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Worm gears with worm wheels made from Hostaform®
COPYRIGHT: All rights reserved, in particular for reproduction and copying, and for distribution as well as for translation. No part of this publication may be reproduced or processed by means of electronic systems, reproduced or distributed (by photocopying, microfilm or any other process), without written permission by Ticona. © 2004 Ticona GmbH, Kelsterbach NOTICE TO USERS: To the best of our knowledge, the information contained in this publication is accurate, however we do not assume any liability whatsoever for the accuracy and completeness of such information. The information contained in this publication should not be construed as a promise or guarantee of specific properties of our products. Further, the analysis techniques included in this publication are often simplifications and, therefore, approximate in nature. More vigorous analysis techniques and prototype testing are strongly recommended to verify satisfactory part performance. Anyone intending to rely on any recommendation or to use any equipment, processing technique or material mentioned in this publication should satisfy themselves that they can meet all applicable safety and health standards. It is the sole responsibility of the users to investigate whether any existing patents are infringed by the use of the materials mentioned in this publication. Properties of molded parts can be influenced by a wide variety of factors including, but not limited to, material selection, additives, part design, processing conditions and environmental exposure. Any determination of the suitability of a particular material and part design for any use contemplated by the user is the sole responsibility of the user. The user must verify that the material, as subsequently processed, meets the requirements of the particular product or use. The user is encouraged to test prototypes or samples of the product under the harshest conditions to be encountered to determine the suitability of the materials. Material data and values included in this publication are either based on testing of laboratory test specimens and represent data that fall within the normal range of properties for natural material or were extracted from various published sources. All are believed to be representative. These values alone do not represent a sufficient basis for any part design and are not intended for use in establishing maximum, minimum, or ranges of values for specification purposes. Colorants or other additives may cause significant variations in data values.
We strongly recommend that users seek and adhere to the manufacturer’s current instructions for handling each material they use, and to entrust the handling of such material to adequately trained personnel only. Please call the numbers listed for additional technical information. Call Customer Services at the number listed for the appropriate Material Safety Data Sheets (MSDS) before attempting to process our products. Moreover, there is a need to reduce human exposure to many materials to the lowest practical limits in view of possible adverse effects. To the extent that any hazards may have been mentioned in this publication, we neither suggest nor guarantee that such hazards are the only ones that exist. The products mentioned herein are not intended for use in medical or dental implants. Ticona GmbH Information Service Tel. +49 (0) 180-584 2662 (Germany) +49 (0) 69-305 16299 (Europe) Fax +49 (0) 180-202 1202 (Germany and Europe) e-mail [email protected] Internet www.ticona.com
Contents
1. Introduction
2. Requirements for worm gears
3. Materials
4. Types ofworm gear4.1 Worm
4.2 Worm wheel4.3 Basic tooth profile
5. Important stress characteristics5.1 Sliding speed w of the flanks5.2 Load characteristic c
5.3 Flank pressure k at the pitch point5.4 Tooth breakage force FB5.5 Safety factor S against tooth breakage
6. Design ofworm gears6.1 Load characteristic c
6.2 Flank pressure k
6.3 Shear strength TB
6.4 Flow chart for designing worm wheels
using load characteristic c
6.5 Flow chart for designing worm wheels
using tooth breakage force FB
3 7. Calculation examples7.1 Sliding speed w
3 7.2 Load characteristic c
7.3 Safety factor S against tooth breakage
8. Examples of applications8.1 Cable control mechanism for a truck
spare wheel carrier
8.2 Stirrer arm for a kitchen mixer
8.3 Food slicer8.4 Food slicer with modified
tooth profile8.5 Windshield wiper drive
5
5
5
5
6
6
6
8 9. Explanation of symbols8
9 10. Literature
9
9
9
9
11
11
12
12
13
13
15
15
17
18
19
20
21
22
HostaformAcetal copolymer (POM)
= registered trademark
1. Introduction 2. Requirements forworm gears
Worm gears are characterized by silent running, hightransmission ratio in one step and a relatively compact,lightweight design. This has made them the preferredtype of drive for small electrical appliances such as food
slicers, flour mills, electric knives, carpet and upholsterycleaning equipment, etc. Because worm wheels can bemanufactured at low cost from thermoplastics - especiallypolyacetals such as Hostaform - worm gears are fre
quently used in automotive engineering as drives forwindshield wipers, window cranks, seat and mirror
adjusting mechanisms, etc.
A characteristic feature of worm gear is high slidingspeed between the flanks of the worm and worm wheel.The heat thereby generated places a limit on the trans
missible output of worm gears. In this brochure, we giveload characteristics which can be used as an aid in
designing worm gears with Hostaform worm wheels.
Worm gears may be called on to meet widely varyingrequirements with regard to speed ratio, transmission
output, service temperatures and service life. Whilealmost negligible transmission outputs are required for
electricity meters, speedometers and time switches,outputs of up to about 400 watts may be needed for thestirrer arms of kitchen mixers or for mincers. Drives forfood slicers are normally operated at room temperature,while windshield wiper drives in motor vehicles haveto function at service temperatures ranging from about40 C to +80 C. Some worm gears operate for short
periods at a time only while others must be designed forcontinuous operation. The service life required in each
particular case may be anything from a few to severalhundred operating hours.
In many cases, shock loads have to be absorbed such as
when the mechanism runs up against a stop or when themotor is blocked by outside effetcts.
In the production of long runs, automatic gear assemblyshould be possible. For these gears, once-only lubricationmust generally suffice. As little as possible of the trans
mitted output should be converted into heat. This means
that the friction coefficient of the worm/wormwheel
sliding surfaces must be as low as possible.
3. Materials Fig. l : Coefficient of friction and wear volume of threeHostaform grades, sliding partners: I lostaform/steel
In gears with negligibly low transmitted output, both the
worm and worm wheel are frequently injection moldedfrom Hostaform, the Hoechst acetal copolymer.For the more highly stressed sliding component (worm),a slip-modified grade such as Hostaform C 9021 K
(injection molding grade modified with a special chalk)or Hostaform C 9021 TF (injection molding grademodified with Hostaflon = PTFE) is sometimes used.Combinations with Hostaform C 9021 TF have lowfriction coefficients. Hostaform C 9021 K is additionallycharacterized by high wear resistance (fig. 1).
If different plastics are to be chosen for the worm and
worm wheel, polybutylene terephthalate (PBT), polyethylene terephthalate (PET) and polyamide (PA) are
good sliding partners for Hostaform and its modifications.
For high output gears, the worm is, without exception,made from steel because of the better heat dissipation.Even in combination with steel, Hostaform C 9021 TFhas a low friction coefficient and Hostaform C 9021 K
exhibits markedly less wear. In applications where thetotal slide path during the service life is short (operatinghours are minimal) but increased dimensional stabilityunder heat is required, the glass-fiber-reinforced gradeHostaform C 9021 GV 1/30 may be used to advantageprovided that the resulting level of wear on the steel
worm can be tolerated.
0.5
0.4
0.3
0.2
0.1
ffî^
Range of scatter
1 ^
20
u
e
1t
10
r~i EH1 Hostaform C 9021 (basic grade)2 Hostaform C 9021 K3 Hostaform C 9021 TF
4. Types ofworm gear
4.1 Worm
Depending on the production process and position of the
shaping surfaces of the mold, the worm teeth can havedifferent flank shapes. A distinction is made betweenflank shapes A, N, I and K and the worms are accord
ingly designated ZA, ZN, ZI, and ZK [1] .The preferred
worm shape for electrical appliances is the ZI worm
(involute worm) which is usually rolled or milled directlyonto the motor shaft.
4.2 Worm wheel
Fig. 2: Worm gear with cylindrical worm and differentworm wheels
4.3 Basic tooth profile
The lower strength of Hostaform compared with steelcan be partially offset by suitable design measures. Thusthe tooth thickness of the worm wheel (plastic) shouldbe greater than the tooth thickness of the worm (steel).By way of example, fig. 3 shows the basic tooth profileof a modified tooth system. The tooth thickness sp of theworm wheel on the profile center line is in the presentcase 59% of the pitch p while the tooth space ep accounts
for 41%. In other designs, the tooth thickness/tooth spaceratio may be increased up to 65 : 35. This design measure
increases the sustainable blocking moment or tooth
breakage force under shock stress. Modification of thebasic tooth profile usually also involves a change inthe half flank angle a which is reduced from 20 to 15
or 10
Fig. 3: Basic tooth profile of involute gear tooth
system before and after modification
Unmodified basic profile
1 Globoidal worm wheel2 Semi-globoidal worm wheel3 Helical gear
profilereferenceline
effectiveflank area
Injection molded worm wheels are usually designed as
helical gears (fig. 2). With these, the advantages of easymanufacture and assembly must be weighed against thedisadvantage of the theoretical point contact between theworm and worm wheel which gives rise to high pressureloadings. Lower pressure loadings are obtained with linearcontact such as occurs with globoidal (concave) wormwheels. These worm wheels can be injection molded in
one piece in (more expensive) split molds or in two
pieces and then assembled. Semigloboidal worm wheelswhich offer relatively straightforward production and
part-linear contact are a compromise solution.
Modified basic profile
ÜH
\w//m
( p = .T-m
*p"0.59p ep=0.41p
5. Importantstress characteristics
Fig. 4: Design characteristics of a worm gear
5.1 Sliding speed w of the flanks
The peripheral speed v of the worm on the reference
circle is
v _dmi JE nt
_
dmi ni ["mm!60 19.1
.
s
Thus the sliding speed w of the flanks is
(1)
w ==v
_
dmt nt framlcos /m 19.1 cos ym l. s
.
(2)
where
dmi worm reference circle diameter [mm]ni worm speed [I/mm]ym lead angle on reference circle []
(fig. 4)
5.2 Load characteristic c
In design calculations for worm gearing, the load characteristic c is used. This is defined as
cfz b p fz b m Jt
[N/mm2] (3)
whereF2 peripheral force on the worm wheel [N]fz tooth number coefficient (fig. 6)b effective width [mm]
For globoidal worm wheels, the effective width has been
specified as the chord to the outside diameter dai of the
worm at the height of the reference circle dmi (fig. 7) :
b =yd2ai - d2mi [mm] (4)
wheredai worm tip circle diameter [mm]dmi worm reference circle diameter [mm]p axial pitch of the worm [mm]
= transverse pitch of the worm wheel
m module [mm] ,m =
cos ymwhere mn real pitch moduleym lead angle on reference circle (fig. 4)
This definition has been adopted for the semi-globoidalworm wheel and for the helical gear.
Fig. 5: Diagram showing load distribution in a worm
gear
Fig. 6: Tooth number coefficient fz as a function of thetooth number Z2 of the worm wheel
J3
3C
1I
0.60 20 30 40 50 60 70 80
Number of teeth z2 on the worm wheel
7: Effective width b on globoidal and helicalworm wheels
In the case of the worm wheel bearing where the wormwheel runs on a steel shaft, losses of 5-8% can be
expected, depending on lubrication.
The efficiency rjL2 is thus
rjL2 = 0.92 -0.95
For the tooth system efficiency r)z, the following applies(fig- 8)
Jfc =tan 7 r
tan(ym+ Q) (8)
globoidal worm wheel helical worm wheel
where
ym lead angle on reference circletan o = ,M coefficient of friction between the worm and
worm wheel
For the peripheral force on the worm wheel, the follow
ing applies (fig. 5):
fi-1* '""'[N] (5)d2 n2 771,2
whereP2 effective power on the worm wheel [W]d2 pitch circle diameter of the worm wheel [mm]??L2 efficiency of the worm wheel bearingn2 speed of the worm wheel [1/min]
The effective power P2 on the wheel requires a drive
power on the worm PI of
P.-/*- [N]ï/ges.
(6)
where
%es. overall transmission efficiency of the gearing.
In calculating the overall transmission efficiency of the
gears in question here, it is necessary to take into account
the tooth system losses, expressed as the tooth systemefficieny fjz, the worm bearing losses rju and wormwheel bearing losses 7jL2:
%s. = î?z ' 1lU 1/L2 (7)
The worm bearing losses with sintered iron or sinteredbronze bearings or with deep groove ball bearings maybe estimated as 3-5% per bearing point [2], so that
r?u = 0.90 -0.94
Fig. 8 shows the tooth system efficiency rj z as a functionof the lead angle on the reference circle ym for a frictioncoefficient range of // = 0.05-0.3. The smallest coefficient
corresponds approximately to the initial situation whenthere is sufficient lubricant between the flanks. The
Fig. 8: Tooth system efficiency 1J2 as a functionof the lead angle on the reference circle ym for variouscoefficients of friction ft,
3 5 10 20
Lead angle on reference circle ym
30
coefficient of friction JÀ = 0.3 applies broadly to the drysliding combination POM/steel. The friction coefficientsof the POM grades modified with PTFE, lubricants or
wear-reducing additives come in the middle of the range
GM = 0.15 -0.25).
5.3 Flank pressure k at the pitch point
The flank pressure k is obtained from Hertz's law andrelated to the curvature conditions of the flanks at the
pitch point, which vary according to the different flank
shapes.
For flank shape I (involute worm) [1], the followingapplies:
k =F,
b-da-fj [N/mm2] (9)
(10)
(11)
where
Fa peripheral force [N]b effective width [mm]d2 pitch circle diameter of the worm wheel [mm]fj transverse pressure angle factor for ZI worms
fj = cos ygsin yg
yglead angle on the base cylinder
cos yg = cos ym cos a m
half flank angle of the basic profile (fig. 3)
5.4 Tooth breakage force FB
An important criterion for the serviceability of worm
gears is their resistance to tooth fracture. This is particularly important for gears in which shock stress is likelyto occur. This is the case, for example, in mincer drivessince in addition to meat, pieces of bone may get into themachine and can block the drive. Another example is thewindshield wiper drive, which is sometimes called uponto free a solidly frozen wiper in winter. There are alsodrives which run up against a stop such as window crankdrives in motor vehicles, which switch off the motor
when the stop is contacted.
Overloading trials have shown that the tooth breakageforce FB is the product of the shear strength TB of thematerial and the sum of the shear-stressed areas of the
engaged worm wheel teeth:
FB=TB-2At [N] (12)
where
TTB shear strength (fig. 12) [N/mm2]A shear-stressed area of a tooth [mm2]t number of engaged teeth
2* At= Ages.
Fig. 9: Diagram of the engaged worm wheel teeth to
determine shear-stressed areas 2"At
The number of engaged teeth is calculated as follows
(fig- 9):
t =<P
(13)
where
<p = - [] pitch angle
z2 number of teeth on the worm wheel;
this number is rounded off to the nearest lower wholenumber.
y. is calculated as follows:
ra2 2mn roy. = 2 arc cos
(14)
where
ra2 worm wheel tip circle radius [mm]mn real pitch module [mm]
The formula for calculating the shear-stressed area oftooth is:
A = b sm [mm2] (15)
where
sm average width of the shear-stressed surface [mm]b length of the arc (fig. 9) [mm]
The width of the shear-stressed surface varies over thetooth height in accordance with the change in tooththickness with tooth height and also in accordance withthe inclination of the individual tooth to the longitudinalaxis of the worm (pitch angle <p). The average shear-stressed surface width is about 85% of the tooth thick
ness s on the pitch circle.
sm = 0.85 -s- 0.85 -y-jr [mm] (16)
The arc length b of the individual shear-stressed surfacesis calculated as follows (fig. 9):
b = 2-A-ral [mm] (17)
1 (18)where À ~ arc coslid
whereA angle in radian measure
1 side adjacent to A (fig. 9) [mm]rai worm tip circle radius [mm]
Since the shear-stressed surfaces of the individual teethare symmetric with respect to the center line, only thearea of half the shear-stressed surfaces need be calculated.With an even number t of engaged teeth, then t/2 shear-stressed surface areas must be calculated. If, as in fig. 9,there is an odd number of worm teeth engaged with the
worm then shear-stressed surface areas must becalculated.
6. Design ofworm gears
6.1 Load characteristic c
Fig. 10 shows the limit curves for the load characteristic c
in helical gears and semi-globoidal worm wheels madefrom Hostaform as a function of sliding speed w. Thecurves were determined in tests on actual worm gearsand indicate the limitation on transmitted output in con
tinuous operation arising mainly from heat development.For the sake of comparison, the limit curve for bronze
worm wheels is also shown. Gears operating for short
periods at a time only tend to permit a higher loadcharacteristic but it should be remembered that because
plastics have low thermal conductivity, the flank tem
perature can very quickly rise to unacceptably highvalues.
6.2 Flank pressure k
We know from helical gears that wear of the tooth flanksin dry running operation depends primarily on flank
pressure. The limit curve shown in fig. 11 for practicallytested gears gives an indication as to the values up to
which excessive wear is avoided. For the sake of comparison, the flank pressure range for bronze worm wheelsis given.
For an odd number of gear teeth t:
lu = a - ra2 cos (x (p) [mm]
a r2 + rmi (fig. 4) [mm]
t-1
(19a)
For x, use 0 to2
For an even number of teeth t:
Ig = a - ra2 cos |x + -^- } [mm] (19b)
For x, use 0 tot-2
2'
6.3 Shear strength TB
Fig. 12 shows the shear strength determined in the over
loading trials described in section 5.4 as a function of
temperature. The product of the shear strength TB andthe shear-stressed area A of the engaged teeth enables an
estimation of tooth breakage force FB, formula (12), to bemade.
5.5 Safety factor S against tooth breakage
The safety factor S against tooth breakage is the ratio ofthe tooth breakage force FB to the maximum peripheralforce F2 max.
S = -FB-T2 max.
(20)
10 m/s
Fig. 10: Limit curves of permissibleload characteristic c as a function of the
sliding speed w of the tooth flanks
Sliding speed w
5 6
Sliding speed w
Fig. 11: Limit curves of flank pressurek as a function of the sliding speed wof the tooth flanks, determined on
practically tested gears
60
N/mm2
50
|
J/5
40
30
20
10
Calculation example,section 7.
Fig. 12: Shear strength TB determinedon Hostaform worm wheels as a function of temperature
10 20 30 40 50 60 70 80 90 100 110
CTemperature
10
6.4 Flow chart for designing worm wheels using load 6.5 Flow chart for designing worm wheels using toothcharacteristic c breakage force FB
Given:
Output PI [W] or torque MI [N m]
Speeds n^ n2 [1/min]
Calculation of peripheral force F2on the worm wheel:
F2 =_P, m/n2 ??i 19.1 103
4-2 ' H][N]
?i = ??Li Wzriz see fig. 8 (assume // = 0.1)7?u 0.90 to 0.94
P2 19.1 Iffor F2 = j [N]d2 n2 ^u
7?L2 = 0.92 to 0.95
^F^Mr^rVn^-jg! ^
Determination of c valui
c= , .
Fz[N/mm2]fz b m n j
fz see fig. 6
b = ]/d2al - d2ml [mm]
Determination of sliding speed w:
dmi ' niw =
19.1 cos yr[mm/s ]
Comparison of calculated c value with
Czul. f (w) see fig. 10
Given:
Output PI [W] or torque Mi [N m]
Speeds ni; n2 [1/min]
Calculation of peripheral force F2on the worm wheel:
as in flow chart 6.4
1Determination of the number t of
engaged teeth:
f-
<P =
9
360^z2
T r*2 - 2mnx = 2 arc cos
ra2
Calculation of the total shear-stressed area Ages.:(odd number of t)
Ages. = 2 At = sm S b [mm2]
..m
r -,
sm = 0.85 - n LmmJ
2 b according to section 5.4:
b = 2 A ral [mm]
A = arc cos - []
1 = a ra2 cos (x <p) [mm]
, t-1x rrom 0 to -
Calculation of tooth breakage force FBFB = Ages. fz [N] TB see fig. 12
Comparison of tooth breakage force FBwith peripheral force F2.
11
7. Calculation example
In a newly designed windshield wiper drive, it is necessaryto ascertain whether the worm wheel can be producedfrom Hostaform.
The worm gear has the following specifications
Real pitch module
Lead angle on reference circle
Module
Reference circle diamter of worm
Tip circle diameter of worm
Pitch circle diameter of worm wheel
Tip circle diameter of worm wheel
Width of worm wheel
Number of worm teeth
Number of worm wheel teeth
Speed of worm wheel
Torque on worm wheel
Maximum torque on worm wheel (starting up)
mn = 1.25 mm
v 9C7m ~
m =
nip
cos yn
1.25 mm
0.9877= 1.265 mm
dmi = 7.85 mm
dai = 10.35 mm
d2 = 80 mm
da2 = 82.5 mm
B = 10 mm
zi = l
z2 = 64
nz = 70 min"1
M2 = 6 Nm
M2 max.= 60 Nm
It is assumed that the worm wheel temperature can rise to a maximum of 75 C.
To assess whether Hostaform is suitable, the load charac- The worm speed isteristic c of the gear is calculated and compared with the
permissible value for Hostaform worm wheels. In addi
tion, it is necessary to determine the sliding speed w. It isalso important to examine whether the maximum torqueon the worm wheel can cause tooth fracture as the gearstarts up.
n, = n2zi
= -j1 70 min-1
= 4480 min-1
7.1 Sliding speed w
The sliding speed is calculated according to formula (2):
Hence
w =
w =
dml ' HI
19.1 COS yn
7.85 mm 4480
19.1 s cos 9
= 1864.2^
-1.86-ïs
12
7.2 Load characteristic c
For the load characteristic, formula (3) is used:
F2
2-M2,
c =
fz
' b jt m
The peripheral force on the worm wheel F2 is calculatedfrom the torque on the worm wheel M2 and the pitch
. , j. d2circle radius to give
F2 =
2
2-M2
_
2 6000 N mm
80mm
= 150 N
The load-bearing width b is calculated using formula (4):
b = j/d2al-d2ml
= 1/(10.35 mm)2 - (7.85 mm)2
= 6.74 mm
From fig. 6, the tooth number coefficient for z2 = 64 can
be read off as fz = 1.45.
Thus the load characteristic
c =150 N
1.45 6.74 mm ,T 1.265 mm
= 3.86N/mm2
From fig. 10, it can be seen that the calculated loadcharacteristic is slightly above that for helical gears(3.5 N/mm2) but below that for semi-globoidal wormwheels.
7.3 Safety factor S against tooth breakage
The safety factor S against tooth breakage is the ratio ofthe tooth breakage force FB to the maximum peripheralforce F2 max.
S =FB2 max.
The maximum peripheral force F2 max.is calculated from
the maximum worm wheel torque M2 max. at start-up.
"2 max.
U2
^
2 60 OOP N mm
80mm
= 1500 N
The tooth breakage force FB is calculated as the productof the total shear-stressed area and the shear strength TBof the material at the maximum occurring temperature of75 C. The shear strength is (fig. 12):
TZ = 40 N/mm2
To calculate the individual shear-stressed areas, firstof all the number of the engaged teeth t is determined,formula (13):
x ,360C
t = where <P =~<p z2
360
64
= 5.62
For x, formula (14) aplies:
Ta2 2mnx = 2 arc cos
= 2 arc cos
ra2
41.25 mm 2.5 mm
41.25 mm= 2 20.05
= 40.1
Thus
t =40.1e
5.62
= 7.13
As explained in section 5.4, it is necessary to round off to
the nearest lower number, i. e. there are 7 engaged teeth.The total shear-stressed area, which is the sum of all theshear-stressed areas of the individual teeth, is calculatedas
2i A; Ages. sm ' 2* b
where the average tooth thickness sm, formula (16) is:
sm = 0.85m
= 0.85 0.5 1.265 mm jt
= 1.7 mm
13
The arc length b~ must be determined separately for theindividual teeth:
b = 2 A rai (formula 17)
where A = arc cos - (fig. 9; formula 18)ral
in which
1 = a ra2 cos (x <f) (formula 19a)
Since there is an odd number of engaged teeth (t = 7),
the value given to x should be 0 to -
,i. e. 0 to 3.
Center tooth 1 (x = 0):
1[ = a ra2 cos (0 <p)a = r2 + rm!
= 43.93 mm
1, = 43.93 mm - 41.25 mm 1
= 2.68 mm
A = arc cosrai
= arc cos2.68
5.17
= 58.8 A 1.026 rad
Hence
b = AI dai= 1.026 10.35 mm= 10.62 mm
Teeth 2 + 3 (x = 1):
k/3 = a - ra2 cos (1 <p)= 43.93 mm - 41.25 mm cos 5.62= 43.93 mm - 41.05 mm= 2.88 mm
2.88/ 2/3 = arc cos -y^-
= 56.1 0.979 rad
bz/3 = A 2/3 dai= 0.979 10.35 mm= 10.13 mm
In the same way, the arc lengths for the shear-stressedareas of teeth 4 and 5 (x = 2) are calculated as
b4/5 = 8.64 mm
and for teeth 6 + 7 as
be// = 5.51 mm
The total shear-stressed area Ages. is
AgeS. = 1.7 mm (10.62 mm + 2 10.13 mm
+ 2 8.64 mm + 2 5.51 mm)= 100.61 mm2
With a shear strength at 75 C of 40 N/mm2 (fig. 12),the tooth breakage force (formula 12) is:
FB = r B I At
= 40 N/mm2 100.61 mm2
= 4024.4 N
Hence the safety factor against tooth breakage(formula 20) is:
S =FB
*2 max.
_
4024.4 N
1500 N= 2.7
Conclusion:Hostaform C can be used for the worm wheel if thewheel is designed as a semi-globoidal type. Since thecalculated load characteristic is only slightly higher thanthat for helical gears, it would seem to be worth tryinga helical gear design in order to take advantage of the
more straightforward assembly. There is a sufficientfactor of safety against tooth breakage despite the 10 times
higher starting torque with this type of gear.
14
8. Examples ofapplications
Photo 1
8.1 Cable control mechanism for a truck spare wheelcarrier
Photo 1 shows a worm and a worm wheel made from
high-impact Hostaform T 1020 for the manually operatedcable of a spare wheel carrier mechanism on a truck.The maximum spare wheel weight is 1500 N. The largestpossible winding radius on the cable drum correspondsto the pitch circle radius of the worm wheel -~ so that
the peripheral force on the worm wheel is also
F2 = 1500 N
A drawing of the worm and worm wheel with the tooth
system specifications is shown in fig. 13.
The load-bearing width b of the semi-globoidal wormwheel is composed of a curved and a straight part. Thecurved part is calculated according to formula (4).
b = 0.5 ^d2,! - d2,ml= 0.5 ]/(87 mm)2 - (80 mm)2= 17.1 mm
With the tooth number coefficient for z2 = 20 (fig. 6)
fz = 0.65
the load characteristic
c = 10.7 N/mm2.
The sliding speed w is negligibly small.
High-molecular-weight Hostaform T 1020 was chosenbecause it met the requirements for adequate toughnessright down to low temperatures and for good wear
properties even when the gear flanks were contaminatedwith dirt, as a practical trial showed.
15
Fig. 13: Worm and worm wheel with tooth system specifications for the manually operated cable controlmechanism of a truck spare wheel carrier
1I
;: G
1'
**
^\.'AvX
jn
/,//,1 //
:-*
1T^xt,>>v///f
V22i,^*f'r-**"""1
30 -
^
s
*~
A J1 T1J <
O^ *fc <*e. *
1 11I 1*
1
~
J
i
9 is> oc
1
Worm wheel tooth specificationsNumber of teeth
Flank angleFlank direction
Pitch
Helix angleTransverse module
Real pitch modulePitch circle diameter
Tip circle diameter
Root circle diameter
Center distance
Tooth tip clearance
20
40
left
12.566
534'20"
3.981
80
87
69.4
60
R 1.5Worm tooth specifications
Number of teeth
Flank angleFlank direction
Pitch
Average lead angleAxial profile moduleReal pitch modulePitch circle diameter
Tip circle diameter
Root circle diameter
Center distance
Tooth tip clearance
1
40
left
12.566
534'20"
4
3.981
42
50
32.4
60
0.8
16
Photo 2
8.2 Stirrer arm for a kitchen mixer
Photo 2 shows the stirrer arm of a kitchen mixer with a
nominal drive power rating of 300 W. The worm wheel
torque in the operating range is 0.22 Nm at a speedn2 = 280 min"1. The worm (involute worm) is rolledonto the motor shaft. On the worm side, the shaft runs
in a deep-groove ball bearing and at the other end ina sintered metal bushing. The worm wheel material isHostaform C 9021.
Tooth system specifications:
Worm: Number of teeth
Module
Real pitch module
Lead angleLead direction
Referencecircle diameter
Tip circle diameter
z, = 2
m = 1.3302 mm
mn = 1.25 mm
7m = 20
left
dmi = 7.31 mm
dai = 10 mm
Worm wheel: Number of teeth z2 = 50
Pitch circle diameter d2 = 66.51 mm
Wheel width
Stress characteristics:
Sliding speed of the flanks
Load-bearing width
Load characteristic
Flank pressure
B = 11 mm
w = 2. 85 m/s
b = 6.8 mm
c = 1.6 N/mm2
k = 0.55 N/mm2
17
Photo 3
8.3 Food sheer
Photo 3 shows two designs for a food slicer gear: the one
on the right has a motor mounting made from a zinc
diecasting while the mounting on the left is made from
a glass-fiber-reinforced thermoplastic. The motor shaftruns in two cup bearings made from sintered metal.The axial thrust is absorbed by a steel ball pressed intothe motor shaft which pushes against a brass plate.The worm wheel is made from Hostaform C 13021 inthe form of a helical gear. Drive power rating is 100 W.
Worm wheel: Number of teeth z2 = 38
Tooth system specifications
Worm: Number of teeth
Module
Lead angleLead direction
Referencecircle diameter
Tip circle diameter
z, = 2
m = 1.07061 mm
ym = 20 55' 30"
right
dmi = 5.6 mm
dai = 7.6 mm
Addendummodification coefficient = 0.1451
Pitch circle diameter
Tip circle diameter
Wheel width
Stress characteristics:
Sliding speed of the flanks
Load-bearing width
Load characteristic
d2 = 40.683 mm
daa = 42.4 mm
B = 7 mm
w = 2.2 m/s
b = 5.2 mm
c = 1.7N/mm2
18
Photo 4
8.4 Food sucer with modified tooth profile
Photo 4 shows the worm gear for a food slicer with a
drive power rating of 140 W. The worm wheel with inte
grally molded sprocket wheel is injection moulded fromHostaform C 9021. The tooth system with a real pitchmodule mn = 1 mm has been modified so that the tooththickness of the worm is reduced in favor of the tooththickness of the worm wheel. The tooth thickness of the
worm wheel on the pitch circle is 65% of pitch p andthat of the worm 35%.
Half flank angleLoad characteristic
Sliding speed
a =10
c =4.47N/mm2
w= 2.7 m/s.at a
19
Photo 5
8.5 Windshield wiper drive
Photo 5 shows a windshield wiper drive in which themotor power is transmitted via two worm wheels madefrom Hostaform C 9021 to a third spur wheel. Owing to
the opposite lead direction of the two worms, the axialforces cancel each other out. Hence it is sufficient to
mount the motor shaft in two radial bearings.
Tooth system specifications
Worms: Number of teeth
Module
Lead angleLead direction
Referencecircle diameter
Tip circle diameter
Z, = 2
m = 0.8263 mm
v = 14 5/m J.T.J
right and left
dm = 6.39 mm
da = 8 mm
Stress characteristics:
This is a two-speed motor and so there are two loadcharacteristics at the two different sliding speeds:
n, =1860 min-'
M, = 700 N m
Ci =1.82 N/mm2
c2 =1.3 N/mm2
The load-bearing width b is
b = 4.8 mm
n2 = 2900 min-'
M2 = 500 N m
at wi = 0.64 m/s
at w2 = 1.0 m/s
Worm wheels: Number of teeth z2 = 29
Pitch circle diameter d2 = 23.963 mm
Wheel width B = 7 mm
Center distance a = 15 mm
20
9. Explanation ofsymbols
On the basis of [2] and [5], the following symbols are
used:
Symbol
a
Ages.
A,
b
b
B
c
d2
dal
da2
dml
fj
fz
FB
F2
1
k
m
mn
M2
ni
n2
P
P,
P2
T2
Unit
mm
mm2
mm2
mm
mm
mm
N/mm2
mm
mm
mm
mm
N
N
N/mm2
mm
mm
Nm
1/min
1/min
mm
W
W
mm
Explanation
center distance
total shear-stressed area ofall engaged teeth
shear-stressed area of one tooth
effective width of the worm wheel
arc length of the shear-stressedsurface
width of the worm wheel
load characteristic
pitch circle diameter of theworm wheel
tip circle diameter of the worm
tip circle diameter of the wormwheel
reference circle diameterof the worm
transverse pressure angle factorfor ZI worms
tooth number coefficient
tooth breakage force
peripheral force on the wormwheel
flank shape of the ZI worm
flank pressure at the pitch pointmodule (axial module)real pitch module
torque on the worm wheel
worm speedworm wheel speedworm pitch (axial pitch)drive power on the worm
effective power on the wormwheel
pitch circle radius of the wormwheel
Symbol Unit Explanation
ri
Ta2
S
t
V
W
X
Zi
Z2
rg
7m
?2
>?ges.
flLl
flL2
K
A
H
Q
TB
<F
ni n
if\tt
mm
mm
mm
mm
m/s
m/s
rad
N/mm2
tip circle radius of the worm
tip circle radius of the wormwheel
tooth thicknes of the worm wheeltooth on the pitch circle
average width of the shear-stressed surface
safety factor
number of engaged teeth
peripheral speed of the worm on
the reference circle
sliding speed between the flanks
factor
number of worm teeth
number of worm wheel teeth
half flank angle of the basic profilelead angle on the base cylinderlead angle on the referent circle
tooth system efficiencyoverall efficiency
efficiency of the worm bearings
efficiency of the worm wheel
bearings
angle of tooth engagement arc
center angle to the half arc lengthfriction coefficient
angle of incline
shear strength
pitch angleindex for worm
index for worm wheel
index for even number of gearteeth
index for odd number of gear teeth
21
10. Literature
[1] DIN 3975 Begriffe und Bestimmungsgrößen für
Zylinderschneckengetriebe mit Achswinkel 90.
[2] A. K. Thomas und W. Charchut: Die Tragfähigkeitder ZahnräderCarl Hanser Verlag, Munich 1971.
[3] B. Klein: Wirkungsgrad und Selbsthemmungan Schneckengetriebenant. Antriebstechnick 19 (1980) 9.
[4] R. Debrunner: Wirkungsgrade von Klein-Schnecken-
getrieben und ihre Beeinflussungsfaktorenant. Antriebstechnik 19 (1980) 11.
[5] H. Schmidt: Schneckengetriebe mit Schneckenräder aus Hostaformant. Antriebstechnik 24 (1985) 3.
Technical plasticsDesign Calculations Applications
Publications so far in this series:
A. Technical plasticsA. 1.1 Grades and properties - HostaformA. 1.2 Grades and properties - HostacomA. 1.4 Grades and properties - Hostalen GUR
A. 1.5 Grades and properties - Celanex,Vandar, Impet
A.2.1 Calculations principlesA.2.2 Hostaform - Characteristic values and
calculation examplesA.2.3 Hostacom - Characteristic values and
calculation examples
B. Design of technical mouldingsB.I.I Spur gears with gearwheels made from
Hostaform, Celanex and Hostalen GUR
B.2.2 Worm gears with worm wheels made from
HostaformB.3.1 Design calculations for snap-fit joints in
plastic partsB.3.2 Fastening with metal screws
B.3.3 Plastic parts with integrally moulded threads
B.3.4 Design calculations for press-fit jointsB.3.5 Integral hinges in engineering plasticsB.3.7 Ultrasonic welding and assembly of
technical plastics
C. Production of technical mouldingsC.2.1 Hot runner system - Indirectly heated,
thermally conductive torpedoC.2.2 Hot runner system - Indirectly heated,
thermally conductive torpedoDesign principles and examples of mouldsfor processing Hostaform
C.3.1 Machining HostaformC.3.3 Design of mouldings made from
engineering plasticsC.3.4 Guidelines for the design of mouldings
in engineering plasticsC.3.5 Outsert moulding with Hostaform
22
In this technical information brochure, Hoechst aims to
provide useful information for designers who want to
exploit the properties of engineering polymers such as
Hostaform. Our technical service team will also bepleased to advise you on materials, design and processing.
This information is based on our present state of knowledge and is intended to provide general notes on our
products and their uses. It should not therefore be con
strued as guaranteeing specific properties of the productsdescribed or their suitability for a particular application.Any existing industrial property rights must be observed.The quality of our products is guaranteed under our
General Conditions of Sale.
Applications involving the use of the Hoechst materialsHostaform are developments or products of the plasticsprocessing industry. Hoechst as manufacturers of thestarting material will be pleased to give the names of processors of plastics for technical applications.
© Copyright by Hoechst Aktiengesellschaft
Issued in August 19967 1st edition
23
Hostaform®, Celcon®
polyoxymethylene copolymer (POM)
Celanex®
thermoplastic polyester (PBT)
Impet®
thermoplastic polyester (PET)
Vandar® thermoplastic polyester alloys
Riteflex®
thermoplastic polyester elastomer (TPE-E)
Vectra®
liquid crystal polymer (LCP)
Fortron®
polyphenylene sulfide (PPS)
Celstran®, Compel® long fiber reinforced thermoplastics (LFRT)
GUR®
ultra-high molecular weight polyethylene (PE-UHMW)
EuropeTicona GmbHInformation ServiceTel.: +49 (0) 180-5 84 26 62 (Germany) +49 (0) 69-30 51 62 99 (Europe)Fax: +49 (0) 180-2 02 12 02eMail: [email protected]: www.ticona.com
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