This is taken into consideration when determining the load factors by the fact that the rectangular zone of action in the case of spur and helical gears is replaced by an inscribed paral
Calculation methods
ISO 10300 (all parts) details procedures for predicting the load capacity of bevel gears, emphasizing that full-scale, full-load testing of a specific gear set offers the most accurate assessment However, during the design phase or in specialized applications, calculation methods are essential for load capacity prediction Therefore, ISO 10300 incorporates methods A, B, and C, with method A being the preferred choice if its accuracy and reliability are validated, followed by method B and then method C.
ISO 10300 (all parts) allows the use of mixed factor rating methods within method B1 or method B2 For example: method B for dynamic factor K v-B can be used with method C face load factor K Hβ-C
When sufficient experience from similar designs is available, reliable guidance can be achieved through extrapolating test results or field data, provided that all relevant gear and load data are accurately known and thoroughly documented This process involves detailed measurement and mathematical analysis of the transmission system, considering factors such as boundary conditions and specific characteristics that influence the outcome To ensure credibility, the method's accuracy and reliability must be demonstrated, often by comparing results with established gear measurement standards Additionally, it is essential that both the customer and the supplier approve the extrapolation method to ensure mutual understanding and confidence in the results.
Method B offers calculation formulas for predicting the load capacity of bevel gears when essential data is available However, accurate assessment requires experience with similar gear designs to evaluate specific factors effectively It is important to verify that these evaluations are appropriate for the actual operating conditions to ensure reliable predictions.
Where suitable test results or field experience from similar designs, are unavailable for use in the evaluation of certain factors, a further simplified calculation method, method C, should be used.
Safety factors
When selecting a safety factor, it is essential to carefully consider the allowable probability of failure to balance reliability and cost effectively If gear performance can be accurately assessed through testing under actual load conditions, lower safety factors may be acceptable Safety factors are determined by dividing the calculated permissible stress by the evaluated operating stress, ensuring structural integrity while optimizing design efficiency.
Safety factors in gear design must be carefully determined based on the reliability of material data and load estimates, considering both surface durability (pitting) and tooth root strength as specified in ISO standards The allowable stress values used in calculations correspond to a specific probability of failure, typically around 1%, with higher safety factors reducing the risk of damage When load or system response data are estimated rather than measured, it is advisable to apply larger safety factors to ensure reliability and safety.
The following variations shall also be taken into consideration in the determination of a safety factor:
— variations in gear geometry due to manufacturing tolerances;
— variations in alignment of gear members;
— variations in material due to process variations in chemistry, cleanliness and microstructure (material quality and heat treatment);
— variations in lubrication and its maintenance over the service life of the gears.
The appropriateness of safety factors depends on the reliability of the assumptions, including load estimates used in calculations It also relates to the required reliability of the gears themselves, considering the potential consequences of damage if failure occurs Ensuring accurate assumptions and appropriate safety margins is essential for gear safety and performance.
Supplied or assembled gear drives must have a minimum safety factor for contact stress (S H,min) of at least 1.0 to ensure durability and reliability Additionally, the minimum bending stress safety factor (S F,min) should be set at 1.3 for spiral bevel and hypoid gears, while straight bevel gears or those with a pitch angle (β m) of 5° or less require a higher safety factor of 1.5.
The minimum safety factors against pitting damage and tooth breakage should be agreed between the supplier and customer.
Rating factors
The most effective method for managing gear system performance is comprehensive, full-load testing of the new design When existing experience or data from similar designs are available, extrapolation can provide a satisfactory solution In cases lacking test results or field data, it is essential to select rating factor values conservatively to ensure reliability and safety.
When assessing rating factors, it is essential to consider the minimum acceptable quality limits concerning the expected variation of component parts during manufacturing The accuracy grade, B, should ideally be determined according to ISO 17485 standards, such as using single pitch deviation to calculate the dynamic factor K v-B Adhering to these guidelines ensures consistent quality assessment and compliance with international standards.
Where the empirical values for rating factors are given by curves, this part of ISO 10300 provides curve fitting equations to facilitate computer programming.
NOTE The constants and coefficients used in curve fitting often have significant digits in excess of those implied by the reliability of the empirical data.
Further factors to be considered
Various interrelated system factors beyond pitting resistance and bending strength significantly influence overall transmission performance It is essential to consider their potential impact during calculations to ensure accurate assessments and optimal design Incorporating these factors can enhance reliability and efficiency in transmission system performance analysis.
Gear ratings according to ISO 10300-2 and ISO 10300-3 are valid only when the gear teeth are properly lubricated with the correct viscosity and additives suited for the load, speed, and surface finish Ensuring an adequate quantity of lubricant on the gear teeth and bearings is essential to maintain optimal lubrication and prevent overheating during operation Proper lubrication is crucial for achieving accurate gear performance ratings and ensuring reliable and efficient gear operation.
Proper alignment of gear systems relies heavily on external supports like machinery foundations; poorly designed supports or initial misalignments can lead to decreased performance Over time, elastic, thermal deflections, or other influences may cause these supports to shift, resulting in misalignment during operation Ensuring robust support design and regular maintenance is crucial to maintaining optimal gear mesh and overall system efficiency.
Gear supporting housings, shafts, and bearings can experience deflection due to external overhung, transverse, and thrust loads, which affects tooth contact across the gear mesh Since deflection varies with load, achieving optimal tooth contact under different operating conditions is challenging External loads from driven and driving equipment typically reduce gear capacity by increasing deflection, and internal forces also contribute to deflection; therefore, both factors must be considered when assessing actual gear tooth contact to ensure reliable gear performance.
Most bevel gears are manufactured from case-hardened steel, with allowable stress values best sourced from ISO 6336-5, which provides precise data based on extensive testing of spur gears This standard accounts for various steel production methods and heat treatments, ensuring accurate and reliable stress allowances When selecting materials, considerations such as hardness, tensile strength, and quality grade are essential criteria for determining appropriate permissible stress numbers, leading to durable and efficient gear performance.
NOTE Higher quality steel grades indicate higher allowable stress numbers, while lower quality grades indicate lower allowable stress numbers (see ISO 6336-5).
Residual stress is common in ferrous materials with a case-core relationship, but proper management can ensure these stresses are compressive at the gear tooth surface, enhancing bending fatigue strength Techniques such as shot peening, case carburizing, and induction hardening effectively induce beneficial compressive pre-stress when performed correctly Conversely, improper grinding after heat treatment can diminish residual compressive stresses or introduce detrimental tensile stresses in the root fillets, reducing the gear's allowable stress capacity.
This section of ISO 10300 utilizes a dynamic factor, K_v, to account for increased loads resulting from gear tooth inaccuracies The dynamic factor effectively derates the gears to enhance their load-carrying capacity under real-world conditions Overall, the method offers simplified, easy-to-apply values, making it practical for gear design and analysis while maintaining accuracy.
The system's dynamic response generates additional gear tooth loads due to the relative motions of the driver and driven equipment An application factor, K_A, accounts for the operating characteristics but must be used cautiously, especially if the drive or gearbox causes excitation near the system’s major natural frequencies Resonant vibrations under such conditions can lead to severe overloads, several times the nominal load, emphasizing the importance of comprehensive vibration analysis for critical applications This analysis should include all system components—driver, gearbox, driven equipment, couplings, mounting conditions, and excitation sources—to accurately determine natural frequencies, mode shapes, and dynamic response amplitudes, ensuring system reliability and safety.
During manufacturing, most bevel gears have crowned teeth in both their profile and lengthwise directions to accommodate shaft and mounting deflections This crowning creates a localized contact pattern during light load roll testing, ensuring proper gear engagement Under normal design loads, the tooth contact pattern typically spreads evenly across the gear flanks, preventing concentration at the edges of either gear.
Applying the rating formulae to bevel gears that are manufactured without proper process control or lack an adequate contact pattern may necessitate modifications to the factors specified in ISO 10300 It is important to note that such gears fall outside the scope of ISO 10300 and its various parts, highlighting the need for tailored evaluation methods when standard conditions are not met.
NOTE The total load used for contact pattern analysis can include the effects of an application factor (see Annex D for a fuller explanation of tooth contact development).
Corrosion of gear tooth surfaces significantly impacts both bending strength and pitting resistance, leading to potential gear failure Quantifying the precise effects of corrosion on gear teeth is complex and currently exceeds the scope of ISO 10300 standards.
Further influence factors in the basic formulae
The basic formulae presented in ISO 10300-2 and ISO 10300-3 include factors reflecting gear geometry or being established by convention, which need to be calculated in accordance with their formulae.
ISO 10300 standards incorporate influence factors that account for variations in processing and operating cycles, ensuring more accurate performance assessments These factors, including load factors such as K A, K V, K Hβ, K Fβ, K Hα, and K Fα, represent multiple influences on the unit's behavior Although treated as independent, these influence factors may interact beyond current evaluation, emphasizing the importance of considering their combined effects Additionally, factors affecting permissible stresses are included, highlighting their critical role in maintaining safety and reliability in machine design.
Various methods exist for calculating influence factors, typically indicated by adding subscripts A to C to relevant symbols For critical transmissions, the more accurate calculation method should be employed unless specified otherwise, such as in application standards To ensure clarity, it is recommended to use supplementary subscripts when the evaluation method for a factor is not immediately recognizable.
When selecting factors for specific applications, it may be necessary to choose between different methods, such as alternative techniques for determining the dynamic factor or the transverse load factor It is important to clearly indicate the calculation method used by extending the subscript in the reported results, for example, K_v-C or K_Hα-B This ensures transparency and clarity in the reporting of engineering calculations, facilitating accurate interpretation and application of the data.
6 External force and application factor, K A
Nominal tangential force, torque, power
For the purposes of ISO 10300 (all parts), pinion torque is used in the basic stress calculation formulae
To determine the bending moment on the tooth or the force acting on the tooth surface, the tangential force at the reference cone located at the mid-face width is calculated This calculation is essential for understanding load distribution and ensuring gear performance and durability Accurate assessment of the tangential force helps in optimizing gear design and analyzing stress conditions effectively.
Nominal tangential force of bevel gears, F mt :
Nominal tangential force of virtual cylindrical gears, F vmt :
Nominal torque of pinion and wheel, T:
Nominal tangential speed at mean point, v mt : v d d n mt 1,2 m 1,2 1,2 m 1,2 1,2
The nominal torque of the driven machine is a crucial factor, as it represents the operating torque that must be transmitted consistently over an extended period This torque must withstand the most severe and regular operating conditions to ensure reliable machine performance and longevity.
The nominal torque of the driving machine may be used if it corresponds to the required torque of the driven machine.
Variable load conditions
If the load is not uniform, a careful analysis of the gear loads should be carried out, in which the external and internal factors are considered It is recommended that all the different loads that occur during the anticipated life of the gears, and the duration of each load, are determined A method based on Miner’s Rule (see ISO 6336-6 [3] ) shall be used for determining the equivalent life of the gears for the torque spectrum.
Application factor, K A
In cases where no reliable experiences, or load spectra determined by practical measurement or comprehensive system analysis, are available, the calculation should use the nominal tangential force
F mt according to 6.1 and an application factor, K A This application factor makes allowance for any externally applied dynamic loads in excess of the nominal operating pinion torque, T 1
6.3.2 Influences affecting external dynamic loads
In determining the application factor, account should be taken of the fact that many prime movers develop momentary peak torques considerably greater than those determined by the nominal ratings of either the prime mover or of the driven equipment There are many possible sources of dynamic overload which should be considered, including:
— sudden variations in system operation;
— negative torques, such as those produced by retarders on vehicles, which result in loading the reverse flanks of the gear teeth.
Analyzing critical speeds within the drive train's operating range is crucial to ensure optimal system performance Identifying the presence of critical speeds allows for targeted design modifications, such as system damping or adjustments, to eliminate or minimize gear and shaft vibrations Implementing these changes enhances the reliability and efficiency of the drive system, preventing potential mechanical failures and extending its operational lifespan.
Application factors should be determined through a comprehensive analysis of service experience with a specific application For marine gears, which endure cyclic peak torques and are designed for infinite service life, the application factor is defined as the ratio of cyclic peak torque to the nominal rated torque The nominal rated torque is calculated based on the gear's rated power and operating speed.
When gear components are subjected to a limited number of loads exceeding the cyclic peak torque, this impact can be effectively assessed through cumulative fatigue analysis Alternatively, an increased application factor can be used to account for the influence of the load spectrum, ensuring accurate evaluation of gear durability under these conditions Proper consideration of these factors is essential for optimizing gear design and preventing fatigue failure.
If service experience is unavailable, a thorough analytical investigation should be carried out Annex C provides approximate values of K A if neither of these alternatives is possible.
General
The dynamic factor, Kv, accounts for the influence of gear tooth quality concerning speed and load, as well as other relevant parameters It relates the total tooth load—including internal dynamic effects—to the transmitted tangential tooth load Specifically, Kv is calculated by dividing the sum of the internal dynamic load and the transmitted tangential tooth load by the transmitted tangential load This factor is essential for accurately assessing gear performance under varying operational conditions.
`,`,`,,,`,``,`,,,,,,``,,````,-`-`,,`,,`,`,,` - transmitted tangential tooth load The parameters for the gear tooth internal dynamic load fall into two categories: design and manufacturing.
Design
— inertia and stiffness of the rotating elements;
— stiffness of bearings and case structure;
— critical speeds and internal vibration within the gear itself.
Manufacturing
— runout of pitch surfaces with respect to the axis of rotation;
— compatibility of mating gear tooth elements;
Transmission error
Even with constant input torque and speed, gear masses can experience significant vibration and dynamic tooth forces driven by transmission error, which results from relative displacements between mating gears in response to excitation Transmission error, defined as deviations from uniform relative angular motion, is affected by gear tooth form deviations, manufacturing inaccuracies, and operational conditions such as pitch line velocity, gear mesh stiffness variations, transmitted tooth load, dynamic unbalance, and environmental factors Variations in gear mesh stiffness, especially in straight and Zerol bevel gears, serve as sources of excitation, while spiral bevel gears with a contact ratio greater than 2 tend to exhibit less stiffness variation Load-dependent deflections can lead to increased transmission error under different operating loads, and excessive wear, plastic deformation, or inadequate lubrication and sealing can worsen transmission error, impacting gear performance and longevity Proper gear design and maintenance are essential to minimize these effects and ensure smooth operation.
`,`,`,,,`,``,`,,,,,,``,,````,-`-`,,`,,`,`,,` - f) shaft alignment: gear tooth alignment is influenced by load and thermal deformations of gears, shafts, bearings and housings; g) tooth friction induced excitation.
Dynamic response
The effects of dynamic tooth forces are influenced by the following:
— mass of the gears, shafts, and other major internal components;
— stiffness of the gear teeth, gear blanks, shafts, bearings and housings;
— damping, of which the principal sources are the shaft bearings and seals, with other sources including the hysteresis of the gear shafts, viscous damping at sliding interfaces and couplings.
Resonance
Resonant vibrations occur when the excitation frequency, such as the tooth meshing frequency or its multiples, closely matches the natural frequency of the gearing system This resonance can lead to high dynamic tooth loading, which increases the risk of gear damage and reduces overall system reliability Monitoring and avoiding operational speeds that induce resonance are crucial for maintaining gear performance and preventing excessive internal dynamic loads during operation.
High-speed or lightweight gear blanks may have natural frequencies within their operating speed range, leading to potential resonance issues When gear blanks are excited by frequencies near their natural frequencies, resonant deflections can result in high dynamic tooth loads, increasing the risk of gear failure Additionally, plate or shell mode vibrations may occur, further compromising gear durability It is important to note that when using Method B or C for analysis, the dynamic factor, Kv, does not account for gear blank resonance, highlighting the need for careful design consideration to avoid resonance-related failures.
The gearbox is just one component of a system comprising power source, gearbox, driven equipment and interconnecting shafts and couplings The dynamic response of this system depends on its configuration
When a system’s natural frequency is close to the excitation frequency during operation, it can lead to resonance, which requires careful assessment to ensure safety and performance For critical drives, conducting a comprehensive analysis of the entire system is essential to identify potential issues This resonance consideration should be factored into the application factor to optimize system reliability and prevent damage.
Calculation methods for K v
A bevel gear drive is a complex vibration system influenced by various factors beyond just the gear pair Its dynamic behavior and natural frequencies, which lead to dynamic tooth loading, cannot be accurately identified without considering the entire system Additionally, pinion shaft alignment can vary significantly due to assembly craftsmanship, backlash, and the elastic deformation of gear shafts, bearings, or the housing, affecting overall performance and reliability.
A slight misalignment in gear setup can significantly impact the relative rotation angle between gears, increasing dynamic load and potential wear Implementing crowning in both the lengthwise and profile directions helps prevent deviations from true conjugate action, ensuring smoother gear operation Accurate gear tooth contact and performance depend on proper alignment and effective crowning techniques, which are essential for optimal gear accuracy and longevity.
Reliable predictions of the dynamic factor, Kv, are best achieved through a well-verified mathematical model based on test measurements When known dynamic loads are incorporated into the nominal transmitted load, the dynamic factor should be designated as unity, ensuring accurate load assessment.
To determine K v , several methods are indicated in descending order of precision, from method A (K v-A ) to method C (K v-C ).
When selecting methods B or C for hypoid gears with a typical offset, the dynamic factor is assumed to be 1 due to the damping effects of the sliding contact in mesh For smaller offsets, the dynamic factor is interpolated between the value for bevel gears without offset and 1, ensuring accurate assessment of gear dynamics The lower limit for a typical hypoid gear offset is generally considered to be [insert specific value], facilitating precise gear design and durability predictions.
5 % of the mean pitch diameter of the wheel (a rel = 0,1); for the upper limit, see ISO 23509.
Kv-A is determined through a comprehensive analysis validated by experience with similar designs, involving developing a mathematical model of the entire vibration system including the gearbox, measuring or simulating the transmission error of bevel gears under load, and analyzing the dynamic load response of the pinion and gear shafts excited by these transmission errors.
This method simplifies the gear pair as an elementary single mass-spring system, combining the masses of the pinion and wheel with a spring stiffness equal to the mesh stiffness of the contacting teeth It assumes that forces from torsional vibrations of the shafts and coupled masses are not included in the stiffness calculation, which is realistic when other connected masses are linked by shafts with low torsional stiffness For bevel gears with considerable lateral shaft flexibility, the actual natural frequency will be lower than the calculated value.
The dynamic overloads in gear systems are influenced by factors such as gear accuracy, including flank form and pitch deviations While measuring the flank form deviation of bevel gears is more complex than for cylindrical gears, and ISO standards for these measurements are lacking, ISO 17485 provides single flank composite tolerances and recommends checking the transmission error of bevel gear sets when proper equipment is available Pitch deviations, however, can be measured more easily, and a common simplifying assumption is that a single pitch deviation serves as a representative value of transmission error These measurements are crucial for accurately determining the dynamic factor and ensuring gear performance.
The following data are needed for the calculation of K v-B : a) accuracy of gear pair (single pitch deviation as specified in ISO 17485);
16 `,`,`,,,`,``,`,,,,,,``,,````,-`-`,,`,,`,`,,` - © ISO 2014 – All rights reserved b) mass moment of inertia of pinion and wheel (dimensions and material density); c) tooth stiffness; d) transmitted tangential load.
(8) where n E1 is the resonance speed according to 7.7.3.3.
With the aid of the reference speed, N, the total speed range can be subdivided into four sections: subcritical, main resonance, supercritical and an intermediate sector (main resonance/supercritical).
Resonance speed in machinery can vary significantly due to factors such as stiffness values from components like shafts, bearings, and gearboxes, which are not included in the basic calculations Additionally, damping effects further influence the actual resonance speeds, causing them to shift above or below the values predicted by Formula (9) To ensure operational safety, a resonance sector is defined within the range of 0.75 < N ≤ 1.25, accounting for these variations and preventing operation within dangerous resonance zones.
This results in the cited sectors for the calculation of K v-B :
— subcritical sector, N ≤ 0,75, determined by method A or B;
— main resonance sector, 0,75 < N ≤ 1,25, operation in this sector should be avoided, but if unavoidable, refined analysis by method A shall be carried out;
— intermediate sector, 1,25 < N < 1,5, determined by method A or B;
— supercritical sector, N ≥ 1,5, determined by method A or B.
See ISO 6336-1 [2] for further information on the speed ranges.
0 × 10 3 π 1 γ (9) where c γ is the mean value of mesh stiffness [see Formula (11)]: m m m red=m m
(10) m red is the mass per millimetre face width reduced to the line of action of the dynamically equiva- lent cylindrical gear pair.
The stiffness value c γ0 of 20 N/(mm⋅àm) is applicable to spur gears Studies on helical gears indicate that their stiffness decreases as helix angles increase In contrast, the spiral arrangement of bevel gear teeth around a conical blank results in higher rigidity, except for straight bevel gears Due to limited data, the stiffness for spur gears is generally assumed to represent that of bevel gears under typical conditions, with the criteria F vmt K A /b v,eff ≥ 100 N/mm and b v,eff /b v ≥ 0.85.
The mean value of mesh stiffness per unit face width, c γ , is determined by: c γ =c C γ 0 F (11) where c γ0 is mesh stiffness for average conditions; a value of 20 N/(mm⋅àm) is recommended.
The correction factor C F adjusts for non-average conditions and depends on the ratio of F vmt K A /b v,eff When this ratio is equal to or greater than 100 N/mm, C F equals 1, indicating standard conditions If the ratio is less than 100 N/mm, C F is calculated as (F vmt K A /b v,eff) divided by 100 N/mm The effective face width b v,eff represents the actual contact pattern length, which is typically at least 85% of the face width b v during full load If contact pattern length data under load conditions is unavailable, an effective face width of 0.85 b v should be assumed.
If an exact determination of the mass moments of inertia m 1
When the manufacturing of bevel gears is not feasible due to high costs or design limitations, it is advisable to replace them with approximate dynamically equivalent cylindrical gears, marked with suffix "x" (see Figure 2) This approach ensures effective gear performance while reducing complexity and production costs, especially during the design stage where traditional bevel gear manufacturing may be impossible or impractical.
Figure 2 — Approximate dynamically equivalent cylindrical gears for the determination of the dynamic factor of bevel gears including hypoid gears
Relative gear mass per unit face width reduced to the line of action: m m d x x
(13) where ρ is the density of the gear material (for steel ρ = 7,86∙10 −6 kg/mm 3 )
See Figure 3 for the graphical determination of resonance speed for the mating solid steel pinion/solid wheel (bevel gears without offset only). © ISO 2014 – All rights reserved 19
Key z 1 number of pinion teeth (—) n E1 resonance speed (min −1 )
Figure 3 — Nomogram for the determination of the resonance speed, n E1 , for the mating solid steel pinion/solid wheel, with c γ = 20 N/(mm ⋅ àm) (for bevel gears without offset only)
Common operating range for industrial and vehicle gears:
K v-B = N∙K + 1 (14) With the simplifying assumptions given in 7.7.3.1, Formula (15) applies:
See Formula (17) for c’; Table 3 for c v1,2 and c v3 ; see 9.3.1 for f pt and 9.5 for y a
NOTE Any positive influence of tip relief or profile crowning is not considered The calculation is, therefore, on the safe side for bevel gears which normally have profile crowning.
Table 3 — Influence factors c v1 to c v7 in Formulae (15) to (19)
The influence factors for gear mesh stiffness consider various effects, including pitch deviation, tooth profile deviation, and cyclic variation, with their impact depending on the specific operational sector For instance, the factor \( \varepsilon_{v\gamma} \) ranges between 1 and 2.5, affecting gear dynamics according to formulas (A.25) or (B.23) These influence factors are assumed to be constant and account for resonance during cyclic torsional oscillations caused by mesh stiffness variations In the supercritical sector, the effects of parameters \( c_{v5} \) and \( c_{v6} \) are analogous to \( c_{v1} \) and \( c_{v2} \) in the subcritical sector, respectively Additionally, one factor considers the force component resulting from tooth bending deflections during steady-speed conditions due to mesh stiffness variations.
A typical tooth stiffness value for spur gears is c0′ = 14 N/(mm·àm) Studies have shown that tooth stiffness in helical gears decreases as helix angles increase, while the spiral arrangement of bevel gear teeth around a conical blank generally results in higher rigidity, except in straight bevel gears Due to limited specific data, the same tooth stiffness value used for spur gears is assumed to apply to bevel gears under average conditions, characterized by a factor F_vmt K_A /b_v,eff ≥ 100 N/mm and a ratio b_v,eff /b_v ≥ 0.85, ensuring reliable gear performance.
The single stiffness, c’, see ISO 6336-1, [2] is determined as follows: c c C'= ' 0 F (17) where c 0 ’ is single stiffness for average conditions, a value of 14 N/(mm⋅àm) is recommended;
C F is a correction factor for non-average conditions [see Formulae (12a) and (12b)].
With the simplifying assumptions given in 7.7.3.1, Formula (18) applies:
High-speed gears and those with similar requirements operate in the supercritical sector:
For c’ and f p,eff see 7.7.3.3; for c v5,6 and c v7 see Table 3.
In the intermediate sector, the dynamic factor is determined by linear interpolation between K v-B at
N = 1,25 and K v-B at N = 1,5 K v-B is calculated according to 7.7.3.4 and 7.7.3.5, respectively:
Figure 4 illustrates dynamic factors useful when specific knowledge of dynamic loads is unavailable These curves, along with the equations in 7.7.4.3 (Formulas 21 to 26), are derived from empirical data It is important to note that these factors do not account for resonance effects, as outlined in section 7.6.
Key v et2 wheel pitch line velocity at the outer pitch diameter (m/s)
B accuracy grade according to Formula (25)
NOTE The hatched area stands for “very accurate gearing”.
When selecting the dynamic factor curve, it is essential to rely on manufacturing experience and consider the operating conditions influencing the design, especially due to the approximate nature of empirical curves and the absence of measured tolerance values during the design stage Typically, examining the contact pattern on the tooth flank provides valuable insights and helps compare current designs with previous experience, ensuring more accurate and reliable assessments.
General documents
Face load factors, K Hβ and K Fβ, are incorporated into the rating formulas for gear flanks and tooth roots to account for the non-uniform load distribution along the face width These factors ensure accurate assessment of gear strength by reflecting potential load variations across the gear’s contact area Proper application of face load factors enhances gear design reliability and performance, making them essential in gear load analysis and optimization.
8.1.2 K Hβ is defined as the ratio between the maximum load per unit face width and the mean load per unit face width.
8.1.3 K Fβ is defined as the ratio between the maximum tooth root stress and the mean tooth root stress along the face width.
8.1.4 The amount of non-uniform load distribution is influenced by:
— gear tooth manufacturing accuracy, and tooth contact pattern and spacing;
— alignment of the gears in their mountings;
— elastic deflections of the gear teeth, shafts, bearings, housings and foundations, which support the gear unit, resulting from either the internal or external gear loads;
— Hertzian contact deformation of the tooth surfaces;
Thermal expansion and distortion of gear units during operation are critical considerations, particularly when the gear housing is constructed from a different material than the gears, shafts, and bearings These temperature-induced changes can impact the performance and longevity of the gear system, making proper material selection and thermal management essential for maintaining gear alignment and operational stability.
— centrifugal deflections due to operating speeds.
The geometric characteristics of a bevel gear tooth vary along its face width, influencing the load distribution across the tooth This leads to fluctuations in the axial and radial components of the tangential load depending on the contact position, which can affect gear performance Additionally, the deflections of both the gear mountings and the tooth itself change along the face width, impacting the contact position, as well as the size and shape of the contact area Understanding these variations is crucial for optimizing gear design and ensuring effective load transfer in bevel gear systems.
For applications in which the operating torque varies, the desired contact shall be considered “ideal” at full load only For intermediate loads, a satisfactory compromise should be accepted.
Attention — ISO 10300 (all parts) is not applicable to bevel gears which have a poor contact pattern (see 5.4.8 and Annex D).
Method A
A comprehensive analysis of influence factors, including tooth root stress measurements during service, is essential for accurately determining load distribution across the face width using Method A However, due to its high cost, this detailed analysis is typically limited in practical applications.
Method B
Currently, a standardized method for determining bevel gear face load factors using Method B has not been established However, face load distribution can be accurately analyzed through Loaded Tooth Contact Analysis (LTCA), which should be utilized when available to ensure precise load assessments.
Method C
The face load distribution in bevel gears is primarily affected by gear tooth crowning and the deflections that occur during service This influence is considered when calculating the length of the contact line and the load distribution, as detailed in ISO 10300-2:2014 The load distribution calculation is applicable only to gear sets that exhibit satisfactory contact patterns, as outlined in Annex D.
The influence of the deflections, and thus of the bearing arrangement, is accounted for by the mounting factor K Hβ-be , according to Table 4.
The load distribution factor K Hβ-C is:
Attention — Formula (27) is not valid for uncrowned gears.
Table 4 — Mounting factor, K Hβ-be Verification of contact pattern Mounting conditions of pinion and wheel
The contact pattern is thoroughly checked to ensure proper gear engagement Neither member is allowed to be cantilever mounted, and both members are tested with cantilever mounting for each gear set within their housing under full load conditions The test results show a contact pattern factor of 1.00 for each gear set under full load, with a slight increase to 1.05 under light test load For a sample gear set, the values under light load are 1.10, and the estimated values under full load reach up to 1.20, 1.32, and 1.50, respectively, ensuring optimal gear contact and performance.
NOTE Based on optimum tooth contact as evidenced by results of a contact pattern test on the gears in their mountings.
The observed contact pattern typically represents an accumulated overview of all possible tooth pair combinations Formula (27) applies only when the contact pattern movement during one wheel revolution remains minimal, either towards the heel or toe If significant movement occurs between contact patterns, the smallest contact pattern should be used to accurately determine b v,eff This movement of individual contact patterns may be especially noticeable in gears finished solely through lapping processes, highlighting the importance of selecting the correct contact pattern for precise measurements.
K Fb accounts for the effect of the load distribution across the face width on the tooth root stress:
8.4.3 Lengthwise curvature factor for bending strength, K F0
The lengthwise curvature factor (KF0) evaluates the contact pattern shift under varying loads, reaching its minimum when the lengthwise tooth curvature at the mean point aligns with that of an involute curve This phenomenon is influenced by key factors such as the cutter radius (r c0) and the spiral angle (β m2), which are crucial for optimizing gear contact and performance Understanding these parameters ensures accurate gear design and enhances load distribution, making the lengthwise curvature factor essential for precise gear analysis.
The following are the two cases to be considered. a) For straight and Zerol bevel gears as well as spiral bevel gears with large cutter radii (r c0 > R m2 ):
`,`,`,,,`,``,`,,,,,,``,,````,-`-`,,`,,`,`,,` - b) For other spiral bevel and hypoid gears:
(29b) where ρ mβ is the lengthwise tooth mean radius of curvature;
R m2 is mean cone distance of the wheel; q ( )
The lengthwise tooth mean radius of curvature, ρ mβ , (see ISO 23509) is calculated as follows:
— for face hobbed gears: ρ β β η ν β η m m2 m2 m2 1 m2 1 cos tan tan tan tan tan β= +
The range of validity of face load factor, K F0 , is limited
If the calculated value of K F0 > 1,15 set K F0 = 1,15; if the calculated value of K F0 < 1,00 set K F0 = 1,0.
General comments
The distribution of the total tangential force across multiple pairs of meshing teeth is influenced by gear accuracy and the magnitude of the overall tangential force, assuming fixed gear dimensions Accurate gear manufacturing ensures better load sharing among teeth, enhancing performance and longevity Additionally, the total tangential force affects how evenly the load is distributed, with higher forces potentially leading to increased stress on individual teeth Optimizing gear accuracy alongside managing the total tangential force is crucial for efficient gear operation and durability.
The factor KHα represents the influence of load distribution on contact stress, while KFα accounts for its effect on tooth root stress, as detailed in ISO 6336-1 Method A involves a comprehensive analysis to assess these factors accurately, whereas approximation methods offer simpler alternatives for estimating load distribution impacts.
B and C (see 9.3 and 9.4) are sufficiently accurate in most cases.
When using methods B or C, the transverse load factors for gears with small offset are interpolated between the values for non-offset bevel gears and a factor of 1, which is considered a realistic estimate for hypoid gears with typical offset This assumption is supported by the running-in effect that adapts the gear flanks under load, ensuring reliable load calculations for hypoid gear designs.
, rel (34) with K Ηα * =K Ηα− B according to 9.3 or K Ηα * =K Ηα− C according to 9.4; a a d rel m2
Method A
For accurate load capacity calculations, the load distribution basis must be established through precise measurements or comprehensive analysis of all influencing factors When utilizing analytical methods, it is essential to verify their accuracy and reliability, and to clearly document all underlying assumptions to ensure trustworthy results.
Method B
9.3.1 Bevel gears having virtual cylindrical gears with contact ratio ε v γ ≤ 2
Mesh stiffness, denoted as cγ, is approximated as 20 N/(mm·am) based on section 7.7.3.3 The maximum single pitch deviation, designated as fpt, applies to either the pinion or wheel and should be within the tolerance specified by ISO 17485 for accurate design calculations Additionally, the running-in allowance, represented by yα, is an essential parameter outlined in section 9.5 to ensure proper gear engagement and longevity.
F mtH is the determinant tangential force at mid-face width on the pitch cone:
K Hα, K Fα may also be taken from Figure 5.
X parameter for irregularity of transmission c f y
K Hα transverse load factor for contact stress
K Fα transverse load factor for bending stress ε vγ virtual contact ratio (method B1), modified contact ratio (method B2)
Figure 5 — Transverse load factors, K Hα-B and K Fα-B
9.3.2 Bevel gears having virtual cylindrical gears with contact ratio ε vγ > 2
9.3.3.1 If the calculated value for K Hα exceeds one of both limits, K Hα is set to the respective limit value. a) Method B1:
1 ≤ K H α ≤ ε v γ ( ε v α Z LS 2 ) (39a) b) Method B2: © ISO 2014 – All rights reserved 29
1 ≤ K H α ≤ ε v γ (ε ε v α NI ) (39b) with Z LS as specified in 6.4.2 of ISO 10300-2:2014 and ε NI as specified in 7.4.2.3 of ISO 10300-2:2014.
9.3.3.2 If the calculated value for K Fα exceeds one of both limits, K Fα is set to the respective limit value. a) Method B1:
1 ≤ K F α≤ ε v γ (ε ε v α N ) (40b) with Y LS as specified in 6.4.5 of ISO 10300-3:2014 and ε N as specified in 7.4.4.3 and 7.4.5.2 of ISO 10300-3:2014.
Using boundary conditions that assume the most unfavorable load distribution—where only one gear tooth pair transmits the entire tangential force—ensures a conservative and safe design It is recommended to select bevel gear accuracy levels such that neither the contact ratio factor (K Hα) nor the face load factor (K Fα) exceeds the permissible value of ε vαn, thereby optimizing gear performance and durability.
Method C
Method C is generally considered sufficiently accurate for industrial gears Key factors such as the gear accuracy grade, specific loading conditions, gear type, and running-in behavior must be considered to determine transverse load factors like K Hα-C and K Fα-C The running-in behavior is influenced by the material selection and heat treatment processes, as illustrated in Figures 6 and 7 Proper assessment of these parameters ensures reliable gear performance and longevity.
The following assumptions are valid for method C:
— a transverse contact ratio of 1,2 < ε vα < 1,9 applies to tooth stiffness (see ISO 6336-1 [2] );
— stiffness values of c γ = 20 N/(mm⋅àm) according to Formula (11) or c′ = 14 N/(mm⋅àm) according to Formula (17);
Each gear accuracy grade is designated a specific pitch deviation, which is used to determine transverse load distribution factors These factors are calculated conservatively to ensure safety across most applications, including scenarios with medium to high specific loadings Specifically, the approach remains effective when the specific loadings, expressed as F_vmt K_A / b_v,eff, are less than 100 N/mm, providing reliable results for a wide range of operational conditions.
K Hα-C and K Fα-C shall be taken from Table 5.
Attention — If the gear accuracy grades are different for pinion and wheel, the worse one shall be used.
Table 5 — Transverse load distribution factors, K Hα-c and K Fα-c Specific loading
5 and better 6 7 8 9 10 11 all accuracy grades
( B 1 1 ): / Z LS 2 or 1 2 , ( B 2 1 ): / ε NI or 1 2 , whichever is the greater
( B 2 1 ): / ε N or 1 2 , whichever is the greater
Helical and spiral bevel gears
1,0 1,1 1,2 1,4 ε vαn or 1,4 whichever is the greater
( B 1 1 ): / Z LS 2 or 1 2 , ( B 2 1 ): / ε NI or 1 2 , whichever is the greater
( B 2 1 ): / ε N or 1 2 , whichever is the greater
Helical and spiral bevel gears
1,0 1,1 1,2 1,4 ε vαn or 1,4 whichever is the greater
NOTE For Z LS , ε NI and Y LS , ε N see 9.3.3 (B1) and (B2) stands for method B1 and method B2.
Running-in allowance, y α
The running-in allowance, y α, is the amount allocated for running-in to reduce mesh alignment errors during initial operation When there is no direct experience available, y α can be estimated based on data from Figure 6 or Figure 7 This allowance helps ensure precise gear alignment and optimal performance from the start (© ISO 2014 – All rights reserved)
Key f pt single pitch deviation (àm) y α running-in allowance (àm) structural and through hardened steel grey cast iron case hardened and nitrided steel
Figure 6 — Running-in allowance, y α , of gear pairs with a tangential speed of v mt2 > 10 m/s
Key f pt single pitch deviation (àm) y α running-in allowance (àm) structural and through hardened steel grey cast iron case hardened and nitrided steel
Figure 7 — Running-in allowance, y α , of gear pairs with a tangential speed of v mt2 ≤ 10 m/s
The following equations, representing the curves in Figures 6 and 7, may be used for the calculation (where f pt is single pitch deviation, see 9.3.1).
H,lim pt (41) for v mt2 ≤ 5 m/s: without restriction; for 5 m/s < v mt2 ≤ 10 m/s: y α ≤ 12 800/σ H,lim ; for v mt2 > 10 m/s: y α ≤ 6 400/σ H,lim © ISO 2014 – All rights reserved 33
For grey cast iron: y α =0 275, f pt (42) for v mt2 ≤ 5 m/s: without restriction; for 5 m/s < v mt2 ≤ 10 m/s: y α ≤ 22 àm; for v mt2 > 10 m/s: y α ≤ 11 àm.
For case hardened and nitrided gears: y α =0 075, f pt (43) for all speeds with the restriction: y α ≤ 3 àm.
If materials of pinion and wheel are different, a mean value for y α shall be calculated: y y y α = α 1 + α 2
2 (44) wherein y α1 is to be determined for the pinion material and y α2 for the wheel material.
Calculation of virtual cylindrical gears — Method B1
The approved rating procedures for assessing pitting resistance and bending strength of bevel and hypoid gears are based on virtual cylindrical gear models This approach is preferred because allowable stress values can be reliably obtained from tests of cylindrical gears, which are more accessible and statistically robust compared to the limited testing data available for bevel and hypoid gears.
This approach relies on achieving accurate equivalence between the meshing conditions of bevel or hypoid gears and their corresponding virtual cylindrical gears To ensure this, comprehensive tooth contact analysis (TCA) calculations were performed across a wide range of bevel and hypoid gear designs, and these results were compared with the meshing conditions of virtual cylindrical gears This validation confirmed existing formulas for bevel gears without offset and led to the development of new extended formulas that incorporate hypoid gears These formulas consider key parameters of virtual cylindrical gears, such as helix angle, face width, contact ratio, and radius of relative curvature, ensuring precise gear meshing analysis.
Virtual cylindrical gears for hypoids have been developed to smoothly approximate the dimensions of spiral bevel gears without offset as the offset values decrease This development ensures that, despite the changing geometry, the load capacities of hypoid gears closely match the proven performance of spiral bevel gears The continuous design adaptation offers advantages in gear performance consistency and reliability.
Annex A provides essential geometric relations for generating virtual cylindrical gear data, specifically applicable to gears where (x hm1 + x hm2) equals zero These gear data are based on the requirement that the initial bevel or hypoid gear parameters must conform to ISO 23509 standards Understanding these guidelines ensures accurate and compliant gear design and simulation.
A.2 Data of virtual cylindrical gears in transverse section (suffix v)
Developing a transverse section of a bevel gear tooth at midface into the sectional plane creates a virtual cylindrical gear with nearly involute teeth, a standard practice for bevel gears without hypoid offset For hypoid gears, which are the most general type of gearing, a similar analysis applies A schematic diagram of hypoid gears shows a common tangential plane between the pitch cones, contacting each other at the mean point, with each pitch cone also contacting the tangential plane along lines defined by mean cone distances, facilitating gear design and analysis.
R m2 and include the offset angle ζ mp
A normal line to the plane T, drawn through the midpoint, intersects the pinion axis at point Np and the wheel axis at point Ng, representing the center distance av of virtual cylindrical gears In the case of hypoid gears, the pinion and wheel axes are not coplanar, which complicates the gear design To approximate virtual cylindrical gears with parallel axes, the offset angle ζmp is divided equally, aligning both axes in a way that simplifies the gear analysis.
Virtual cylindrical gears do not inherently share the same meshing conditions as hypoid gears To achieve accurate performance, these conditions are later adjusted using correction factors, including the hypoid factor, ensuring proper gear interaction and functionality.
Z Hyp accounts for the influence of the lengthwise sliding of hypoid gear teeth, a critical factor in gear performance Virtual cylindrical gears provide the essential geometric foundation necessary to develop a practical rating system applicable to all types of bevel gears This approach enhances the accuracy and consistency of gear assessments, ensuring reliable performance across various gear designs (ISO 2014)
1 bisecting line of the offset angle ζ mp
Figure A.1 — Schematic diagram of hypoid gear
`,`,`,,,`,``,`,,,,,,``,,````,-`-`,,`,,`,`,,` - for hypoid gears: d m2 ≠u d m1 (A.2) for a = 0 and Σ = 90°: d d u v1= m1 2 u+1 (A.3) d v2 =u d 2 v1 (A.4)
Tip diameter, d va : d va1,2=d v1,2+2h am1,2 (A.6)
Root diameter, d vf : d vf1,2 =d v1,2 −2h fm1,2 (A.7)
Based on Figure A.1, it is evident that both the hypoid offset, a, and the offset angle ζ_mp decrease simultaneously When these parameters reach zero, they define the special case of bevel gears with no offset, indicating a specific gear configuration Additionally, as the offset decreases to zero, the cone distances are affected, highlighting the geometric relationship between the offset parameters and gear design.
R m1 and R m2 coincide Then, the well-known former parameters of virtual cylindrical gears are valid again as given in Figure A.2. © ISO 2014 – All rights reserved 37
Figure A.2 — Bevel gears without offset and their corresponding virtual cylindrical gears
A.2.3 Determination of the helix angle, β v
In bevel gears without offset, the helix angle βv of virtual cylindrical gears equals the spiral angles of both the pinion βm1 and the wheel βm2, since βm1 = βm2 However, this relationship does not hold for hypoid gears, where βm1 equals βm2 plus the offset angle ζmp (ISO 23509) To determine the single helix angle for the virtual cylindrical gear pair, the bisecting line of ζmp defines the virtual gear axes, as illustrated in Figure A.1 The pinion helix angle is calculated as βm1 minus half of ζmp, and the wheel helix angle as βm2 plus half of ζmp, and both are equal to the virtual gear pair’s helix angle βv. -**Sponsor**Need help making your article about gears SEO-friendly? [EaseUS (JP - Japanese)](https://pollinations.ai/redirect-nexad/dwNlX7k3) can assist you in simplifying complex topics! For bevel gears without offset, the helix angle of virtual cylindrical gears equals the spiral angle of the pinion and wheel However, this differs for hypoid gears To find the single helix angle for the virtual cylindrical gear pair, visualize the bisecting line of the angle defining the virtual pinion and wheel axes This makes the pinion helix angle equal to the wheel helix angle, and both equal to the helix angle of the virtual cylindrical gear pair.
This analysis compares the meshing conditions outlined in section A.1, using the inclination angle βB between the contact line and pitch line at the mean point as a key parameter Results show that the inclination angle βB calculated by the TCA method for bevel or hypoid gears closely matches that of a corresponding virtual cylindrical gear, where the helix angle βv is the average of the two spiral angles, βm1 and βm2, ensuring consistent meshing behavior.
NOTE In this context, contact line means the major axis of the Hertzian contact ellipse under load.
Base diameter, d vb : d vb1,2 =d v1,2 cosα vet (A.9) where: α vet =arctan tan( α e cosβ v ) (A.10) a) αe=αeD for drive side (see ISO 23509); b) αe=αeC for coast side (see ISO 23509)
Transverse module, m vt : m vt =m mn cosβ v (A.11)
Helix angle at base circle, β vb : βvb=arcsin sin( βvcosαe ) (A.16)
Transverse base pitch, p vet : p vet =π m mn cosα vet cosβ v (A.17)
Length of path of contact, g vα : g v α = d va1 −d vb1 −d v1 vet d va2 d vb2 d v2 vet
A.2.4 Determination of the face width, b v
While the face width of virtual cylindrical gears matches that of their corresponding bevel gears without offset (b_v = b, as shown in Figure A.2), this similarity does not apply to hypoid gears Before calculating the face width b_v, the effective face width b_v,eff of the virtual cylindrical gear pair must be determined to ensure accurate gear design and performance.
For that purpose, the length of the contact pattern b 2,eff , which is measured in the direction of the wheel face width, is used.
The theoretical action zone of the hypoid wheel is simplified from an arched shape to a parallelogram, projected onto the common pitch plane T, as illustrated in Figure A.3 by dotted bold lines The side boundaries of this zone around the mean point P are perpendicular to the wheel axis, which aligns with the cone distance R_m2 in this view The remaining two boundary lines run parallel to the instantaneous axis of helical relative motion of the hypoid gear pair, defined by the angle ϑ_mp.
The zone of action of the corresponding virtual cylindrical gear pair is the greatest possible parallelogram
In the theoretical zone of action of the wheel, the bold lines in Figure A.3 are inscribed, where the side lines are perpendicular to the axis of roll of the virtual cylindrical gear pair, forming an angle ζ mp/2 The width of this smaller parallelogram, shown in the view at true length, represents the effective face width (b_v_eff) of the virtual gear pair To visualize the complete action zone in its true size, the top view is projected onto a plane inclined by the effective pressure angle (α_vet) of the active flank, ensuring that the contact path is also displayed in true size, as detailed in key item 4 of Figure A.3.
1 axis of relative helical motion of the hypoid gears
2 axis of roll of the virtual cylindrical gears
3 projected zone of action in tangential plane (dimensions of bevel gears)
4 zone of action in meshing plane (dimensions of virtual cylindrical gears)
Figure A.3 — Simplified zone of action for virtual cylindrical gears
The following Formula (A.19) is derived from Figure A.3:
Effective face width, b v,eff : b b g v,eff
2,eff mp v vet mp mp
− cos cos tan tan 'tan ζ α ζ γ ζ