Involute gearing - theory

This part contains a summary of some theoretical information and formulas related to the geometry design, calculation of force and performance parameters as well as the strength control of involute gearing. These data were used in the calculations below.

  • Spur gearing external
  • Spur gearing internal
  • Planetary Gearing with Spur and Helical Toothing
  • Spur gear - Rack
  • Content:

    1. Geometry, dimensions
    2. Moment, power, forces, efficiency
    3. Planetary Gearing
    4. Spur gear - Rack
    5. Stress and safety ISO 6336:2006
    6. Stress and safety ANSI/AGMA 2001-D04

    1. Geometry, dimensions.

    Applied formulas (calculation of gearing geometry).

    Some of the most important formulas for calculation of the gearing geometry are specified below. In the formulas, indexes 1 and 2 are used for a pinion and a wheel (which is a pair: a sun and a planet, or a planet and a ring gear). For the internal gearing (ring gear), a negative value for the number of internal gear teeth is used, i.e. also a negative value for the centre distance and diameters.

    In the case of planetary gearing, the individual gears are dependent on each other and the gearing must be handled as a whole, including the respective constraining conditions (see below).

    Basic profile parameters: mn (module, DP for inch calculation), a (pressure angle), ha*, c*, rf* (cutting tool parameters)

    Pinion and wheel parameters: z1, z2 (number of teeth of pinion and wheel), x1, x2 (addendum modification), b (helix angle), b (width of gearing)

    1. Transmission ratio
      i = z2 / z1
    2. Module
      mt = mn / cos(b) ... Transverse
    3. Circular pitch
      p = p • mn ... Circular pitch
      pt = p / cos(b) ... Transverse circular pitch
      ptb = pt • cos(at) ... Base circular pitch
    4. Pressure angle
      at = arctg(tg(a) / cos(b)) ... Transverse pressure angle
      awn = arcinv(2 • (x1 + x2) / (z1 + z2) • tg(a) + inv(a)) ... Pressure angle at the pitch cylinder
      awt = arcinv(2 • (x1 + x2) / (z1 + z2) • tg(a) + inv(at)) ... Transverse pressure angle at the pitch cylinder
    5. Base helix angle
      bb = arcsin(sin(b) • cos(a))
    6. Diameter
      d = z • mt ... Reference diameter
      db = d • cos
      (at) ... Base diameter
      dw = d • cos
      (at) / cos(awt) ... Operating pitch diameter
      da = mn • (z / cos
      (b) + 2 • (ha* + x - DY) ... Tip diameter
      df = d - hf ... Root diameter
    7. Center distance
      a = (z1 + z2) • mn / (2 • cos
      (b)) ... Center distance (pitch)
      av = a + (x1 + x2) • mn ... Center distance (production)
      aw = a • cos
      (at) / cos(atw)  ... Center distance (working)
      DY = (a - aw) / mn + (x1 + x2) ... Unit correction
    8. Contact ratio
      ea = ((da1^2 - db1^2)^0.5 + (da2^2 - db2^2)^0.5 - 2 • aw • sin(awt)) / (2 • p • cos(at) / cos(b)) ... Transverse contact ratio
      eb = b / mn / p • sin(b) ... Transverse overlap ratio
      eg = ea + eb ... Total overlap ratio

    Principle of corrections, use of corrections.

    Approaching and withdrawal of the production tool from the gear center changes the shapes and therefore also properties of the involute toothing. This creates corrected toothing. The illustration shows:

    1. Production tool
    2. Produced gear

    Correction (addendum modification) of toothing enables:

    Example of a tooth profile (z=10, a=20;b=0), where at X=0 the teeth are undercut and the value x=0.7 causes sharpness of teeth.

    Hint: It is recommended to look for more detailed information on possibilities and methods of corrections in specialized literature.

    Recommended values - optimization.

    When determining values of corrections, first it is necessary to fulfill functional requirements for toothing, where the most important items include

    After securing function requirements, it is possible to further optimize corrections in order to improve one or more important toothing parameters. From the frequently used optimizing methods, it is possible to optimize the toothing in order to balance specific slips [5.14 - 5.17] and minimize specific slips [5.18]. For other optimizing processes there is a wide range of recommendations in professional literature, namely the so-called diagrams (charts) of limit corrections, providing a clear view of possibilities and selection of corrections.

    2. Moment, power, forces, efficiency

    Moment, power, forces in gearing

    Torsional moment
    Mk [Nm] = Pw * 9550 / n  .............. (SI units)
    T [lb.in] = PP * 63000 * / n  ........... (Imperial)
    Pw, PP ... power [kW, HP]
    n ........... speed [/min]
     

    Power (Spur gear - Rack)
    Pw = Ft * v / 1000
    Ft ... Tangential force
    v .... Rack speed

    Forces calculation:

    Tangential force
    Ft = Mk * 2000 / dw
    dw ... Operating pitch diameter
    MK ... Torsional moment

    Axial force
    Fa = Ft * tan(b) / cos(awt)
    b ...  Base helix angle
    awt ... Transverse pressure angle at the pitch cylinder

    Radial force
    Fr = Ft * tan(awt)

    Normal force
    Fn = (Ft2 + Fa2 + Fr2)0.5

    Bending moment
    Mo = Fa * (dw / 2000)

    Efficiency and losses.

    The losses in planetary gearing can be divided into losses due to idle run and losses due to loading. The losses due to idle run (lubrication, unloaded mesh, bearings) are difficult to specify analytically and in general, they are significantly lower than losses due to loading. The losses due to loading arise during power transmission and include:

    Losses in gearing

    The loss coefficient can be approximately described according to the formula:
    Spur gearing: zz = 0.5 • f • p • e • (1/z1 +- 1/z2)
    Helical gearing: zz = 0.25 / cos(b) • f • p • e • (1/z1 +- 1/z2)
    with:
    z1, z2 - number of teeth
    f - friction coefficient (0.04 - 0.08)
    e - mesh ratio
    b - helix angle
    Plus sign (+) for external gearing, (-) for internal gearing.

    Losses in bearings

    The loss output can be determined from the relation:
    PVL = w • F • f • r
    with:
    w - angular velocity
    F - resulting bearing loading (carrier, centrifugal force)
    f - friction coefficient (0.001 - 0.005)
    r - mean bearing radius

    Note: The friction coefficient (for both gearing and bearing) is estimated in the calculation based on the selected accuracy level (gearing roughness) and used lubricant.

    For the calculation of loss (efficiency) of the planet gearing, we will use the losses in the helical gearing (carrier stopped) with:

    with:
    zz0/z1 - loss coefficient sun - planet
    zz1/z2 - loss coefficient planet - ring gear

    z0,z2 - number of teeth of the sun and ring gear

    For individual cases of output flow, it applies for the loss calculation:

    z = ir • zr / (ir - 1) Sun => Planet (carrier)
    z = zr Sun => Ring gear
    z = ir • zr / (ir - 1 + zr ) Planet (carrier) => Sun
    z = - zr / (ir - 1) Planet (carrier) => Ring gear
    z = - zr / (ir - 1 - ir • zr) Ring gear => Planet (carrier)
    z = zr Ring gear => Sun

    Note: Detailed and accurate identification of losses in the planetary mechanism is demanding and depends on a detailed knowledge of construction, the used materials and operating conditions. Therefore, it is necessary to consider the values from the calculation as informative.

    3. Planetary Gearing

    Planetary gearings consist of a system of gearwheels and a carrier. The so-called suns are aligned with the carrier and the central axis of the gear mechanism. The planets are the gearwheels mounted in a pivoting way on the carrier and they mesh with the suns or with each other. The planets may have one, two or more gearings. Two or multi-speed planets have more constructional variants with wider possibilities; however, they are more complex and expensive to manufacture.

    You can see an example of a simple planetary gearing with single-speed toothing of the planet below. This basic type of planetary gearing is also handled in complex in this program.

    Simple planetary gearing (differential):

    0 - Sun; 1 - Carrier; 2 - Ring gear; 3 - Planet.

    If all three members of a simple planetary gearing (0, 1, 2) are free, the system is called a differential (2 degrees of freedom), which enables it to compose / decompose two moves into one. That is used e.g. for machine tools (composition) or for a car differential (move decomposition).

    If one of the basic members (0 or 2) is connected to the frame, the system is called planetary gearing (1 degree of freedom), specifically a reductor in case of a drive outwards from the sun or a multiplicator in case of a drive outwards from the carrier. The system with the carrier connected to the frame is called a normal transmission or helical gearing.

     Planetary gearings may be arranged in various ways. The most frequent way includes the serial arrangement, where the total transmission ratio (efficiency) is determined by the product of partial transmission ratios (efficiency). The composed gearings often use the possibility to brake individual members, i.e. gear changing.

    Advantages:

    Disadvantages:

    Use:

    With respect to the above-specified advantages, the use of planetary gearings is popular in a wide range of applications (e.g. motor vehicle transmissions, building machinery, lifting equipment, marine transmissions, turbine reducers, etc.) The combination of planetary gearing with hydraulic or friction gearing is also common.

    Designing and geometrical relations.

    The formulas below use the following indexes.

    For:

    With respect to the possibilities of assembly and functioning of planetary gears, the geometry of the gearwheels cannot be chosen at random. In order to provide proper functioning, the following several conditions must be followed and observed.

    Condition of alignment.

    The planets of the planetary gearings gear with the suns, possibly with other planets. This calculation applies to the joint mesh of a planet with suns (planet, ring gear). As the planet and the ring gear have the same axis, the centre distance between the planet and both suns must be the same.

    It applies for the generally corrected wheels that:
    aw (0,1) = aw (1,2)
    with aw (0,1)=mt • (z0+z1)/2 • COS(alfat)/COS(alfawt(0,1))
    with aw (1,2)=mt • (z1+z2)/2 • COS(alfat)/COS(alfawt(1,2))

    Note: In the program, a violation of this condition is indicated by the cells with calculated centre distance highlighted in red.

    Condition of assembly.

    For simple planets and uniform planet distribution, the following condition must be complied with:

    g = (abs (z0) + abs (z2))/P
    With:
    g - must be a random integer
    P - number of planets
    z - number of teeth

    Note: This condition need not always be achievable (e.g. if a specific transmission ratio is required). This condition may be bypassed by uneven distribution of the planets, which may result in greater demands on production, imbalance of the planet carrier, imbalance of inner forces and increased stress.

    Condition of backlash between neighbouring planets.

    This condition provides minimum backlash between the planets vmin (1-2 mm, 0.05 in).

    Maximum planets P = int(asin((da1+vmin)/(aw • 2)))

    Note: In the program, a violation of this condition is indicated by the cells with number of planets highlighted in red.

    4. Spur gear - Rack

    Geometry

    This is a standard gearing calculation, where the rack is in gear with the pinion. The manufacturing tool profile can be specified both for the pinion and the gear rack.

    It is possible to select the number the pinion teeth, the pressure angle and the tooth slope. Since it has no meaning to correct the gear rack in this case, only the pinion correction is possible (axial distance, gear condition and strength parameter improvement).

    Load selection

    In the calculation it is possible to enter the tangential force, which is the force of the gear rack acting on the pinion, and the speed of the gear rack (pinion circumferential speed). These two values are then used to calculate the power and torque transferred via the pinion. Since it is possible to use the gear rack for a range of various design solution, it is then necessary to calculate (estimate) and transfer the transmission requirements on these two values.

    Strength calculation

    Since there are no standards related to the strength calculation of the pinion in gear with the rack, the ISO6336(ANSI/AGMA2001-D04) standard is used for the strength calculation. The gear rack is replaced by a toothed wheel with a high number of teeth (1 000 teeth).

    Wheel relief coefficient – critical speed

    There is no specific methodology of specifying the critical speed for a gear rack application. As a rough estimate, the calculation of two toothed wheels may be used (the gear rack is substituted by a toothed wheel. For a light rack that is not connected to the structure, use the coefficient sR/h=1; if the rack is connected to the structure, use the value of 20.

    Number of cycles

    To determine the lifetime coefficient (YNT, ZNT), a number of cycles must be known. Enter the number of the pinion and gear rack loading cycles.

    5. Allowable stress and safety for Involute Spur and Helical Gear Teeth  ISO 6336:2006.

    In the following paragraphs, the method of the bearing capacity calculation is described. The ISO6336:2006 standard is used for the calculation. The description includes the key formulas used as well as the notes important to understand the calculation and to operate this application. This text does not replace the full text of the standards used.

    ISO 6336-1:2006 Part 1: Basic principles, introduction and general influence factors

    This part of ISO 6336 presents the basic principles of, an introduction to, and the general influence factors for, the calculation of the load capacity of spur and helical gears.

    Basic relations for the gears load

    Ft = 2000 * T1,2 / d1,2 = 19098 * 1000 * P / (d1,2 * n1,2) = 1000 * P / v
    w1,2 = 2000 * v / d1,2 = n1,2 / 9549

    Ft ... (nominal) transverse tangential load at reference cylinder per mesh
    T1,2 ...  nominal torque at the pinion (wheel)
    d1,2 ... reference diameter of pinion (wheel)
    P ... transmitted power
    n1,2 ... rotation speed of pinion (wheel)
    v ... tangential velocity (without subscript, at the reference circle =tangential velocity at pitch circle)
    w1,2 ... angular velocity of pinion (wheel)

    Application factor KA

    The application factor, KA, is used to modify the value of Ft to take into account loads additional to nominal loads, which are imposed, on the gears from external sources. The empirical guidance values in table B.1 ISO 6336-6 can be used (for industry gears and high speed gears).

    Internal dynamic factor KV

    (The internal dynamic factor KV makes allowance for the effects of gear tooth accuracy grade as related to speed and load.

    There are three calculation methods (B2006), (C2006) a (C1996).)

    The method B is suitable for all the types of spur gears. It is relatively complicated and may give completely unrealistic KV values, if the materials or the degree of accuracy are not properly selected. Therefore it is possible to set the maximum limit for the calculation (pre-set to 5.0). If this limit is exceeded, it is recommended to check the selected material in proportion to the gearing load.

    Internal dynamic factor KV(B)
    N = n1 / nE1

    For N < NS (Subcritical range)
    NS = 0.5 + 0.35 * ( Ft * KA / b )0.5 ...... [ Ft * KA / b < 100 ]
    NS = 0.85 ...... [ Ft * KA / b >= 100 ]
    KV(B) = ( N * K ) + 1
    K = ( CV1 * BP ) + ( CV2 * Bf ) + ( CV3 * BK )
    BP = c' * fpb eff / ( Ft * KA / b )
    Bf  = c' * fta eff / ( Ft * KA / b )
    BK = abs (1 + c' * Ca / ( Ft * KA / b ))

    For Ns < N < 1.15 (Main resonance range)
    KV(B) = ( CV1 * BP ) + ( CV2 * Bf ) + ( CV4 * BK ) + 1

    For N >= 1.5 (Supercritical range)
    KV(B) = ( CV5 * BP ) + ( CV6 * Bf ) + CV7

    For 1.15 < N < 1.5 (Intermediate range)
    KV(B) = KV(N=1.5) + ( KV(N=1.15) - KV(N=1.5)) / 0.35 * (1.5 - N)

    Coeficients

     

    1.0 < eg <=2.0 eg > 2.0
    CV1 0.32 0.32
    CV2 0.34 0.57 / (eg - 0.3)
    CV3 0.23 0.096 / (eg - 1.56)
    CV4 0.90 (0.57 - 0.05 * eg ) / (eg - 1.44)
    CV5 0.47 0.47
    CV6 0.47 0.12 / (eg - 1.74)
    CV7 1.0 < eg <=1.5 0.75
    CV7 1.5 < eg <=2.5 0.125 * sin(p * (eg - 1.74)) + 0.875
    CV7 eg > 2.5 1.0
    Cay1,2 1 / 18 * (sHlim1,2 / 97 - 18.45)2 + 1.5
    Cay 0.5 * (Cay1 + Cay2)
    Ca Ca = Cay

     

    Internal dynamic factor KV(C)

    Method C supplies average values which can be used for industrial transmissions and gear systems with similar requirements in the following fields of application:

    Method C can also generally be used, with restrictions for the following fields of application:

    Method (C2006) is different from (C1996) by adding the coefficient K3.
    Example for input values KA * Ft / b = 100; v = 3 m / s; Q = 7 and straight teeth.

    KV(C..1996)
    KVa,b = 1 + (K1 / ( Ft * KA / b ) + K2) * v * z1 / 100 * (u2 / (1 + u2))0.5 ... [ eb = 0; eb >= 1.0]

    KV(C..2006)
    KVa,b = 1 + (K1 / ( Ft * KA / b ) + K2) * v * z1 / 100 * K3 * (u2 / (1 + u2))0.5 ... [ eb = 0; eb >= 1.0]

    KV = KVa - ea* ( KVa - KVb ) ... [0 < eb < 1.0]

    Coeficients

      K1 (Accuracy grades as speclfled in ISO1328-1) K2
      3 4 5 6 7 8 9 10 11 12 All
    Spur gears eb = 0 2.1 3.9 7.5 14.9 26.8 39.1 52.8 76.6 102.6 146.3 0.0193
    Helical gears eb >=1.0 1.9 3.5 6.7 13.3 23.9 34.8 47.0 68.2 91.4 130.3 0.0087
    SRC = v * z1 / 100 * (u2 / (1 + u2))0.5
    K3 = 2.0 ...... [SRC <= 0.2]
    K3 = -0.357 * SRC + 2.071 ...... [SRC > 0.2]

     

    Main resonance of a gear pair NE1

    nE1 = 30000 / ( p * z1 ) * ( cga  / mred )0.5

    where:

    mred = m*1 * m*2 / ( m*1 + m*2 )
    m*1,2 = J*1,2 / (rb 1,2)2   [kg/mm]
    J*1,2 = J1,2 / b1,2
    cga = c' * (0.75 * ea + 0.25)
    c' = c'th * CM * CR * CB * CE * CFK * cos(
    b)
    c'th = 1 / (0.04723 + 0.15551/zn1 + 0.25791/zn2 - 0.00635*x1 - 0.11654*x1/zn1 - 0.00193*x2 - 0.24188*x2/zn2 + 0.00529*x12 + 0.00182*x22)
    c'th = 1 / (0.04723 + 0.15551/zn1  - 0.00635*x1 - 0.11654*x1/zn1 - 0.00193*x2 + 0.00529*x12 + 0.00182*x22) ... for internal gearing
    CM = 0.8
    CR = 1 + ln(bs / b) / (5 * e(sR/(5 * mn))) ...... [0.2 < bs < 1.2 ]
    CB = 0.5 * (CB1 + CB2); CB1,2 = (1 + 0.5 * (1.2 - hf1,2 / mn)) * (1 - 0.02*(20 - aPn))
    CE = (( 2 * E1 * E2 ) / ( E1 + E2 )) / 206
    CFK = (( Ft * KA / b ) / 100 )0.25...... [ CFK<= 1.0 ]

    zn1,2 = z1,2 / cos(b)3

    Main resonance of gear with idler gears, inner gears and planet gears are calculated by different process. Details in ISO6336-1.

    Coefficient KHb (KFb)

    This coefficient takes into account the effect of the non-uniform distribution of load over the gear face. Uneven load distribution is caused by an elastic deformation of gears and housing, manufacturing deviations and thermal distortion. Methods, principles and assumptions are given in standard ISO6336-1. Because the determination of the coefficient is dependent on a number of factors and primarily on the specific dimensions and design of the gearbox, is for the design purposes selected the coeficient KHb from graphs based on practical experiences. The calculation is in paragraph [18].

    Determination KHb (method C)

    Detail description is in ISO6336-1. Here is just a selection of formulas, information and comments that are related to the calculation KHb.

    a) KHb = (2 * Fby * cgb / (Fm / b))0.5 ...... [ Fby * cgb / (2 * Fm / b) >= 1.0; KHb >= 2.0 ]
    b) KHb = 1 + Fby * cgb / (2 * Fm / b) ...... [ Fby * cgb / (2 * Fm / b)  < 1.0; KHb > 1.0 ]

    where:

    Fm = Ft * KA * KV
    Fby =  Fbx *  yb
    cgb = 0.85 * cga
    b ...... gear width
    yb ... Running-in allowance from graph

    where:

    1) Fbx = Choosing your own values
    2) Fbx = 1.33 * B1 * fsh + B2 * fma ....... [ Fbx >=  Fbxmin ]
    3) Fbx = abs( 1.33 * B1 * fsh - fHb6) ...... [ Fbx >=  Fbxmin ]
    4) Fbx = 1.33 * B1 * fsh + fsh2 + fma + fca +fbe

    where:

    B1, B2 coeficients, table 8, ISO6336-1
    fHb6 ... Helix slope deviation for Q=6, ISO1328-1
    fsh ... Component of equivalent misatignment. It is possible to use several methods (calculation, measurement, estimation). Used formula:
    fsh = Fm / b * 0.023 * (abs(B' + K' * l * s / d12 * (d1 / dsh)4 - 0.3) + 0.3) * (b / d1)2 ... [s / l < 0.3]
    fsh = Fm / b * 0.046 * (abs(B' + K' * l * s / d12 * (d1 / dsh)4 - 0.3) + 0.3) * (bB / d1)2 ... [s / l < 0.3]
    fsh2, fca, fbe ... can be determined by ISO6336-1
    B' = 1.0 ... for both spur and single helical gears, for the total transmitted power
    K' = arrangement coefficient, gray indicates the less deformed helix of a double helical gear

    K' with stiffening without stiffening
    A 0.48 0.80
    B -0.48 -0.80
    C 1.33 1.33
    D -0.36 -0.60
    E -0.60 -1.00

     

    l, s .... picture
    dsh ... shaft diameter
    fma ... mesh misalignment. It is possible to use several methods (calculation, measurement, estimation). Used formula:
    fma = (fHb12 + fHb22)0.5 .

    Determination KHb (Simplified formula)

    a) KHb = Acoef * (2 * Fby * cgb / (Fm / b))0.5 ...... [ Fby * cgb / (2 * Fm / b) >= 1.0; KHb > 1.0 ]
    b) KHb = Acoef * (1 + Fby * cgb / (2 * Fm / b)) ...... [ Fby * cgb / (2 * Fm / b)  < 1.0; KHb > 1.0 ]

    where:

    Fm = Ft * KA * KV
    Fby =  Fb * 0.8 ..... [Fb from ISO 1328]
    cgb = 0.85 * cga
    b ...... gear width

    Acoef = 1.0 ..... Double-sided symmetrically supported gearing
    Acoef = (0.9 + 0.15 * (b1 / d1)2 + 0.23 * (b1 / d1)3) ..... Double-sided non-symmetrically supported gearing
    Acoef = (0.9 + (b1 / d1)2) ..... Overhung gearing

    Determination KHb (Approximation from the table)

    For the preliminary design is possible to use values ​​from these graphs.
    X Axis: Ratio gear width to gear diameter
    Y Axis: Factor KHb ..... [min. value = 1.05]
    Accuracy grade 7


    Not-hardended gears, VHV<370, design type A-F ... calculation paragraph [2.0]


    Hardended gears, VHV<=370, design type A-F ... calculation paragraph [2.0]

    Coefficient KFb

    KFb = ( KHb )NF
    NF = (b / h)2 / (1 + b / h + (b / h)2) ...... [když b / h < 3; pak b / h = 3] ([if b / h < 3; then b / h = 3])

    The smaller of the values b1/h1, b2/h2 is to be used as b/h.

    Coefficient KHa (KFa)

    KHa = KFa = eg / 2 * (0.9 + 0.4 * (cga * (fpb - ya)) / (FtH / b)) ...... [eg <= 2.0]
    KHa = KFa = 0.9 + 0.4 * (2.0 * (eg - 1.0) / eg)0.5 * cga * (fpb - ya)  / (FtH / b) ...... [eg > 2.0]

    Pro: (For:)
    KHa > eg / ( ea * Ze2) ...... KHa = eg / ( ea * Ze2)
    KHa < 1.0 ...... KHa = 1.0

    Pro: (For:)
    KFa > eg / (0.25 * ea + 0.75) ...... KFa = eg / (0.25 * ea + 0.75)
    KFa < 1.0 ...... KFa = 1.0

    fpb = fpt (ISO1328-1)

    ya ... Material: St, St(cast), V, V(cast), GGG(perl.), GGG(bai.), GTS(perl.)
    ya = fpb * 160 /
    σHlim [ v < 5m/s ]
    ya <= 12800 / σHlim [ 5m/s < v <= 10m/s ]
    ya <= 6400 / σHlim [ v > 10m/s ]

    ya ... Material: GG, GGG(ferr.)
    ya = fpb
    0.275 [ v < 5m/s ]
    ya <= 22 [ 5m/s < v <= 10m/s ]
    ya <= 11 [ v > 10m/s ]

    ya ... Material: Eh, IF, NT(nitr.), NV(nitr.), NV(nitrocar.)
    ya = fpb
    0.075 [ ya <= 3 ]
     

    ISO 6336-2:2006 Part 2: Calculation of surface durability (pitting)

    This part of ISO 6336 specifies the fundamental formulae for use in the determination of the surface load capacity of cylindrical gears with involute external or internal teeth. It includes formulae for all influences on surface durability for which quantitative assessments can be made. lt applies primarily to oil-lubricated transmission, but can also be used to obtain approximate values for (slow-running) grease-lubricated transmissions, as long as sufficient lubricant is present in the mesh at all times.

    Safety factor for surface durability (against pitting), SH

    Calculate SH separately for pinion and wheel:

    SH1,2 =  σHG1,2 /  σH1,2 > SHmin

    Contact stress σH

    σH1 = ZB * σH0 * (KA * KV * KHb * KHa)0.5
    σH2 = ZD * σH0 * (KA * KV * KHb * KHa)0.5

    The nominal contact stress at the pitch point σH0
    σH0 = ZH * ZE * Ze * Zb * (Ft / (b * d1) * (u + 1) / u)0.5

    Permissible contact stress σHP: Method B

    σHP = ZL * ZV * ZR * ZW * ZX * ZNT * σHlim / SHmin = σHG / SHmin

    The pitting stress limit σHG
    σHG = σHP * SHmin

    Zone factor ZH

    ZH = (2 * cos(bb) * cos(awt) / (cos(at)2 * sin(awt)))0.5

    Single pair tooth contact factors ZB and ZD

    M1 = tan(awt) / ((((da1 / db1)2 - 1.0)0.5 - 2 * p / z1) * (((da2 / db2)2 - 1.0)0.5 - (ea - 1.0) * 2 *  p / z2))0.5
    M2 = tan(awt) / ((((da2 / db2)2 - 1.0)0.5 - 2 * p / z2) * (((da1 / db1)2 - 1.0)0.5 - (ea - 1.0) * 2 *  p / z1))0.5

    Spur gears with, ea > 1.0
    ZB = 1.0 ... [ M1<= 1.0 ]
    ZB = M1 .... [ M1 > 1.0 ]
    ZD = 1.0 ... [ M2<= 1.0 ]
    ZD = M2 .... [ M2 > 1.0 ]

    Helical gears with, eb >= 1.0
    ZB = ZD = 1.0

    Helical gears with, eb < 1.0
    ZB = M1 - eb * (M1 - 1.0) ... [ ZB >= 0 ]
    ZD = M2 - eb * (M2 - 1.0) ... [ ZD >= 0 ]

    (For internal gears, ZD shall be taken as equal to 1.0)

    Elasticity factor ZE

    ZE = (p * ((1.0 - n12) / E1 + (1 - n22) / E2))-0.5

    where

    n1,2 ... Poisson's ratio
    E1,2 ... modulus of elasticity

    Contact ratio factor Ze

    Ze = ((4.0 - ea) / 3 * (1.0 - eb) + eb / ea)0.5  ... [ 0 <= eb < 1.0 ]
    Ze = (1.0 / ea)0.5  ... [ eb >= 1.0 ]

    Helix angle factor, Zb

    Zb = 1 / (cos(b))0.5

    Life factor ZNT

    X axis ... number of cycles
    Y axis ... ZNT

    Lubricant factor ZL

    ZL = CZL + 4 * (1.0 - CZL) / (1.2 + 80 / n50)2 = CZL + 4 * (1.0 - CZL) / (1.2 + 134 / n40)2
    CZL = 0.83 ... [ σHlim < 850 ]
    CZL = σHlim / 4375 + 0.6357 ... [ 850 <= σHlim <= 1200 ]
    CZL = 0.91 ... [ 1200 < σHlim ]
    n50 ( n40) ... Nominal viscosity in 50°C (40°C) [mm2/s]


    Diagram viscosity / temperature for viscosity index VI = 50

    Velocity factor ZV

    ZV = CZV + 2 * (1.0 - CZV) / (0.8 + 32 / v)0.5
    CZV = CZL + 0.02

    Roughness factor, ZR

    ZR = (3 / Rz10)CZR
    CZR = 0.15 ... [ σHlim < 850 ]
    CZR = 0.32 - 0.0002 * σHlim ... [ 850 <= σHlim <= 1200 ]
    CZR = 0.08 ... [ 1200 < σHlim ]
    Rz10 = Rz * (10 / rred)(1/3)
    rred = (r1 * r2) / (r1 + r2)
    r1,2 = 0.5 * db1,2 * tan(awt)

    Work hardening factor, ZW

    The work hardening factor, ZW takes account of the increase in the surface durability due to meshing a steel wheel (structural steel, through-hardened steel) with a hardened or substantially harder pinion with smooth tooth flanks.

    Pinion Surface-hardened, Gear through-hardened
    ZW = 1.2 * (3 / RzH)0.15 ... [ HB < 130 ]
    ZW = (1.2 - (HB - 130) / 1700) * (3 / RzH)0.15 ... [ 130 <= HB <= 470 ]
    ZW = (3 / RzH)0.15 ... [ HB > 470 ]

    ZW for static stress
    ZW = 1.05 ... [ HB < 130 ]
    ZW = 1.05 - (HB - 130) / 680 ... [ 130 <= HB <= 470 ]
    ZW = 1.0 ... [ HB > 470 ]
    RzH = Rz1 * (10 / rred)0.33 * (Rz1 / Rz2)0.66) / ( n40 * v / 1500)0.33) ... [ 3 <= RzH <=16 ]

    Through-hardened pinion and gear
    ZW = 1.0 ... [ HB1/HB2 < 1.2 ]
    ZW = 1.0 + A * (u - 1.0) ... [ 1.2 <= HB1/HB2 <= 1.7 ]
    ZW = 1.0 + 0.00698 * (u - 1.0) ... [ 1.7 < HB1/HB2 ]
    A = 0.00898 * HB1/HB2  - 0.00829

    ZW for static stress
    ZW = 1.0

    ISO 6336-3:2006 Part 3: Calculation of tooth bending strength.

    This part of ISO 6336 specifies the fundamental formulae for use in tooth bending stress calculations for involute external or internal spur and helical gears with a rim thickness sR > 0.5 * ht for external gears and sR >1.75 * mn for internal gears.

    Safety factor for bending strength (safety against tooth breakage), SF

    Calculate SF separately for pinion and wheel:

    SF1,2 =  σFG1,2 /  σF1,2 >= SFmin

    Tooth root stress σF

    σF = σF0 * KA * KV * KFb * KFa

    The nominal tooth root stress σF0
    σF0 = Ft / (b * mn) * YF * YS * Yb * YB * YDT

    Permissible bending stress σFP : Method B

    σFP = σFlim * YST * YNT * YdrelT * YRrelT * YX / SFmin = σFG / SFmin

    Tooth root stress limit σFG
    σFG = σFP * SFmin

    The form factor, YF : Method B

    YF = (6 * hFe / mn * cos(aFen)) / ((sFn / mn)2 * cos(an))

    Tooth root normal chord sFn ; radius of root fillet rF ; bending moment arm hFe

     

    Dimensions and basic rack profile of the teeth (finished profile)
    A...without undercut
    B...with undercut

    auxiliary values
    E = p / 4 * mn - hfP * tan(an) + spr / cos(an) - (1 - sin(an) * rfP / cos(an)
    spr = pr - q
    spr = 0 when gears are not undercut
    rfPv = rfP ... external gears
    rfPv = rfP + mn * (x0 + hfp/mn - rfP/mn)1.95 / (3.156 * 1.036z0) ... internal gears
    x0 ... the pinion-cutter shift coefficient
    z0 ... the number of teeth of the pinion cutter
    G = rfPv / mn - hfP / mn + x
    H = 2 / zn * (p / 2 - E / mn) - T
    T = p / 3 ... external gears
    T = p / 6 ... internal gears
    q = 2 * G / zn * tan(q) - H

    Determination of normal chordal dimensions of tooth root critical section for Method B
    A...external gears
    B...internal gears

    Tooth root normal chord sFn
    sFn / mn = zn * sin(p/3 - q) + (3)0.5 * (G / cos(q) - rfPv / mn) ... external gears
    sFn / mn = zn * sin(p/6 - q) + (G / cos(q) - rfPv / mn) ... internal gears

    Radius of root fillet rF
    rF  / mn = rfPv / mn + 2 * G2 / (cos(q) * (zn * cos(q)2 - 2 * G))

    Bending moment arm hFe
    hFe / mn = 0.5 * ((cos(ge) - sin(ge) * tan(aFen)) * den / mn - zn * cos(p/3 - q) - G / cos(q) + rfPv / mn)) ... external gears

    hFe / mn = 0.5 * ((cos(ge) - sin(ge) * tan(aFen)) * den / mn - zn * cos(p/6 - q) - (3)0.5 * (G / cos(q) - rfPv / mn))) ... internal gears

    Parameters of virtual gears

    bb = arcsin(sin(b) * cos(an))
    zn = z / (cos(bb))3
    ean= ea / (cos(bb))2
    dn = mn * zn
    pbn = p * mn * cos(an)
    dbn = dn * cos(an)
    dan = dn + da - d
    den = 2 * z / abs(z) * ((((dan / 2)2 - (dbn / 2)2)0.5 - p * d * cos(b) * cos(an) * (ean - 1) / abs(z))2 + (dbn / 2)2)0.5

    *The number of teeth z is positive for external gears and negative for internal gears

    aen = arccos(dbn / den)
    ge = (0.5 * p + tan(an) * x) / zn + inv(an) - inv(aen)
    aFen = aen - ge

    Stress correction factor YS

    The stress correction factor YS is used to convert the nominal tooth root stress to local tooth root stress.
    YS = (1.2 + 0.13 * L) * qs(1 / (1.21 + 2.3 / L))
    L = SFn / hFe
    qs = SFn / (2 * rF)

    Stress correction factor for gears with notches in fillets YSg

    YSg = 1.3 * YS / (1.3 - 0.6 * (tg / rg)0.5) ... [ (tg / rg)0.5 < 2.0 ]

    Helix angle factor Yb

    Yb = 1 - eb * b / 120 ... [if b > 30; b = 30]

    Rim thickness factor YB

    external gears
    YB = 1.0 ... [sR / ht >= 1.2]
    YB = 1.15 * ln(8.324 * mn / sR) ... [0.5 < sR / ht < 1.2]

    internal gears
    YB = 1.0 ... [sR / mn >= 3.5]
    YB = 1.6 * ln(2.242 * ht / sR) ... [1.75 < sR / mn < 3.5]

    Deep tooth factor YDT

    YDT = 1.0 ... [ean <= 2.05] or [accuracy grade > 4]
    YDT = -0.666 * ean + 2.366 ... [2.05 < ean <= 2.5] and [accuracy grade <= 4]
    YDT = 0.7 ... [ean > 2.5] and [accuracy grade <= 4]

    Life factor YNT

    X axis ... number of cycles
    Y axis ... YNT

    Relative notch sensitivity factor YdrelT for reference stress

    YdrelT = Yd / YdT = (1 + (r' * c*)0.5) / (1 + (r' * cT*)0.5)
    c* = cP* * (1 + 2 * qs)
    cP* = 1 / 5 = 0.2
    cT* = cP* * (1 + 2 * 2.5)

    Material: GG [σB=150MPa], GG, GGG(ferr.)[σB=300MPa]
    r' = 0.31

    Material: NT, NV
    r' = 0.1005

    Material: St, V, GTS, GGG(perl.), GGG(bai.)
    r' = MAX(MIN(13100 / Rp0.2(2.1) - (MAX(600;Rp0.2)-600)(0.35) / 1600;0.32);0.0014)

    Material: Eh, IF(root)
    r' = 0.003

    Relative notch sensitivity factor YdrelT for static stress

    Material: St, V, GGG(perl.), GGG(bai.)
    YdrelT = (1 + 0.82 * (YS - 1) * (300 / σ0.2)(1/4)) / (1 + 0.82 * (300 / σ0.2)(1/4))

    Material: Eh, IF, IF(root)
    YdrelT = 0.44 * YS + 0.12

    Material: NT, NV
    YdrelT = 0.20 * YS + 0.60

    Material: GTS
    YdrelT = 0.075 * YS + 0.85

    Material: GG, GGG(ferr.)
    YdrelT = 1.0

    Relative surface factor YRrelT for reference stress

    Rz < 1 mm

    Material: V, GGG(perl.), GGG(bai.), Eh, IF
    YRrelT = 1.12

    Material: St
    YRrelT = 1.07

    Material: GG, GGG(ferr.), NT, NV
    YRrelT = 1.025

    1mm < Rz < 40 mm

    Material: V, GGG(perl.), GGG(bai.), Eh, IF
    YRrelT = 1.674 - 0.529 * (Rz + 1)0.1

    Material: St
    YRrelT = 5.306 - 4.203 * (Rz + 1)0.01

    Material: GG, GGG(ferr.), NT, NV
    YRrelT = 4.299 - 3.256 * (Rz + 1)0.0058

    Relative surface factor YRrelT for static stress

    YRrelT = 1.0

    Size factor YX

    YX = 1.0 ... All materials for static stress

    YX ... Material: St, St(cast), V, V(cast), GGG(perl.), GGG(bai.), GTS(perl.)
    YX = 1.0 ... [ mn <= 5 ]
    YX = 1.03 - 0.006 * mn ... [ 5 < mn < 30 ]
    YX = 0.85 ... [ mn >= 30 ]

    YX ... Material: Eh, IF(root), NT, NV, NT(nitr.), NV(nitr.), NV(nitrocar.)
    YX = 1.0 ... [ mn <= 5 ]
    YX = 1.05 - 0.01 * mn ... [ 5 < mn < 25 ]
    YX = 0.80 ... [ mn >= 25 ]

    YX ... Material: GG, GGG(ferr.)
    YX = 1.0 ... [ mn <= 5 ]
    YX = 1.075 - 0.015 * mn ... [ 5 < mn < 25 ]
    YX = 0.70 ... [ mn >= 25 ]

    ISO 6336-5 Strength and quality of materials.

    This part of ISO 6336 describes contact and tooth-root stresses, and gives numerical values for both limit stress numbers. It specifies requirements for material quality and heat treatment and comments on their influences on both limit stress numbers.

    The allowable stress numbers, σH lim, and the nominal stress numbers, σF lim, can be calculated by the following equation:

    a) σHlim = A * x + B
    b) σFlim = A * x + B

    where x is the surface hardness HBW or HV and A, B are constants

    Requirements for material quality and heat treatment

    The three material quality grades ML, MQ and ME, stand in relationship to
    - ML stands for modest demands on the material quality and on the material heat treatment process during gear manufacture.
    - MQ stands for requirements that can be met by experienced manufacturers at moderate cost.
    - ME represents requirements that must be realized when a high degree of operating reliability is required

    In this calculation, except σHlim and σFlim, are proposed other material parameters that are necessary for calculating the gearing. The values of Ro, E and Poisson constant are commonly available. For the proposal of the tensile strength Rm and yield strength Rp0.2 was used information from the ISO 1265 and specialized literature. Parameters for the time-strength curves were obtained from ISO6336-2 and 3. These curves can be seen in a small graph in the calculation.
    All calculated values are design and based on empirical experience. The exact values for a concrete material you can obtain from your manufacturer or from material tests.

    Hardness notice

    Values HB for HB<=450 steel ball, HB>450 carbide ball
    Values HB used recalculation HB=HV-HV/20
    Values HRC used recalculation  HRC=(100*HV-14500)/(HV+223)

    6. Allowable stress and safety for Involute Spur and Helical Gear Teeth ANSI/AGMA 2001-D04

    Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth ANSI/AGMA 2001-D04

    dynamic factor, Kv

    Kv =  (C / (C + vt))−B
    C = 50 + 56 * (1.0 − B) ... [ 6 ≤ Av ≤ 12 ]
    B = 0.25 * (Av − 5.0)0.667

    vt max = [C + (14 − Av)]2

    Overload factor, Ko

    The empirical guidance values from table B.1 ISO 6336-6 are used.

    Elastic coefficient, Cp

    Cp = (1 / p * (((1 - mP2) / EP) + ((1 - mG2) / EG)))0.5  ... [lb/in2]0.5
    mP and mG is Poisson’s ratio for pinion and gear, respectively; EP and EG is modulus of elasticity for pinion and gear [lb/in2].

    Surface condition factor, Cf

    Cf = 1.0

    Hardness ratio factor, CH

    Through hardened gears
    CH = 1.0 + A * (mG - 1.0)
    A = 0.00898 *(HBP / HBG) - 0.00829
    HBP is pinion Brinell hardness number [HB]; HBG is gear Brinell hardness number,[HB].
    This equation is valid for the range 1.2 ≤ HBP / HBG ≤ 1.7 For HBP / HBG < 1.2, A = 0.0 HBP / HBG > 1.7, A = 0.00698

    Surface hardened/through hardened values
    CH = 1.0 + B * (450 - HBG)
    B = 0.00075 * (2.71828)-0.0112 * (fp)
    fp is surface finish of pinion, microinches, Ra
    if fp>64 ... CH = 1.0

    Load distribution factor, Km

    Km = f (Cmf, Cmt)
    Km = Cmf

    Face load distribution factor, Cmf

    Cmf = 1.0 + Cmc * (Cpf * Cpm + Cma * Ce)
    Cmc is 1.0 for gear with unmodified leads; Cmc is 0.8 for gear with leads properly modified by crowning or lead correction.
    Cpf = F / (10 * d) − 0.025 ... [F<=1.0]
    Cpf = F / (10 * d) − 0.0375 + 0.0125 * F ... [1.0<F<=17.0]
    Cpf = F / (10 * d) − 0.1109 + 0.0207 * F − 0.000228 * F2 ... [17.0<F<=40.0]
    Cpm = 1.0 ... [S1 / S < 0.175]
    Cpm = 1.1 ... [S1 / S >= 0.175]
    Cma = A + B * F + C * F2

      A B C
    1…Open gearing 0.247 0.0167 -0.0000765
    2…Commercial enclosed gearboxes 0.127 0.0158 -0.0001093
    3…Precision enclosed gearbox 0.0675 0.0128 -0.0000926
    4…Extra precision enclosed gearbox 0.038 0.0102 -0.0000822

    Ce = 0.8 ... [gearing is adjusted at assembly; gearing is improved by lapping]
    Ce = 1.0 ... [for all other conditions]

    Reliability factor, KR

    KR = 1.50 [Fewer than one failure in 10 000]
    KR = 1.25 [
    Fewer than one failure in 1000]
    KR = 1.00 [
    Fewer than one failure in 100]
    KR = 0.85 [
    Fewer than one failure in 10]
    KR = 0.70 [
    Fewer than one failure in 2]

    Rim thickness factor, KB

    KB = 1.6 * ln(2.242 / mB) ... [for mB<1.2]
    KB = 1.0 ... [for mB>=1.2]

    mB = tR / ht
    tR is gear rimthickness below the tooth root [in]; ht is gear tooth whole depth [in]

    Pitting resistance

    The contact stress number formula for gear teeth is:

    sc = Cp (Wt * Ko * Kv * Ks * Km * Cf / (d * F * I))0.5

    Allowable contact stress number
    The relation of calculated contact stress number to allowable contact stress number is:

    sc ≤ (sac * ZN * CH) / (KT * SH * KR)

    Pitting resistance power rating
    The pitting resistance power rating is:

    Pac = (p * np * F / 396 000) * I / (Ko * Kv * Ks * Km * Cf) * ((d * sac * ZN CH) / (Cp * SH * KT * KR))2

    Safety coefficient for surface durability

    SH = sac / sc * (ZN * CH) / (KT * KR)

    Bending strength

    The fundamental formula for bending stress number in a gear tooth is:

    st = Wt * Ko * Kv * Ks * (Pd * Km * KB / (F * J))

    Allowable bending stress number
    The relation of calculated bending stress number to allowable bending stress number is:

    st ≤ (sat * YN) / (SF * KT * KR)

    Bending strength power rating
    The bending strength power rating is:

    Pat = (p * np * F / 396 000) * (F * J) / (Ko * Kv * Pd * Ks * Km * KB) * (sat * YN) / (SF * KT * KR)

    Safety coefficient for bending strength

    SF = sat / st * YN / (KT * KR)

    Transmitted tangential load
    Wt = 33000 * P / vt = 2 * T / d = 396000 * P / (p * np * d)

    P is transmitted power [hp]; T is transmitted pinion torque [lb*in]; vt is pitch line velocity at operating pitch diameter, [ft/min]
    vt = p * np * d / 12

    Allowable stress numbers, sac and sat ANSI / AGMA 2001-D04

    This part of ANSI / AGMA 2001-D04 describes the allowable stress numbers sac and sat, for pitting resistance and bending strength.

    Allowable stress numbers in this standard are determined or estimated from laboratory tests and accumulated field experiences. They
    are based on unity overload factor, 10 million stress cycles, unidirectional loading and 99 percent reliability. For service life other than 10
    million cycles, the allowable stress numbers are adjusted by the use of stress cycle factors YN and ZN

    The allowable stress numbers sac and sat can be calculated by the following equation:

    a) sac = A * x + B
    b) sat = A * x + B

    where x is the surface hardness HBW and A, B are constants

    Requirements for material quality and heat treatment.

    These requirements are specified in this standard and are divided in three material quality grades 1,2 an 3.

    In this calculation, except sac and sat, are proposed other material parameters that are necessary for calculating the gearing. The values of p, E and Poisson constant are commonly available. For the proposal of the tensile strength Rm and yield strength Rp0.2 was used information from the ISO 1265 and specialized literature. All calculated values are design and based on empirical experience. The exact values for a concrete material you can obtain from your manufacturer or from material tests.

    Hardness notice

    Values HB for HB<=450 steel ball, HB>450 carbide ball
    Values HB used recalculation HB=HV-HV/20
    Values HRC used recalculation  HRC=(100*HV-14500)/(HV+223)