GB Samp Calculation

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    Sample Calculations:

    Design Example of Helical Gear

    [Note: Some nomenclature may not match with the figure.

    Ref book for Calculation : Design of Machine Elements by M. F. Spotts]

    Problem (Typical)

    A helical gear reduction unit has to transmit 30 Nm input torque at 1500 rpm with a total

    reduction of about 37 to 40. At starting the torque may go as high as 200% and also thereis medium shock loads during operation. The material for pinion is EN 19A and for gear

    wheel it is EN 18A. The gear box may be an ordinary industrial class unit preferably with

    uncorrected gears. The bearing life should not be below 40,000 hours.

    Soln.- Considering the helix angle for helical teeth around 12 degree we consider that

    the pinion teeth will be minimum 16.Avoiding tooth haunting and looking into size optimization (from experience) we

    consider that two stage gear box with first stage reduction not more than 5 may be

    considered. Let the pinions and gears have following teeth

    171 Z and 812 Z at the first stage and 163 Z and 1314 Z at the second stage.

    Therefore, transmission ratios are:

    76.417

    81

    1

    21

    Z

    Zi ; 19.8

    16

    131

    3

    42

    Z

    Zi and

    01.3919.876.416

    131

    17

    81

    3

    4

    1

    221

    Z

    Z

    Z

    Ziiit

    Design of first stage gear set:

    Module on the basis of bending strength:

    The Lewiss Formula for module calculation.

    3cos2

    ZYcS

    Tm

    vo

    n

    EN 19A: Ultimate Strength MPaSu 940

    Yield Strength MPaSy 600

    At 300-340 BHN (Hardened and Tempered)

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    EN 18A: Ultimate Strength MPaSu 860

    Yield Strength MPaSy 550

    At 250-300 BHN (Hardened and Tempered)

    For hardened and tempered (BHN up to 350, up to which hobbing can be used in

    finishing cut), the allowable design strength in Lewis formula may be taken as

    approximately 1/3 rd. of ultimate strength. Now considering EN 19A for pinion and EN18A for gear allowable design strengths.

    For pinion, MPaSop 310

    And For gear, MPaSog 285

    We consider the helix angle in first stage 1 such that, 98.0cos 1 . This intension is tomake the centre distance multiple of 10 or 5mm for uncorrected gears.

    i.e., Centre distance in first stage

    z

    n mmZZ

    A 50cos2 1

    211

    NmNmofT 6030%200 Now formative number of teeth become 18 and 86 for gear and pinion

    and corresponding Lewiss for factors 308.0pY and 44.0gY

    The product MPaYS pop 5.95308.0310

    And MPaYS gog 12544.0285 Therefore, the pinion is weaker and it is to be designed.

    Now, 2562111 o . With such a helix angle the width factor may betaken as 16

    vC , the velocity factor is taken for very accurate gears by hobbing and fine finished

    VCv

    6

    6

    Assuming that the pinion will have pcd around 50 mm V is approximately 4m/sec .

    Therefore,

    6.046

    6

    vC

    3

    1

    1cos2

    ZYcS

    Tm

    vop

    n

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    mmn 002.017308.0166.01031

    98.06023

    6

    Usually for Industrial gear unit no less than 2.5 mm module is considered. Therefore, let

    the module is 2.5 mm.

    Now pcd ppd becomes 43.37.mm and pitch line velocity is less than we have initially

    assumed. Therefore, without recalculating the module we proceed further for other load

    calculation.

    Probable Dynamic Load :

    sec/414.3sec/1060

    150037.43

    60 3mm

    NdV

    p

    637.0414.36

    66

    6

    V

    Cv

    t

    v

    tdd FC

    FCF1

    kNd

    TF

    pp

    t 77.237.43

    106022 3

    637.0

    11

    v

    dC

    C 1.563

    kNFCF tdd 35.477.2563.1

    Wear load Capacity:

    For helical gear

    1

    2cos

    bKQdF

    pp

    w

    Where

    21

    211

    4.1

    sin

    EEK es and

    i

    iQ

    1

    2

    d1= pinion diameter and i=d2/d1 ; b=face width.

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    Considering

    Pinion teeth are hardened up to = 350 BHN and gear teeth up to 300 BHN;

    GPaEE 21021 for steel

    From table

    MPaK 64.1

    Calculating

    65.1Q

    Therefore,

    kN

    bKQd

    F

    pp

    w 9.4)98.0(

    )105.216()1037.43(65.1)1064.1(

    cos 2

    336

    12

    Comparison with probable dynamic load and wear load capacity:

    From the calculated values

    dw FF

    Therefore the design is satisfactory.

    Second stage Gear Design

    In the specific design, which is not detailed in this example exercise, the normalmodule for first stage is increased to 3 mm. For second stage the normal module is

    calculated following the same procedure as in for first stage. It is finally taken as 4 mm.

    for second stage. However, the gear data are as follows.

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    Sl.No.

    Description

    First Stage Second Stage

    Pinion Gear Pinion Gear

    1. Z , Number of Teeth 17 81 16 131

    2. Profile o20 Involute Full Depth, Un corrected

    3.nm , Normal module 3 mm 4 mm

    4. , Helix Angle 256211 o 256211 o RH LH LH RH

    5.na mf = nm0.1

    Addendum Height (mm)

    3.0 4.0

    6.nd mf = nm25.1

    Dedendum Height (mm)

    3. 75 5.0

    7.pd , Pitch Circle

    Diameter (PCD) (mm)

    52.04 247.96 65.306 534.69

    8. ad , Addendum or Tip

    Circle Diameter (mm)

    58.04 253.96 73.30 542.70

    9.dd , Dedendum or root

    Circle Diameter (mm)

    44.54 240.46 55.30 524.70

    10. b , Face width. (mm) 63 58 68 63

    11. Material EN 19A EN 18A EN 19A EN 18A

    12. Surface Hardness (BHN)(Through Hardened)

    350 300 350 300

    p and g may be added to subscript of Nomenclature to indicate pinion and

    gear respectively. Similarly 1 and 2 can be added to indicate stage of Gear.

    Tooth Loads :

    Fig. 3.1 Shows a helical gear pair in contact. Fig. 3.2 shows all forces acting ontooth of a helical gear when axes are parallel.

    Fig. 3.1 : Plan view of a Helical Gear Pair in Contact

    ppd gpd

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    nF cos

    t

    tn

    FF

    tF aF rF

    ntn

    tn

    n FF

    F

    sec.seccos

    nntnnr FFF sin.sec.secsin.

    ntr FF tan.sec

    tansin tna FFF

    Fig. 3.2 : Forces on Helical Gear Tooth.

    may be as high as 30o.. For herring bone and double helical gear it is 23o to as

    high as 45o

    Tooth height

    Addendum 1m Same as spur gear.

    Dedendum 1.25m Pitch circle dia

    Addendum circle dia = P.C.D. + 2aDedendum circle dia = P.C.D. 2a

    Centre distance

    External:

    1750cos2

    51753

    cos2

    o

    npg mZZ

    mm.180

    Internal: npg

    mZZ

    cos2

    cos

    536.13

    Let o50.13

    Stresses in Shaft- Design Verification :

    In gear unit design the size of the gear shaft usually biased by the sizes of gears, bearing

    layout and centre distances. Particularly in case of shaft integral with the pinion there is

    n f

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    little scope of pre-designing the shaft. In such cases maximum stresses in the shaft are

    estimated identifying critical sections. Then a factor of safety can be estimated using the

    following formula, which is base on maximum shear stress theory under combined,

    bending, torsion and direct normal stresses.

    22

    4 maen

    y

    fm

    s

    y

    S

    Sk

    f

    S

    Where, yS = Yield strength of shaft material

    enS = Endurance strength of shaft material

    m = Mean (average) stress at considered section due to axial load.

    a = Maximum alternating stress at considered section due to bending.

    m = Maximum shear stress at considered section due to torsion.

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    Intermediate Shaft :

    mminareensionsAll dim

    178

    53

    PinionStagend.2 GearStagest.1

    50

    3rVL FR

    3tHL FR HRrt RFF 22

    23 aa FF VRR

    23 aa MM

    3rF

    HLR 2rF HRR

    23 aa MM

    Nm72

    3tF

    VLR

    2tF

    VRR

    Nm187

    planeverticalin

    forcesduetomomentBending

    planehorizontalin

    forcestoduemomentBending

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    Plan View with gears on lower Housing

    Elevation (3rd. Angle Projection) with both housings

    A typical 3 stage Gear Box (with Cast Housing)

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    Pictorial view of a typical 3 stage Gear Box (with Cast Housing)

    A typical 3 stage Gear Box (Fabricated Housing- single piece)

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    A typical 2 stage Gear Box (Fabricated Housing- single piece)(Note: Similar housing may be used for the given problem)

    Gear Arrangement inside the above type Gear Box