Lecture 15 – WORM GEARS

Contents 1. Worm gears –an introduction 2. Worm gears - geometry and nomenclature 3. Worm gears- tooth force analysis 4. Worm gears-bending stress analysis 5. Worm gears-permissible bending stress 6. Worm gears- contact stress analysis 7. Worm gears- permissible contact stress 8. Worm gears -Thermal analysis WORM GEARS – INTRODUCTION Worm gears are used for transmitting power between two non-parallel, non-intersecting shafts. High gear ratios of 200:1 can be got.

Fig.1. Single enveloping Fig. 2. Double enveloping worm gear. worm gear.

Fig.3 showing the cut section of a worm gearbox with Fins and fan for cooling. WORM GEARS – GEOMETRY AND NOMENCLATURE

a. The geometry of a worm is similar to that of a power screw. Rotation of the worm simulates a linearly advancing involute rack, Fig.4. b. The geometry of a worm gear is similar to that of a helical gear, except that the teeth are curved to envelop the worm. c. Enveloping the gear gives a greater area of contact but requires extremely precise mounting.

1. As with a spur or helical gear, the pitch diameter of a worm gear is related to its circular pitch and number of teeth Z by the formula

2. When the angle is 90° between the nonintersecting shafts, the worm lead angle λ is equal to the gear helix angle ψ. Angles λ and ψ have the same hand.

3. The pitch diameter of a worm is not a function of its number of threads, Z1.

4. This means that the velocity ratio of a worm gear set is determined by the ratio of gear teeth to worm threads; it is not equal to the ratio of gear and worm diameters.

5. Worm gears usually have at least 24 teeth, and the number of gear teeth plus worm threads should be more than 40:

Z1 + Z2 > 40 (3) 6. A worm of any pitch diameter can be made with any number of threads and any axial pitch. 7. For maximum power transmitting capacity, the pitch diameter of the worm should normally be related to the shaft center distance by the following equation

8. Integral worms cut directly on the shaft can, of course, have a smaller diameter than that of shell worms, which are made separately. 9. Shell worms are bored to slip over the shaft and are driven by splines, key, or pin. 10. Strength considerations seldom permit a shell worm to have a pitch diameter less

than d1 = 2.4p + 1.1 (5) 11. The face width of the gear should not exceed half the worm outside diameter. b ≤ 0.5 da1 (6)

= 22

pd (1)π

Z

1 2

2 1

ω Z= (2)ω Z

1

0.875 0.875C Cd (4)3.0 1.7

≤ ≤

12. Lead angle λ, Lead L, and worm pitch diameter d1 have the following relation-ship in connection with the screw threads.

13. To avoid interference, pressure angles are commonly related to the worm lead angle as indicated in Table 1. Table 1. Maximum Worm Lead Angle and Worm Gear Lewis Form Factor for Various Pressure angles.

Pressure Angle Φn (Degrees)

Maximum Lead Angle λ

(degrees)

Lewis formfactor y

Modified Lewis form

factor Y

14.5

15

0.100

0.314

20

25

0.125

0.393

25

35

0.150

0.473

30

45

0.175

0.550

Table 2. Frequently used standard values of module and axial pitch of worm or circular pitch of gear p in mm :

Module m mm

2.0

2.5

3.15

4.0

5.0

6.3

Axial pitch p mm

6.283

7.854

9.896

12.566

15.708

19.792

Module m mm

8

10

12.5

16

20

Axial pitch p mm

25.133

31.416

39.270

50.625

62.832

1

Ltanλ= (7)πd

b. Values of addendum and tooth depth often conform generally to helical gear practice but they may be strongly influenced by manufacturing considerations. c. The load capacity and durability of worm gears can be significantly increased by modifying the design to give predominantly “recess action” i.e. the angle of approach would be made small or zero and the angle of recess larger. WORM GEARS – FORCE ANALYSIS

a. The tangential, axial, and radial force components acting on a worm and gear are illustrated in Fig.5 .

b. For the usual 90° shaft angle, the worm tangential force is equal to the gear axial force and vice versa,

F1t = F2a ( 8 ) F2t = F1a ( 9 ) c. The worm and gear radial or separating forces are also equal, F1r = F2r (10) If the power and speed of either the input or output are known, the tangential force acting on this member can be found from equation

1t1000 WF = (11)

V

1. In the following figure, the driving member is a clockwise-rotating right hand worm. 2. The force directions shown can readily be visualized by thinking of the worm as a right hand screw being turned so as to pull the “nut” (worm gear tooth) toward the “screw head”.

3. Force directions for other combinations of worm hand and direction of rotation can be similarly visualized. WORM GEARS – THRUST FORCE ANALYSIS, FIG.6. WORM GEARS – THRUST FORCE ANALYSIS , FIG. 7.

The thread angle λ of a screw thread corresponds to the pressure angle φn of the worm, we can apply the force, efficiency, and self-locking equations of power screw directly to a worm and gear set. These equations are derived below with reference to the worm and gear geometry. Figs. 8 to 10 show in detail the forces acting on the gear. Components of the normal tooth force are shown solid. Components of the friction force are shown with the dashed lines.

Fig.10 illustrates the same directions of rotation as figure a but with the torque direction reversed (i.e., gear driving). Then contact shifts to the other side of the gear tooth, and the normal load reverses. The friction force is always directed to oppose the sliding motion. The driving worm is rotating clockwise: Combining eqns. (11) with (12), we have: Combining eqns. (11) with (13) and (12) with (13), we have:

2t 1a n n n

1t 2a n n n

2r 1r n n

F =F =F cosφ cosλ-f F sinλ (11)

F =F =F cosφ sinλ+f F cosλ (12)

F =F =F sinφ (13)

n

n

F cosφ cosλ - f sinλ2t = (14)F cosφ sinλ + f cosλ1t

WORM GEARS – KINEMATICS The relationship between worm tangential velocity, gear tangential velocity, and sliding velocity. WORM GEARS– EFFICIENCY Efficiency η is the ratio of work out to work in. For the usual case of the worm serving as input member, The overall efficiency of a worm gear is a little lower because of friction losses in the bearings and shaft seals, and because of “churning” of the lubricating oil. WORM GEARS– FRICTION ANALYSIS The coefficient of friction, f, varies widely depending on variables such as the gear materials, lubricant, temperature, surface finishes, accuracy of mounting, and sliding velocity. The typical coefficient of friction of well lubricated worm gears are given in Fig.11.

n2r 1r 2t

n

n1t

n

sinφF =F =Fcosφ cosλ - f sinλ

sinφ =F (15)cosφ sinλ + f cosλ

2

1

V = tanλ (16)V

n

n

n

n

F V cos cos - f sin2t 2t tanF V cos sin f cos1t 1cos - f tan (17)cos f cot

φ λ λη = = λ

φ λ + λ

φ λη =

φ + λ

WORM GEARS – KINEMATICS The sliding velocity Vs is related to the worm and gear pitch line velocities and to the worm lead angle by WORM GEARS – FRICTION FORCE ANALYSIS a. Eqn. 18 shows that with a sufficiently high Coefficient of friction, the gear tangential force becomes zero, and the gearset “self-locks” or does not “over-haul.” b. With this condition, no amount of worm torque can produce motion. c. Self-locking occurs, if at all, with the gear driving. d. This is desirable in many cases and helps in holding the load from reversing, similar to a self- locking power screw. The worm gear set self-locks if this force goes to zero, which happens if A worm gear set can be always overhauling or never overhauling, depending on the selected value coefficient of friction (i.e., λ and to a lesser extent on φn).

1 2s

V VV = = (18)cosλ sinλ

1t n n nF F cos sin - f F cos (19)= φ λ λ

nf cosφ tanλ (20)≥

WORM GEARS – BENDING AND SURFACE FATIGUE STRENGTHS Worm gear capacity is often limited not by fatigue strength but by cooling capacity. The total gear tooth load Fd is the product of nominal load Ft and factors accounting for impact from tooth inaccuracies and deflections, misalignment, etc.). Fd must be less than the strength the bending fatigue and surface fatigue strengths Fb and Fw The total tooth load is called the dynamic load Fd, the bending fatigue limiting load is called strength capacity Fb, and the surface fatigue limiting load is called the wear capacity Fw. For satisfactory performance, Fb ≥ Fd (21) and Fw ≥ Fd (22) The “dynamic load” is estimated by multiplying the nominal value of gear tangential force by velocity factor “Kv” given in the following figure. Adapting the Lewis equation to the gear teeth, we have Where, [σb] is the permissible bending stress in bending fatigue, in MPa, Table 3 Table 3. Permissible stress in bending fatigue, in MPa

Material of the gear

[σb] MPa

Centrifugally cast Cu-Sn bronze

23.5

Aluminium alloys Al-Si alloy

11.3

Zn alloy

7.5

Cast iron

11.8

b – is the face width in mm ≤ 0.5 da1 p – is the axial pitch in mm, Table 2 m – is module in mm, Table 2 y – is the Lewis form factor, Table 1 Y – is modified Lewis form factor, Table 1

2vd 2t 2t

6.1+VF =F K =F (23)6.1

⎛ ⎞⎜ ⎟⎝ ⎠

b b bF =[ ] bpy = [ ] bmY (24) σ σ

By assuming the presence of an adequate supply of appropriate lubricant, the following equation suggested by Buckingham may be used for wear strength calculations Fw – Maximum allowable value of dynamic load with respect to surface fatigue. dg - Pitch diameter of the gear. b - Face width of the gear. Kw - A material and geometry factor with values empirically determined from the Table 4. Table 4. Worm Gear Wear Factors Kw

Material

Kw (MPa)

Worm

Gear

λ<10°

λ<25°

λ>25°

Steel, 250 BHN

Bronze

0.414

0.518

0.621

Bronze

0.552

0.690

0.828

Hardened steel (Surface 500 BHN) Chill-cast

Bronze

0.828

1.036

1.243

Cast iron

Bronze

1.036

1.277

1.553

WORM GEARS –THERMAL CAPACITY The continuous rated capacity of a worm gear set is often limited by the ability of the housing to dissipate friction heat without developing excessive gear and lubricant temperatures. Normally, oil temperature must not exceed about 200ºF (93oC) for satisfactory operation. The fundamental relationship between temperature rise and rate of heat dissipation used for journal bearings, does hold good for worm gearbox. Where H – Time rate of heat dissipation (Nm/sec) CH – Heat transfer coefficient (Nm/sec/m2/ºC) A – Housing external surface area (m2) To – Oil temperature (º C) Ta – Ambient air temperature (º C)

w w2F =d bK (25)

( )H 0 aH = C A T -T (26)

Surface area of A for conventional housing designs may be roughly estimated from the Equation: Where A is in m2 and C (the distance between the shafts) is in m. Housing surface area can be made far greater than the above equation value by incorporating cooling fins. Rough estimates of C can be taken from the following figure. WORM GEARS–DESIGN GUIDELINES Table 5. Recommended pressure angles and tooth depths for worm gearing

Lead angle λ in degrees

Pressure angle φn in degrees

Addendum ha in mm

Dedendum hf in mm

0-15

14.5

0.3683 p

0.3683 p

15-30

20

0.3683 p

0.3683 p

30-35

25

0.2865 p

0.331 p

35-40

25

0.2546 p

0.2947 p

40-45

30

0.2228 p

0.2578 p

1.7A=14.75 C (27)

Table 6. Efficiency of worm gear set for f = 0.05

Helix angle Ψ in O

Efficiency η in %

Helix angleΨ in O

Efficiencyη in %

Helix angleΨ in O

Efficiency η in %

1.0

25.2

7.5

71.2

20.0

86.0

2.5

46.8

10.0

76.8

25.0

88.0

5.0

62.6

15.0

82.7

30.0

89.2

Table 7. Minimum number of teeth in the worm gear

Pressure angle φn

14.5o

17.5o

20o

22.5o

25o

27.5o

30o

Z2 minimum

40

27

21

17

14

12

10

Table 8. Maximum lead angle for normal pressure angle

Normal Pressure angle φn

14.5o

20o

25o

30o

Maximum lead angle λmax

16 o

25 o

35 o

45 o