Chapter 10 Helical Gears, Bevel Gears, and Wormgearing

The Big Picture

• The geometry of helical gears, bevel gears, and wormgearing was described in Chapter 8 (p. 289).

• The principles of stress analysis of gears were discussed in Chapter 9 for spur gears. Much of that information is applicable to the types of gears discussed in this chapter.

In the chapter, you acquire the skills to perform the necessary analyses to design safe gear drives that use helical gears, bevel gears, and wormgearing and that demonstrate long life.

• Much was introduced in chapter 8 and 9 about the kinematics of gears and about the stress analysis and design of spur gears. That information is important to the objectives of this chapter, in which we extend the application of those concepts to the analysis and design of helical gears,bevel gears, and wormgearing.

The basic geometry of helical gears was described in chapter 8-7. The force system on helical gear teeth was also described, and that system on helical gear teeth was also described, and that system is important to you understanding of the stresses and modes of potential failure for helical gears that we discuss in this chapter.

• In chapter 9, you learned how to analyze spur gears for bending strength and resistance to pitting of the surface of the teeth.

• This chapter modifies that same approach for application to the special geometry of helical gears. So we need to refer to chapter 9 from time to time.

• Similarly, the geometry of bevel gears was described in section8-8, and wormgearing was described in section 8-9 and 8-10.

• This chapter includes information about stresses in bevel gears and wormgearing.

• Figure 10-1 shows an example of a large,commercially available,double-reduction, parallel-shaft reducer employing helical gears.

• Figures 10-2 shows another form of helical gear reducer in which the drive motor is mounted above the reducer, and the shaft of the driven machine is inserted directly through the hollow output shaft of the reducer. This allows the reducer to be supported by the frame of the driven machine.

• Refer to figure 8-22 for drawings and photographs of bevel gear systems. pp290

• Refer to figure 8-24 for a photograph if a commercially available wormgear reducer. pp295

• The lubrication of the gears is very important because sliding action between the worm threads and the wormgear teeth inherently occurs, causing frictional heat to be generated.consistent lubrication of the gear mesh is critical to the performance, efficiency, and life of the system.

• This chapter presents more information about these factors in wormgearing system design. The design for strength and pitting resistance is presented for each type of gear,with an analysis of the geometry and the forces exerted on the gears.

10-1 objectives of this chapter

• describe the geometry of helical gears and compute the dimensions of key features.

• Compute the forces exerted by one helical gear on its mating gear.

• Compute the stress due to bending in helical gear teeth and specify suitable materials to withstand such stresses.

• Design helical gears for surface durability.• Describe the geometry of bevel gears and compute the

dimensions of key features.• Analyze the forces exerted by one bevel gear on another

and show how those forces are transferred to the shafts carrying the gears.

• Design and analyze bevel gear teeth for strength and surface durability.

• Describe the geometry of worms and wormgears.

• Compute the forces created by a wormgear drive system and analyze their effect on the shafts carrying the worm and the wormgear.

• Compute the efficiency of wormgear drives.

• Design and analyze wormgear drives to be safe for bending strength and wear.

Reference 3,4,7,8 and 17-22 are recommended for general guidelines for design and application of helical gears,bevel gears, and wormgearing.

10-2 Forces on helical gear teeth

• Figure 10-3 shows a photograph of two helical gears in mesh and designed to be mounted on parallel shafts. pp.396

• Figure 10-4 for a representation of the force system that acts between the teeth of two helical gears in mesh. pp396

斜齿轮受力分析

受力简图的平面表示方法

We defined following forces:

• WN is the true normal force that acts perpendicular to the face of the tooth in the plane normal to the surface of the tooth. The normal plane is shown in Part(d) of figure 10-4. pp.396

• We seldom need to use the value of WN because its three orthogonal components.

• Normal pressure angle:n;

• Transverse pressure angle: t;

• Helix angle:

For helical gears, the helix angle and one of the other two are specified. The third angle can be computed from:

1)-(10 costantan tn

Wt is the tangential force that acts in the transverse plane and tangent to the pitch circle of the helical gear and that causes the torque to be transmitted from the driver the driven gear. Therefore, this force is often called the transmitted force.

If the torque being transmitted (T) and the size of the gear (D) are known, we can compute Wt as follows:

2)-(10 )2/(DTWt

斜齿轮传动的啮合区

斜齿轮齿面上的接触线

If the power being transmitted(P) and the rotational speed (n) are known,torque (T) can be computed as follows:

• For the unit-specific situation where power is expressed in horsepower and the rotational speed is in r·p·m, the torque in lb·in is

3)-(10 )/( nPT

6)-(10 /)(33000W

5)-(10 )])(/[()(126000)]2/)(/[()(63000W

as expressed be alsocan force l tangentiaThen the

4)-(10 /)(63000

t

t

tvP

DnPDnP

nPT

• The value of the tangential load is the most fundamental of the three orthogonal components of the true normal force. The calculation of the bending stress number and the pitting resistance of the gear teeth depends on Wt.

• Wr is the radial force that acts toward the center of the gear perpendicular to the pitch circle and to the tangential force.It tends to the two gears apart. As can be seen in figure10-4(c),

teethhelical for the angle pressure where

7)-(10 tan

t transverse

WW ttr

• Wx is the axial force that acts parallel to the axis of the gear and causes a thrust load that must be resisted by the bearings carrying the shaft. With the tangential force known, the axial force is computed from

8)-(10 tantx WW

10-3 stresses in helical gear teeth

• We will use the same basic equation for computing the bending stress number for helical gear teeth as we did for spur gear teeth in chapter 9, given in equation(9-15) and repeated here:

15)-(9 vBmsodt

t KKKKKFJ

pWs

Figures 10-5,10-6 and 10-7 show the values for the geometry factor,J,for helical gear teeth with 15°,20°,and 22° normal pressure angles,respectively. The K factors are the same as those used for spur gears.

10-4 pitting resistance for helical gear teeth

• Pitting resistance for helical gear teeth is evaluated using the same procedure as that discussed in chapter 9 for spur gears. Equation (9-25) is repeated here:

25)-(9 IFD

KKKKWCS

p

vmsotpc

All of the factors are the same for helical gears excepted the geometry factor for pitting resistance,I. The values for Cp are found in table 9-10. Note that the other K factors have the same values as the K factors discussed and identified in section 10-3.

• Because of the larger variety of geometry features needed to define the form of helical gears, it is not reasonable to reproduce all of the necessary tables of values or the complete formulas for computing I. Values change with the gear ratio,the number of teeth in the pinion, the tooth form,the helix angle, and the specific values for addendum,whole depth, and fillet radius. See references 6 and 13 for extensive discussions about the procedures. To facilitate problem solving in this book, tables 10-1 and 10-2 give a few values for I.

• Figure 9-9:Steel,through-hardened,Grades1 and 2

• Tables 9-3: Steel, case-hardened,Grades 1,2 and 3; flame- or induction-hardened, carburized, or nitrided

• Table 9-4: Cast iron and bronze

For design,when the computed contact stress number is known, a material must be specified that has as allowable contact stress number,Sac, greater than Sc. Design values for Sac can be found from the following:

10-5 design of helical gears

• Examples problem 10-2 illustrates the procedure to design helical gears.

10-6 forces on straight bevel gears

• Review section 8-8 and figure8-22 for the geometry of bevel gears. Also see references 1,5, and 12.

• Because of the conical shape of bevel gears and because of the involute-tooth form, a three-component set of forces acts on bevel gear teeth. Using notation similar to that for helical gears, we will compute the tangential force, Wt; the radial force, Wr; and the axial force, Wx. It is assumed that the three forces act concurrently at the midface of the teeth and on the pitch cone (see figure 10-8). Although the actual point of application of the resultant force is a little displaced from the middle, no serious errors results.

直齿圆锥齿轮传动的几何参数

直齿圆锥齿轮的受力

• The tangential force acts tangential to the pitch cone and is the force that produces the torque on the pinion and the gear. The torque can be computed from the known power transmitted and the rotational speed:

nPT /63000Then, using the pinion, for example, the transmitted load is

end. large itsat tooth theof linepitch the

tomeasured is d,diameter,pitch hat theremember t

10)-(10 sin)2/(2/r

from computed becan r of valuethe

pinion theof radius r where

9)-(10 /

m

m

m

Fd

mean

rTW mtp

• The radial load acts toward the center of the pinion, perpendicular to its axis,causing bending of the pinion shaft. Thus,

gear. theof radiusmean the

toequal axial thefrom distance at the actsit

becauseshaft on themoment bending a produces

alsoit bearings.shaft on the load thrust a causesit

gear. mating thefromaway it push totending

pinion, theof axis the toparallel acts load axial the

11)-(10 costan trp WW

direction. and magnitude

both in gear theandpinion on the forces the

between ipsrelationsh for the 8-10 figure refer to

pinion.for that dsubstitute isgear for thegeometry

theif pinion, for the hereshown equations same the

by calculated becan gear on the forces for the valuethe

12)-(10 sintanW

Thus,

xp tW

10-7 bearing forces on shafts carrying bevel gears

• Because of the three-dimensional force system that acts on bevel gears, the calculation of the forces on shaft bearings can be cumbersome. An example is worked out here to show the procedure. In order to obtain numerical data, the arrangement shown in figure 10-9 is proposed for the bevel gear pair that was the subject of example problem 8-6 and 10-3. The locations for the bearings are given with respect to the apex ( 顶点 ) of the two pitch cones where the shaft axes intersect.

• Note that both the pinion and the gear are straddle-mounted; that is, each gear is positioned between the supporting bearings. This is the most preferred arrangement because it usually provides the greatest rigidity and maintains the alignment of the teeth during the power transmission. Care should be exercised to provide rigid mountings and stiff shafts when using bevel gears.

• The arrangement of figure 10-9 is designed so that the bearing on the right resists the axial thrust load on the pinion, and the lower bearing resists the axial thrust load on the gear.

10-8 bending moments on shafts carrying bevel gears

• Because there are forces acting in two planes on bevel gears, as discussed in the preceding section, there is also bending in two planes. The analysis of the shearing force and bending moment diagrams for the shafts must take this into account.

Figures 10-11 and 10-12 show the resulting diagrams for the pinion and the gear shafts, respectively, for the gear pair used for example problems 9-6, 10-3, and 10-4. Notice that the axial thrust load on each shaft provides a concentrated moment to the shaft equal to the axial load times the distance that it is offset from the axis of the shaft. Also notice that the maximum bending moment for each shaft is the resultant of the moments in the two planes.

• On the pinion shaft, the maximum moment is 240 lb·in at E, where the lines of action for the radial and tangential forces intersect the shaft. Similarly, on the gear shaft, the maximum moment is 404 lb·in at F. these data are used in the shaft design ( as discussed in chapter 12).

10-9 stresses in straight bevel gear teeth

• The stress analysis for bevel gear teeth is similar to that already presented for spur and helical gear teeth. The maximum bending stress occurs at the root of the teeth in the fillet. This stress can be computed from

13)-(10 0

v

msdtt K

KKK

FJ

PWS

The terms have all been used before. But there are minor differences in the manner of evaluating the factors, so they will be reviewed here.

Tangential force, Wt

• Contrary to the way Wt was computed in the preceding section, we compute it here by using the diameter of the gear at its large end, rather than the diameter at the middle of the tooth. This is more convenient, and the adjustment for the actual force distribution on the teeth is made in the value of the geometry factor, J. then

(in) end large itsat pinion theofdiameter pitch d

(rpm)pinion theof speed n

(hp) smittedpower tranP

pinion(in) theof radiuspitch r

in)(lb ed transmittT

where

14)-(10 2/

1)(63000

P

rotational

torque

dn

P

r

TW

pt

Dynamic factor,Kv

• The values for the dynamic factor for bevel gears are different from those for spur or helical gears. Factors affecting the dynamic factor include the accuracy of manufacture of the gear teeth (quality number Q); the pitch line speed,vt; the tooth load; and the stiffness of the teeth. AGMA Standard 2003-A86 recommends the following procedure for computing Kv for the bending strength calculation and Cv for pitting resistance:

)(1085

125

)2(

8u

where

15)-(10

5.0

uK

EEs

vK

KKC

z

GpatQ

u

tz

zvv

• If the equation for u results in a negative value, use u=0. As a check on the specification of an appropriate value for the quality number, a minimum value for Cv should be computed from

)333/(tan2 1

min tv vC

The value of the result of the inverse tangent calculation must be in radians. If the actual value of Cv is less than Cvmin, a higher number should be specified.

Size factor,Ks

• Use the values from table 9-7.

Load-distribution factor, Km

• Values are highly dependent on the manner of mounting both the pinion and the gear. The preferred mounting is called straddle mounting, in which the gear is between its supporting bearings. Figure 10-9 shows straddle mounting of both the pinion and the gear. In addition, stiff, short shafts are recommended to minimize shaft deflections that cause misalignment of the gear teeth.

Geometry factor,J

• Use figure 10-13 if the pressure angle is 20° and the shaft angle is 90°.

The computed value for stress from equation(10-13) can be compared with the allowable bending stress number from tables 9-3 and 9-4 and figure 9-8. A life factor,YN, or a reliability factor,KR, can be applied as discussed in chapter 9 if the design life is different from 107 cycles or if the desired reliability is different from 0.99.

10-10 design of bevel gears for pitting resistance

• The approach to the design of bevel gears for pitting resistance is similar to that for spur gears. The failure mode is fatigue of the surface of the teeth under the influence of the contact stress between the mating gears.

• The contact stress,called the Hertz stress,Sc, can be computed from

10.-9 tableand

23)-equation(9in shown as same theis andt coefficien elastic theis

Cfactor also.Themeanings, same thehave d andF, Wt, termsthe

section. preceding in the stress bending thecomputingfor

, ,,K as same theare ,,Cfactor the

16)-(10

p

00

0

used

KandKCandC

C

CC

FDI

WCCS

mvmv

v

mtbPc

• For steel or cast iron gears.

)1030(E

gear iron cast a andgear steel afor 2100

)1030(E gearscast for two 1960

)1030(E gears steel for two 2300

6

6

6

psi

C

psiC

psiC

p

p

p

4.-9 in table values theuse iron,cast for

steel. is material theif 3-9 or table 9-9 figure from Sac,

number, stresscontact allowable with thecompared be should

16)-0equation(1 from computedstress,contact hertz the

14.-10 figure from found becan and

durability surfacefor factor geometry theis I

factor thegears. helical andspur for as stresscontact

allowable same theof use theallows 634.0 sin bCgu

Practical consideration for bevel gearing

• Factors similar to those discussed for spur and helical gears should be considered in the design of systems using bevel gears. The accuracy of alignment and the accommodation of thrust loads discussed in the example problems are critical. Figures 10-15 and 10-16 show commercial applications.

10-11 forces,friction, and efficiency in wormgear sets

• See chapter 8 for the geometry of wormgear sets. Also see references 2,14,15, and 17.

• The force system acting on the worm/wormgear set is usually considered to be made of three perpendicular components as was done for helical and bevel gears. There are a tangential force, a radial force, and an axial force acting on the worm and the wormgear. We will use the same notation here as in the bevel gear system.

• Figure 10-17 shows two orthogonal views (front and side) of a worm/wormgear pair, showing only the pitch diameter of the gears. The figure shows the separate worm and wormgear with the forces acting on each.

Note that because of the 90°orientation of the two shafts,

17)-(10 rWrG

tWxG

xWtG

WW

WW

WW

Of course, the directions of the paired forces are opposite because of the action/reaction principle.

the tangential force on the wormgear is computed first and is based on the required operating conditions of torque,power, and speed at the output shaft.

Coefficient of friction,• Friction plays a major part in the operation of a wormgear

set because there is inherently sliding contact between the worm threads and the wormgear teeth. The coefficient of friction is dependent on the materials used, the lubricant, and the sliding velocity. Based on the pitch line speed of the gear, the sliding velocity is

19)-(10 cos/v

worm, theof speed linepitch on the based

18)-(10 sin/

s

tW

tGs

v

vv

• The AGMA(see reference 15) recommends the following formulas to estimate the coefficient of friction for a hardened steel worm (58HRC minimum),smoothly ground, or polished, or rolled, or with an equivalent finish, operating on a bronze wormgear. The choice of formula depends on the sliding velocity.

• Note: Vs must be in ft/min in the formulas;

• 1.0ft/min = 0.0051 m/s.

velocity sliding theversus

t coefficien theofplot a is 18-10 figure

21)-(10 0.012103.0

min/10 v:speedhigher

20)-(10 124.0

)/051.0min(/10 v:speed low

0.150

0v:Condition

S

)110.0(

S

)074.0(

S

S

450.0

645.0

S

S

V

V

e

ft

e

smft

Static

Output torque from wormgear drive,T0

• In most design problems for wormgear drives,the output torque and the rotating speed of the output shaft will be known from the requirements of the driven machine. Torque and speed are related to the output power by

Gn

PT

)(63000 00

• By referring to the end view of the wormgear in figure 10-17, you can see that the output torque is

system. drivemgear worm/worain acting

forces thecompute toused becan procedure following the

)2/(0

then

DWrWT GrGGtG

Procedure for calculating the forces on a worm/wormgear set • Given:• Output torque,T0,in lb·in• Output speed,nG,in rpm• Pitch diameter of the wormgear,DG, in inches• Lead angle,• Normal pressure angle,n• Compute:

• 2 ．失效形式• 蜗杆传动的主要失效形式有：点蚀（齿面接触疲劳破坏）、齿根折断（轮齿弯曲

疲劳）、齿面胶合及过度磨损等。• 由于材料和结构上的原因，蜗杆螺旋齿部分的强度总是高于蜗轮轮齿的强度，所

以失效经常发生在蜗轮轮齿上。因此，一般只对蜗轮轮齿进行承载能力计算。• （ 1 ）点蚀• 在闭式传动中，多因齿面胶合或点蚀而失效。因此，通常是按齿面接触疲劳强度

进行设计，而按齿根弯曲疲劳强度进行校核。• （ 2 ）折断• 而在开式传动中，多发生轮齿折断或齿面磨损，因此应以保证齿根弯曲疲劳强度

作为开式传动的主要设计准则。• （ 3 ）胶合• 由于蜗杆与蜗轮齿面间有较大的相对滑动，从而增加了产生胶合和磨损失效的可

能性，因此，蜗杆传动的承载能力往往受到抗胶合能力的限制。尤其在润滑不良等条件下，蜗杆传动因齿面胶合而失效的可能性更大。在闭式传动中，由于散热较为困难，通常应作热平衡核算。

• （ 4 ）磨损• 由上述蜗杆传动的失效形式可知，蜗杆、蜗轮的材料不仅要求具有足够的强

度，更重要的是要具有良好的磨合和耐磨性能。因此，必须选用适当的材料。

• Forces analysis

当不计摩擦力的影响时，各力的大小可按下列各式计算：

coscos

2

coscoscoscos

tan

2

2

2

221

221

2

221

1

121

nn

t

n

an

trr

ta

at

d

TFFF

FFF

d

TFF

d

TFF

式中， Tl 、 T2 分别为蜗杆及蜗轮上的公称转矩； dl 、 d2 分别为蜗杆及蜗轮的分度圆直径。

20. referencein shown is equations theoft developmen complete the

teeth. wormgear and threads wormmeshing theoflocation at the

forcefriction theand wormgear on the force driving

l tangentiaboth the of scomponenet theusing derived were

23)-(10 and 22)-10equations(17),-0equation(1 using

n,observatioby obtained becan wormon the forces the

23)-(10 sincoscos

sinW

22)-(10 sincoscos

cossincos

/2

rG

0

n

ntG

n

ntGxG

GtG

W

WW

DTW

Friction force,Wf

• The friction force, Wf ,acts parallel to the face of the worm threads and the gear teeth and depends on the tangential force on the gear, the coefficient of friction, and the geometry of the teeth:

24)-(10 ))(cos(cos n

tGf

WW

Power loss due to friction, PL

Power loss is the product of the friction force and the sliding velocity at the mesh. That is ,

33000fS

L

WP

In this equation, the power loss is in hp,s is in ft/min, and Wf is in in·lb.

Input power,Pi

• The input power is the sum of the output power and the power loss due to friction:

Li PPP 0

Efficiency,

• Efficiency is defined as the ratio of the output power to the input power:

iPP /0

• 闭式蜗杆传动的功率损耗一般包括三部分：啮合损耗1、轴承摩擦损耗2及浸入油池中的零件搅油时的溅油损耗3。因此总效率为• （6-29）

• 在蜗杆传动中，轴承摩擦及溅油这两项功率损耗不大，一般可取23=0.95~0.96。一般以啮合损耗的效率1计算为主。当蜗杆主动时，啮合损耗的效率1按下式计算：• （6-30）• 式中，为普通圆柱蜗杆分度圆柱上的导程角；v为当量摩擦角，其值可根据蜗杆和蜗轮间的相对滑动速度vs从附表7-8中选取。• 蜗杆和蜗轮间的相对滑动速度vs由图6-22得• （6-31）• 式中，v1和v2分别为蜗杆、蜗轮分度圆的圆周速度。

321

)tan(

tan1

v

sincos21 vv

vs

)tan(

tan)96.0~95.0(

v321

在设计之初，为了近似地求出蜗轮轴上的扭矩 T2，值可如下估取

蜗杆头数 z1

1 2 4 6

总效率 0.7 0.8 0.9 0.95

Factors affecting efficiency

• As can be seen in equation(10-24), the lead angle, the normal pressure angle, and the coefficient of friction all affect the efficiency. The one that has the largest effect, and the one over which the designer has the most control, is the lead angle,. The larger the lead angle, the higher the efficiency, up to approximately =45°. Now,looking back to the definition of the lead angle,note that the number of threads in the worm has a major effect on the lead angle. Therefore, to obtain a high efficiency, use multiple-threaded worms. But there is a disadvantage to this conclusion. More worm threads require more gear teeth to achieve the same ratio, resulting in a larger system overall. The designer is often forced to compromise.

Self-locking wormgear sets

• Self-locking is the condition in which the worm drives the wormgear,but, if torque is applied to the gear shaft, the worm does not turn.

• It is locked!• The locking action is produced by friction force between t

he worm threads and the wormgear teeth, and this is highly dependent on the lead angle.

• It is recommended that a lead angle no higher than about 5.0° be used in order to ensure that self-locking will occur. This lead angle usually requires the use of a single-threaded worm; the low lead angle results in a low efficiency,possibly as low as 60% or 70%.

10-12 stress in wormgear teeth

• We present here an approximate method of computing the bending stress in the teeth of the wormgear. Because the geometry of the teeth is not uniform across the face width, it is not possible to generate an exact solution. However, the method given here should predict the bending stress with sufficient accuracy to check a design because most worm/wormgear systems are limited by pitting ,wear, or thermal considerations rather than strength.

• Only the wormgear teeth are analyzed because the worm threads are inheretly stronger and are typically made from a stronger material.

The stress in the gear teeth can be computed from

29)-(10 gear theof speed line 12/v

28)-(10 )1200/(1200K

and

27)-(10 /W

from estimated becan load dynamic the

26)-(10 /Pcospcospitch cricular P

gear theof width faceF

4)-10 table(seefactor form lewisy

gear teeth on the load W

where

25)-(10

tG

v

d

dn

d

pitchnD

v

KW

normal

dynamic

yFP

W

GG

tG

vtG

n

d

Only one value is given for the lewis form factor for a given pressure angle because the actual value is very difficult to calculate precisely and does not vary much with the number of teeth. The actual face width should be used, up to the limit of two-thirds of the pitch diameter of the worm.

• The computed value of tooth bending stress from equation (10-25) can be compared with the fatigue strength if the material of the gear. For manganese gear bronze, use a fatigue strength of 17000 psi; for phosphor gear bronze, use 24 000 psi. for cast iron, use approximately 0.35 times the ultimate strength, unless specific data are available for fatigue strength.

10-13 surface durability of wormgear drives

• AGMA Standard 6034-A87(see reference 15) gives a method for rating the surface durability of hardened steel worms operating with bronze gears. The ratings are based on the ability of the gears to operate without significant damage from pitting or wear.

The procedure calls for the calculation of a rated tangential load,WtR,from

21)-10 figure (fromfactor C

20)-10 figure (fromfactor correction

.D 0.67 of maximum a toup wormgear theofwidth

face actual theuse inches.in width,face F

inchesin wormgear, theofdiameter D

19)-10 figure (fromfactor C

where

30)-(10

v

W

e

G

s

8.0

velocity

ratioC

effective

pitch

material

CCFDCW

m

vmeGstR

• 2 ．蜗杆传动的热平衡计算• 由于蜗杆传动效率低，所以工作时发热量大。如果产生的热量不能

及时散逸或排出，必将导致温度明显升高。温度的升高会使润滑油稀释，从而加大摩擦表面的接触机会，增加啮合摩擦损耗。同时，温度过高还会导致胶合的发生。所以，对蜗杆传动而言，必须进行热平衡计算，以保证油温处于允许的范围内。

• 由于摩擦损耗的功率 Pf=P（ 1- ），其产生的热流量为• H1=1000P （ 1- ） w

（ 6-33 ）• 式中， P为蜗杆传递的功率， Kw 。• 以自然冷却方式，从箱体外壁散发到周围空气中去的热量为• H2=dS （ t0-ta ） w

（ 6-34 ）

• 式中， d为箱体的表面传热系数，可取 d=(8.15~17.45( W/(m2 )℃ ，当周围空气流通良好时，取偏大值； S为散热面积， m2 ； t0 为润滑油工作温度； ta为室温。

• 根据单位时间内的发热量H1 等于同时间内的散热量 H2 的条件，可求得工作油温为

• ℃（ 6-35 ）

• 或保持正常工作温度所需要的散热面积为• m2

（ 6-36 ）• 润滑油工作温度 t0 一般限制在 60~70℃，最高不应超过 80℃。当 t0>80℃或有

效的散热面积不足时，为提高散热能力须采取以下措施。• 1 ）加散热片以增大散热面积（图 6-23 ）；• 2 ）在蜗杆轴端加装风扇以加速空气的流通（图 6-23 ）；• 3 ）在传动箱内装循环冷却管路（图 6-24 ）。

加散热片和风扇的蜗杆传动

图 6-24 装有循环冷却管路的蜗杆传动

)1)(( Ff PPP

在蜗杆轴端加装风扇会增加功率损耗，因此总的功率损耗为

式中， PF 为风扇消耗的功率，可按下式估算。

5

3

10

5.1 FF

vP

式中， vF 为风扇叶轮的圆周速度，可按下式计算：

60000FF

F

nDv

• 其中， DF为风扇叶轮外径， mm ； nF为风扇叶轮转速， rpm 。• 由摩擦消耗的功率所产生的热流量为

)1)((10001 FPPH散发到空气中的热流量为

))(( 021'

2 add ttSSH 式中， Sl 、 S2 分别为风冷面积及自然冷却面积， m2 ； d’

为风冷时的表面传热系数，按附表 7-9 选取。

Conditions on the use of equation (10-30)

• 1. The analysis is valid only for a hardened steel worm(58 HRC minimum) operating with gear bronze specified in AGMA Standard 6034-A87.

• The classes of bronzes typically used are tin bronze, phosphor bronze, maganese bronze, and alumimum bronze. The material factors,Cs, is dependent on the method of casting the bronze, as indicated in figure 10-19. The values for Cs can be computed from the following formulas.

Sand-cast bronzes:

1000C

in, 5.2Dfor

31)-(10 )(log545.476636.1189C

in, 5.2D

s

G

10s

G

GD

for

S

Ptt

da

)1(10000

)(

)1(1000

0 ad tt

PS

Static-chill-cast or forged bronzes:

1000C

in, 0.8Dfor

32)-(10 )(log825.455651.1411C

in, 0.8D

s

G

10s

G

GD

for

Centrifugally Cast Bronzes:

1000C

in, 25Dfor

33)-(10 )(log750.179291.1251C

in, 25D

s

G

10s

G

GD

for

• 2. The wormgear diameter is the second factor in determining Cs. The mean diameter at the midpoint of the working depth of the gear teeth should be used. If standard addendum gears are used, the mean diameter is equal to the pitch diameter.

• 3. Use the actual face width,F, of the wormgear as Fe if F<0.667 (DW). For larger face widths,use Fe=0.67(DW),because the excess width is not effective.

4. Ratio correction factor,Cm, can be computed from the following formulas.

• For gear ratios,mG,from 6 to 20

36)-(10 00658.01483.1

76mfor

35)-(10 )514556(0107.0C

76 to2 ,mratios,gear for

34)-(10 46.0)7640(0200.0

G

5.0m

G

5.02

Gm

GmG

GGm

mC

mm

from

mmC

5. The velocity factor depends on the sliding velocity,vs,computed from equation(10-18) or (10-19). Values for Cv can be computed from the following formulas.

• For vs from 0 to 700ft/min

38)-(10 31.13 )571.0( sv vC

• For vs from 700 to 3000ft/min

• For vs >3000ft/min

37)-(10 659.0 )0011.0( Svv eC

39)-(10 52.65 )774.0( sv vC

• 6. The proportions of the worm and the wormgear must confirm to the following limits defining the maximum and minimum pitch diameters of the worm in relation to the center distance,C,for the gear set. All dimensions are in inches.

41)-(10 0.3/D Minimum

40)-(10 6.1/D 875.0

W

875.0W

C

CMaximum

• The shaft carrying the worm must be sufficiently rigid to limit the deflection of the worm at the pitch point to the maximum value of

xP005.0

wormgear. theofP,pitch,circular the toequaly numericall iswhich

worm, theofpitch axial theis P xwhere

• 8. When you are analyzing a given wormgear set, the value of the rated tangential load,WtR,must be greater than the actual tangential load,Wt,for satisfactory life.

• 9. Rating given in this section are valid only for smooth system such as fans or centrifugal pumps driven by an electric or hydraulic motor operating under 10 hours per day. More severe conditions,such as shock loading,internal combustion engine drives,or longer hours of operation,require the application of a service factor. Reference 15 lists several such factors based on field experience with specific types of equipment. For problems in this book, factors from table9-6 can be used.