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Computer-aided design of multispeed gearboxes
Technicalnote Computer-aided design of multispeed gearboxes I Hfiseyin Filiz and Mustafa Dfilger
A practical method for designing gears in multispeed gearboxes is developed and automated with an interactive and highly user-oriented computer program. The basic principles of obtaining kinematic-arrangement diagrams are discussed. A general model for the arrangement of gears and their shafts is suggested. An algorithm is developed for the design of all of the gears in gearboxes.
Keywords: gearbox design, mechanical-engineeringdesign
The main purpose of multispeed gearboxes is the provision of a series of stepped speeds that meet the requirements of the systems in which they are used. Careful consideration must be given to the design of geartrains, since they introduce into the system extra inertia and elasticity. In general, a design study of gearboxes includes the following steps: the obtaining of a kinematic-arrangement diagram for a specified number of shafts and speeds, the determination of the sizes of the gears in each transmission that are necessary to transfer the power safely according to the kinematic-arrangement diagram obtained in the first step. The design process of the gearbox is considered to be completed if, in addition to the above, the shaft sizes, the types and sizes of the bearings, and the details of the housing are determined. However, these are all dependent on the results of the first two steps. Kinematicarrangement diagrams have received considerable attention, and the ways of obtaining these are discussed in detail in the literature1 5. The sizing of gears has also Department of Mechanical Engineering, University of Gaziantep, Gaziantep, Turkey Paper received: 19 December 1990. Revised: 19 February 1993
been studied in detail, but separately from kinematic analysis 6. In this study, kinematic-arrangement diagrams are reviewed, and, among the various choices, the one that will best result in a small gearbox is discussed. An automated design method which takes the speed, strength and wear requirements into account is developed. Gears are sized by the use of the AGMAapproach. The following discussions basically apply to machine tools, but they can also be applied to some other types of application.
KINEMATIC-ARRANGEMENT DIAGRAM A general kinematic-arrangement diagram (KAD)is shown in F i g u r e 1. It includes the data, which are the number of transmission groups, the transmission ratios, the number of transmissions in each group, the relative order of the groups in the train of the transmissions, the characteristics of each group, and the relationship between the transmission ratios. On the diagrams, shafts are represented by horizontal lines at equal distances from each other. The last line represents the output shaft, and it contains the output speeds. In determining the output speeds, a geometricprogression method is generally used for machine-tool gearboxes 1,2. This method is accepted as the most advantageous, since it provides an even distribution of the output speeds for a given speed range 3'4. A constant progression ratio is used between two consecutive spindle speeds. In practical applications, standard progression ratios are used, and they are obtained by using the properties of decimal preferred numbers as suggested by Koenigsberger 1. Standard progression ratios ~b are 1.06, 1.12, 1.41, 1.58, 1.78 and 2. The selection of one of these is governed by the number of speed steps and by the speed distribution for the given speed range. For a greater number of speed steps, a smaller progression ratio is used, but one must be aware that this increases the cost of the gearbox.
0010-4485/93/110720-07 © 1993 Butterworth-Heinemann Ltd 720
Computer-Aided Design Volume 25 Number 11 November 1993
Computer-aided design of multispeed gearboxes: I H Filiz and M D~ilger Group
Shaft
Number of Transmission
rlem
1
P, S~
I
\ . . . . . . . . .
;,~p
S,~P ~
~,
S, ¢9P,-1
P,
2 2
I
x, . . . . . . . . .
S2cP P1
Sa ~ P1P2
3
$299 2P ,
S~2PI
P2
~.
$2 ~(Pz- 1) P1
$3 ~(P~- 1)P1P2
P,
r+l
Sr
S , ~ (e-2)P,~ e_,
S r ~ PI I'2 • ...Pr-I
S,~(e-1)P,
P~ • Pr-,
Figure 1 General kinematic-arrangementdiagram
As can be seen in Fioure 1, the total number of groups in each gearbox is equal to the number of shafts minus 1, and the number of speed steps z may be expressed in terms of the number of simple transmissions in each group z = P1 P2 P 3 . . . Pr
(1)
It is accepted as good practice to select a number for z which has factors of 2 and 3, since the gearboxes with uniform groups with p = 2 and p = 3 transmissions in each group provide the minimum number of transmissions in all groups. When the transmission ratios of each group are set up, they can be arranged in any order to obtain the same output speeds by changing the number of transmissions in each group. If there are r groups, there will be r! different arrangements, i.e. z = P1 P2 P 3 . . . P,
= P2
= P3 Pz P I . . . P, = ...
Px P a . . . P, (2)
If there are t groups, with an equal number of transmissions in each, the number of design options is reduced to r!/t!. The arrangement order of the main, first, second, etc. groups also introduces r! options for each kinematic diagram. Therefore, the total number of options is (r!)2/t!. For example, for the structure z = 12 = 2 x 2 x 3, in which r = 3 and t = 2, the number of design options is 18. It is stated by Koenigsberger 1 that it is preferable to have the largest transmission range in the last group for reasons of tooth strength and space requirements. Acherkan 2 suggests that the transmission ratio must be reduced to a smaller and smaller value as the train approaches the spindle. The transmission ratio between the spindle and the preceding shaft is taken to be equal to the minimum limiting value. Hence, the selection between these many choices must be based on these two considerations. Karsli 3'4 and Filiz 4 describe how all the kinematic arrangements are obtained, and how the most suitable one is selected taking into consideration the above suggestions and the limitations on the minimum
Computer-Aided Design Volume 25 Number 11 November 1993
721
Computer-aided design of multispeed gearboxes: I H Filiz and M D01ger
and m a x i m u m transmission ratios. They a u t o m a t e d the calculations involved in obtaining kinematic arrangements of gearboxes by a c o m p u t e r p r o g r a m called KAD. With this program, they obtained the recommended options which satisfied the limitations set by the designer. They also calculated gear-teeth numbers in each transmission using Gray's m e t h o d 7 and the m e t h o d suggested by Reddy 8. With KAD, the designer is given an opportunity to analyse the kinematic-arrangement diagram of an existing gearbox and c o m p a r e the results with the solution provided by the program. The input data and limitations are given below for a typical currently available milling machine, for the sake of the completeness of the discussion and for the following analysis.
Input values: n u m b e r of speeds n u m b e r of shafts shafts with one transmission input speed, rev/min m i n i m u m speed, rev/min m a x i m u m speed, rev/min progression ratio
= 12 = 5 = 1 = 1400 = 42.0 = 1988.02 = 1.42
Limitations: m i n i m u m n u m b e r of teeth total n u m b e r of teeth error on gear ratio, % error on output speed, % m i n i m u m transmission ratio m a x i m u m transmission ratio
=18 = 120 =_1 =+2 = 0.25 = 2.20
The results are given in Table 1.
MODELLING GEARTRAINS
AND LIMITATIONS
OF
After the creation Of KAD, which provides the required speed ratios, comes the determination of gear sizes. In this study, only spur gears are considered. Figure 2 shows a schematic view of a 3-shaft, 6-speed gearbox. The first g r o u p has two transmissions, and the second g r o u p has three transmissions. Shaft A is directly connected to an electric motor, and is hence the input shaft. All the shafts are parallel, but they do not necessarily lie in the same plane. If the plane of shafts A and B is taken as the datum, the relative positions of three shafts might be rearranged by changing the angle of the plane of shafts A and B and shafts B and C. In the case of n shafts, the n u m b e r of angles that determine the relative positions of the shafts isn-2. Shafts are considered to be stepped at sections at which gears are located. The increasing or decreasing order of shaft diameters at each step is as indicated in Figure 2. The axial dimensions of the shafts, together with the steps, must be specified by the designer. The meshing of gears is assumed to take place via the sliding of any of the meshing gears or by the implementation of positive clutches (not shown in Figure 2). The main limitation in all gearboxes is that the centre distance of shafts in a g r o u p of transmissions must be fixed. Because the sums of the gear-teeth numbers for every mesh in a transmission g r o u p are practically the same, all the gears in the same g r o u p must be cut with the same module m. That is, gears 1, 2, 3 and 4 have the same module to hold fixed the centre distance CA.B between shafts A and B. Similarly,
Table 1 KADcomputer output for milling machine Group number
Speed number
Power of progression ratio
Required ratio
Teeth number
Found ratio
Error, %
1
1
0
0.714
25:35
0.7143
-0.040
2
3
0 1 2
0.500 0.710 1.008
20:40 25:35 30:30
0.5000 0.7143 1.0000
- 0.000 -0.604 - 0.813
3
2
0 3
0.346 0.991
18:52 35:35
0.3462 1.0000
- 0.004 0.939
4
2
0 6
0.250 2.050
18:72 60:30
0.2500 2.0000
- 0.000 2.420
Speed number 1 2 3 4 5 6 7 8 9 10 11 12
722
Required speed, rev/min
Found speed, rev/min
Error, %
43.23 61.39 87.17 123.79 175.78 249.61 354.44 503.30 714.69 1014.86 1441.10 2048.37
42.37 61.81 86.54 125.00 178.57 245.00 346.15 494.51 692.31 1000.00 1428.57 2000.00
- 0.08 -0.69 0.73 0.98 1.58 O.17 2.34 1.73 3.15 1.44 0.84 2.25
Computer-Aided Design Volume 25 Number 11 November 1993
- -
Computer-aided design of multispeed gearboxes: I H Filiz and M Dfilger
2
_~A
=
5
m
4
m - -
6
CAB
D
1
8
Figure2 Gearbox model
gears 5, 6, 7, 8, 9 and 10 must be cut with the same module, but this is not necessarily equal to the module of gears 1, 2, 3 and 4 for the centre distance CR.c to be held fixed. Limitations of the kinematic scheme are the teeth numbers and transmission ratios. For dynamic performance, the minimum teeth number should be 18, and the total number of teeth 6 should not exceed 120. The transmission ratios for a group of transmissions I are restricted to be in the range 0.25-2.40. The error on the gear ratios exceeding + 1% is not reliable from the speed-progression point of view 4. In some applications in which very fine transmission ratios are essential, slight changes are allowed in gear-teeth numbers. In such cases, to keep the gears on the same shafts, gear profiles must be shifted, and thus nonstandard gears are produced.
DETERMINATION
OF GEAR SIZE
The data available after the kinematic-arrangement diagram are the speeds of shafts at each transmission, and the teeth numbers of gears in each group. To start with the design, the amount of the torque transmitted by each gear must be evaluated. Since it is assumed that there is negligible power loss, the torque transmitted at high speeds is smaller than that at low speeds. Hence, the minimum transmission ratio is selected to obtain the maximum torque transmission. The speed of shaft i can be written as ni=nmSl
(3)
where n,, is the speed of the input shaft, and Si is the transmission ratio. To obtain the minimum speed, St must be a minimum, which results in the maximum torque transmission. There are many different transmissions
in a gearbox kinematic-arrangement diagram. Each transmission has different loading conditions. Although the power does not vary, each transmission carries different torques at the specified output speeds. Therefore, each transmission must be considered as a separate design unit with its own loading conditions. Taking this fact into account, the facewidths and the modules of gears in each transmission are determined by using the AGMA approach, as explained below.
C O M P U T E R P R O G R A M FOR SIZING GEARS Because of the large number of interdependent variables, the design process does not consist in inserting numbers into certain formulas that will produce a complete set of results; it is rather a trial-and-error process, which is tedious in terms of manual calculations. A design algorithm is developed, and the process is automated with a computer program whose simplified flowchart is given in Fioure 3. The input includes the output data of the KAD program, the safety factor, and the materials properties of the gears. If this automated design algorithm is applied to the first group, the facewidth F 1 and the module m x are determined by assuming a value for module m, and calculating the facewidth F using the following equation: F=
nK°KmWt KaKbKcKfSe.mJKv
(4)
where n is the safety factor, Wt is the tangential load acting on the gear, K o is the overload factor, K m is the load-distribution factor, K a is the surface-finish factor, K b is the size factor, K c is the reliability factor, Kf is the load factor, Se is the endurance factor, d is the AGMA geometry factor, and K v is the velocity factor. A database !s _prepared for all the values of the K factors, which are taken from AGMA Information Sheet 225.01 l° and
Computer-Aided Design Volume 25 Number 11 November 1993
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Computer-aided design of multispeed gearboxes: I H Filiz and M DOIger
References 11 and 12. The process of calculating the facewidth is terminated when it is ~2 between 3rim and 5rim. The wear aspect of gears is another important factor to be considered. Hence, the gear must be checked against the surface durability by calculating the safety factor from
the following equations:
S CLCT ~2
CCHCRCp # KvFdpI
n =
(5)
WtKoKm Sc = 2.76HB - 70
(~
DISK
INPUT SPEED INPUT POWER i FACTOR [IF SAFETY TENSILE STRENGTH I BRINELL HARDNESS I
NUMBER OF GROUP (k)
C ~ > VDU =
I
NUMBER OF TRANSMISSIONS I IN EACH GROUP (t) I
I J°0
i -1 TRIAL
I ':°
I
2
(7)
mG+ 1
CALCULATE FACTOR OF
I
SAFETY FOR SURFACE
I
EXAMPLE
VALUE FOR mI
I i
I
CALCULATE f i FDR FATIGUE STRENGTH REQUIREMENT
I J=
sin ~b cos q5 mG
where q5 is the pressure angle, and mG = dG/dp, where dG is the pitch diameter of the gears. All the C factors are in the database of the program, and are taken from AGMA Information Sheet 225.011° and References l l and 12. F 2 and m 2 are determined in the same manner. Since a single module must be used for these two transmissions owing to the fixed centre distance, the larger of m 1 and mz must be selected and replaced with the nearest standard module. This is the principal module mp of the first group. In the same manner, by applying the design algorithm once for each transmission in the second group, the principal module of the second group is determined. This process terminates after the principal module of the last group is determined.
ASSUME
I i=
(in megapascals, where Sc is the surface-contact strength, CL is the life factor, Ca- is the temperature factor, CH is the hardness-ratio factor, C, is the reliability factor, C o is the elasticity coefficient, and dv is the pitch diameter of the pinions) and I =
I
(6)
.xj
[INCREASE (mi ) ]
L
1
+1[ DURABILITY REQUIREMENT
1
SELECT MAXIMUMOF rolAND REPLACE IT WITHTHE NEAREST STANDARDMODULE(mp)
CALCULATE NEhN FACTOR OF SAFETY
PRINCIPLEMODULE (rap FACTOROF SAFETY FACEWIDTHS
The example is given as an alternative solution for the gears used in the main drive gearbox of the milling machine whose KAD has already been created. The gear-data input to the program is as follows:
Gear-material specification: yield strength, M P a ultimate strength, M P a Brinell hardness, HB Young's modulus, MPa Poisson's ratio
= 800 = 880 = 410 = 207000 =0.3
Design requirements: = 2.5 factor of safety = 25 pressure angle, deg = 99 reliability, % operating temperature, deg C = 75 overload factor = 1 life factor = 1
I
MODULE " ~ FACEWlDTH DIAMETER
Manufacturing requirements: high-precision ground teeth
Characteristic of supports: Figure 3 Flowchart of computer program for sizing gears
724
accurate mountings
Computer-Aided Design Volume 25 Number 11 November 1993
Computer-aided design of multispeed gearboxes: I H Filiz and M D01ger Table 2
Gear-sizing computer output
Input speed, rev/min = 1400 Input power, k W = 4 Group number
Mesh number
Teeth number
1
1
25:35
2
1 2 3
20:40 25:35 30:30
3
1 2
18:52 35:35
4
1 2
18:72 60:30
Angle of planes of shafts 2 and 3, deg = 0.0 Angle of planes of shafts 3 and 4, deg = 20.0 Angle of planes of shafts 4 and 5, deg = 0.0 W h a t now (enter ? for help) : G o W h a t next (enter ? for help) : Old Old data are now in progress W h a t next (enter ? for help) : G o Gear design starts Gear design has been completed Do you want to see facewidths ? (Y/N)
:Y
Grub number
Mesh number
Teeth number
Module, m m
Facewidth, mm
Pitch diameter, mm
Safety factor
1 1 2 3 1 2 1 2
25:35 20:40 25:35 30:30 18:52 35:35 18:72 60:30
2.25 2.50
26 38 29 29 46 38 68 48
56.3:78.8 50.0:100.0 62.5:87.5 75.0:75.0 63.0:182.0 122.5:122.5 90.0:360.0 300.0:150.0
2.5 2.5 2.5 2.8 2.5 3.5 2.5 4.8
1 2
3 4
3.50 5.0
The output, together with kinematic data, is given in Table 2.
CONCLUSIONS The determination of the module and the facewidth of a gear which has a certain number of teeth is primarily dependent on the torque transmitted and the speed. Although the torque increases as the speed decreases, this increase in torque may be countered by the decrease in the dynamic load. Therefore, the designer should carry out the design of gears at all output speeds, especially for those gearboxes having a large number of output speeds. The computer program presented in this paper allows the user to carry out the design of gears at each output speed separately. After specifying the speed number, the program starts to design gears at that speed. The logical chains among the variables are created within the program. As noticed in the results, the safety factors of some transmissions are larger than the original safety factor. This is because the design modules for these transmissions are replaced with the principal module
Dr I Hi~seyin Filiz was born in Gaziantep, Turkey. He gained a BSc and an MSc in mechanical engineering at the Middle East Technical University, Turkey, in 1973 and 1975. In 1981, he earned a PhD from the University of Manchester Institute of Science and Technology, UK. His research interests are in the design of CNC machine tools, the design of machine elements, and CAB.
Mustafa Di~lger was born in Bayburt, Turkey, in 1963. He gained a BSc in mechanical engineering and an MSc at the Middle East Technical University, Turkey, in 1987 and 1989. His research interests are in the design of CNC machine tools.
Computer-Aided Design Volume 25 Number 11 November 1993
725
Computer-aided design of multispeed gearboxes: I H Filiz and M Diilger
which is determined for the weakest transmission to satisfy the fixed-centre-distance requirement. For the purpose of production, the modules determined are replaced with the nearest standard modules.
5 6 7
REFERENCES 8 1 2 3 4
726
Acherkan, N Machine Tool Design - - Vol 3 Mir Publishers (1973) Koenigsberger, F Design Principles of Metal Cutting Machine Tools Pergamon Press, UK (1964) Karsli, S 'Computer aided design of gearbox kinematical arrangement diagrams' MSe Thesis Middle East Technical University, Gaziantep, Turkey (Sep 1985) Karsli, S and Filiz, I H 'Computer aided design of gearbox
9 10
11 12
kinematical arrangement diagrams' Proc. lASTED Int, Syrup. Computers & Their Applications for Development Taormina, Italy (3-5 Sep 1986) pp 174-179 Diilger, M 'Computer aided design of gearboxes for machine tools' MSc Thesis Middle East Technical University, Gaziantep, Turkey (Jun 1989) Chironis, N P Gear Design Application McGraw Hill, USA (1962) Sanger, D J 'A note on maximum speed ranges and distribution of speeds of three shaft gear trains' lnt. J. Mach. Tool. Des. & Res. Vol 12 (1972) pp 55-63 Reddy, C Y 'Selection of number of teeth for machine tool gearboxes' J. Mach. Tool Des. & Res. Vol 14 (1974) pp 125-134 TOS Milling Machine Catalogue 'AGMA Information Sheet 255.01' American Gear Manufacturers Association, USA (1969) Dudley, D W Gear Handbook McGraw-Hill, USA (1962) Shigley, J E and Mitchell, L D Mechanical Engineering Design (4th Ed.) McGraw-Hill, Japan (1983)
Computer-Aided Design Volume 25 Number 11 November 1993
A practical method for designing gears in multispeed gearboxes is developed and automated with an interactive and highly user-oriented computer program. The basic principles of obtaining kinematic-arrangement diagrams are discussed. A general model for the arrangement of gears and their shafts is suggested. An algorithm is developed for the design of all of the gears in gearboxes.
Keywords: gearbox design, mechanical-engineeringdesign
The main purpose of multispeed gearboxes is the provision of a series of stepped speeds that meet the requirements of the systems in which they are used. Careful consideration must be given to the design of geartrains, since they introduce into the system extra inertia and elasticity. In general, a design study of gearboxes includes the following steps: the obtaining of a kinematic-arrangement diagram for a specified number of shafts and speeds, the determination of the sizes of the gears in each transmission that are necessary to transfer the power safely according to the kinematic-arrangement diagram obtained in the first step. The design process of the gearbox is considered to be completed if, in addition to the above, the shaft sizes, the types and sizes of the bearings, and the details of the housing are determined. However, these are all dependent on the results of the first two steps. Kinematicarrangement diagrams have received considerable attention, and the ways of obtaining these are discussed in detail in the literature1 5. The sizing of gears has also Department of Mechanical Engineering, University of Gaziantep, Gaziantep, Turkey Paper received: 19 December 1990. Revised: 19 February 1993
been studied in detail, but separately from kinematic analysis 6. In this study, kinematic-arrangement diagrams are reviewed, and, among the various choices, the one that will best result in a small gearbox is discussed. An automated design method which takes the speed, strength and wear requirements into account is developed. Gears are sized by the use of the AGMAapproach. The following discussions basically apply to machine tools, but they can also be applied to some other types of application.
KINEMATIC-ARRANGEMENT DIAGRAM A general kinematic-arrangement diagram (KAD)is shown in F i g u r e 1. It includes the data, which are the number of transmission groups, the transmission ratios, the number of transmissions in each group, the relative order of the groups in the train of the transmissions, the characteristics of each group, and the relationship between the transmission ratios. On the diagrams, shafts are represented by horizontal lines at equal distances from each other. The last line represents the output shaft, and it contains the output speeds. In determining the output speeds, a geometricprogression method is generally used for machine-tool gearboxes 1,2. This method is accepted as the most advantageous, since it provides an even distribution of the output speeds for a given speed range 3'4. A constant progression ratio is used between two consecutive spindle speeds. In practical applications, standard progression ratios are used, and they are obtained by using the properties of decimal preferred numbers as suggested by Koenigsberger 1. Standard progression ratios ~b are 1.06, 1.12, 1.41, 1.58, 1.78 and 2. The selection of one of these is governed by the number of speed steps and by the speed distribution for the given speed range. For a greater number of speed steps, a smaller progression ratio is used, but one must be aware that this increases the cost of the gearbox.
0010-4485/93/110720-07 © 1993 Butterworth-Heinemann Ltd 720
Computer-Aided Design Volume 25 Number 11 November 1993
Computer-aided design of multispeed gearboxes: I H Filiz and M D~ilger Group
Shaft
Number of Transmission
rlem
1
P, S~
I
\ . . . . . . . . .
;,~p
S,~P ~
~,
S, ¢9P,-1
P,
2 2
I
x, . . . . . . . . .
S2cP P1
Sa ~ P1P2
3
$299 2P ,
S~2PI
P2
~.
$2 ~(Pz- 1) P1
$3 ~(P~- 1)P1P2
P,
r+l
Sr
S , ~ (e-2)P,~ e_,
S r ~ PI I'2 • ...Pr-I
S,~(e-1)P,
P~ • Pr-,
Figure 1 General kinematic-arrangementdiagram
As can be seen in Fioure 1, the total number of groups in each gearbox is equal to the number of shafts minus 1, and the number of speed steps z may be expressed in terms of the number of simple transmissions in each group z = P1 P2 P 3 . . . Pr
(1)
It is accepted as good practice to select a number for z which has factors of 2 and 3, since the gearboxes with uniform groups with p = 2 and p = 3 transmissions in each group provide the minimum number of transmissions in all groups. When the transmission ratios of each group are set up, they can be arranged in any order to obtain the same output speeds by changing the number of transmissions in each group. If there are r groups, there will be r! different arrangements, i.e. z = P1 P2 P 3 . . . P,
= P2
= P3 Pz P I . . . P, = ...
Px P a . . . P, (2)
If there are t groups, with an equal number of transmissions in each, the number of design options is reduced to r!/t!. The arrangement order of the main, first, second, etc. groups also introduces r! options for each kinematic diagram. Therefore, the total number of options is (r!)2/t!. For example, for the structure z = 12 = 2 x 2 x 3, in which r = 3 and t = 2, the number of design options is 18. It is stated by Koenigsberger 1 that it is preferable to have the largest transmission range in the last group for reasons of tooth strength and space requirements. Acherkan 2 suggests that the transmission ratio must be reduced to a smaller and smaller value as the train approaches the spindle. The transmission ratio between the spindle and the preceding shaft is taken to be equal to the minimum limiting value. Hence, the selection between these many choices must be based on these two considerations. Karsli 3'4 and Filiz 4 describe how all the kinematic arrangements are obtained, and how the most suitable one is selected taking into consideration the above suggestions and the limitations on the minimum
Computer-Aided Design Volume 25 Number 11 November 1993
721
Computer-aided design of multispeed gearboxes: I H Filiz and M D01ger
and m a x i m u m transmission ratios. They a u t o m a t e d the calculations involved in obtaining kinematic arrangements of gearboxes by a c o m p u t e r p r o g r a m called KAD. With this program, they obtained the recommended options which satisfied the limitations set by the designer. They also calculated gear-teeth numbers in each transmission using Gray's m e t h o d 7 and the m e t h o d suggested by Reddy 8. With KAD, the designer is given an opportunity to analyse the kinematic-arrangement diagram of an existing gearbox and c o m p a r e the results with the solution provided by the program. The input data and limitations are given below for a typical currently available milling machine, for the sake of the completeness of the discussion and for the following analysis.
Input values: n u m b e r of speeds n u m b e r of shafts shafts with one transmission input speed, rev/min m i n i m u m speed, rev/min m a x i m u m speed, rev/min progression ratio
= 12 = 5 = 1 = 1400 = 42.0 = 1988.02 = 1.42
Limitations: m i n i m u m n u m b e r of teeth total n u m b e r of teeth error on gear ratio, % error on output speed, % m i n i m u m transmission ratio m a x i m u m transmission ratio
=18 = 120 =_1 =+2 = 0.25 = 2.20
The results are given in Table 1.
MODELLING GEARTRAINS
AND LIMITATIONS
OF
After the creation Of KAD, which provides the required speed ratios, comes the determination of gear sizes. In this study, only spur gears are considered. Figure 2 shows a schematic view of a 3-shaft, 6-speed gearbox. The first g r o u p has two transmissions, and the second g r o u p has three transmissions. Shaft A is directly connected to an electric motor, and is hence the input shaft. All the shafts are parallel, but they do not necessarily lie in the same plane. If the plane of shafts A and B is taken as the datum, the relative positions of three shafts might be rearranged by changing the angle of the plane of shafts A and B and shafts B and C. In the case of n shafts, the n u m b e r of angles that determine the relative positions of the shafts isn-2. Shafts are considered to be stepped at sections at which gears are located. The increasing or decreasing order of shaft diameters at each step is as indicated in Figure 2. The axial dimensions of the shafts, together with the steps, must be specified by the designer. The meshing of gears is assumed to take place via the sliding of any of the meshing gears or by the implementation of positive clutches (not shown in Figure 2). The main limitation in all gearboxes is that the centre distance of shafts in a g r o u p of transmissions must be fixed. Because the sums of the gear-teeth numbers for every mesh in a transmission g r o u p are practically the same, all the gears in the same g r o u p must be cut with the same module m. That is, gears 1, 2, 3 and 4 have the same module to hold fixed the centre distance CA.B between shafts A and B. Similarly,
Table 1 KADcomputer output for milling machine Group number
Speed number
Power of progression ratio
Required ratio
Teeth number
Found ratio
Error, %
1
1
0
0.714
25:35
0.7143
-0.040
2
3
0 1 2
0.500 0.710 1.008
20:40 25:35 30:30
0.5000 0.7143 1.0000
- 0.000 -0.604 - 0.813
3
2
0 3
0.346 0.991
18:52 35:35
0.3462 1.0000
- 0.004 0.939
4
2
0 6
0.250 2.050
18:72 60:30
0.2500 2.0000
- 0.000 2.420
Speed number 1 2 3 4 5 6 7 8 9 10 11 12
722
Required speed, rev/min
Found speed, rev/min
Error, %
43.23 61.39 87.17 123.79 175.78 249.61 354.44 503.30 714.69 1014.86 1441.10 2048.37
42.37 61.81 86.54 125.00 178.57 245.00 346.15 494.51 692.31 1000.00 1428.57 2000.00
- 0.08 -0.69 0.73 0.98 1.58 O.17 2.34 1.73 3.15 1.44 0.84 2.25
Computer-Aided Design Volume 25 Number 11 November 1993
- -
Computer-aided design of multispeed gearboxes: I H Filiz and M Dfilger
2
_~A
=
5
m
4
m - -
6
CAB
D
1
8
Figure2 Gearbox model
gears 5, 6, 7, 8, 9 and 10 must be cut with the same module, but this is not necessarily equal to the module of gears 1, 2, 3 and 4 for the centre distance CR.c to be held fixed. Limitations of the kinematic scheme are the teeth numbers and transmission ratios. For dynamic performance, the minimum teeth number should be 18, and the total number of teeth 6 should not exceed 120. The transmission ratios for a group of transmissions I are restricted to be in the range 0.25-2.40. The error on the gear ratios exceeding + 1% is not reliable from the speed-progression point of view 4. In some applications in which very fine transmission ratios are essential, slight changes are allowed in gear-teeth numbers. In such cases, to keep the gears on the same shafts, gear profiles must be shifted, and thus nonstandard gears are produced.
DETERMINATION
OF GEAR SIZE
The data available after the kinematic-arrangement diagram are the speeds of shafts at each transmission, and the teeth numbers of gears in each group. To start with the design, the amount of the torque transmitted by each gear must be evaluated. Since it is assumed that there is negligible power loss, the torque transmitted at high speeds is smaller than that at low speeds. Hence, the minimum transmission ratio is selected to obtain the maximum torque transmission. The speed of shaft i can be written as ni=nmSl
(3)
where n,, is the speed of the input shaft, and Si is the transmission ratio. To obtain the minimum speed, St must be a minimum, which results in the maximum torque transmission. There are many different transmissions
in a gearbox kinematic-arrangement diagram. Each transmission has different loading conditions. Although the power does not vary, each transmission carries different torques at the specified output speeds. Therefore, each transmission must be considered as a separate design unit with its own loading conditions. Taking this fact into account, the facewidths and the modules of gears in each transmission are determined by using the AGMA approach, as explained below.
C O M P U T E R P R O G R A M FOR SIZING GEARS Because of the large number of interdependent variables, the design process does not consist in inserting numbers into certain formulas that will produce a complete set of results; it is rather a trial-and-error process, which is tedious in terms of manual calculations. A design algorithm is developed, and the process is automated with a computer program whose simplified flowchart is given in Fioure 3. The input includes the output data of the KAD program, the safety factor, and the materials properties of the gears. If this automated design algorithm is applied to the first group, the facewidth F 1 and the module m x are determined by assuming a value for module m, and calculating the facewidth F using the following equation: F=
nK°KmWt KaKbKcKfSe.mJKv
(4)
where n is the safety factor, Wt is the tangential load acting on the gear, K o is the overload factor, K m is the load-distribution factor, K a is the surface-finish factor, K b is the size factor, K c is the reliability factor, Kf is the load factor, Se is the endurance factor, d is the AGMA geometry factor, and K v is the velocity factor. A database !s _prepared for all the values of the K factors, which are taken from AGMA Information Sheet 225.01 l° and
Computer-Aided Design Volume 25 Number 11 November 1993
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Computer-aided design of multispeed gearboxes: I H Filiz and M DOIger
References 11 and 12. The process of calculating the facewidth is terminated when it is ~2 between 3rim and 5rim. The wear aspect of gears is another important factor to be considered. Hence, the gear must be checked against the surface durability by calculating the safety factor from
the following equations:
S CLCT ~2
CCHCRCp # KvFdpI
n =
(5)
WtKoKm Sc = 2.76HB - 70
(~
DISK
INPUT SPEED INPUT POWER i FACTOR [IF SAFETY TENSILE STRENGTH I BRINELL HARDNESS I
NUMBER OF GROUP (k)
C ~ > VDU =
I
NUMBER OF TRANSMISSIONS I IN EACH GROUP (t) I
I J°0
i -1 TRIAL
I ':°
I
2
(7)
mG+ 1
CALCULATE FACTOR OF
I
SAFETY FOR SURFACE
I
EXAMPLE
VALUE FOR mI
I i
I
CALCULATE f i FDR FATIGUE STRENGTH REQUIREMENT
I J=
sin ~b cos q5 mG
where q5 is the pressure angle, and mG = dG/dp, where dG is the pitch diameter of the gears. All the C factors are in the database of the program, and are taken from AGMA Information Sheet 225.011° and References l l and 12. F 2 and m 2 are determined in the same manner. Since a single module must be used for these two transmissions owing to the fixed centre distance, the larger of m 1 and mz must be selected and replaced with the nearest standard module. This is the principal module mp of the first group. In the same manner, by applying the design algorithm once for each transmission in the second group, the principal module of the second group is determined. This process terminates after the principal module of the last group is determined.
ASSUME
I i=
(in megapascals, where Sc is the surface-contact strength, CL is the life factor, Ca- is the temperature factor, CH is the hardness-ratio factor, C, is the reliability factor, C o is the elasticity coefficient, and dv is the pitch diameter of the pinions) and I =
I
(6)
.xj
[INCREASE (mi ) ]
L
1
+1[ DURABILITY REQUIREMENT
1
SELECT MAXIMUMOF rolAND REPLACE IT WITHTHE NEAREST STANDARDMODULE(mp)
CALCULATE NEhN FACTOR OF SAFETY
PRINCIPLEMODULE (rap FACTOROF SAFETY FACEWIDTHS
The example is given as an alternative solution for the gears used in the main drive gearbox of the milling machine whose KAD has already been created. The gear-data input to the program is as follows:
Gear-material specification: yield strength, M P a ultimate strength, M P a Brinell hardness, HB Young's modulus, MPa Poisson's ratio
= 800 = 880 = 410 = 207000 =0.3
Design requirements: = 2.5 factor of safety = 25 pressure angle, deg = 99 reliability, % operating temperature, deg C = 75 overload factor = 1 life factor = 1
I
MODULE " ~ FACEWlDTH DIAMETER
Manufacturing requirements: high-precision ground teeth
Characteristic of supports: Figure 3 Flowchart of computer program for sizing gears
724
accurate mountings
Computer-Aided Design Volume 25 Number 11 November 1993
Computer-aided design of multispeed gearboxes: I H Filiz and M D01ger Table 2
Gear-sizing computer output
Input speed, rev/min = 1400 Input power, k W = 4 Group number
Mesh number
Teeth number
1
1
25:35
2
1 2 3
20:40 25:35 30:30
3
1 2
18:52 35:35
4
1 2
18:72 60:30
Angle of planes of shafts 2 and 3, deg = 0.0 Angle of planes of shafts 3 and 4, deg = 20.0 Angle of planes of shafts 4 and 5, deg = 0.0 W h a t now (enter ? for help) : G o W h a t next (enter ? for help) : Old Old data are now in progress W h a t next (enter ? for help) : G o Gear design starts Gear design has been completed Do you want to see facewidths ? (Y/N)
:Y
Grub number
Mesh number
Teeth number
Module, m m
Facewidth, mm
Pitch diameter, mm
Safety factor
1 1 2 3 1 2 1 2
25:35 20:40 25:35 30:30 18:52 35:35 18:72 60:30
2.25 2.50
26 38 29 29 46 38 68 48
56.3:78.8 50.0:100.0 62.5:87.5 75.0:75.0 63.0:182.0 122.5:122.5 90.0:360.0 300.0:150.0
2.5 2.5 2.5 2.8 2.5 3.5 2.5 4.8
1 2
3 4
3.50 5.0
The output, together with kinematic data, is given in Table 2.
CONCLUSIONS The determination of the module and the facewidth of a gear which has a certain number of teeth is primarily dependent on the torque transmitted and the speed. Although the torque increases as the speed decreases, this increase in torque may be countered by the decrease in the dynamic load. Therefore, the designer should carry out the design of gears at all output speeds, especially for those gearboxes having a large number of output speeds. The computer program presented in this paper allows the user to carry out the design of gears at each output speed separately. After specifying the speed number, the program starts to design gears at that speed. The logical chains among the variables are created within the program. As noticed in the results, the safety factors of some transmissions are larger than the original safety factor. This is because the design modules for these transmissions are replaced with the principal module
Dr I Hi~seyin Filiz was born in Gaziantep, Turkey. He gained a BSc and an MSc in mechanical engineering at the Middle East Technical University, Turkey, in 1973 and 1975. In 1981, he earned a PhD from the University of Manchester Institute of Science and Technology, UK. His research interests are in the design of CNC machine tools, the design of machine elements, and CAB.
Mustafa Di~lger was born in Bayburt, Turkey, in 1963. He gained a BSc in mechanical engineering and an MSc at the Middle East Technical University, Turkey, in 1987 and 1989. His research interests are in the design of CNC machine tools.
Computer-Aided Design Volume 25 Number 11 November 1993
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Computer-aided design of multispeed gearboxes: I H Filiz and M Diilger
which is determined for the weakest transmission to satisfy the fixed-centre-distance requirement. For the purpose of production, the modules determined are replaced with the nearest standard modules.
5 6 7
REFERENCES 8 1 2 3 4
726
Acherkan, N Machine Tool Design - - Vol 3 Mir Publishers (1973) Koenigsberger, F Design Principles of Metal Cutting Machine Tools Pergamon Press, UK (1964) Karsli, S 'Computer aided design of gearbox kinematical arrangement diagrams' MSe Thesis Middle East Technical University, Gaziantep, Turkey (Sep 1985) Karsli, S and Filiz, I H 'Computer aided design of gearbox
9 10
11 12
kinematical arrangement diagrams' Proc. lASTED Int, Syrup. Computers & Their Applications for Development Taormina, Italy (3-5 Sep 1986) pp 174-179 Diilger, M 'Computer aided design of gearboxes for machine tools' MSc Thesis Middle East Technical University, Gaziantep, Turkey (Jun 1989) Chironis, N P Gear Design Application McGraw Hill, USA (1962) Sanger, D J 'A note on maximum speed ranges and distribution of speeds of three shaft gear trains' lnt. J. Mach. Tool. Des. & Res. Vol 12 (1972) pp 55-63 Reddy, C Y 'Selection of number of teeth for machine tool gearboxes' J. Mach. Tool Des. & Res. Vol 14 (1974) pp 125-134 TOS Milling Machine Catalogue 'AGMA Information Sheet 255.01' American Gear Manufacturers Association, USA (1969) Dudley, D W Gear Handbook McGraw-Hill, USA (1962) Shigley, J E and Mitchell, L D Mechanical Engineering Design (4th Ed.) McGraw-Hill, Japan (1983)
Computer-Aided Design Volume 25 Number 11 November 1993