By Connie Reynolds,2014-05-06 12:00
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    Shigley: Chapter 13


I. Background

    a. Prime movers

    ? Power: P = T? = CTn

     Where P = Power; T = Torque: ? = angular velocity

     n rpm; C Conversion Constant

     For Horsepower

     ?Tn P?W?5252

    ? where T - ft-lb; n rpm; W horsepower

     For kilowatts

    ?TTns sP?? 95491000

     Where T - N-m; P kilowatts; - rad/sec s

    b. Drive Train

    ? Typical prime mover speeds

    a. I C engine - 6000rpm

    b. Electric motor - 1800 to 3600 rpm

    c. Gas Turbine - 12,000 to 15,000 rpm

    ? Typically prime mover output speed is much higher than

    application requirements

    ? Usually the application torque requirement is higher than the

    prime mover output

    ? A drive train or transmission is required to make the prime mover

    torque/speed output match the application’s need

    ? Prime movers are typically high speed low torque


    a. Types of gears

    ? Spur gears Gear teeth are parallel to the axis of rotation. For

    power transfer between parallel shafts.

    Rev 3/30/07 1

? Helical gears Gear teeth are inclined to the axis of rotation.

    Allows more continuous tooth engagement with less noise (vs. spur

    gears). Helical gears will have thrust loads. Allows power transfer

    between parallel and non-parallel shafts. See fig. 13-2

    ? Bevel gears -- Gear teeth are formed on a conical surface and are

    used mainly to transmit power with intersecting shafts. See figure


    ? Worm gears -- A worm gear set consists of a worm (resembling a

    screw thread) and the gear (a specialized helical gear) usually on

    shafts intersecting at 90 degrees. A worm gear gives very high gear

    reduction ratios. See figure 13-4.

    b. Nomenclature -- See figure 13- 5

    ? Pitch circle a theoretical circle on which all calculations are

    based. See figure 13-5.

    ? Pitch diameter, d - diameter of pitch circle (in.) [for SI mm]

    ? Pinion smaller of two mating gears; the larger is called the gear ? Circular pitch, p = sum of tooth thickness and width of space

    ? Diametral pitch, P - ratio of the number of teeth to the pitch


    ? Addendum, a radial distance between the pitch circle and the top

    of the gear tooth.

    ? Dedendum, b -- radial distance between the bottom land and the

    pitch circle.

    ? Backlash -- the amount by which the width of a tooth space

    exceeds the thickness of the engaging tooth measured on the pitch


    ? Basic Gear Geometry Equations:

    a. P = N/d Eqn. 13-1

    m?d/N modulus for S I units Eqn. 13-2 b.

    c. p = d/N = m Eqn. 13-3

    d. pP = Eqn. 13-4

    Rev 3/30/07 2

    e. Where N is the number of teeth; m is module (mm);

    f. d pitch diameter (in) or for SI (mm); p circular pitch (in)

    g. P diametral pitch, (teeth/in)

c. Conjugate Gear Action

    ? On mating gears, when the tooth profiles are designed so as to

    produce a constant angular velocity ratio, these gears are said to

    have conjugate action

    ? The standard tooth profile that provides conjugate gear action is

    the involute profile (There are others, but not used often)

    d. Involute Properties

    ? Generating a involute tooth profile see figures 13-7, 13- 8

    ? Base circle - circle on which the involute is generated

    e. Fundamentals [Spur Gears]

    ?rd122V?r? Pitch line velocity or ?? ??r?1122?rd211

    Where r’s are the gear pitch circle radii

     d’s are the gear pitch circle diameters

     ’s are the gear angular velocities

    ? See figure 13-9and figure 13-10

    ? Gear pitch circles are tangent at the pitch point

    ? Pressure line (line of action) the direction in which the resultant

    force acts between the gear

    ? Base circles are tangent to the pressure line

    ? Addendum height is 1/P (P is the diametral pitch)

    ? Dedendum height is 1.25/P to 1.35/P (P is the diametral pitch)

    ? E Base pitch is related to circular pitch

    ?pcos? bcp

    Rev 3/30/07 3



    Where is the pressure angle the angle between the pressure

    line and a line joining the centers of the two gears

    ? See example 13 - 1

    f. Contact Ratio

    ? Tooth contact begins and ends at the two addendum circles

    qL? Generally want more than one tooth in contact at the same time tab Eqn. 13- 8&9 ??mcppcos?? Contact ratio

    Where m is the contact ratio; q is the sum of the arc of ct

    approach and arc of recess; L is the length of the line of ab

    action (see fig. 13-15 for visual of these terms) g. Interference

    ? See figure 13-16 for interference action in gear teeth

    ? Sometimes undercutting is used to eliminate interference; this can

    weaken the tooth

    ? Equation for the minimum number of teeth on a pinion and gear

    without interference, N


    2k22N?m?m???1?2msin? Eqn. (13-11) P2??1?2msin???

    k = 1 for full depth threads; 0.8 for stub teeth

     = pressure angle

    m = N /N the ratio of gear teeth to pinion teeth Gp

h. The Forming of Gear Teeth

    ? See figures 13-17; 13-18; and 13-19

    a. Milling ??

    b. Shaping

    c. Hobbing

    ? Finishing can be important to gears operating at high speed

    Rev 3/30/07 4


    i. Straight Bevel Gears

    NNPG? See figure 13-20 for geometry tan?? Or Eqn. 13-14 ??NNGP? Pitch angle - see figure 13-20

    ? Tredgold’s approximation tan

    ? Figure 13-20 shows that he shape of the teeth, when projected on

    the back cone, is the same as a spur gear having a radius equal

    to the back-cone distance r b

    N’?2?r/p b

    where N’ is the virtual number of teeth of this imaginary

    gear and p is the circular pitch at the large end of the teeth

    j. Parallel Helical Gears

    ? Helical gears used to transmit power between parallel shafts

    ? The helix angle is the same on both gears

    ? One gear must have a right-hand helix and the other left-handed

    ? Helical gears subject the shaft to radial and thrust loads

    ? Because of the nature of the contact between helical gear the

    contact ratio is of minor importance [it is the contact area (~ face

    width) that becomes significant]

    ? Herringbone gear a gear with a double helix to avoid thrust

    loads. Like two helical gears of opposite hand mounted side by

    side on the same shaft.

    ????? Equations:

    ?pcos? Eqn. 13-16 ntp

    Where is the normal circular pitch np

     is the transverse circular pitch tp

    ptp? Eqn. 13-17 xtan?

    where is the axial pitch xp

    Rev 3/30/07 5






    PtP Eqn. 13-18 ?ncos?

    Where is the normal diametral pitch nP

    ?tanncos Eqn. 13-19 ??tan?t

    where is the pressure angle in the normal direction ?ntan

    and is the pressure angle in the transverse direction ?ttan

    ? See example 13-2

    k. Worm Gears

    ? A worm gear set consists of a worm and a gear

    ? The worm looks like a thread on a bolt

    ? The nomenclature is shown on figure 13-24

    a. Axial pitch on the worm xp

    b. Transverse circular pitch (circular pitch) of the gear tp

    ? The pitch diameter of the gear Gd

    pNGt Eqn. 13-25 ?Gd?

    ? Since the worm pitch diameter does not depend on the number

    of teeth, it can have any value

    ? The pitch diameter of the worm should be ??0.8750.875CC ` ` Eqn. 13-26 ??dW??3.01.7

     Where C is the center distance between worm and gear

    ??? Worm lead, L, and lead angle,,

    ?? Eqn. 13-27 NxWL?p

    ??Ltan?? Eqn. 13-28 ?dW

    Where L is the lead; d is the worm pitch diameter; N is the WW

    number of “starts” on the worm thread [I think].

    l. Tooth Systems

    m. Gear Trains


    ??Rev 3/30/07 6






    Nd22 Eqn. 13-29 ?n?n322nNd33? The speed of a driven: gear #3

    where n = revolutions or rpm; N = number of teeth

    and d = the pitch diameter

    ? A gear train of 6 gears (shown in figure 13-27) where gear 2 in

    the input (drive gear) and gear 6 is the output (driven gear).

    NNN352n?n 62NNN346

    n. Force Analysis Spur Gearing

    ? Forces on two mating spur gears (figures 13-32 & 13-33)

    ddtT?F?W Eqn. (a) & (b) 32t22

    tF?W= tangential force from gear 3 acting on gear 2 32t

    WVt? Eqn. 13-35 H33,000

    WV??dn/12 where V is in ft/min; is the transmitted load in t

    lb; H is in horsepowerf

    360(10)W?Hor in SI units Eqn. 13-36 t?dn

    where W= transmitted load, kN t??

     H = power, kW

     D = gear diameter, mm

     n = speed, rpm

    ? Example 13-6

    ? Force Analysis -- Bevel Gearing

    a. Equations:

    T Eqn. 13-37 W?trav

    T = the torque

    Rev 3/30/07 7

    = the pitch radius at the midpoint of the tooth for the av

    gear under consideration

     Eqn. 13-38 W?Wtan?cos?rtR

     Eqn. 13-38 W?Wtan?sin?at

    See figure 13-35 for terms

    ? Example 13-7

    ? Force Analysis Helical Gearing

    o Equations:

    ? ]W?Wtan? rtt

    ? W?Wtan? at

    Wt? W? Eqn. 13-40 cos?cos?n

    ? Reference figure for terms

    ? Example 13-8

    ? Force Analysis -- Worm Gearing

    o Equations (with friction):

    xW?Wcos?sin??fcos? n

    yW?Wsin? Eqn. 13-43 n

    zW?Wcos?cos??fsin? n

    Where worm and gear forces are opposite

    xW??W?W WtGa

    yW??W?W Eqn. 13-42 WrGr

    z W??W?W WaGt? Frictional force Wf

    fWGt Eqn. 13-44 W?fW?ffsin??cos?cos?n

    ? Worm and Gear tangential forces

    Rev 3/30/07 8

    ???cossin?fcosn Eqn. 13-45 W?WWtGtfsin??cos?cos?n? Efficiency

    ??cos?ftann? Eqn. 13-46 ?cos??fcot?n

    ? Sliding velocity, V S

    V?Vcos? Eqn. 13-47 SW

    where V is the pitch line velocity W

    ? Coefficient of friction is dependent on the relative or

    sliding velocity

    ? See table 13-6 for typical worm gear efficiencies

    ? See figure 13-42 for representative values of f

    ? Example problem 13-10

    Rev 3/30/07 9

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