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Worm Gears
CHAPTER 11
Worm Gears Chapter Outline 11.1 Introduction 439 11.2 Force Analysis 446 11.3 AGMA Equations 449 11.4 Design Procedure 453 11.5 Conclusions 455 References 456 Further Reading 456 Nomenclature 457
Abstract Worm and wheel gears are widely used for nonparallel, nonintersecting, right angle gear drive system applications where a high transmission gearing ratio is required. In comparison to other gear, belt, and chain transmission elements, worm and wheel gear sets tend to offer a more compact solution. In certain configurations, a worm and wheel gear set can provide sufficiently high friction to be self-locking. This can be a desirable feature if a defined position is required for a gear train, if it is not braked or unpowered. This chapter provides an overview of worms and wheels and outlines a selection procedure.
11.1 Introduction A worm gear is a cylindrical helical gear with one or more threads and resembles a screw thread. A worm wheel or worm gear is a cylindrical gear with flanks cut in such a way as to ensure contact with the flanks of the worm gear. The worm wheel is analogous to a nut that fits on the screw thread of the worm. If the worm is restrained axially within its housing and if the worm is rotated, the worm gear will also rotate. Typical forms for worms and worm gears are shown in Figure 11.1. In a worm and wheel gear set rotary power can be transmitted between nonparallel and nonintersecting shafts. A worm and wheel gear set is typically used when the speed ratio of the two shafts is high, say three or more. Worm and wheel gear sets are used for steering gear, winch blocks, e.g. see Figure 11.2, low-speed gearboxes, rotary tables, and remote valve control. Worm and wheel gear sets are Mechanical Design Engineering Handbook. http://dx.doi.org/10.1016/B978-0-08-097759-1.00011-3 Copyright Ó 2014 Elsevier Ltd. All rights reserved.
439
440 Chapter 11
Figure 11.1 Worm and wheel gear sets.
capable of high-speed reduction and high load applications where nonparallel, non-intersecting shafts are used. The 90 configuration is most common, although other angles are possible. Frictional heat generation is high in worm gears because of the high sliding velocities, so continuous lubrication and provision for heat dissipation are required. The direction of rotation of the worm wheel depends on the direction of rotation of the worm and on whether the worm teeth have a right-hand thread or a left-hand thread. The direction of rotation for a worm and wheel gear sets is illustrated is Figure 11.3. Worms usually have just one tooth and can, therefore, produce gearing ratios as high as the number of teeth on the gear wheel. Herein lies the principal merit of worm and wheel gear sets. In comparison to other gear sets, which are typically limited to a gear ratio of up to 10:1, worm
Figure 11.2 Possible outline winch configuration incorporating a worm and wheel gear set.
Worm Gears 441 Right hand Thrust bearing
Driver
Driver
Left hand
Driver
Driver
Figure 11.3 Rotation and hand relations for worm and wheel gear sets. After Boston Gear Division.
and wheel gear sets can achieve gear ratios of up to 360:1, although most manufacturers quote ranges between 3:1 and 100:1. Ratios above 30:1 generally have one thread on the worm, while ratios below 30:1 tend to have a worm with multiple threads (sometimes referred to as starts). The gear ratio for a worm and wheel gear set is given by mG ¼
NG NW
(11.1)
where mG ¼ gear ratio, NG ¼ number of teeth in the worm gear, and NW ¼ number of threads in the worm. A particular merit of worm and wheel gear sets is their ability to self-lock. If a worm set is self-locking it will not back drive and any torque applied to the worm gear will not rotate the worm. A self-locking worm and wheel gear set can only be driven forward by rotation of the worm. Such a gear set can, therefore, be used to hold a load. This is commonly exploited in a car jack. Whether a worm and wheel gear set will be self-locking depends on frictional contact between the worm and the worm wheel flanks. There are two types of worm and wheel gear sets, depending on whether the teeth of one or both wrap around each other. • •
Single enveloping worm and wheel gear sets, see Figure 11.4, and Double enveloping worm and wheel gear sets, see Figure 11.5.
442 Chapter 11 Pitch diameter d w Helix
Worm Root diameter
Pitch cylinder
Helix angle ψ w Axial pitch lpitch
Lead L Worm gear
Pitch diameter dG
Lead angle λ
Figure 11.4 Nomenclature for a single enveloping worm and wheel gear set.
Worm
3 4 1 2 5
Gear driven by worm
Figure 11.5 Double enveloping worm and wheel gear set.
Worm Gears 443 Rotation of gear A
Line 2 Line 1
Rotation of worm
Line 2 A
Line 1
Center section AA
Figure 11.6 Lines of contact for a worm and wheel gear set.
As the worm rotates through the worm gear, lines of contact either roll or progress from the tip to the root of the worm gear teeth. At any instant in time, there may be two or three teeth in contact and transmitting power as illustrated in Figure 11.6. Some of the key geometric features and dimensions for a worm gear are illustrated in Figure 11.7. The helix angle on a worm is usually high, and the helix angle on the worm wheel is low. Normal convention is to define a lead angle, l, on the worm and a helix angle, jG, on the
FW
do d
a
lpitch dr
ht
b
c c C
FG
Dt Dr Do
Figure 11.7 Worm gear dimensions.
Dm
444 Chapter 11 worm gear. For a 90 configuration, l ¼ jG. The distance that a point on the mating worm gear moves axially in one revolution of the worm is called the lead, L. The following relationships apply to the lead, L, and lead angle, l: L ¼ lpitch NW ¼ tan l ¼
pdG NW NG
L pdW
(11.2) (11.3)
where L ¼ lead (mm), lpitch ¼ worm axial pitch (mm), NW ¼ number of teeth on the worm, dG ¼ pitch diameter of the worm gear (mm), NG ¼ number of teeth on the worm gear, l ¼ lead angle ( ), and dW ¼ pitch diameter of the worm (mm). The worm lead angle and the worm helix angle, jW, are related by l ¼ 90 jW. The lead angle will vary from the root to the outside diameter of the worm, as indicated in Figure 11.8. Generally, self-locking occurs for lead angles below 6 . However, care is necessary in relying solely on self-locking to brake or sustain a load, as vibration has been Lead angle at root diameter Lead angle at outside diameter
Figure 11.8 Variation of the lead angle on a worm gear. Figure after AGMA.
Worm Gears 445 known to result in a reduction of the frictional contact between the worm and wheel for lead angles below 6 and resulting movement or failure of the device. The axial pitch of the worm and the transverse circular pitch of the wheel will be equal for a 90 set configuration, d ¼ mN
(11.4)
The worm can have any pitch diameter, as this is not related to the number of teeth. General guidance for optimum power capacity indicates that the pitch diameter, d, of the worm should fall in the following ranges (AGMA 6022-C93): C 0:875 C 0:875 dmax 1:6 1:07
(11.5)
C 0:875 C 0:875 dmin 3 2
(11.6)
where: C ¼ center distance (mm), d ¼ worm pitch diameter (mm), dmax ¼ maximum worm pitch diameter (mm), and dmin ¼ minimum worm pitch diameter (mm). Dudley (1984) using Imperial units, see also Radzevich (2012), recommends C 0:875 dz 2:2
(11.7)
The pitch diameter of the worm gear, dG, is related to the center distance C and the pitch diameter of the worm, by dG ¼ 2C d
(11.8)
The addendum, a, and dedendum, b, are given by a ¼ 0:3183lpitch
(11.9)
b ¼ 0:3683lpitch
(11.10)
The face width of a worm gear, Figure 11.7, is limited by the worm diameter. The ANSI/ AGMA 6034-B92 recommendation for the minimum face width, for a pitch exceeding 4.06 mm, is given by
446 Chapter 11 FG ¼ 0:67d
(11.11)
The tooth forms for worm and wheel gear sets are not involutes. They are manufactured as matched sets. The worm is subject to high stresses and is normally made using a hardened steel such as AISI 1020, 1117, 8620, 4320 hardened to HRC 58e62, or a medium carbon steel such as AISI 4140 or 4150 induction or flame hardened to a case of HRC 58e62 (Norton, 2006). They are typically ground or polished to a roughness of Ra ¼ 0.4 mm. The worm gear needs to be of softer material that is compliant enough to run-in and conform to the worm under the high sliding running conditions. Sand cast or forged bronze is commonly used. Cast iron and polymers are sometimes used for lightly loaded, low-speed applications. An analysis of the forces associated with a worm and wheel gear set can be undertaken readily, and this is outlined in Section 11.2. Such information is critical in order to enable suitable bearings to be selected for both shafts. Worm and wheel gear sets tend to fail due to pitting and wear (see Maitra (1994) and Dudley (1984)). The American Gear Manufacturer Association power ratings, based on wear and pitting resistance, are presented in Section 11.3 and an associated design procedure in Section 11.4.
11.2 Force Analysis The force exerted on a worm by a gear is illustrated in Figure 11.9 where, for the time being, friction has been neglected. The resultant force, W, will have three components: Wx ¼ W cos fn sin l
(11.12)
Wy ¼ W sin fn
(11.13)
Wz ¼ W cos fn cos l
(11.14)
where Wx ¼ force component in the tangential direction on the worm (N), Wy ¼ force component in the radial direction on the worm (N), Wz ¼ force component in the axial direction on the worm (N), l ¼ lead angle ( ), and fn ¼ normal pressure angle of the worm thread at the mean diameter ( ). Standard pressure angles for worm and wheel gear sets are 14.5, 17.5, 20, 22.5, 25, 27.5, and 30 . The higher the pressure, the higher the tooth strength; albeit at the expense of higher friction, bearing loads, and bending stresses in the worm. The minimum number of worm gear teeth, Nmin, as a function of the pressure angle is listed in Table 11.1.
Worm Gears 447 y
Wy
φ
fWsin λ
W t
φ
n
λ
Wx
fWcos λ x
Wz
Wf =fW
Pitch helx z nW
Pitch cylinder
Figure 11.9 Pitch cylinder of the worm, showing the forces exerted on the worm by the worm gear. Image after Shigley (1986).
As the forces on the worm and worm gear are equal and opposite, the tangential, radial, and axial forces are given by WWt ¼ WGa ¼ Wx
(11.15)
WWr ¼ WGr ¼ Wy
(11.16)
WWa ¼ WGt ¼ Wz
(11.17)
where WWt ¼ tangential force component acting against the worm (N), WWr ¼ radial force component acting against the worm (N), Table 11.1: Suggested minimum number of teeth for the worm (AGMA). Pressure angle ( )
Nmin
14.5 17.5 20 22.5 25 27.5 30
40 27 21 17 14 12 10
448 Chapter 11 WWa ¼ axial force component acting against the worm (N), WGt ¼ tangential force component acting against the gear (N), WGr ¼ radial force component acting against the gear (N), and WWa ¼ axial force component acting against the gear (N). Introducing a coefficient of friction, f, to account for the sliding motion experienced in the motion between a worm thread and the wheel teeth surfaces, Wx ¼ Wðcos fn sin l þ f cos lÞ
(11.18)
Wy ¼ W sin fn
(11.19)
Wz ¼ Wðcos fn cos l f sin lÞ
(11.20)
From Eqns (11.15e11.18) Wf ¼ f W ¼
f WGt f sin l cos fn cos l
(11.21)
WWt ¼ WGt
cos fn sin l þ f cos l f sin l cos fn cos l
(11.22)
The efficiency of a worm and wheel gear set can be defined by h¼
WWt; without friction WWt; with friction
(11.23)
h¼
cos fn f tan l cos fn þ f cot l
(11.24)
or
A typical value for the coefficient of friction for worm gears is f z 0.05. The variation of efficiency with helix angle is given in Table 11.2. Experiments have shown that efficiency for a worm and wheel gear set is a function of the sliding velocity. Taking VG as the pitch line velocity of the gear and VW as the pitch line velocity of the worm, the sliding velocity, Vs, is given by vector addition: VW ¼ VG þ Vs
(11.25)
or Vs ¼
VW cos l
(11.26)
Worm Gears 449 Table 11.2: Worm and wheel efficiency (taking f [ 0.05). j
l
F
fn
h
1 2.5 5 7.5 10 15 20 25 30
1 2.5 5 7.5 10 15 20 25 30
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
20 20 20 20 20 20 20 20 20
24.7 45.0 61.9 70.7 76.1 82.2 85.6 87.5 88.7
where VG ¼ pitch line velocity of the gear (m/s), VW ¼ pitch line velocity of the worm (m/s), and VS ¼ sliding velocity (m/s).
11.3 AGMA Equations Worm sets are generally rated by their capacity to handle a particular level of input power, output power, or allowable torque at a particular speed for the input or output shaft. The AGMA power rating is based on pitting and wear resistance, as this is the usual failure mode for worm sets. The AGMA rating (ANSI/AGMA 6034-B92) is based on 10 h of continuous operation under a uniform load. The input-power rating, Pinput, is given by Pinput ¼ Poutput þ Ploss
(11.27)
where Ploss is the power lost due to friction in the mesh (kW). The output power is given by Poutput ¼ where n ¼ rotational speed of the worm (rpm), Wtg ¼ worm gear tangential force (N), Poutput ¼ output power (kW), mG ¼ gear ratio, and dg ¼ mean gear diameter (mm).
nWtg dg 1:91 107 mG
(11.28)
450 Chapter 11 The power lost is given by Ploss ¼
Vt Wf 1000
(11.29)
where Ploss ¼ lost power (kW), Vt ¼ sliding velocity at the mean worm diameter (m/s), and Wf ¼ friction force (N). The AGMA tangential load on a worm gear is given by Wt ¼
Cs Cm Cv d g0:8 F 75:948
(11.30)
f Wt cos l cos fn
(11.31)
where Cs ¼ materials factor, dg ¼ mean diameter of the gear (mm), F ¼ effective face width (mm), Cm ¼ ratio correction factor, and Cv ¼ velocity factor. The friction force can be determined by Wf ¼ where f ¼ coefficient of friction, Wt ¼ tangential load on the worm gear tooth (N), l ¼ lead angle ( ), and fn ¼ normal pressure angle of the worm thread at the mean diameter ( ). The sliding velocity at the mean worm diameter can be determined by Vt ¼
ndm 19;098 cos l
(11.32)
where n ¼ rotational speed of the worm (rpm), and dm ¼ mean worm diameter (mm). Values for the ratio correction factor, the velocity factor, and materials factors can be found from tables provided in the ANSI/AGMA 6034-B92 standard. The following equations for
Worm Gears 451 the ratio correction factor, the velocity factor, and materials factors provide approximations to the values given in the tables. The ratio correction factor Cm is a function of the gear ratio, mG. For 3 mG 20 0:5 Cm ¼ 0:02 m2G þ 40mG 76 þ 0:46
(11.33)
0:5 Cm ¼ 0:0107 m2G þ 56mG þ 5145
(11.34)
Cm ¼ 1:1483 0:00658mG
(11.35)
Cv ¼ 0:659e0:2165Vt
(11.36)
Cv ¼ 0:652Vt0:571
(11.37)
Cv ¼ 1:098Vt0:774
(11.38)
0:645 f ¼ 0:124eð2:233V t Þ
(11.39)
0:45 f ¼ 0:103eð1:185V t Þ
(11.40)
For 20 mG 76
For mG > 76
For 0 Vt 3.556 m/s
For 3.556 Vt 15.24 m/s
For Vt > 15.24 m/s
For Vt ¼ 0, take f ¼ 0.15. For 0 Vt 0.0508 m/s
For Vt > 0.0508 m/s
The materials factor Cs depends on the method of casting. For C 76.2 mm, an initial estimate for the materials factor can be obtained from Cs ¼ 720 þ 0:000633C3
(11.41)
This value can be compared to the values obtained for the relevant means of casting, as indicated by the following relationships and the smaller value used.
452 Chapter 11 For sand cast gears then: for dm < 63.5 mm, Cs ¼ 1000 for dm > 63.5 mm, Cs ¼ 1859:104 476:5454 log10 dm
(11.42)
For chill cast bronze gears then: for dm < 203.2 mm, Cs ¼ 1000 for dm > 203.2 mm Cs ¼ 2052:011 455:8259 log10 dm
(11.43)
For centrifugally cast gears then: for dm < 635 mm, Cs ¼ 1000 for dm > 635 mm, Cs ¼ 1503:811 179:7503 log10 dm
(11.44)
The efficiency, in percent, for worm gearing is given by Poutput 100 Pinput
(11.45)
nWt dm 100 1:91 107 mG Pinput
(11.46)
h¼ Substituting for the output power h¼ where
Poutput ¼ rated output power (kW), Pinput ¼ rated input power (kW), n ¼ rotational speed of the worm (rpm), Wt ¼ tangential load on the worm gear (N), dm ¼ mean diameter of the gear (mm), mG ¼ gear ratio.
Worm Gears 453
11.4 Design Procedure An outline design procedure for a worm and wheel gear set using the AGMA equations is listed next. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Define the number of starts. Define the center distance, C. Determine a suitable worm gear diameter. Determine the lead. Determine the lead angle. Determine the maximum recommended face width, F. Determine the material factor, Cs. Determine the ratio correction factor, Cm. Determine the tangential velocity, Vt. Determine the velocity factor, Cv. Determine the tangential load, Wt. Determine the coefficient of friction, f. Determine the friction force, Wf. Determine the output power, Poutput. Determine the power lost in mesh, Ploss. Determine the rated input power, Pinput. Estimate the efficiency of the gear set, h. Determine the output torque, Tq. Establish whether the power rating and output torque are sufficient for the application. If not, alter the number of starts, worm diameter, center distance, etc. to provide suitable power rating and output torque.
Example 11.1 Develop a design for a worm and wheel gear set. The prime mover is an electric motor running at 1470 rpm. A reduction ratio of 50:1 is required. The peak torque required is 850 N m. Assume sand cast gears. Solution Input speed ¼ 1470 rpm. Ratio 50:1. Output ¼ 29.4 rpm. Sand cast gears. If a single start worm is specified, then a 50 tooth worm gear will be needed to give a ratio of 50:1.
454 Chapter 11 Fifty teeth are above the minimum recommended (see Table 11.1). An estimate for the center distance needs to be made. If the center distance between the worm and wheel is taken as 140 mm, then from Eqns (11.5) and (11.6), the pitch diameter range is found to be between 20.72 and 70.55 mm. A midvalue of 50 mm is selected here. From Eqn (11.8), dG ¼ 2C d ¼ ð2 140Þ 50 ¼ 230:0 mm The lead, from Eqn (11.2), is L¼
pdG NW 1 ¼ p 230:0 ¼ 14:45 mm NG 50
The lead angle, from Eqn (11.3), L 1 1 14:45 l ¼ tan ¼ tan ¼ 5:256 pdw p 50 Generally, self-locking occurs for lead angles below 6 . This is less than 6 so the worm set will be self-locking. The face width can be determined from Eqn (11.11) FG ¼ 0:67d ¼ 0:67 50 ¼ 33:5 mm The materials factor for sand cast gears is given by Eqn (11.42), if the mean worm diameter is greater than 63.5 mm. Here, the mean worm diameter is 50 mm so Cs ¼ 1000 mG ¼ 50: From Eqn (11.34), Cm ¼ 0:7896 The tangential velocity at the mean worm diameter can be determined from Eqn (11.32), Vt ¼
nd 1470 50:0 ¼ ¼ 3:865 m=s 19; 098 cos l 19;098 cos 5:256
The velocity factor, from Eqn (11.37), is Cv ¼ 0:3013 From Eqn (11.30), the tangential load is given by Wt ¼
1000 0:7896 0:3013 2300:8 33:50 ¼ 8134 N 75:948
Worm Gears 455 The coefficient of friction is given by Eqn (11.40) 0:45 0:45 m ¼ 0:103eð1:185Vt Þ ¼ 0:103eð1:1853:865 Þ ¼ 0:01167
The friction force is given by Wf ¼
f Wt 0:01167 8134 ¼ ¼ 101:5 N cos l cos f cos 5:256 cos 20
The rated output power, Eqn (11.28), is Poutput ¼
nWtg dg 1470 8134 230 ¼ ¼ 2:880 kW 7 1:91 10 mG 1:91 107 50
The lost power, Eqn (11.29), is Ploss ¼
Vt Wf 3:865 101:5 ¼ ¼ 0:392 kW 1000 1000
The input-power rating is 2.880 þ 0.392 ¼ 3.272 kW. The efficiency of the gear set is 2.880/3.272 ¼ 0.881 ¼ 88.1%. The output torque is given by Tq ¼ Wt
d 0:230 ¼ 8134 ¼ 935:4 N m 2 2
This torque exceeds the requirement, suggesting that the design is suitable. If the value was below that required, then an alternative center distance could be explored and, if necessary, the number of starts could be increased.
11.5 Conclusions Worm and wheel gears are usually used for nonparallel nonintersecting right angle gear drive system applications, where a high gear ratio is required. They can also be used for medium-speed reductions. The worm is generally the driving member. The high transmission ratio leads to a compact solution for many applications in comparison to other types of gearing. For certain arrangements, self-locking is possible, and this can provide an attribute for some applications where a set position is desirable if the drive train is not braked or powered.
456 Chapter 11
References Books and Papers Dudley, D.W., 1984. Handbook of Practical Gear Design. McGraw Hill. Maitra, G.M., 1994. Handbook of Gear Design, second ed. Tata McGraw Hill. Norton, R.L., 2006. Machine Design, third ed. Pearson. Radzevich, S.P., 2012. Dudley’s gear handbook of practical gear design and manufacture, 2nd ed. CRC Press. Shigley, J.E., 1986. Mechanical Engineering Design, first metric ed. McGraw Hill.
Further Reading Shigley, J.E., Mischke, C.R., Budynas, R.G., 2004. Mechanical Engineering Design, seventh ed. McGraw Hill. Townsend, D.P., 1992. Dudley’s Gear Handbook, second ed. McGraw Hill.
Standards AGMA Design manual for cylindrical wormgearing. ANSI/AGMA Standard 6022-C93. Reaffirmed 2008. AGMA Practice for enclosed cylindrical wormgear speed reducers and gearmotors. ANSI/AGMA Standard 6034-B92. Reaffirmed 2005. BS 721-1:1963. Specification for worm gearing. Imperial units. BS 721-2:1983. Specification for worm gearing. Metric units. BS ISO TR 10828:1997. Worm gears. Geometry of worm profiles. PD ISO/TR 14521:2010. Gears. Calculation of load capacity of wormgears.
Web Sites At the time of going to press the world-wide-web contained useful information relating to this chapter at the following sites: www.ashokaengineering.com/ www.bandhgears.co.uk www.bellgears.co.uk www.bostongear.com/products/open/worms.html www.brentwingearcompany.co.uk www.davall.co.uk www.delroyd.com www.gearcutting.com www.gearmanufacturer.net/ www.girard-transmissions.com www.hewitt-topham.co.uk/ www.hopwoodgear.com www.hpcgears.com www.huco.com www.mmestrygears.com www.muffettgears.co.uk/ www.qtcgears.com/ www.rarodriguez.co.uk www.traceygear.com www.wmberg.com
Worm Gears 457
Nomenclature The following symbols have been used in this chapter. Generally, preferred SI units have been stated: a b c C Cm Cs Cv d dg dG dm dmax dmin do dr dW Dm Do Dr Dt f F FG FW ht lpitch L mG n NG Nmin NW Pinput Ploss Poutput
addendum (mm) dedendum (mm) clearance (mm) center distance (mm) ratio correction factor materials factor velocity factor worm pitch diameter (mm) mean diameter of the gear (mm) pitch diameter of the worm gear (mm) mean diameter of the worm gear (mm) maximum pitch diameter of the worm (mm) minimum pitch diameter of the worm (mm) worm outer diameter (mm) worm root diameter (mm) pitch diameter of the worm (mm) mean gear diameter (mm) worm gear outer diameter (mm) worm gear root diameter (mm) worm gear throat diameter (mm) coefficient of friction effective face width (mm) worm gear face width (mm) worm face width (mm) full depth of worm thread (mm) worm axial pitch (mm) lead (mm) gear ratio rotational speed of the worm (rpm) number of teeth on the worm gear minimum number of worm gear teeth number of threads on the worm rated input power (kW) lost power (kW) rated output power (kW)
458 Chapter 11 Ra Tq VG Vs Vt VW W Wf WGr WGt Wt Wtg WWa WWa WWr WWt Wx Wy Wz fn h l j jG jW
roughness (mm) torque (N m) pitch line velocity of the gear (m/s) sliding velocity (m/s) sliding velocity at the mean worm diameter (m/s) pitch line velocity of the worm (m/s) resultant force (N) friction force (N) radial force component acting against the gear (N) tangential force component acting against the gear (N) tangential load on the worm gear (N) worm gear tangential force (N) axial force component acting against the gear (N) axial force component acting against the worm (N) radial force component acting against the worm (N) tangential force component acting against the worm (N) force component in the tangential direction on the worm (N) force component in the radial direction on the worm (N) force component in the axial direction on the worm (N) normal pressure angle of the worm thread at the mean diameter ( ) efficiency lead angle ( ) helix angle ( ) helix angle ( ) worm helix angle ( )
Worm Gears Chapter Outline 11.1 Introduction 439 11.2 Force Analysis 446 11.3 AGMA Equations 449 11.4 Design Procedure 453 11.5 Conclusions 455 References 456 Further Reading 456 Nomenclature 457
Abstract Worm and wheel gears are widely used for nonparallel, nonintersecting, right angle gear drive system applications where a high transmission gearing ratio is required. In comparison to other gear, belt, and chain transmission elements, worm and wheel gear sets tend to offer a more compact solution. In certain configurations, a worm and wheel gear set can provide sufficiently high friction to be self-locking. This can be a desirable feature if a defined position is required for a gear train, if it is not braked or unpowered. This chapter provides an overview of worms and wheels and outlines a selection procedure.
11.1 Introduction A worm gear is a cylindrical helical gear with one or more threads and resembles a screw thread. A worm wheel or worm gear is a cylindrical gear with flanks cut in such a way as to ensure contact with the flanks of the worm gear. The worm wheel is analogous to a nut that fits on the screw thread of the worm. If the worm is restrained axially within its housing and if the worm is rotated, the worm gear will also rotate. Typical forms for worms and worm gears are shown in Figure 11.1. In a worm and wheel gear set rotary power can be transmitted between nonparallel and nonintersecting shafts. A worm and wheel gear set is typically used when the speed ratio of the two shafts is high, say three or more. Worm and wheel gear sets are used for steering gear, winch blocks, e.g. see Figure 11.2, low-speed gearboxes, rotary tables, and remote valve control. Worm and wheel gear sets are Mechanical Design Engineering Handbook. http://dx.doi.org/10.1016/B978-0-08-097759-1.00011-3 Copyright Ó 2014 Elsevier Ltd. All rights reserved.
439
440 Chapter 11
Figure 11.1 Worm and wheel gear sets.
capable of high-speed reduction and high load applications where nonparallel, non-intersecting shafts are used. The 90 configuration is most common, although other angles are possible. Frictional heat generation is high in worm gears because of the high sliding velocities, so continuous lubrication and provision for heat dissipation are required. The direction of rotation of the worm wheel depends on the direction of rotation of the worm and on whether the worm teeth have a right-hand thread or a left-hand thread. The direction of rotation for a worm and wheel gear sets is illustrated is Figure 11.3. Worms usually have just one tooth and can, therefore, produce gearing ratios as high as the number of teeth on the gear wheel. Herein lies the principal merit of worm and wheel gear sets. In comparison to other gear sets, which are typically limited to a gear ratio of up to 10:1, worm
Figure 11.2 Possible outline winch configuration incorporating a worm and wheel gear set.
Worm Gears 441 Right hand Thrust bearing
Driver
Driver
Left hand
Driver
Driver
Figure 11.3 Rotation and hand relations for worm and wheel gear sets. After Boston Gear Division.
and wheel gear sets can achieve gear ratios of up to 360:1, although most manufacturers quote ranges between 3:1 and 100:1. Ratios above 30:1 generally have one thread on the worm, while ratios below 30:1 tend to have a worm with multiple threads (sometimes referred to as starts). The gear ratio for a worm and wheel gear set is given by mG ¼
NG NW
(11.1)
where mG ¼ gear ratio, NG ¼ number of teeth in the worm gear, and NW ¼ number of threads in the worm. A particular merit of worm and wheel gear sets is their ability to self-lock. If a worm set is self-locking it will not back drive and any torque applied to the worm gear will not rotate the worm. A self-locking worm and wheel gear set can only be driven forward by rotation of the worm. Such a gear set can, therefore, be used to hold a load. This is commonly exploited in a car jack. Whether a worm and wheel gear set will be self-locking depends on frictional contact between the worm and the worm wheel flanks. There are two types of worm and wheel gear sets, depending on whether the teeth of one or both wrap around each other. • •
Single enveloping worm and wheel gear sets, see Figure 11.4, and Double enveloping worm and wheel gear sets, see Figure 11.5.
442 Chapter 11 Pitch diameter d w Helix
Worm Root diameter
Pitch cylinder
Helix angle ψ w Axial pitch lpitch
Lead L Worm gear
Pitch diameter dG
Lead angle λ
Figure 11.4 Nomenclature for a single enveloping worm and wheel gear set.
Worm
3 4 1 2 5
Gear driven by worm
Figure 11.5 Double enveloping worm and wheel gear set.
Worm Gears 443 Rotation of gear A
Line 2 Line 1
Rotation of worm
Line 2 A
Line 1
Center section AA
Figure 11.6 Lines of contact for a worm and wheel gear set.
As the worm rotates through the worm gear, lines of contact either roll or progress from the tip to the root of the worm gear teeth. At any instant in time, there may be two or three teeth in contact and transmitting power as illustrated in Figure 11.6. Some of the key geometric features and dimensions for a worm gear are illustrated in Figure 11.7. The helix angle on a worm is usually high, and the helix angle on the worm wheel is low. Normal convention is to define a lead angle, l, on the worm and a helix angle, jG, on the
FW
do d
a
lpitch dr
ht
b
c c C
FG
Dt Dr Do
Figure 11.7 Worm gear dimensions.
Dm
444 Chapter 11 worm gear. For a 90 configuration, l ¼ jG. The distance that a point on the mating worm gear moves axially in one revolution of the worm is called the lead, L. The following relationships apply to the lead, L, and lead angle, l: L ¼ lpitch NW ¼ tan l ¼
pdG NW NG
L pdW
(11.2) (11.3)
where L ¼ lead (mm), lpitch ¼ worm axial pitch (mm), NW ¼ number of teeth on the worm, dG ¼ pitch diameter of the worm gear (mm), NG ¼ number of teeth on the worm gear, l ¼ lead angle ( ), and dW ¼ pitch diameter of the worm (mm). The worm lead angle and the worm helix angle, jW, are related by l ¼ 90 jW. The lead angle will vary from the root to the outside diameter of the worm, as indicated in Figure 11.8. Generally, self-locking occurs for lead angles below 6 . However, care is necessary in relying solely on self-locking to brake or sustain a load, as vibration has been Lead angle at root diameter Lead angle at outside diameter
Figure 11.8 Variation of the lead angle on a worm gear. Figure after AGMA.
Worm Gears 445 known to result in a reduction of the frictional contact between the worm and wheel for lead angles below 6 and resulting movement or failure of the device. The axial pitch of the worm and the transverse circular pitch of the wheel will be equal for a 90 set configuration, d ¼ mN
(11.4)
The worm can have any pitch diameter, as this is not related to the number of teeth. General guidance for optimum power capacity indicates that the pitch diameter, d, of the worm should fall in the following ranges (AGMA 6022-C93): C 0:875 C 0:875 dmax 1:6 1:07
(11.5)
C 0:875 C 0:875 dmin 3 2
(11.6)
where: C ¼ center distance (mm), d ¼ worm pitch diameter (mm), dmax ¼ maximum worm pitch diameter (mm), and dmin ¼ minimum worm pitch diameter (mm). Dudley (1984) using Imperial units, see also Radzevich (2012), recommends C 0:875 dz 2:2
(11.7)
The pitch diameter of the worm gear, dG, is related to the center distance C and the pitch diameter of the worm, by dG ¼ 2C d
(11.8)
The addendum, a, and dedendum, b, are given by a ¼ 0:3183lpitch
(11.9)
b ¼ 0:3683lpitch
(11.10)
The face width of a worm gear, Figure 11.7, is limited by the worm diameter. The ANSI/ AGMA 6034-B92 recommendation for the minimum face width, for a pitch exceeding 4.06 mm, is given by
446 Chapter 11 FG ¼ 0:67d
(11.11)
The tooth forms for worm and wheel gear sets are not involutes. They are manufactured as matched sets. The worm is subject to high stresses and is normally made using a hardened steel such as AISI 1020, 1117, 8620, 4320 hardened to HRC 58e62, or a medium carbon steel such as AISI 4140 or 4150 induction or flame hardened to a case of HRC 58e62 (Norton, 2006). They are typically ground or polished to a roughness of Ra ¼ 0.4 mm. The worm gear needs to be of softer material that is compliant enough to run-in and conform to the worm under the high sliding running conditions. Sand cast or forged bronze is commonly used. Cast iron and polymers are sometimes used for lightly loaded, low-speed applications. An analysis of the forces associated with a worm and wheel gear set can be undertaken readily, and this is outlined in Section 11.2. Such information is critical in order to enable suitable bearings to be selected for both shafts. Worm and wheel gear sets tend to fail due to pitting and wear (see Maitra (1994) and Dudley (1984)). The American Gear Manufacturer Association power ratings, based on wear and pitting resistance, are presented in Section 11.3 and an associated design procedure in Section 11.4.
11.2 Force Analysis The force exerted on a worm by a gear is illustrated in Figure 11.9 where, for the time being, friction has been neglected. The resultant force, W, will have three components: Wx ¼ W cos fn sin l
(11.12)
Wy ¼ W sin fn
(11.13)
Wz ¼ W cos fn cos l
(11.14)
where Wx ¼ force component in the tangential direction on the worm (N), Wy ¼ force component in the radial direction on the worm (N), Wz ¼ force component in the axial direction on the worm (N), l ¼ lead angle ( ), and fn ¼ normal pressure angle of the worm thread at the mean diameter ( ). Standard pressure angles for worm and wheel gear sets are 14.5, 17.5, 20, 22.5, 25, 27.5, and 30 . The higher the pressure, the higher the tooth strength; albeit at the expense of higher friction, bearing loads, and bending stresses in the worm. The minimum number of worm gear teeth, Nmin, as a function of the pressure angle is listed in Table 11.1.
Worm Gears 447 y
Wy
φ
fWsin λ
W t
φ
n
λ
Wx
fWcos λ x
Wz
Wf =fW
Pitch helx z nW
Pitch cylinder
Figure 11.9 Pitch cylinder of the worm, showing the forces exerted on the worm by the worm gear. Image after Shigley (1986).
As the forces on the worm and worm gear are equal and opposite, the tangential, radial, and axial forces are given by WWt ¼ WGa ¼ Wx
(11.15)
WWr ¼ WGr ¼ Wy
(11.16)
WWa ¼ WGt ¼ Wz
(11.17)
where WWt ¼ tangential force component acting against the worm (N), WWr ¼ radial force component acting against the worm (N), Table 11.1: Suggested minimum number of teeth for the worm (AGMA). Pressure angle ( )
Nmin
14.5 17.5 20 22.5 25 27.5 30
40 27 21 17 14 12 10
448 Chapter 11 WWa ¼ axial force component acting against the worm (N), WGt ¼ tangential force component acting against the gear (N), WGr ¼ radial force component acting against the gear (N), and WWa ¼ axial force component acting against the gear (N). Introducing a coefficient of friction, f, to account for the sliding motion experienced in the motion between a worm thread and the wheel teeth surfaces, Wx ¼ Wðcos fn sin l þ f cos lÞ
(11.18)
Wy ¼ W sin fn
(11.19)
Wz ¼ Wðcos fn cos l f sin lÞ
(11.20)
From Eqns (11.15e11.18) Wf ¼ f W ¼
f WGt f sin l cos fn cos l
(11.21)
WWt ¼ WGt
cos fn sin l þ f cos l f sin l cos fn cos l
(11.22)
The efficiency of a worm and wheel gear set can be defined by h¼
WWt; without friction WWt; with friction
(11.23)
h¼
cos fn f tan l cos fn þ f cot l
(11.24)
or
A typical value for the coefficient of friction for worm gears is f z 0.05. The variation of efficiency with helix angle is given in Table 11.2. Experiments have shown that efficiency for a worm and wheel gear set is a function of the sliding velocity. Taking VG as the pitch line velocity of the gear and VW as the pitch line velocity of the worm, the sliding velocity, Vs, is given by vector addition: VW ¼ VG þ Vs
(11.25)
or Vs ¼
VW cos l
(11.26)
Worm Gears 449 Table 11.2: Worm and wheel efficiency (taking f [ 0.05). j
l
F
fn
h
1 2.5 5 7.5 10 15 20 25 30
1 2.5 5 7.5 10 15 20 25 30
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
20 20 20 20 20 20 20 20 20
24.7 45.0 61.9 70.7 76.1 82.2 85.6 87.5 88.7
where VG ¼ pitch line velocity of the gear (m/s), VW ¼ pitch line velocity of the worm (m/s), and VS ¼ sliding velocity (m/s).
11.3 AGMA Equations Worm sets are generally rated by their capacity to handle a particular level of input power, output power, or allowable torque at a particular speed for the input or output shaft. The AGMA power rating is based on pitting and wear resistance, as this is the usual failure mode for worm sets. The AGMA rating (ANSI/AGMA 6034-B92) is based on 10 h of continuous operation under a uniform load. The input-power rating, Pinput, is given by Pinput ¼ Poutput þ Ploss
(11.27)
where Ploss is the power lost due to friction in the mesh (kW). The output power is given by Poutput ¼ where n ¼ rotational speed of the worm (rpm), Wtg ¼ worm gear tangential force (N), Poutput ¼ output power (kW), mG ¼ gear ratio, and dg ¼ mean gear diameter (mm).
nWtg dg 1:91 107 mG
(11.28)
450 Chapter 11 The power lost is given by Ploss ¼
Vt Wf 1000
(11.29)
where Ploss ¼ lost power (kW), Vt ¼ sliding velocity at the mean worm diameter (m/s), and Wf ¼ friction force (N). The AGMA tangential load on a worm gear is given by Wt ¼
Cs Cm Cv d g0:8 F 75:948
(11.30)
f Wt cos l cos fn
(11.31)
where Cs ¼ materials factor, dg ¼ mean diameter of the gear (mm), F ¼ effective face width (mm), Cm ¼ ratio correction factor, and Cv ¼ velocity factor. The friction force can be determined by Wf ¼ where f ¼ coefficient of friction, Wt ¼ tangential load on the worm gear tooth (N), l ¼ lead angle ( ), and fn ¼ normal pressure angle of the worm thread at the mean diameter ( ). The sliding velocity at the mean worm diameter can be determined by Vt ¼
ndm 19;098 cos l
(11.32)
where n ¼ rotational speed of the worm (rpm), and dm ¼ mean worm diameter (mm). Values for the ratio correction factor, the velocity factor, and materials factors can be found from tables provided in the ANSI/AGMA 6034-B92 standard. The following equations for
Worm Gears 451 the ratio correction factor, the velocity factor, and materials factors provide approximations to the values given in the tables. The ratio correction factor Cm is a function of the gear ratio, mG. For 3 mG 20 0:5 Cm ¼ 0:02 m2G þ 40mG 76 þ 0:46
(11.33)
0:5 Cm ¼ 0:0107 m2G þ 56mG þ 5145
(11.34)
Cm ¼ 1:1483 0:00658mG
(11.35)
Cv ¼ 0:659e0:2165Vt
(11.36)
Cv ¼ 0:652Vt0:571
(11.37)
Cv ¼ 1:098Vt0:774
(11.38)
0:645 f ¼ 0:124eð2:233V t Þ
(11.39)
0:45 f ¼ 0:103eð1:185V t Þ
(11.40)
For 20 mG 76
For mG > 76
For 0 Vt 3.556 m/s
For 3.556 Vt 15.24 m/s
For Vt > 15.24 m/s
For Vt ¼ 0, take f ¼ 0.15. For 0 Vt 0.0508 m/s
For Vt > 0.0508 m/s
The materials factor Cs depends on the method of casting. For C 76.2 mm, an initial estimate for the materials factor can be obtained from Cs ¼ 720 þ 0:000633C3
(11.41)
This value can be compared to the values obtained for the relevant means of casting, as indicated by the following relationships and the smaller value used.
452 Chapter 11 For sand cast gears then: for dm < 63.5 mm, Cs ¼ 1000 for dm > 63.5 mm, Cs ¼ 1859:104 476:5454 log10 dm
(11.42)
For chill cast bronze gears then: for dm < 203.2 mm, Cs ¼ 1000 for dm > 203.2 mm Cs ¼ 2052:011 455:8259 log10 dm
(11.43)
For centrifugally cast gears then: for dm < 635 mm, Cs ¼ 1000 for dm > 635 mm, Cs ¼ 1503:811 179:7503 log10 dm
(11.44)
The efficiency, in percent, for worm gearing is given by Poutput 100 Pinput
(11.45)
nWt dm 100 1:91 107 mG Pinput
(11.46)
h¼ Substituting for the output power h¼ where
Poutput ¼ rated output power (kW), Pinput ¼ rated input power (kW), n ¼ rotational speed of the worm (rpm), Wt ¼ tangential load on the worm gear (N), dm ¼ mean diameter of the gear (mm), mG ¼ gear ratio.
Worm Gears 453
11.4 Design Procedure An outline design procedure for a worm and wheel gear set using the AGMA equations is listed next. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Define the number of starts. Define the center distance, C. Determine a suitable worm gear diameter. Determine the lead. Determine the lead angle. Determine the maximum recommended face width, F. Determine the material factor, Cs. Determine the ratio correction factor, Cm. Determine the tangential velocity, Vt. Determine the velocity factor, Cv. Determine the tangential load, Wt. Determine the coefficient of friction, f. Determine the friction force, Wf. Determine the output power, Poutput. Determine the power lost in mesh, Ploss. Determine the rated input power, Pinput. Estimate the efficiency of the gear set, h. Determine the output torque, Tq. Establish whether the power rating and output torque are sufficient for the application. If not, alter the number of starts, worm diameter, center distance, etc. to provide suitable power rating and output torque.
Example 11.1 Develop a design for a worm and wheel gear set. The prime mover is an electric motor running at 1470 rpm. A reduction ratio of 50:1 is required. The peak torque required is 850 N m. Assume sand cast gears. Solution Input speed ¼ 1470 rpm. Ratio 50:1. Output ¼ 29.4 rpm. Sand cast gears. If a single start worm is specified, then a 50 tooth worm gear will be needed to give a ratio of 50:1.
454 Chapter 11 Fifty teeth are above the minimum recommended (see Table 11.1). An estimate for the center distance needs to be made. If the center distance between the worm and wheel is taken as 140 mm, then from Eqns (11.5) and (11.6), the pitch diameter range is found to be between 20.72 and 70.55 mm. A midvalue of 50 mm is selected here. From Eqn (11.8), dG ¼ 2C d ¼ ð2 140Þ 50 ¼ 230:0 mm The lead, from Eqn (11.2), is L¼
pdG NW 1 ¼ p 230:0 ¼ 14:45 mm NG 50
The lead angle, from Eqn (11.3), L 1 1 14:45 l ¼ tan ¼ tan ¼ 5:256 pdw p 50 Generally, self-locking occurs for lead angles below 6 . This is less than 6 so the worm set will be self-locking. The face width can be determined from Eqn (11.11) FG ¼ 0:67d ¼ 0:67 50 ¼ 33:5 mm The materials factor for sand cast gears is given by Eqn (11.42), if the mean worm diameter is greater than 63.5 mm. Here, the mean worm diameter is 50 mm so Cs ¼ 1000 mG ¼ 50: From Eqn (11.34), Cm ¼ 0:7896 The tangential velocity at the mean worm diameter can be determined from Eqn (11.32), Vt ¼
nd 1470 50:0 ¼ ¼ 3:865 m=s 19; 098 cos l 19;098 cos 5:256
The velocity factor, from Eqn (11.37), is Cv ¼ 0:3013 From Eqn (11.30), the tangential load is given by Wt ¼
1000 0:7896 0:3013 2300:8 33:50 ¼ 8134 N 75:948
Worm Gears 455 The coefficient of friction is given by Eqn (11.40) 0:45 0:45 m ¼ 0:103eð1:185Vt Þ ¼ 0:103eð1:1853:865 Þ ¼ 0:01167
The friction force is given by Wf ¼
f Wt 0:01167 8134 ¼ ¼ 101:5 N cos l cos f cos 5:256 cos 20
The rated output power, Eqn (11.28), is Poutput ¼
nWtg dg 1470 8134 230 ¼ ¼ 2:880 kW 7 1:91 10 mG 1:91 107 50
The lost power, Eqn (11.29), is Ploss ¼
Vt Wf 3:865 101:5 ¼ ¼ 0:392 kW 1000 1000
The input-power rating is 2.880 þ 0.392 ¼ 3.272 kW. The efficiency of the gear set is 2.880/3.272 ¼ 0.881 ¼ 88.1%. The output torque is given by Tq ¼ Wt
d 0:230 ¼ 8134 ¼ 935:4 N m 2 2
This torque exceeds the requirement, suggesting that the design is suitable. If the value was below that required, then an alternative center distance could be explored and, if necessary, the number of starts could be increased.
11.5 Conclusions Worm and wheel gears are usually used for nonparallel nonintersecting right angle gear drive system applications, where a high gear ratio is required. They can also be used for medium-speed reductions. The worm is generally the driving member. The high transmission ratio leads to a compact solution for many applications in comparison to other types of gearing. For certain arrangements, self-locking is possible, and this can provide an attribute for some applications where a set position is desirable if the drive train is not braked or powered.
456 Chapter 11
References Books and Papers Dudley, D.W., 1984. Handbook of Practical Gear Design. McGraw Hill. Maitra, G.M., 1994. Handbook of Gear Design, second ed. Tata McGraw Hill. Norton, R.L., 2006. Machine Design, third ed. Pearson. Radzevich, S.P., 2012. Dudley’s gear handbook of practical gear design and manufacture, 2nd ed. CRC Press. Shigley, J.E., 1986. Mechanical Engineering Design, first metric ed. McGraw Hill.
Further Reading Shigley, J.E., Mischke, C.R., Budynas, R.G., 2004. Mechanical Engineering Design, seventh ed. McGraw Hill. Townsend, D.P., 1992. Dudley’s Gear Handbook, second ed. McGraw Hill.
Standards AGMA Design manual for cylindrical wormgearing. ANSI/AGMA Standard 6022-C93. Reaffirmed 2008. AGMA Practice for enclosed cylindrical wormgear speed reducers and gearmotors. ANSI/AGMA Standard 6034-B92. Reaffirmed 2005. BS 721-1:1963. Specification for worm gearing. Imperial units. BS 721-2:1983. Specification for worm gearing. Metric units. BS ISO TR 10828:1997. Worm gears. Geometry of worm profiles. PD ISO/TR 14521:2010. Gears. Calculation of load capacity of wormgears.
Web Sites At the time of going to press the world-wide-web contained useful information relating to this chapter at the following sites: www.ashokaengineering.com/ www.bandhgears.co.uk www.bellgears.co.uk www.bostongear.com/products/open/worms.html www.brentwingearcompany.co.uk www.davall.co.uk www.delroyd.com www.gearcutting.com www.gearmanufacturer.net/ www.girard-transmissions.com www.hewitt-topham.co.uk/ www.hopwoodgear.com www.hpcgears.com www.huco.com www.mmestrygears.com www.muffettgears.co.uk/ www.qtcgears.com/ www.rarodriguez.co.uk www.traceygear.com www.wmberg.com
Worm Gears 457
Nomenclature The following symbols have been used in this chapter. Generally, preferred SI units have been stated: a b c C Cm Cs Cv d dg dG dm dmax dmin do dr dW Dm Do Dr Dt f F FG FW ht lpitch L mG n NG Nmin NW Pinput Ploss Poutput
addendum (mm) dedendum (mm) clearance (mm) center distance (mm) ratio correction factor materials factor velocity factor worm pitch diameter (mm) mean diameter of the gear (mm) pitch diameter of the worm gear (mm) mean diameter of the worm gear (mm) maximum pitch diameter of the worm (mm) minimum pitch diameter of the worm (mm) worm outer diameter (mm) worm root diameter (mm) pitch diameter of the worm (mm) mean gear diameter (mm) worm gear outer diameter (mm) worm gear root diameter (mm) worm gear throat diameter (mm) coefficient of friction effective face width (mm) worm gear face width (mm) worm face width (mm) full depth of worm thread (mm) worm axial pitch (mm) lead (mm) gear ratio rotational speed of the worm (rpm) number of teeth on the worm gear minimum number of worm gear teeth number of threads on the worm rated input power (kW) lost power (kW) rated output power (kW)
458 Chapter 11 Ra Tq VG Vs Vt VW W Wf WGr WGt Wt Wtg WWa WWa WWr WWt Wx Wy Wz fn h l j jG jW
roughness (mm) torque (N m) pitch line velocity of the gear (m/s) sliding velocity (m/s) sliding velocity at the mean worm diameter (m/s) pitch line velocity of the worm (m/s) resultant force (N) friction force (N) radial force component acting against the gear (N) tangential force component acting against the gear (N) tangential load on the worm gear (N) worm gear tangential force (N) axial force component acting against the gear (N) axial force component acting against the worm (N) radial force component acting against the worm (N) tangential force component acting against the worm (N) force component in the tangential direction on the worm (N) force component in the radial direction on the worm (N) force component in the axial direction on the worm (N) normal pressure angle of the worm thread at the mean diameter ( ) efficiency lead angle ( ) helix angle ( ) helix angle ( ) worm helix angle ( )