v min = − es v = −(T + λ) / 2 H/k es v = (T + λ) c v min = − es v = −(T + λ)
Fundamental deviation es v àm at pitch diameter D
D mm Relative to tooth thickness S for externals Relative to space width E for internals For d e f h js k H u 3 −20 −14 −6 0 0
0 a + (T + λ )/2 relative to tolerance class considered; for T + λ , see 9.1 b + (T + λ ) relative to tolerance class considered; for T + λ , see 9.1
9 Space width and tooth thickness tolerances
Total tolerance T + λ
ISO 4156 specifies four classes of total tolerance (T + λ) for space width and tooth thickness, based on combinations of tolerance units (i) derived from ISO 286-1 These tolerance classes are detailed in Table 6, which shows the corresponding tolerance unit (i) combinations, ensuring precise control of gear dimensions.
Table 6 — Total space width and tooth thickness tolerance (T + λ )
Spline tolerance class Total tolerance (T + λ) àm
7 T + = λ (40 ⋅ + i d 160 ⋅ i E ) where d 0,45 3 0,001 i = ⋅ D + ⋅ D for D u 500 mm (1) d 0,004 2,1 i = ⋅ + D for D > 500 mm (2)
D is the pitch diameter, in millimetres;
E is the basic space width, in millimetres;
S is the basic tooth thickness, in millimetres.
Deviation allowance, λ
The deviation allowance, which encompasses the total index deviation, profile deviation, and helix deviation, significantly impacts the effective fit of an involute spline While individual deviations may have limited effect due to areas with more than minimum clearance tolerating form, helix, or index errors without compromising the fit, it is unlikely that these errors reach their maximum levels simultaneously on the same spline Therefore, these deviations are aggregated statistically, with 60% of the combined total used to assess their influence on the spline fit This approach provides a practical basis for calculating the deviation allowance to ensure proper spline performance.
In the following subclauses, the values of F p , F α and F β are referenced to the datum of the effective spline axis See ISO 4156-3.
Total pitch deviation, F p
Total pitch deviation measures the cumulative pitch error between the two most extreme opposite pitch errors within any sector of a half circumference This value is essential for assessing the overall accuracy of the pitch performance in a given segment To accurately calculate the total pitch deviation, the formulae provided in Table 7 should be applied, ensuring precise and reliable results in pitch deviation analysis.
Spline tolerance class Total pitch deviation
Where L is the length of the arc:
Total profile deviation, F α
Total profile deviation is defined as the absolute difference between the maximum positive and negative deviations from the theoretical tooth profile, measured normal to the gear flanks A positive deviation indicates deviation toward the gear space, while a negative deviation signifies deviation toward the tooth, as illustrated in Figure 17 The calculations for total profile deviation (F α) are performed using the formulas provided in Table 8, with the results expressed in micrometers for precise measurement and analysis.
Spline tolerance class Total profile deviation
Where ϕ f is the tolerance factor: f m 0,012 5 m z ϕ = + ⋅ ⋅ (6)
The permissible positive deviation on external splines and the negative deviation on internal splines from the design profile, within the central one-third of the flank depth to the form diameter, must not exceed one-third of the calculated values This ensures adherence to manufacturing tolerances and maintains proper spline function, as illustrated in Figure 17.
`,,```,,,,````-`-`,,`,,`,`,,` - a Space (internal) b Tooth (external) c Reference profile d Positive profile deviation e Negative profile deviation f Centre third
Total helix deviation, F β
Total helix deviation is defined as the absolute difference between the two extreme deviations from the theoretical direction, measured perpendicular to the flank along the entire length of the spline This measurement ensures precise assessment of the helix's accuracy The calculation of total helix deviation, denoted as F β, utilizes specific formulae provided in Table 9 Expressed in micrometres, this parameter is essential for evaluating spline quality and ensuring manufacturing specifications are met. -**Sponsor**Looking to revamp your article and ensure it's SEO-friendly? [Blogify](https://pollinations.ai/redirect-nexad/8CRLmb9g) can help! This AI-driven platform effortlessly transforms complex text, like your technical explanation of total helix deviation, into engaging and optimized content With Blogify, you can quickly generate coherent paragraphs containing the core meaning of your article while adhering to SEO rules, streamlining your content creation process and boosting your online visibility Maximize your content's potential with Blogify's versatile content conversion and AI-powered optimization features, designed for content creators like you.
Spline tolerance class Total helix deviation
7 F β = ⋅ 2 b + 10 b is the spline length, in millimetres.
Machining tolerance, T
Machining tolerance (T) is calculated as the difference between the total tolerance (T + λ) and the deviation allowance (λ), expressed as (T + λ) − λ According to clause 6, dividing the total class tolerance between machining tolerance and deviation allowance provides general guidance for manufacturing processes However, specific design requirements or particular spline manufacturing methods may necessitate an alternative division to meet precise specifications.
Effective clearance tolerance, T v
If T v is necessary, it is recommended that it be made equal to T.
Use of effective and actual dimensions for space width and tooth thickness
It is limited by maximum actual space width E max and minimum actual tooth thickness S min (see Figure 18 and Table 10)
9.8.2 Maximum material (minimum effective clearance)
It is limited by minimum effective space width E v min and maximum effective tooth thickness S v max (see Figure 18 and Table 10)
The effective space width, E v max, and minimum tooth thickness, S v min, define the core dimensional limits for gear manufacturing, as illustrated in Figure 18 and detailed in Table 10 To ensure precise gear performance, maintaining these tolerances is crucial; therefore, additional inspection procedures are necessary when tight control over the effective tolerance band is required.
Figure 18 — Graphical display of space width and tooth thickness tolerance zones according to inspection methods
Table 10 — Relationship between parameters and control method
Minimum material Minimum effective clearance
Parameters S min /E max S v max /E v min S v min /E v max
Tolerances
Table 11 — Tolerances for minor diameter internal spline, D ii and for major diameter external spline, D ee
Tolerances on D ii àm for modules m
Tolerances on D ee àm for modules m mm m u 0,75 0,75 < m < 2 m W 2 m u 0,75 0,75 < m < 2 m W 2
Adjustment to minor diameters (D ie ), form diameters (D Fe ) and major diameters (D ee ) of
When fundamental deviations d, e, and f are applied to external splines, it is essential to adjust the major, form, and minor diameters accordingly For JS and K class splines, only the minor and form diameters require adjustment, as detailed in the formulae and notes provided in Table 1.
Radii
External splines can be produced using pinion-type shaper cutters, hobs, or by cutting without generating motion with tools shaped to match tooth contours, while cold forming also creates external splines, typically with a fillet root design Internal splines are commonly manufactured through broaching, form cutting, or generation with shaper cutters, with each method producing unique fillet contours even when full-tip radius tools are used The fillets of generated splines are curves related to prolate epicycloids for external splines and prolate hypocycloids for internal splines, reflecting the specific cutting processes involved.
These fillets feature a minimum radius of curvature at the point where they are tangent to the external spline's minor diameter circle or the internal spline's major diameter circle The radius of curvature then increases rapidly, reaching its maximum as the fillet approaches the involute profile This design ensures smooth transitions in spline geometry, enhancing functionality and durability in mechanical applications.
The minimum fillet radius values in Table 12 serve as the baseline for stress calculations and are essential for defining the minimum radii of curvature These values are derived from the fillet radii displayed on standard rack profiles When cutting internal and external splines using a pinion-type shaper cutter, the tool design must be tailored to match the specific dimensions of the internal splines to ensure accurate machining.
Table 12 — Minimum root radius of internal or external splines
Minimum root radius mm for α D = 30° flat root α D = 30° fillet root α D = 37,5° α D = 45°
Profile shifts
External splines with a standard fundamental deviation can be manufactured using standard cutters that conform to the basic rack by radially shifting the reference line away from the spline pitch circle This method preserves the involute profile of the spline teeth, with different sections of the same involute being utilized depending on the radial displacement The extent of this displacement is calculated as 0.5 es v / tan α D, ensuring precise and consistent spline production.
The magnitude of fundamental deviation (es v) significantly influences the tooth form, particularly affecting the circular tip thickness at the major diameter of the external spline As the fundamental deviation varies, the tip thickness at this critical point also changes When applying fundamental deviations, it is essential to verify the tip thickness at the maximum diameter of the external spline to ensure proper gear performance and structural integrity.
A tip thickness of less than 0,25 m should be avoided Tip thickness is computed as follows, considering
( ) ee ee max min / inv D inv D ee
The tip thickness of the gear, referred to as See, is influenced by various parameters such as the pressure angle at the pitch diameter (α D) and the pressure angle at the maximum major diameter (α Dee) The relationship between these angles and the tip thickness can be mathematically expressed, with equations like ee b eemax cos D D α = D (9) Additionally, the inverse of the pressure angle at the pitch diameter is calculated using the formula inv α D = tan α D − α D ⋅ π /180 (10), and similarly for α Dee, the inverse is determined by inv α Dee = tan α Dee − α Dee ⋅ π /180 (11) These formulas are essential for precise gear design and ensuring optimal gear performance.
S min is the minimum actual tooth thickness b) Undercut fillet at the root circle (interference)
Selecting a fit with looseness can cause generating tools to produce undercut fillets on the external spline To prevent issues, it is essential to perform calculations using proper tool data to identify potential undercutting If undercutting is detected, additional calculations or layout assessments are necessary to determine whether the degree of undercutting is acceptable for the intended application.
Low tooth numbers and pressure angles introduce the risk of undercutting, which occurs with basic rack generation when
Eccentricity and misalignment
Design assembly may require applying a tolerance to the position of the effective spline axis and the functional axis of the part To ensure clarity and standardization, ISO 1101 symbols should be used to represent these tolerances effectively.
ISO 4156 does not recommend any values, but they shall instead be defined according to the assembly requirements
When assembly requires looseness or interference and the mating spline axes are misaligned, it is necessary to increase the minimum effective clearance This can be achieved by reducing the effective and actual tooth thickness of the external spline, ensuring the splines can properly assemble despite misalignment Proper clearance adjustment facilitates smoother assembly and reliable spline performance.
The eccentricity of major and minor diameters does not interfere with the form diameters of the mating spline, even at maximum clearance However, if misalignment occurs during assembly, it may be necessary to reduce the external spline's major diameter or increase the internal spline's minor diameter to ensure proper fit and function.
Basic dimensions
Spline data is used for engineering and manufacturing purposes.
Combination of types
Flat root side fit splines can be combined with fillet root splines to achieve a larger radius, which helps in reducing stress concentrations and facilitating manufacturing processes Such spline root combinations are often acceptable as viable design options, offering enhanced flexibility and performance in mechanical components These designs optimize stress control and manufacturing efficiency, making them suitable choices for various engineering applications.
The major diameter of a flat root internal spline, along with the radius tangent point diameter if necessary, should be positioned between the form diameter and the maximum major diameter of the fillet root internal spline Ensuring proper placement of these dimensions is essential for accurate spline design and optimal mechanical performance This alignment helps in achieving precise fitment and reliable load distribution in spline applications Properly defining these key diameters is crucial for manufacturing accuracy and achieving desired functionality in mechanical systems.
Designation
The mating parts of straight cylindrical involute splines (metric modules, side fit) shall be designated in the following order:
Number of teeth = z (preceded by the number)
Module = m (preceded by the value)
Fit class, external spline = k; js; h; f; e; d
EXAMPLE Mating splines, 24 teeth; module 2,5; pressure angle 30° fillet root; tolerance class 5; fit
Drawing data
Using standardized uniform specifications for detail drawings of splines ensures clear and complete information, reducing misunderstandings Following the recommended arrangement of dimensions and data, as illustrated in figure 19 and Table 13, helps maintain consistency The number of X in the specifications indicates the typical number of decimal places required Typically, this charted approach to spline specifications eliminates the need for a graphical representation of the spline teeth, simplifying the design process.
Figure 19 — Internal and external spline drawing dimensions
Table 13 — Spline terms and symbols
Internal spline ISO 4156 External spline ISO 4156
Number of teeth z Number of teeth z
Pressure angle α D XX° Pressure angle α D XX°
Pitch diameter D XX,XXXX a Pitch diameter D XX,XXXX a
Base diameter D b XX,XXXX a Base diameter D b XX,XXXX a
Major diameter D ei XX,XX max Major diameter D ee XX,XX h
Form diameter D Fi XX,XX min Form diameter D Fe XX,XX max
Minor diameter D ii XX,XX H Minor diameter D ie XX,XX min
This article provides key specifications for mechanical components, including space width (b), tooth thickness (b), and various maximum and minimum measurements such as actual and effective energies (E max, E v max, E v min, E min) and shear stresses (S v max, S max, S v min, S min) Precise measurement between balls or pins is essential, ensuring optimal assembly and performance Adhering to these parameters is crucial for maintaining component integrity, safety, and efficiency in mechanical systems.
M Ri XX,XXX max a Measurement over balls or pins
M Re XX,XXX max aux a
Measurement between balls or pins
M Ri XX,XXX min aux a Measurement over balls or pins
Ball or pin diameter D Ri X,XXX Ball or pin diameter D Re X,XXX
Fillet radius See Table 2 Fillet radius See Table 2 a Calculated dimensions b See 9.8.
The values presented in Tables 14 to 17 are based on an engagement length equal to half the pitch diameter For other engagement lengths, it may be necessary to adjust the deviation allowance to maintain proper fit and performance.
Table 14 — Deviation allowances λ — Modules 0,25 to 1
Tolerance class Tolerance class Tolerance class Tolerance class z
Table 15 — Deviation allowances λ — Modules 1,25 to 2
Tolerance class Tolerance class Tolerance class Tolerance class z
Table 16 — Deviation allowances λ — Modules 2,5 to 5
Tolerance class Tolerance class Tolerance class Tolerance class z
Table 17 — Deviation allowances, λ — Modules 6 to 10
Tolerance class Tolerance class Tolerance class z
ISO 4156-2 specifies drawing data for spline fit class H/h, offering essential guidelines for accurate representation The article includes example calculations demonstrating how to determine this data, serving as a useful reference for professionals Additionally, it provides a comprehensive guide for calculating drawing parameters for non-tabulated spline fit classes, ensuring precise and standardized spline design.
NOTE Unless otherwise stated, all formulae are provided in Table 1
Minimum major diameter (flat root) — not tabulated but necessary for calculating maximum major diameter
D ei min = m ⋅ (z + 1,5) = 1,0 ⋅ (25 + 1,5) = 26,50 Maximum major diameter D ei max = D ei min + (T + λ ) / tan α D
D ei max = 26,50 + 0,138 / tan(30°) = 26,74 Form diameter D Fi min = m ⋅ (z + 1) + 2 ⋅ c F c F = 0,1 ⋅ 1,0 = 0,1
Minor diameter D ii min = D Fe max + 2 ⋅ c F
From Table 2 h s = 0,6 ⋅ m = 0,6 ⋅ 1,0 = 0,6 From Table 5 es v = 0 (see Table 1, Note 2)
D ii = 24,09 H11 (+0,130/0) Minimum effective space width E v min = 0,5 ⋅ π ⋅ m = 0,5 ⋅ π ⋅ 1,0
E v min = 1,571 Maximum actual space width E max = E v min + (T + λ )
From 9.1 class 5 (T + λ ) = 16 ⋅ i d + 64 ⋅ i E From above formula i d = 1,3408 and i E = 0,5247
E max = 1,571 + 0,055 = 1,626 Minimum actual space width E min = E v min + λ
NOTE λ ҏcan be calculated from total pitch deviation (F p ) total profile deviation (F α ) and total helix deviation (F β ), see 9.2; or else can be obtained from Table 14
Total pitch deviation F p = 3,55 ⋅ L + 9 (formula from Table 7) where 1,0 25 39,269 908 17
Total profile deviation F α = 2,5 ⋅ ϕ f + 16 (formula from Table 8) where ϕ f = m + 0,012 5 ⋅ ⋅ = m z 1,0 0,0125 1,0 25 1,312 50 + ⋅ ⋅ =
Total helix deviation F β = b + 5 (formula from Table 9) where b = length of spline (assume to be one half of the pitch diameter)
E min = 1,571 + 0,022 = 1,593 Max effective space width E v max = E max − λ
E v max = 1,626 − 0,022 = 1,604 Measuring ball or pin diameter D Ri (formulae taken from ISO 4156–3:2005, 8.5.2) i cos D b inv D
From ISO 3 R40 no series D Ri = 1,800 Maximum measurement between balls or pins M Ri max (formulae taken from ISO 4156-3:2005, 8.6.1.2)
From above formula D Ri = 1,800 i Ri b inv E inv D D
For odd numbers of teeth Ri max b Ri i cos 90 cos
= − = ° Minimum measurement between balls or pins M Ri min
= − = ° Fillet radius ρ Fi = 0,2m (formula from Table 2)
In this example, a length of spline of spline of 25,0 mm has been used to calculate total helix deviation and hence the deviation allowance
NOTE Unless otherwise stated all formulae are provided in Table 1
Minimum major diameter (fillet root) — not tabulated but necessary for calculating maximum major diameter
D ei min = m ⋅ (z + 1,8) = 1,0 × (25 + 1,8) = 26,80 Maximum major diameter D ei max = D ei min + (T + λ ) / tan α D
D ei max = 26,80 + 0,138 / tan(30°) = 27,04 Form diameter D Fi min = m ⋅ (z + 1) + 2 ⋅ c F c F = 0,1 ⋅ 1,0 = 0,1
D Fi min = 1,0 ⋅ (25 + 1) + 2 ⋅ 0,1 = 26,20 Minor diameter D ii min = D Fe max + 2 ⋅ c F
From Table 5 es v = 0 (see Table 1, Note 2)
D ii = 24,09 H11 (+0,130/0) Min effective space width E v min = 0,5 ⋅ π ⋅ m = 0,5 ⋅ π ⋅ 1,0
E v min = 1,571 Max actual space width E max = E v min + (T + λ )
E max = 1,571 + 0,138 = 1,709 Min actual space width E min = E v min + λ
NOTE λ can be calculated from total pitch deviation (F p ) total profile deviation (F α ) and total helix deviation (F β ), see 9.2, or λ can be obtained from Table 14
Total pitch deviation F p = 7,1 ⋅ L + 18 (formula from Table 7) where 1,0 25 39,269 908 17
Total profile deviation F α = 6,3 × ϕ f + 40 (formula from Table 8) where ϕ f = m + 0,012 5 × × = m z 1,0 0,012 5 1,0 25 1,312 50 + × × =
6,3 1,312 50 F α = ì + 40 48,27 = àm 0,048 mm = Total helix deviation F β = ⋅ 2 b + 10 (formula from Table 9) where b = length of spline (assume to be 25,0 mm)
E min = 1,571 + 0,049 = 1,620 Max effective space width E v max = E max − λ
E v max = 1,709 − 0,049 = 1,660 Measuring ball or pin diameter D Ri (formulae taken from ISO 4156-3:2005, 8.5.2) i cos D b inv D
DE ∩ = ⋅ E α + D ⋅ α where inv α D = 2 tan α D − α D × 360 × π ° = tan 30 ( ) 30 2
From ISO R40 no series D Ri = 1,800
Maximum measurement between balls or pins M Ri max (formulae taken from ISO 4156-3:2005, 8.6.1.2)
From above formula D Ri = 1,800 i Ri b inv E inv D D
For odd numbers of teeth Ri max b Ri i cos 90 cos
Min measurement between balls or pins M Ri min
= − = ° Fillet radius ρ Fi = 0,4 × m (formula from Table 2)
NOTE Unless otherwise stated all formulae are provided in Table 1
Major diameter ee max ( 1 ) v tan D
Maximum minor diameter (flat root) — not tabulated but necessary for calculating minimum minor diameter
Minimum minor diameter ( ) ie min ie max tan D
D = − = ° Max effective tooth thickness S v max = S + es v
Min actual tooth thickness S min = S v max − ( T + λ )
S min = 1,571 − 0,034 = 1,537 Max actual tooth thickness S max = S v max − λ
NOTE λ can be calculated from total pitch deviation (F p ) total profile deviation (F α ) and total helix deviation (F β ), see 9.2 , or λ can be obtained from Table 14
Total pitch deviation F p = 2,5 × L + 6,3 (formula from Table 7) where 1,0 25 39,269 908 17
Total profile deviation F α = 1,6 × ϕ f + 10 (formula from Table 8) where ϕ f = m + 0,012 5 × × = m z 1,0 0,012 5 1,0 25 1,312 50 + × × =
Total helix deviation F β = 0,8 ⋅ b 4 + (formula from Table 9) where b = length of spline (assume to be one half of the pitch diameter)
S max = 1,571 − 0,016 = 1,555 Min effective tooth thickness S v min = S min + λ
S v min = 1,537 + 0,016 = 1,553 Measuring ball or pin diameter D Re (formulae taken from ISO 4156-3:2005, 8.5.1)
DE ∩ = p − S × α + D × α where p b = × π × m cos α D = 1,0 × π × cos 30 ( ) ° = 2,720 70 and inv α D = 2 tan α D − α D × 360 × π ° = tan 30 ( ) ° − 30 ° × 360 2 × π °
From ISO R40 no series D Re = 1,900
Max measurement over balls or pins M Re max (formulae taken from ISO 4156-3:2005, 8.6.1.1)
From above formula D Re = 1,900 e Re b inv S inv D D
For odd numbers of teeth Re max b Re e cos 90 cos
Min measurement over balls or pins M Re min
Fillet radius ρ Fe = 0,2 × m (formula from Table 2)
NOTE Unless otherwise stated all formulae are provided in Table 1
Major diameter ee max ( 1 ) v tan D
= ⋅ + + α From Clause 10, adjustment is made to the major, form and minor diameters for fundamental deviation e
From Table 2 h s = 0,6 ⋅ m = 0,6 ⋅ 1,0 = 0,6 From Table 5 es v = −0,040
D Fe max = 23,83 Maximum minor diameter (fillet root) — not tabulated but necessary for calculating minimum minor diameter
Minimum minor diameter ( ) ie min ie max tan D
D = − = ° Max effective tooth thickness S v max = S + es v
S v max = 1,571 + (−0,040) = 1,531 Min actual tooth thickness S min = S v max − ( T + λ )
S min = 1,531 − 0,086 = 1,445 Max actual tooth thickness S max = S v max − λ
NOTE λ can be calculated from total pitch deviation (F p ) total profile deviation (F α ) and total helix deviation (F β ), see
9.2 , or λ can be obtained from Table 14
Total pitch deviation F p = × 5 L + 12,5 (formula from Table 7) where 1,0 25 39,269 908 17
Total profile deviation F α = × 4 ϕ f + 25 (formula from Table 8) where ϕ f = m + 0,012 5 × × = m z 1,0 0,0125 1,0 25 1,312 50 + × × =
Total helix deviation F β = 1,25 × b + 6,3 (formula from Table 9) where b = length of spline (assumed to be one half of the pitch diameter)
S max = 1,531 − 0,033 = 1,498 Min effective tooth thickness S v min = S min + λ
S v min = 1,445 + 0,033 = 1,478 Measuring ball or pin diameter D Re (formulae taken from ISO 4156-3:2005, 8.5.1)
DE ∩ = p − S × α + D × α where p b = × π × m cos α D = 1,0 × π × cos 30 ( ) ° = 2,720 70 and inv α D = tan 2
From ISO R40 no series D Re = 1,900 Maximum measurement over balls or pins M Re max (formulae taken from ISO 4156-3:2005, 8.6.1.1)
For odd numbers of teeth Re max b Re e cos 90 cos
= + = ° Min measurement over balls or pins M Re min
= + = ° Fillet radius ρ Fe = 0,4 × m (formula from Table 2)
NOTE Unless otherwise stated all formulae are provided in Table 1
Major diameter ee max ( 1 ) v tan D
From Clause 10, no adjustment is applied to the major diameter, but adjustment is applied to the form and minor diameters for fundamental deviation js
D Fe max = 23,93 Maximum minor diameter (flat root) — not tabulated but necessary for calculating minimum minor diameter
Min minor diameter ( ) ie min ie max tan D
D = − = ° Max effective tooth thickness S v max = S + es v
S v max = 1,571 + 0,028 = 1,599 Minimum actual tooth thickness S min = S v max − ( T + λ )
S min = 1,599 − 0,055 = 1,544 Max actual tooth thickness S max = S v max − λ
NOTE λ can be calculated from total pitch deviation (F p ) total profile deviation (F α ) and total helix deviation (F β ), see 9.2, or λ can be obtained from Table 14
Total pitch deviation F p = 3,55 × L + 9 (formula from Table 7) where 1,0 25 39,269 908 17
Total profile deviation F α = 2,5 × ϕ f + 16 (formula from Table 8) where ϕ f = m + 0,012 5 × × = m z 1,0 0,012 5 1,0 25 1,312 50 + × ⋅ =
Total helix deviation F β = b + 5 (formula from Table 9) where b = length of spline (assume to be one half of the pitch diameter)
S max = 1,599 − 0,022 = 1,577 Min effective tooth thickness S v min = S min + λ
S v min = 1,544 + 0,022 = 1,566 Measuring ball or pin diameter D re (formulae taken from ISO 4156-3:2005, 8.5.1)
DE ∩ = p − S × α + D × α where p b = × π × m cos α D = 1,0 × π × cos 30 ( ) ° = 2,720 70 and inv α D = 2 tan α D − α D × 360 × π ° = tan 30 ( ) 30 2
From ISO R40 no series D Re = 1,900 Maximum measurement over balls or pins M Re max (formulae taken from ISO 4156-3:2005, 8.6.1.1)
From above formula D Re = 1,900 e Re b inv S inv D D
For odd numbers of teeth Re max b Re e cos 90 cos
Minimum measurement over balls or pins M Re min
= + = ° Fillet radius ρ Fe = 0,2 × m (formula from Table 2)
[1] ISO 3, Preferred numbers — Series of preferred numbers