Tiêu Chuẩn Iso 09083-2001.Pdf

Microsoft Word ISO 9083 E doc Reference number ISO 9083 2001(E) © ISO 2001 INTERNATIONAL STANDARD ISO 9083 First edition 2001 07 15 Calculation of load capacity of spur and helical gears — Application[.]

Design, specific applications

Gear designers must recognize that application-specific requirements vary significantly, requiring careful evaluation of key factors such as material allowable stresses, load repetition frequency, and the potential consequences of failure rates It is essential to incorporate an appropriate factor of safety tailored to each application's demands, ensuring reliable and safe gear performance.

Effective gear design must incorporate considerations to prevent fractures caused by stress risers in the tooth flank, tip chipping, and failures of the gear blank through the web or hub Utilizing general machine design principles is essential to analyze and mitigate these potential failure points, ensuring enhanced durability and reliability of the gear system Proper stress analysis and design optimization can significantly reduce the risk of gear failure due to these common issues.

Any variances in the calculation method must be reported in the calculation statement If a more refined calculation method is needed or if adhering to clause 4.1 restrictions is impractical, factors can be evaluated based on the basic or alternative standards Additionally, factors derived from reliable experience or test data may be used in place of standard individual factors, following criteria outlined in Method A of ISO 6336-1:1996, section 4.1.8.

Rating calculations must strictly adhere to this International Standard, ensuring that stresses, safety factors, and related parameters are classified and evaluated in accordance with its guidelines, maintaining consistency and compliance across all assessments.

This International Standard applies to wheel blanks, shaft/hub connections, shafts, bearings, housings, threaded connections, foundations, and couplings that meet specific accuracy, load capacity, and stiffness requirements These standards serve as the basis for calculating gear load capacities, ensuring reliable performance and safety Adhering to these guidelines helps optimize gear design, improve durability, and ensure compatibility across mechanical components.

This International Standard outlines a method primarily intended for gear recalculations, which can also be employed to determine gear load capacities through an iterative process The method involves selecting an initial load and calculating the corresponding safety factors against pitting for the pinion (S H1) and the gear (S H2) If these safety factors exceed the minimum requirements (S H min), the load is increased; if they are below, the load is decreased This iterative process continues until the safety factors precisely match the minimum required values The same approach applies to evaluating safety factors against tooth breakage (S F1 and S F2), ensuring for both gears that safety is maintained at the minimum acceptable level.

This International Standard applies to various gear types, including external and internal involute spur, helical, and double helical gears For double helical gears, it is assumed that the total tangential load is evenly distributed between the two helices; however, if external axial forces cause uneven load distribution, this must be considered The two helices are treated as separate single helical gears operating in parallel Additionally, the standard covers planetary and other gear trains with multiple transmission paths.

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The range of transverse contact ratios for actual spur and helical gear pairs is between 1.2 and 2.5, influencing parameters such as C_a, C_g, K_v, K_Hb, K_Fb, K_Ha, and K_Fa Helix angles are recommended to be less than or equal to 30°, which affects C_A, C_g, K_v, and K_Hb There are no restrictions on basic racks.

This International Standard is applicable when s R , the thickness of the wheel rim under the tooth roots of internal and external gears, is > 3,5m n

These include steels (affects Z E ,sH lim, sFE,K v , K H b and K F b ) For materials and their abbreviations used in this International Standard, see Table 2 For other materials, see ISO 6336-1, ISO 6336-2, ISO 6336-3 and ISO 6336-5

Through-hardening steel, alloy or carbon, through-hardened (s B W800 N/mm 2 ) V

Case-hardened steel, case hardened Eh

Steel, flame or induction hardened IF

Nitriding steel, nitrided NT (nitr.)

Through-hardening and case-hardening steel, nitrided NV (nitr.)

Through-hardening and case-hardening steel, nitrocarburized NV (nitrocarb.)

Calculation procedures are applicable to gears that are spray or oil-bath lubricated with lubricants approved by the gear manufacturer or designer These procedures assume the continuous availability of an adequate amount of approved lubricant during operation to maintain proper gear functioning Additionally, cooling provisions are essential to ensure that gear temperatures do not exceed the calculated limits, as excessive heat can impair lubricant film formation and affect gear performance.

Provided that sufficient lubricant is available to the mesh, grease lubrication of slow speed auxiliaries is not excluded.

1) For all practical purposes, it may be assumed that the proportions of the basic rack of the tools are equal to those of the basic rack of the gear.

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Safety factors

It is necessary to distinguish between the safety factor relative to pitting,S H , and the safety factor relative to tooth breakage,S F

For a given application, adequate gear load capacity is demonstrated by the computed values ofS H andS F being equal to or greater than the valuesS H min andS F min , respectively.

Choice of the value of a safety factor should be based on the degree of confidence in the reliability of the available data and the consequences of possible failures.

When evaluating gear failure risks, it is essential to consider that the allowable stress values used in calculations are valid for a specific probability of failure; for example, ISO 6336-5:1996 provides material stress values applicable at a 1% probability of damage Additionally, ensuring the specified quality and effectiveness of quality control throughout all manufacturing stages is crucial to prevent failures Accurate specification of service conditions and external factors is also vital for reliable gear design and operation Moreover, gear tooth breakage is often regarded as a more significant hazard than pitting, highlighting the importance of addressing potential tooth failure modes in gear safety assessments.

Therefore, the chosen value for S F min should to be greater than the value chosen for S H min For calculation of actual safety factor see 6.1.5 (S H , pitting) and 7.1.4 (S F , tooth breakage).

It is recommended that the minimum values of the safety factors should be agreed upon between the purchaser,the manufacturer and the classification authority.

Input data

The following data shall be available for the calculations: a) gear data: a,z 1 ,z 2 ,m n ,d 1 ,d 2 ,d a1 ,d a2 ,b,x 1 ,x 2 ,an,b,ẻ a ,ẻ b (see ISO 53:1998, ISO 54:1996); b) cutter basic rack tooth profile: h a0 ,r a0 (see ISO 53:1998); c) design and manufacturing data:

The analysis includes critical parameters such as gear tooth surface roughness (C a1, C a2), contact pattern breadth (f pb), and minimal mounting distances (S H min, S F min) It also considers surface roughness values (Ra 1, Ra 2), surface finish quality (Rz 1, Rz 2), along with materials used, their hardness, and heat treatment specifications to ensure optimal gear performance Additionally, gear accuracy grades, bearing span length (l), gear positioning relative to bearings, and key dimensions like pinion shaft diameter (d sh) are evaluated When applicable, modifications such as helix crowning and end relief are incorporated to enhance gear contact and load distribution Power data parameters are also factored into the overall assessment for comprehensive gear system analysis.

PorTorF t ,n 1 ,v 1 , details of driving and driven machines.

Requisite geometrical data can be calculated according to national standards.

Information to be exchanged between manufacturer and purchaser should include data specifying material preferences, lubrication, safety factor and externally applied forces due to vibrations and overloads (application factor).

Numerical equations

The units listed in clause 3 shall be used in all calculations Information that will facilitate the use of this International Standard is provided in annex C of ISO 6336-1:1996.

General

The influence factors, K v , K H a , K H b , K F a and K F b , are all dependent on the tooth load Initially, this is the applied load (nominal tangential load multiplied by the application factor).

The analysis involves calculating interdependent factors successively to ensure accuracy First, determine K_v using the applied tangential load F_t divided by the product of K_A; next, calculate K_Hb or K_Fb based on the recalculated load F_t as the product of K_A and K_v; finally, derive K_Ha or K_Fa (Method B) by considering the original tangential load F_t multiplied by the combined factors K_A, K_v, and K_Hb, ensuring a comprehensive assessment of the load effects.

When a gear drives multiple mating gears or is double helical, the load factor should be adjusted by replacing K_A with K_A K_g It is recommended to determine the mesh load factor, K_g, through direct measurement when possible, or alternatively, it can be estimated based on existing literature for accurate gear design and performance assessment.

This clause simplifies all influencing factors based on specific assumptions outlined in clause 4, including that the pinion tooth number (z1) is less than 50 and that the gears are either of solid disc type or equipped with heavy rims.

When details are substantially different from any of the above, refer to ISO 6336-1.

Nominal tangential load, F t , nominal torque, T, nominal power, P

The nominal tangential load (F_t) is calculated in the transverse plane at the reference cylinder, based on the input torque to the driven machine during its heaviest regular operating condition Alternatively, the prime mover’s nominal torque can be used if it matches the torque requirements of the driven machine, or another appropriate basis can be selected to ensure accurate load assessment.

Non-uniform load, non-uniform torque, non-uniform power

When loads are non-uniform, it is essential to evaluate not only peak loads and their cycle counts but also intermediate loads and their occurrence frequencies This load variation is known as a duty cycle and can be modeled through a load spectrum Considering the cumulative fatigue effects of the entire duty cycle is crucial for accurate gearset rating The ISO TR 10495 provides a method for calculating the impact of such variable loads on gear performance.

Maximum tangential load, F t max , maximum torque, T max , maximum power, P max

The maximum tangential load (F t max), along with the corresponding torque (T max) and power (P max), define the peak operating conditions within the variable duty range To prevent damage, these limits should be controlled by a properly calibrated safety clutch Knowing F t max, T max, and P max is essential for ensuring safety against pitting damage and sudden tooth breakage caused by static stress limits, as detailed in section 5.5.

Application factor, K A

The factor K A adjusts the nominal load, F t, to account for additional gear loads from external sources, which are influenced by the characteristics of the driving and driven machines These external forces depend heavily on the system's masses and stiffness, including shafts and couplings, ensuring accurate load compensation for reliable gear operation.

It is recommended that the purchaser and manufacturer/designer agree on the value of the application factor with the accord of the classification authority.

K A should be determined through precise measurements and a thorough analysis of the system, or based on trustworthy operational experience in the specific application area (see 5.3), ensuring accurate and reliable results.

In the absence of reliable data obtained through the procedures outlined in section 5.5.2, or even during the initial design phase, designers can utilize guideline values for KA provided in Annex C to ensure safety and compliance.

Internal dynamic factor, K v

The dynamic factor links the total tooth load, accounting for internal dynamic effects in a "multi-resonance" system, to the transmitted tangential tooth load This standardized approach utilizes Method B from ISO 6336-1:1996, with specific modifications to ensure accurate and reliable calculations in gear design.

This procedure assumes that each gear pair functions as an elementary single mass-spring system, combining the masses of the pinion and wheel with their mesh stiffness It simplifies analysis by considering each gear pair as a single-stage unit, disregarding the influence of other stages in a multi-stage gear system This assumption holds true primarily when the torsional stiffness, measured at the base radius, is the dominant factor governing gear behavior.

According to ISO 2001, the shaft common to both a wheel and a pinion must have a mesh stiffness greater than that of the gears themselves to ensure proper performance For shafts that are exceptionally stiff, refer to section 5.6.3 and annex B.1, which outline the specific procedures for addressing very stiff shafts Adhering to these guidelines helps optimize gear mechanism reliability and efficiency.

Forces caused by torsional vibrations of the shafts and coupled masses are not covered by K v These forces should be included with other externally applied forces (e.g with the application factor).

Multiple mesh gear trains exhibit several natural frequencies, which can be higher or lower than those of a single gear pair with one mesh When operating in the supercritical range, it is recommended to analyze the system using Method A, as outlined in ISO 6336-1:1996, section 6.3.1, to ensure accurate vibration and load assessments.

The specific load for the calculation ofK A is (F t K A )/b.

If (F t K A )/bu100 N/mm, thenF m /b= 100 N/mm.

When the specific load F_t K_A /bis is less than 50 N/mm, there is a significant risk of vibration, especially in spur or helical gears of coarse quality grade operating at high speeds, particularly under certain conditions involving separation of working tooth flanks.

5.6.2 Calculation of the parameters required for evaluation of K v

5.6.2.1 Calculation of the reduced mass, m red a) Calculation of the reduced mass,m red , of a single-stage gear pair

(6) where m red is the reduced mass of a gear pair, i.e of the mass per unit facewidth of each gear, referred to its base radius or to the line of action;

J* 1,2 are the polar moments of inertia per unit facewidth; r b1,2 are the base radii (= 0,5d b1,2 ) b) Calculation of reduced mass,m red , of a multi-stage gear pair

See clause B.1. c) Calculations of reduced mass,m red , of gears of less common designs

For detailed information on specific gear configurations, refer to clause B.1 This includes pinion shafts with diameters at mid-tooth depth approximately equal to the shaft diameter, as well as two rigidly connected, coaxial gears The section also covers scenarios where one large wheel is driven by two pinions, the design and function of planetary gears, and the role of idler gears in gear mechanisms.

5.6.2.2 Determination of the resonance running speed (main resonance) of a gear pair a) Resonance running speed,n E1 , of the pinion:

- g ´ p in min - 1 (7) withc g from annex A b) Resonance ratio,N

The ratio of pinion speed to resonance speed, the resonance ratio,N, is determined as follows: red 1 1 E1

Resonance running speed may deviate from the calculated value in equation (8) due to unaccounted stiffness from shafts, bearings, and housings, as well as damping effects To ensure safety, the resonance range is carefully defined to accommodate these factors.

At loads such that (F t K A )/bis less than 100 N/mm, the lower limit of resonance ratioN S is determined: ắ if (F t K A )/b< 100 N/mm, then t A

5.6.2.3 Gear accuracy and running-in parameters,B p , B f ,B k

B p , B f andB k are non-dimensional parameters used to take into account the effect of tooth deviations and profile modifications on the dynamic load 2) pb eff p

2) The amountC a of tip relief may only be allowed for gears of accuracy grades in the range 0 to 6 as specified in ISO 1328- 1:1995.

- ¢ (14) with c¢ as given in annex A;

C a design amount for profile modification (tip relief at the beginning and end of tooth engagement) A value

C ay from running-in shall be substituted forC a in equation (14) in the case of gears without a specified profile modification.C ay can be obtained from Table 3.

Effective base pitch and profile deviations are measured after running-in, with values off pb eff and f f eff calculated by subtracting estimated running-in allowances (y p and y f) Specifically, pb eff is determined as the greater of pb1 - p1 or pb2 - p2, while f eff is the greater of f1 - y f or f2 - y f These calculations ensure accurate assessment of deviations once the system stabilizes post-operation.

5.6.2.4 Running-in allowance, y = a) For St, V 3) : p pb

I (18) b) For Eh, IF, NT (nitr.), NV (nitr.), NV (nitrocarb.) 3) p= = 0,075 pb y y = f (19) f= 0,075 f y f = (20)

5.6.3 Dynamic factor in the subcritical range ( NuN S )

Resonances in gear systems can occur when the tooth mesh frequency aligns with N= 1/2 or N= 1/3 However, the risk of such resonances is minimal in precision helical or spur gears, especially when they feature proper profile modifications Using gears that meet ISO 1328-1:1995 accuracy grade 6 or higher significantly reduces the likelihood of resonance-related issues, ensuring optimal gear performance and longevity.

A low contact ratio or low-quality straight spur gears can result in a K_v value that is comparable to that in the main resonance-speed range To ensure optimal gear performance and longevity, it is recommended to modify the design or operating parameters when such conditions are present.

3) See Table 2 for an explanation of the abbreviations used.

Resonances at N= 1/4, 1/5, , are seldom troublesome because the associated vibration amplitudes are usually small.

In gear pairs with unequal shaft stiffness, when the torsional stiffness of the stiffer shaft (c) is comparable to the tooth stiffness, the tooth contact frequency can resonate with natural frequencies, especially within the range of 0.2 to 0.5 This phenomenon occurs when the ratio c/rb² is of the same order of magnitude as c, leading to potential dynamic load increases exceeding those predicted by standard calculations.

C v1 andC v2 allow for pitch and profile deviations whileC v3 allows for the cyclic variation of mesh stiffness. See Table 3.

A value C ay resulting from running-in shall be substituted for C a in equation (14) in the case of gears without a specified profile modification The value ofC ay is obtained from Table 3.

See annex A for single tooth stiffnessc´.

5.6.4 Dynamic factor in the main resonance range ( N S x can be verified using the relevant equation.

It is recommended that the values used forf ma be verified by inspection checks, such as the tooth contact pattern in the working attitude.

Refer to annex B for application to planetary gears.

Mesh misalignment caused by manufacturing tolerances can be minimized through different gear assembly methods When assembling gears without modifications or adjustments, the misalignment factor (ma) is set at 1.0, with the condition that \(f_Hf\) exceeds 27 For gear pairs with provisions for adjustment—such as lapping, running-in under light load, adjustable bearings, or modifications to helix angle—and those that are suitably crowned, the misalignment factor is reduced to 0.5, requiring \(f_Hf\) to be greater than 28 Additionally, gear pairs with well-designed end relief exhibit an intermediate misalignment factor of 0.7, with the condition that \(f_Hf\) exceeds 29, thus ensuring better gear alignment and performance.

Of a pair of gears, the larger of the valuesf β of the pair shall be substituted in equations (27) to (29).

5.7.2.3 Running-in allowance, y β , running-in factor,k β

The amounty > represents the reduction in initial equivalent misalignment achieved through running-in after startup The factor k β characterizes the residual equivalent misalignment following the running-in process The use of k > in calculations is valid only when k > is proportional to F > x, ensuring accurate assessment of misalignment adjustments This relationship is particularly relevant for parameters such as St and V, where x is a key variable in evaluating system alignment Proper understanding of these factors helps optimize machinery performance and prolong operational lifespan.

According to ISO 2001 standards, the maximum permissible yaw rate varies based on vehicle speed: there are no restrictions when speed is between 0 to 5 m/s; when speed exceeds 5 m/s up to 10 m/s, the upper limit is 25,600 deg/sec, corresponding to a yaw angle of 80 arc minutes; and for speeds exceeding 10 m/s, the limit decreases to 12,800 deg/sec, with a yaw angle of 40 arc minutes The limit value for \(s_H_{lim}\) is specified in ISO 6336-5:1996 For materials such as Eh, IF, NT (nitrided), and NV (nitrided), specific standards and limitations apply to ensure safety and performance in engineering applications.

The upper speed limit is set at y ≥ 6 mm, corresponding to a force F > 40 mm, across all speed ranges When the pinion material differs from the wheel material, specific factors apply: for the pinion, y > 1 and k > 1; for the wheel, y > 2 and k > 2 These parameters must be determined separately to ensure accurate performance and safety standards.

The mean of the values: β β β y y

= + (32) is used for the calculation.

5.7.2.4 Determination of the face load factor, K H > -C1

5.7.2.4.1 Gears with unmodified helices a) Spur and single helical gears 4) :

> > é ù ổ ử ờ ổ ử ổỗ - ửữỳ ỗ ữ ờ ỗ ữ ố ứỳ p ố ứ ở ố ứ ỷ k k (33) b) Double helical gears 4) 5) :

In standard analyses, it is assumed that the entire torque is applied at one shaft end However, if torque is introduced at both shaft ends or between the helices of a double helical gear, a more precise and comprehensive analysis is required to accurately assess the gear's performance and stress distribution.

The value of K_H > pertains to the more severely stressed helix located closer to the torqued end of the pinion, where tangential load is evenly distributed between the two helices This occurs when the gap width is small relative to the face width (B-2b B )u0,5b B For calculations of K_H >, half the tooth width—including half the gap width—is used, resulting in larger values Therefore, for double helical gears with a substantial gap width, method C2 of ISO 6336-1 is recommended, as outlined in section 5.7.3.

5.7.2.4.2 Gears with modified helices a) Spur and single helical gears 4) ắ with partial helix modification 6) (with compensation for torsional deflection only):

> p > ổ ử ổỗ ữ ốố ứ ỗ - ửữứ k k (35) ắ with full helix modification (with compensation for torsional and bending deflections): ma

> k >U (36) b) Double helical gears 4) 5) ắ with full helix modification 7) (with compensation for torsional and bending deflections): ma

The validity of equations (33) to (37) depends upon compliance with 5.7.2.1, a) to f).

This method is based on fundamental standards and is designed to account for factors such as mesh alignment influences, elastic deformations of the pinion, and manufacturing inaccuracies, ensuring accurate and reliable gear performance.

The total mesh misalignment after running in, denoted as K H, is calculated considering both systematic and random errors Systematic error, represented by f sh, accounts for mesh misalignment caused primarily by pinion shaft deflection and other mechanical deflections that can be accurately evaluated in both magnitude and direction Conversely, random error, indicated by f ma, reflects mesh misalignment due to manufacturing tolerances, where the actual misalignment's direction and amount cannot be precisely determined but are confined within specified tolerance ranges based on gear accuracy grades.

6) Torsional deflection can be almost completely compensated for by means of a linear tooth trace or helix angle modification.

In addition, crowning is necessary when compensation of bending deflection is required.

Full modification of both helices is essential for optimal gear performance, as partial helix angle adjustments to compensate for torsional deflection are inadequate for double helical gears positioned symmetrically between bearings Helix angle modifications can nearly eliminate torsional and bending deflections, but often adjusting only the helix near the torque input end is sufficient, since the deflections of the other helix tend to counteract each other This approach should be verified to ensure effective compensation and reliable gear operation.

Helix correction is a lead modification used to compensate for systematic errors in measurements, aiming to match the calculated deflection for specific loads; however, due to varying loads and evaluation inaccuracies, residual effects on the correction parameter often remain Crowning serves as an effective lead modification to counteract the random misalignment components, providing a robust defensive strategy Because the misalignment factor can vary in either direction, crowning should be symmetrically applied relative to the center of the face width to ensure optimal accuracy.

For a more precise and comprehensive assessment, adherence to ISO 6336-1 is recommended, especially if the design deviates from the requirements specified in clause 4 or if factors such as external loads causing elastic deformations (e.g., belts, chains, couplings), elastic deformations of the wheel and shaft, manufacturing inaccuracies or deformations of the gear case, bearing clearances and deflections, alternative arrangements from those depicted in Figure 2, or any other manufacturing or deformation issues significantly affect gear mesh alignment.

When, by this method, a value of K H > greater than 2,0 is calculated, the true value will usually be less However if the calculated value ofK H > is greater than 1,5, the design should be reconsidered (e.g shaft stiffness increased, bearing positions changed, helix accuracy improved).

The specific loading for the calculation ofK H > is (F t K A K v )/b.

If (F t K A K v )/bu100 N/mm, thenF m /b= 100 N/mm. β β

= + C applies when K H β u2 (38) with the value ofc C taken from annex A.

Note thatbis the smaller of the facewidths of pinion and wheel measured at the pitch circles Chamfers or rounding of tooth ends shall be ignored (For double helical gears,b= 2b B )

5.7.3.3 Mesh misalignment after running-in, F > y y= x y

F > x is the mesh misalignment before running-in (see 5.7.4); y > is the running-in allowance (see 5.7.2.3).

5.7.4 Mesh misalignment before running-in, F > x

The mesh alignment before running-in, F > x , is the absolute value of the sum of manufacturing deviations, and pinion and shaft deflections, measured in the plane of action.

For gear pairs without verification of the favourable position of the contact pattern 8) :

F > x = 1,33B 1 f sh +B 2 f ma (40) withB 1 andB 2 taken from Table 4.

For gear pairs with verification of the favourable position of the contact pattern (e.g by adjustment of bearings):

F > x= |1,33 1 sh- H 5 > | (41) where f H > 5 is the maximum helix slope deviation for ISO accuracy grade 5 (see ISO 1328-1:1995).

By subtracting f H > 5, allowance is made for the compensatory roles of elastic deformation and manufacturing deviations.

Table 4 — Constants for use in equation (40)

3 Central crowning only C > = 0,5 (f ma +f sh ) a 0,5 0,5

4 b Helix correction only Corrected shape calculated to match torque being analysed

5 Helix correction plus central crowning

6 End relief Appropriate amountC I(II) , see annex D

0,7 0,7 a For appropriate crowning,C > , see annex D. b Predominantly applied for applications with constant load conditions. c Valid for very best practice of manufacturing, otherwise higher values appropriate.

For gear pairs without helix correction or crowning, the minimum value for K H > is 1,25; for gear pairs with appropriate helix correction and crowning, the minimum value forK H > is 1,10 A favourable contact pattern shall be verified.

8) With a favourable position of the contact pattern, the elastic deformations and the manufacturing deviations compensate each other (see Figure 1, compensative roles).

Figure Position of contact pattern Determination of F > x a) Contact pattern lies towards mid bearing span F > x in accordance with equation (41)

(compensative) b) Contact pattern lies away from mid bearing span F > x in accordance with equation (40)

(augmentative) c) Contact pattern lies towards mid bearing span F >x in accordance with equation (40) ẵKÂẵ ´l´s/d 1 2 (d 1 /d sh ) 4 uB *

F >x in accordance with equation (41) ẵKÂẵ ´l´s/d 1 2 (d 1 /d sh ) 4 >B *

(compensative) d) Contact pattern lies away from mid bearing span F > x in accordance with equation (41) ẵKÂẵ ´l´s/d 1 2 (d 1 /d sh ) 4 WB * -0,3 (augmentative)

F > x in accordance with equation (41) ẵKÂẵ ´l´s/d 1 2 (d 1 /d sh ) 4 x in accordance with equation (40)

(augmentative) f) Contact pattern lies away from the bearing span F > x in accordance with equation (41)

NOTE Figures a) to d) show the most common mounting arrangement with pinion between bearings Figures e) to f) show the overhung pinion.

T* Input or output torqued end, not dependent on direction of rotation.

B* B* = 1 for spur and single helical gears; B* = 1,5 for double helical gears, the peak load intensity occurs on the helix near to the torqued end.

Figure 1 — Rules for determination of F > x with regard to contact pattern position

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Stiffening is assumed when the ratio of withs/l is between 0.6 and 1.0, specifically when withs/l < 0.3 and when d1/dsh is greater than 1.15 Conversely, no stiffening is expected when d1/dsh is less than 1.15 Additionally, minimal or no stiffening occurs during the sliding of a pinion on a shaft, or when using feather keys and similar fittings Similarly, shrink fittings typically do not provide significant stiffening.

T* Input or output torqued end, not dependent on direction of rotation.

A dashed line indicates the less deformed helix of a double helical gear.

Determinef sh from the diameter in the gaps of double helical gearing mounted centrally between bearings.

Figure 2 — Constant K¢ to substitute in equations (42) and (43) for calculation of f sh

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For spur and single helical gears:

The calculation of f sh for double helical gears is relative to the helix nearest to the shaft end which is driven or which drives the load.

(43) where b= 2b B ; b B is the width of one helix.

In equations (42) and (43),K¢,sandlare according to Figure 2.

Face load factor, K F b

In gear design, 'b' represents the smaller face width of the pinion and wheel measured at the pitch circles, excluding chamfers or rounded tooth ends For double helical gears, the effective width is taken as the helix width, denoted as bB The tooth height 'h' is measured from the tip to the root and is calculated using the formula h = (dₐ - d𝓕) / 2, where dₐ is the addendum diameter and d𝓕 is the root diameter.

Transverse load factors, K H a , K F a

Transverse load factors are crucial for evaluating gear stress, accounting for the non-uniform distribution of transverse loads across multiple contact pairs Specifically, the factor K H is used to assess surface stress, while K F is applied for tooth-root stress The calculation method adheres to Method B of ISO 6336-1:1996, ensuring standardized and accurate assessment of gear loads and stresses.

5.9.2 Determination of the transverse load factors

Equations (48) and (49) assume that base pitch deviations, aligned with the gear accuracy specifications, are evenly distributed around the circumference of the pinion and wheel, reflecting standard manufacturing practices These formulas do not apply when gear teeth are intentionally modified, emphasizing their limitation to typical gear production processes.

In the following equations usec C from annex A andy = from 5.9.4. ắ For gears with total contact ratioẻ C u2: pb

- ổ ử ỗ ữ ố ứ ẻ (48) ắ For gears with total contact ratioẻ C > 2: pb

In equations (48) and (49), the larger of (f pb1 -y = 1 ) and (f pb2 -y = 2 ) is used.

When, in accordance with equations (48) and (49, and

H F 2 H F 2 when K K > , then for K and K , substitute

= = ẻ ẻ ẻ ẻ ẻ ẻ (50) and when K H = < 1,0, and respectively K F = < 1,0, then substitute for K H = , and respectively for K F = , the limit value 1,0.

For optimal performance, it is recommended to select helix gear accuracy levels where the values of K H and K F do not exceed a specified limit, typically denoted as ê To achieve this, it may be necessary to restrict the base pitch deviation tolerances, especially for gears classified under coarse quality grades, ensuring reliable and precise gear operation.

When wind speed is below 5 m/s, there are no restrictions on the parameter y For wind speeds between 5 m/s and 10 m/s, the maximum y value is limited to 12,800/s H lim, corresponding to a p_b less than 80 àm When wind speed exceeds 10 m/s, the upper limit for y decreases to 6,400/s H lim, with p_b restricted to less than 40 àm For Eh, IF, NT (nitr.), and NV (nitr.), y is calculated as 0.075f p_b across all wind speeds, with a minimum y value of 3 mm, corresponding to p_b of 40 àm.

6 Calculation of surface durability (pitting)

Basic formulae

Surface durability is assessed by calculating the contact stress, sH, at the pitch point or the lowest point of single tooth contact, with the higher value used to determine capacity The contact stress, sH, and the permissible contact stress, sHp, must be calculated separately for both the wheel and pinion, ensuring that sH does not exceed sHp for reliable performance.

6.1.2 Determination of contact stress,sH , for the pinion

Contact stress,s H , for the pinion is calculated as follows:

In gear analysis, sH0 represents the nominal contact stress at the pitch point, which is the stress induced in flawless, error-free gears under static nominal torque The facewidth, denoted as b, plays a crucial role, with the value for mating gears determined as the smaller of the facewidths at the pitch circles of the pinion and wheel, excluding any intentional transverse chamfers or tooth-end rounding For double helical gears, the facewidth is doubled, expressed as 2bB Additionally, surface-hardened gear tooth flanks and transition zones, especially those that are unhardened, are not included in the calculation of contact stress, ensuring an accurate assessment of gear contact conditions.

Z B is the single pair tooth contact factor for the pinion (see 6.2).

6.1.3 Determination of contact stress,sH , for the wheel

Contact stress,sH, for the pinion is calculated as

Z D is the single pair tooth contact factor for the wheel (see 6.2).

In gear trains with multiple transmission paths, such as planetary gear systems or split-path configurations, the tangential load is not evenly distributed across individual meshes due to design variations, tangential speed differences, and manufacturing tolerances To accurately account for this, it is essential to modify equations (53) and (55) by replacing the factor K_A with K_C K_A, thereby adjusting the average tangential load per mesh appropriately For detailed guidance on this adjustment, please refer to clause 5.

6.1.4 Determination of permissible contact stress,sHP , for long life

In this International Standard, Method B of ISO 6336-2:1996 is used.

The permissible contact stress (long life) shall be derived from equation (56), with the influence factors sH lim,

According to ISO standards, values such as S H min, Z L, Z V, Z R, Z W, and Z X are calculated following the specified international protocols ISO 6336-2 verifies that the threshold values for these parameters are validated for specific load cycles—namely, 5×10^7 cycles for St, V, and Eh, and 2×10^6 cycles for If and NT (nitriding).

NV (nitr.) and NV (nitrocar.) are critical indicators of gear life, which are likely to be exceeded during the lifespan of a marine gear If these limits are not surpassed, refer to ISO 6336-2 to determine the limited life range Under optimal conditions—considering material quality, lubrication, manufacturing, and operational experience—values of sHP ref from equation (56) can be used in place of sHp Otherwise, for specific material quality MQ, sHp values should be obtained according to ISO 6336-5:1996 using equation (57).

For If, NT (nitr.), NV (nitr.), NV (nitrocar.):

6.1.5 Safety factor for surface durability, S H

S H shall be calculated separately for the pinion and wheel:

S s S s (58) withsHGfor endurance according to equation (57);σHshall be in accordance with equation (53) for the pinion, and with equation (55) for the wheel (see 6.1.1).

NOTE This is the calculated safety factor with regard to contact stress (Hertzian pressure) The corresponding factor relative to torque capacity is equal to the square ofS H

For the minimum safety factor for surface durability,S H min , and probability of failure, see 4.1.3 of ISO 6336-1:1996.

Single pair tooth contact factors, Z B , Z D

When the factors Z_B and Z_D are greater than 1, they are used to convert the contact stress at the pitch point of spur gears to the contact stress at the inner (lowest) limit of the single pair tooth contact—either on the pinion or the wheel This adjustment ensures more accurate assessment of gear contact stresses, especially in internal gears where precise stress analysis is critical for durability and performance.

Z D is always to be taken as unity. b) Spur gears

To optimize gear design, determine M1, the quotient of rrel C at the pitch point rrel B at the inner limit of a single tooth pair contact on the pinion, and M2, the ratio of rrel C to rrel D of the wheel These parameters are essential for assessing gear performance and ensuring proper load distribution Accurate calculation of M1 and M2 helps in minimizing gear stress and increasing operational efficiency Utilizing these quotients enables precise gear analysis, leading to improved durability and smooth operation of the gear system.

(See 6.5.2 for the calculation of the profile contact ratioẻ a )

IfM 2 > 1, thenZ D =M 2 ; ifM 2 u1, thenZ D = 1,0. c) Helical gears withẻ b W1

Z B andZ D are determined by linear interpolation between the values for spur and helical gearing withẻ b W1:

If Z B or Z D are set to unity, the contact stresses calculated using equations (53) or (55) are the values for the contact stress at the pitch cylinder.

The methods outlined in section 6.2 are designed to calculate contact stress when the pitch point is located along the contact path If the pitch point is a critical determinant and lies outside this contact path, then the values of Z_B and/or Z_D, or both, must be determined to accurately assess contact stress.

Reproduction or networking of this content is prohibited without a license from IHS © ISO 2001 When analyzing helical gears, if the helix angle ẻ b is less than 1.0, the gear parameters Z B and Z D should be determined through linear interpolation between the values established for spur gears and those for helical gears with the same ẻ b W1 at the pitch point or adjacent tip circle Accurate gear design requires considering these interpolated values to ensure proper gear meshing and performance. -**Sponsor**Struggling to rewrite technical content and make it SEO-friendly? As a content creator myself, I know the challenge! For high-quality, SEO-optimized articles that save time and money, check out [Article Generation](https://pollinations.ai/redirect-nexad/c526JTpy) It helps you effortlessly create content, perfect for boosting your online presence No more overspending on content creation! 💰

Zone factor, Z H

The zone factor, ZH, is crucial in accounting for the influence of tooth flank curvature on Hertzian pressure at the pitch point It effectively transforms the tangential force at the reference cylinder into the normal force at the pitch cylinder, ensuring accurate load calculations Incorporating ZH into gear analysis helps optimize gear contact stresses and improve overall gear performance.

Elasticity factor, Z E

The elasticity factor, Z_E, accounts for the influence of material properties such as the modulus of elasticity (E) and Poisson's ratio (n) on contact stress Since this international standard applies exclusively to steel gears, Z_E is set to a fixed value for these materials, ensuring accurate stress analysis and adherence to standardized guidelines.

Contact ratio factor, Z ẻ

The contact ratio factor,Z ẻ , accounts for the influence of the transverse contact and overlap ratios on the surface load capacity of cylindrical gears. a) Spur gears:

The conservative value ofZ ẻ = 1,0 may be chosen for spur gears having a contact ratio less than 2,0. b) Helical gears:

6.5.2 Transverse contact ratio,ẻ a ẻ a =g a /p bt (67) with length of path of contact:

= d d d d a g = 1 2 ộờở a1 2 - b1 2 ± a 2 2 - b2 2 ựỳỷ- sina wt (68) and transverse base pitch: t t bt m cos p = p = (69)

The positive sign is used for external gears, the negative sign for internal gears.

Equation (69) is valid only when the contact path is genuinely confined by the tip circle of the pinion and wheel, rather than by undercut tooth profiles Ensuring this condition is crucial for accurate gear analysis and optimal performance Properly defining the contact path helps prevent errors in gear design and guarantees effective load transmission.

See 6.1.2 for the definition of facewidth.

Helix angle factor, Z b

The helix angle factor,Z b , takes account of the influence on surface stress of the helix angle. cos

Allowable stress numbers (contact), I H lim

ISO 6336-5 details the commonly used gear materials and heat treatment methods, emphasizing how gear quality impacts allowable stress values It explains that the allowable stress numbers (sH lim) are derived from test results of standard reference test gears, ensuring reliable performance This guideline helps in selecting appropriate materials and heat treatments to optimize gear strength and durability while adhering to industry standards.

See, too, ISO 6336-5 for requirements concerning material and heat treatment for qualities ML, MQ, ME and MX.Material quality MQ shall be chosen for marine gears, unless otherwise agreed.

Influences on lubrication film formation, Z L , Z v and Z R

According to ISO 6336-2, the ZL factor accounts for the influence of the lubricant’s nominal viscosity (Zv), tooth-flank velocities, and surface roughness (ZR) on lubricant film formation in gear contact zones This standard utilizes Method B from ISO 6336-2:1996 to assess these factors, ensuring accurate consideration of lubrication conditions for gear durability and performance.

Factors shall be determined for the softer material when the hardness of meshing gears is different.

Z L can be calculated using equations (72) to (75):

C (73) b) If 850 N/mm 2 usH limu1 200 N/mm 2 , then

Alternatively,Z L can be calculated from equation (76):

Z L = C ZL + 4(1,0-C ZL )v f (76) where nf= 1/(1,2 + 80/n50) 2 using the viscosity parameters from Table 5.

ISO viscosity class VG 32 a VG 46 a VG 68 a VG 100 VG 150 VG 220 VG 320

Viscosity parameter nf 0,040 0,067 0,107 0,158 0,227 0,295 0,370 a Only for high speed transmission.

Z v can be calculated using equations (77) and (78):

C Zv =C ZL + 0,02 (78) see equations (73) to (75) for the values ofC ZL

Alternatively,Z v can be calculated from equation (79): v = C Z v+ 2 (1,0 C Z v)v p

Z R may be calculated using the following equations:

Rz 1,2 is measured on several tooth flanks The mean roughness Rz 1 (pinion flank) and the mean roughness Rz 2

The surface condition of the wheel flank should be assessed after manufacturing, considering any planned running-in treatments during manufacturing, commissioning, or in-service use when it is safe to assume they will occur If the specified roughness is given as an Ra value (also known as CLA or AA value), an approximation can be used to convert this to other surface roughness parameters This ensures accurate evaluation and compliance with quality standards for wheel flank surfaces.

(also applicable for internal gears,d b then being negative sign)

6.8.4.3 Material dependent index, C ZR a) IfI H lim < 850 N/mm 2 , then

C Z (87) b) If 850 N/mm 2 uIH limu1 200 N/mm 2 , then

Work hardening factor, Z w

The work hardening factor, Z_w, as outlined in ISO 6336-2, accounts for increased surface durability resulting from the meshing of a steel wheel with a significantly harder pinion (approximately 200 HV or more) that has smooth tooth flanks This factor is applicable when the pinion’s hardness exceeds that of the wheel, ensuring wear effects are considered according to the standard Method B of ISO 6336-2:1996 is used to determine this factor, providing a consistent approach for assessing gear contact robustness.

Z (92) where HB is the Brinell hardness of the tooth flanks of the softer gear of the pair.

Size factor, Z X

Statistical evidence indicates that the stress levels leading to fatigue damage decrease as component size increases, due to the greater number of structural weak points Smaller stress gradients and size-related material quality factors, such as forging quality and subsurface defect influence, play a crucial role in this relationship Key parameters affecting fatigue life include material quality, heat treatment details, flank radius curvature, and module size, particularly in surface hardening applications where the hardened layer depth relative to tooth size impacts core support and durability.

For through-hardened gears and for surface-hardened gears with adequate case depth relative to tooth size and radius of relative curvature, the size factor,Z x , is taken to be 1,0.

7 Calculation of tooth bending strength

Basic formulae

According to ISO 6336-3, the maximum tensile stress at the gear tooth root must not surpass the permissible bending stress for the material This guideline is fundamental for assessing the bending strength of gear teeth Ensuring this compliance helps maintain gear durability and prevents premature failure under load Properly evaluating these stresses is essential for reliable gear design and optimal performance.

The actual tooth-root stress, IF, and the permissible bending stress,IFP, shall be calculated separately for pinion and wheel;I F shall be less thanI FP

7.1.2 Determination of tooth root stressσF

Method B of ISO 6336-3:1996 is used in this International Standard.

Tooth root stress,IF, is calculated as follows:

In gear trains with multiple transmission paths, such as planetary gear systems or split-path configurations, the tangential load is not evenly distributed across individual meshes due to design variations, tangential speeds, and manufacturing tolerances To account for this uneven load distribution, it is necessary to replace K_A with K_CK_A in equation (93), ensuring proper adjustment of the average tangential load per mesh.

When designing double helical gears, the face width (b) significantly impacts tooth strength and contact quality If the face width of the gear is larger than that of its mating gear (with a relationship of 2b B), the bending strength should be calculated based on the smaller face width plus an extension not exceeding one module at each end In cases where crown or end relief prevents contact from reaching the gear ends, the smaller face width should be used for both the pinion and gear Face width (b) is measured at the gear's root cylinder, essential for assessing gear durability and performance.

7.1.3 Determination of permissible tooth root stress,I FP

FP ref FE rel T R rel T X

According to ISO 6336-3, the validated values of IF lim and IFE are based on N L = 3×10^6 load cycles, which are typically exceeded in the service life of marine gear If the operational life is shorter, refer to ISO 6336-3 for the limited life range Under optimal conditions, materials, manufacturing, and experience, the IHP ref value from equation (95) can be substituted for IFP, otherwise, the IFP value is calculated using equation (96).

7.1.4 Safety factor for bending strength, S F

The factorS F shall be calculated using the following equation:

S F is calculated separately for pinion and wheel, withIFG calculated in accordance with equation (95) or (96) as appropriate, andIFobtained with equation (93).

More information on the safety factor and probability of failure can be found in ISO 6336-1:1996, 1.3.

Form factor, Y F

Y F is the form factor by means of which the influence of tooth form on nominal bending stress is taken into account.

Y F is relevant to the application of load at the outer limit of single pair tooth contact (method B of ISO 6336-3:1996).

The values of Y F are determined for both spur gears and the virtual spur gears of helical gears Virtual spur gears are characterized by a virtual number of teeth, denoted as z n For detailed calculation methods of z n and other virtual gear parameters, refer to section 7.2.4.

Y F shall be determined separately for wheel and pinion from the following equation (see Figure 3).

Figure 3 — Determination of dimensions of tooth-root chord at the critical section

These equations are applicable to all basic rack tooth profiles, whether with or without undercut, provided specific conditions are met First, the contact point of the 30° tangent must lie on the tooth-root fillet Second, the gear's basic rack profile should include a root fillet to ensure proper engagement Lastly, the teeth are generated using tools like hobs or planer-cutters that feature rack form teeth, ensuring accurate and consistent tooth geometry for effective gear operation.

Figure 4 — Dimensions and basic rack profile of teeth (finished profile)

7.2.2 Parameters required for the determination of Y F

First, determine the auxiliary values E, G and H: pr fP n fP n n n n tan + (1 sin )

= p - - - (99) where s pr =pr-q(see Figure 4); s pr = 0, when gears are not undercut (see Figure 4) fP fP n n

Next, useGandHtogether withG=p/6 as a seed value (on the right hand side) in equation (102). n

Recalculate the value of G using the updated values and repeatedly apply equation (102) until the successive G values show negligible differences, indicating convergence Typically, the function stabilizes after two or three iterations of equation (102) Once convergence is achieved, use the final G value in equations (103), (104), and (105) for further analysis.

Tooth-root normal chord,s Fn :

Bending moment arm,h Fe : bn en en

The form factor of a specialized rack can be approximated by the form factor of an internal gear, providing a useful basis for analysis The rack's profile should be a modified version of the basic rack profile designed to produce a normal profile—comprising tip and root circles—matching that of an exact internal gear counterpart Additionally, the load direction angle, denoted as n, plays a crucial role in ensuring proper gear engagement and performance.

Figure 5 — Parameters for determination of form factor, Y F , of an internal gear

The values to be used in equation (98) are determined as follows:

Tooth-root normal chord,s Fn2 : fP 2 fP 2 pr

Hf P 2 is tool radius (see below).

Fe 2 en 2 fn 2 fP 2 en 2 fn 2 fP 2 n n n n n n n

= -ờờở ỗố - ữứ a ỳỳỷ a - ỗố - ữứ (110) where

HfP2 is tool radius (see below); d en2 is derived from equation (121) adding the subscript 2; d fn2 is derived in the same way asd an [equation (121),note thatd fn2 -d f2 =d n2 -d 2 ].

Obtainh fP2 from equation (111), refer to equation (113) and related information forHfP2. n 2 fn 2 fP 2 2 d d h = - (111)

Root fillet radius,H F2 , tool radius,H fP2:

When the root fillet radius,HF2, is known, it shall be used Otherwise:

(d Nf2 represents the diameter of a circle near the tooth-root, containing the limits of the usable flanks of an internal gear).

If sufficient data are not available, the following approximation may be used:

Ensure that the correct sign is used; see the footnote in Table 1.

7.2.4 Parameters for the virtual gear n 2

= cos d m z d > (118) n n bn= m cos p F = (119) bn= ncos n d d = (120) an= n+ a d d d d - (121)

2 2 2 2 an bn n bn en n cos cos

The value ofzis positive for external gears and negative for internal gears (see clause 3, footnote 2).

Stress correction factor, Y S

The stress correction factor, YS, is essential for converting nominal bending stress into local tooth root stress It must be determined separately for both the pinion and the wheel to ensure accuracy YS is valid within the range of 1 ≤ q_s < 8, providing reliable correction across specified conditions Proper application of YS enhances the precision of gear stress analysis and promotes optimal gear design.

S Fn from equation (103) for external gears, equation (109) for internal gears; h Fe from equation (108) for external gears, equation (110) for internal gears; r F from equation (104) for external gears, equation (113) for internal gears.

Helix angle factor, Y b

The preliminary tooth-root stress of a virtual spur gear is adjusted using the helix factor, Y, to accurately represent that of the corresponding helical gear This conversion accounts for the oblique orientation of the mesh contact lines, resulting in reduced tooth-root stress in helical gears.

Tooth-root reference strength, s FE

ISO 6336-5 specifies standardised sF limits and provides essential guidelines for popular gear materials, ensuring optimal performance and durability It also details the requirements for heat treatment processes and material quality to achieve the desired quality grades ML, MQ, and ME, supporting reliable gear manufacturing and high-quality results.

The quality MQ shall be used for marine gears unless otherwise agreed.

Relative notch sensitivity factor, Y d rel T

Y @ rel T approximately indicates the overstress tolerance of the material in the root fillet region Method B of ISO 6336-3:1996 is used in this International Standard.

= + ¢ (130) where r¢ is the slip-layer thickness taken from Table 6 as a function of the material;

? is the value for the standard reference test gear: ? T * = 1,2;

?* is the relative stress gradient calculated using the following equation: 9)

9) Applies for modulem= 5 mm The influence of size is covered by the factorY X (see 7.8).

? (131) where q s is the notch parameter obtained from equation (125).

Table 6 — Values for slip-layer thicknessr¢

NT (nitr.), NV (nitr.), NV (nitrocar.) 0,100 5

V yield pointI s = 500 N/mm 2 0,028 1 yield pointIs= 600 N/mm 2 0,019 4 limit of proportionalityI0,2= 800 N/mm 2 0,006 4 limit of proportionalityI0,2= 1 000 N/mm 2 0,001 4

Eh, If 0,003 0 a See Table 2 for an explanation of abbreviations used.

Relative surface factor, Y R rel T

The surface factor, denoted as Y R rel T, plays a crucial role in determining the impact of surface condition on tooth-root stress This factor is primarily influenced by the surface roughness at the tooth-root fillets, highlighting the importance of smooth surface finishes for minimizing stress concentrations Understanding how surface roughness affects tooth-root integrity is essential for optimizing gear performance and durability.

The surface condition’s impact on tooth-root bending strength is influenced not only by surface roughness in the fillets but also by the size and shape of defects, such as notches within notches Currently, this complex interaction has not been thoroughly studied, and as a result, isn’t incorporated into the international standard The method used in this context is only applicable when there are no scratches or similar defects deeper than 2×Rz.

NOTE 2´Rzis the preliminary estimated value.

Besides surface texture, factors such as residual compressive stresses from shot peening, grain boundary oxidation, and chemical effects significantly influence tooth bending strength When fillets are shot peened or perfectly shaped, it is advisable to substitute a slightly higher value than that shown in the graph for Y R rel T to account for enhanced strength Conversely, the presence of grain boundary oxidation or chemical effects requires substituting a lower value than indicated by the graph for Y R rel T, reflecting a reduction in strength These considerations are particularly relevant for parameters like V, Eh, and IF when Rz is less than 1 µm, ensuring accurate assessment of mechanical performance.

Y = (132) b) for NT (nitr.), NV (nitr.), NV (nitrocar.) whenRz< 1 àm

Y c) for V, Eh, IF ifRzW1 àm

Size factor, Y X

Y X is employed to account for the influence of size on the probable distribution of weak points within the material structure, considering how stress gradients— which tend to decrease with increasing dimensions— impact material behavior Additionally, this method takes into account factors such as material quality, forging quality, and the presence of defects, ensuring a comprehensive assessment of structural integrity in engineering applications.

Y X is calculated in accordance with Table 7.

Y X = 0,8 a See Table 2 for an explanation of the abbreviations used.

The tooth stiffness parameter is a measure of the required load per 1 mm facewidth, applied along the line of action, to produce a 1 µm deformation in one or more pairs of deviation-free teeth in contact This parameter is essential for understanding gear contact behavior and ensuring optimal load distribution Accurate assessment of tooth stiffness helps in designing gears that can withstand operational stresses and improve durability By evaluating the stiffness, engineers can predict deformations under load, aiding in the development of reliable and efficient gear mechanisms.

Single stiffness (c¢) represents the maximum stiffness of a single-tooth pair in a spur gear, closely approximating the maximum stiffness during contact of a single tooth pair For helical gears, c¢ refers to the maximum normal stiffness to the helix of one tooth pair, providing critical insights into gear contact performance.

Mesh stiffness,c C , is the mean value of stiffness of all the teeth in a mesh.

Method B from ISO 6336-1:1996, used in this International Standard, is applicable in the rangex 1 Wx 2 u2.

For specific loading,F t K A /bW100 N/mm 2 : c¢= 0,8c¢th C R C B cosb (A.1)

A.2.2 Theoretical single stiffness, c¢th th

The tooth deflection can be roughly estimated using the formula F t (F m F tH ) instead of F bt, which represents the load tangent to the base cylinder Conversion between F t and F bt involves relevant factors, but these modifications can often be neglected when compared to other uncertainties such as measurement tolerances.

11) c´ at the outer limit of single pair tooth contact, can be assumed to approximate the maximum value of single stiffness when ẻα > 1,2.

C R = 1 for gears made from solid disc blanks For other gears:

Boundary conditions: whenb s /b< 0,2, substituteb s /b= 0,2; whenb s /b> 1,2, substituteb s /b= 1,2.

C B can be obtained from equation (A.5):

A.2.5 Additional information a) Internal gears: approximate values of the theoretical single stiffness of internal gear teeth can be determined from equations (A.2), (A.3), by the substitution of infinity forz n2 b) Specific load (F t K A /b) < 100 N/mm

100 c = c C C F K b é ù Â Â > ờở ỳỷ (A.6) c) The above is based on steel gear pairs, for other materials and material combinations, refer to ISO 6336-1:1996, clause 9.

For spur gears withẻ = W1,2 and helical gears withbu30°, the mesh stiffness:

Figure A.1 — Wheel blank factor, C R ; mean values for mating gears of similar or stiffer wheel blank design

Special features of less common gear designs

B.1 Dynamic factor, K v, for planetary gears

Gear trains with multiple mesh gears, including idler gears and epicyclic systems like planet and sun gears, exhibit several natural frequencies These frequencies can be higher or lower than those of a simple gear pair with only one mesh, impacting dynamic performance and vibration behavior in gear systems Understanding these natural frequencies is essential for optimizing gear design and reducing noise and mechanical failure risks.

Values of Kv calculated using the formulae in this International Standard are considered unreliable; however, they can serve as useful preliminary assessments It is advisable to re-evaluate these values with more accurate procedures whenever possible.

Method A should be preferred for the analysis of less common transmission designs Refer to 6.1.1 of ISO 6336-1:1996 for further information.

B.1.2 Calculation of the relative mass of a gear with external teeth

B.1.3 Resonance speed determination for less common gear designs

For less common gear designs, resonance speed determination should primarily be performed using Method A, which provides accurate results However, alternative methods can be employed to approximate resonance effects, such as analyzing a pinion shaft with a diameter at mid-tooth depth (d_m1) approximately equal to the shaft diameter, considering two rigidly connected, coaxial gears, or examining systems like a large wheel driven by two pinions Additional approaches include evaluating planetary gear arrangements and the use of idler gears, offering practical approximations when traditional methods are challenging to apply.

B.1.3.2 Pinion shaft diameter equal to diameter at mid-tooth depth, d m1

The torsional stiffness of the pinion shaft is significantly offset by its mass, allowing resonance speeds to be calculated using standard methods Specifically, the resonance speed depends on the mass of the pinion's toothed portion and the typical mesh stiffness This approach ensures accurate determination of critical operational speeds for gear systems, optimizing performance and durability.

B.1.3.3 Two, rigidly connected, coaxial gears

The mass of the larger of the connected gears shall be included.

B.1.3.4 One large wheel driven by two pinions

In gear systems, the mass of the wheel is typically much greater than that of the pinions, allowing each meshing pair to be analyzed independently Specifically, each mesh can be considered as a pair consisting of the first pinion and the wheel, and another pair comprising the second pinion and the wheel.

Planetary gear vibratory behavior is highly complex due to multiple transmission paths involving various stiffnesses beyond just mesh stiffness Simple calculation methods, such as method B for dynamic load factors, tend to be inaccurate but can provide a rough initial estimate when modified appropriately To ensure accuracy, this preliminary estimate should be verified through detailed theoretical analysis, experimental testing, or operational experience.

The reduced mass for the determination of the resonance speed,n E1 , of the sun gear is given by:

* * pla sun red * 2 * 2 pla b sun + sun b pla

The moments of inertia per unit facewidth for the planet gear and sun gear are denoted as J and J sun*, respectively, measured in kg×mm²/mm The radius of the sun gear (r_b_sun) is calculated as half of its diameter (0.5 d_b_sun), while the radius of the planet gear (r_b_pla) is similarly determined as half of its diameter (0.5 d_b_pla) Additionally, the variable p represents the number of planet gears involved in the gear stage under consideration, which is essential for analyzing the gear train’s dynamic behavior.

The value 'm' (red), as determined from equation (B.1), should be used in the calculation of 'N' according to section 5.6.2.2 In this calculation, the mesh stiffness is approximated to that of a single planetary gear, with the mesh stiffness denoted as 'c C' Additionally, the number of teeth on the sun gear (z1) must be incorporated into the computation to ensure accurate results. -**Sponsor**As a content creator, I understand the need for SEO-optimized and coherent articles! If you're looking to revamp your article, [Article Generation](https://pollinations.ai/redirect-nexad/d1EiIej3) can help you effortlessly create high-quality content that complies with SEO rules, saving you time and money With Article Generation, you can easily rewrite and refine your articles to ensure they contain the most important sentences with coherent paragraph structure Perfect for boosting your online presence without breaking the bank!

In planetary gear analysis, the parameter F t in equations (12) to (14) for B p, B f, and B k (see section 5.6.2.3) represents the total tangential load applied to the sun gear, divided by the number of planet gears Additionally, when the planet gear or annulus gear is rigidly connected to the gear case, it influences the load distribution and gear dynamics within the planetary system Understanding these forces and connections is essential for accurate gear design and performance assessment.

In this analysis, the mass of the annulus gear is assumed to be infinite, which simplifies the calculations by making the relative mass equal to the referred mass of the planet gear This approach is essential for accurate modeling of gear dynamics and reduces complexity in the computational process The relevant mathematical expression to determine this is given by \( pla \times red\, 2\, bpla\, m\, J \), where each parameter plays a crucial role in the calculation Understanding this assumption helps in optimizing gear design and ensuring precise performance analysis.

=r (B.2) with the notation as above.

The mass of the annulus gear can be determined similarly to that of an external wheel, ensuring accurate calculations The planet gear's relative mass is calculated according to equation (B.2), facilitating precise design assessments When the annulus gear meshes with multiple planet gears, the procedure described in section B.1.3.4 should be followed to ensure comprehensive analysis and optimal gear performance.

Tiêu đề	Calculation Of Load Capacity Of Spur And Helical Gears — Application To Marine Gears
Trường học	International Organization for Standardization
Chuyên ngành	Marine Gears
Thể loại	Tiêu chuẩn
Năm xuất bản	2001
Thành phố	Geneva

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Số trang	66
Dung lượng	738,7 KB