Qualitative Design of Compact Transmission Mechanisms with
Kinematic analysis of gear boxes and transmissions. [Forbus, Nielsen and .... PIC POC , . PIS POS 90°. POC. = â i=NMaxS. 1. â i PIC. 2. S ynthesis m ethod ...
Qualitative Design of Compact Transmission Mechanisms with Standard Components Jean Christophe FAUROUX
Laboratoire de Recherches et Applications en Mécanique Avancée 1
Introduction General context 3D machine design from written specifications Purpose of this work A software wizzard for preliminary design of transmission mechanisms Type of mechanisms High ratio transmissions Multi stage mechanism Which stage ? In what order ? Where in space ?
[Chakrabarti and Bligh 96] Synthesis of transmission mecanisms with multiple I/O Combination of mechanical modules Orthogonality restrictions
[Kota and Chiou 92] Synthesis method for compound mechanisms Qualitative matrix representation
[Joskowicz and Sacks 93] Kinematic analysis of gear boxes and transmissions
[Forbus, Nielsen and Faltings 91] Qualitative kinematics and dynamics for analysis of complex mechanisms
Previous works :
[Fauroux and Sartor 97] Qualitative synthesis method for exploring domain of feasible solutions
[Fauroux, Sanchez, Sartor and Martins 98] Fuzzy logic evaluation of solution
The idea :
Improving synthesis method for transmission mechanisms Including qualitative analysis of solutions for getting a better characterization
3
1 Basic concepts
Method Architecture Mechanism database
Exploration Generate combinations of mechanisms
Requirements - I/O relative orientations - Efficiency - Speed ratio - Sense of rotation - Stage number
+
Design rule database
Elimination Apply design rules
Sorting Find best solutions sorted by quality
Domain of Domain of potential feasible Geometrical infomation solutions solutions complying with design Mechanical infomations rules
Combinatorial infomation
A method improving creativity Exhaustive exploration of feasible solutions
Ordered list of solutions - CAD models Output
Stage 2
Stage 1
Stage 4
Stage 3 Input
Qualitative solutions are too vague Many equivalent good solutions How to differentiate them ? 4
1 Basic concepts
Qualitative Solutions Qualitative solutions are too vague
Qualitative shape of parts
Relative ordering of parts (connectivity)
Not the final dimensions
Not the final orientations
Demo
Good qualitative solutions are often not easy to differentiate
Qualitative solutions give a good start to designers...
...but need to be enriched in information Example : 1008 solutions and 8 rated N° 1 Which to choose ?
Idea : try to reproduce human mind
5
1 Basic concepts
Standard Orientations Enriching the model by orientating stages 4 standard values : 0° / 90° / 180° / 270°
Strong assumption but corresponding to the vast majority of industrial mechanisms
Permits a fast exploration of mechanism layouts
May be refined later
A given mechanism can take a great number of layouts
2 angles 4 values per angle 16 combinations
Demo
6
0°/0°
0°/90°
0°/180°
0°/270°
90°/180°
90°/270°
180°/180°
180°/270°
90°/0°
Vertical
Horizontal
180°/0°
90°/90°
180°/90°
Horizontal
Com pact 270°/0°
Com pact 270°/90°
Vertical
1 Basic concepts
Standard Orientations
270°/180°
270°/270°
7
1 Basic concepts
Mechanism Compactness Compactness : a way for differentiating solutions Compact = fits the available space
No spoilt space inside No big parts crossing outside
Not Compact
Compact
Space is horizontal Mechanism is vertical Mechanism is totally outside
Output
Available space
Mechanism has good overall orientation Mechanism has good proportions
Available space Output = Specified output
Specified output
Input Input
8
1 Basic concepts
Standard Mechanisms (1/2) Semi dimensioned mechanisms Example : cylindrical gear Diameters have given values > Fixed ratio Pitch diameter or tooth width may be kept undefined
Cylindrical gear sets
Opposite shafts / Shafts on the same side Ratios 1, 2 or 4 Dimensions : 4R, 6R, 10R R
10R
6R
4R
Bevel gear sets
Two shaft settings for reversing sense of rotation Ratios 1, or 2 Big wheels are expensive so ratios are limited
Opposite shafts / Shafts on the same side Ratio 2 gives part interference with shafts on the same side Ratios 3 or 5 Dimensions : 6R or 10R With same overall dimension, internal gear sets reduce more 6R than external ones
10R
Worm gear sets
Great ratios but efficiency sometimes small Two shaft settings 8R Ratios 8, 16 or 32 4 threads / 32 teeth, ratio 8, efficiency 85% 1 thread / 16 teeth, ratio 16, efficiency 75% 1 thread / 32 teeth, ratio 32, efficiency 67% Number of threads ↑ ⇒ ratio ↓ but efficiency ↑ Number of teeth ↑ ⇒ ratio ↑ , dimensions ↑ but efficiency ↓
4R
10
2 Synthesis method
Exploration Combinatorial exploration Global hybrid configuration counter :
Stage nature ( base NMaxM ) Stage orientation ( base 4 )
Global configuration counter Stage nature counter
S1
0 1 2
S2
S3
Stage orientation counter
SNMaxS
O1
O2
O3
Absence of0mechanism 0 0 000° 000° 000° Cylindrical gear and ratio 1 1 1 with shafts1on the same 090°side090° 090° Cylindrical gear and ratio 1 2 2 with opposite 2 shafts180° 180° 180°
270°
270°
270°
ONMaxS
000° 090° 180° 270°
NMaxM Last NMaxM NMaxM of database NMaxM mechanism 11
2 Synthesis method
Elimination Rules The domain of configurations increases greatly Elimination rules can be refined Limitation of the number of stages with orthogonal shafts N MaxS
Good efficiency
C S
with
C = ∏ i
Good speed ratio
U S − U U C U SU
with
U C= ∏ U i
i=1 N MaxS i=1
Good rotation sense
C = Calculated S = Specified
Good absolute orientation of output shafts ZOC= ZOS
with
N MaxS
ZOC= ∏ ℝi ZIC i=1
Correct relative I/O locations
Calculated Output POC
= P IC P OC , P IS P OS 90°
with
1
P OC=
∏
i=N MaxS
ℂi P IC
Qualitative and non dimensional
γ
Input
PIC=PIS
Specified Output P
OS
12
2 Synthesis method
Sorting Criteria (½1/2) Best configurations should be sorted first Five performance functions Mechanism nature quality Power transmission ability Fabrication cost Mounting cost
Overall proportion quality
From 0 (perfect fitting) To infinity (infinite extension along one axis) Overall dimensions of a mechanism configuration
a
PMinS PIS
∣ PMax2
PMaxC
PMin2
POC
B ZIS
∣∣
a b b c − − A B B C
b
Specifications
A
∣
F P=
ZIC PMaxS
ZOC
PMin3 PMax1
C Z P MinC
POS
γ
O X ZOS
c
PMaxC PMax3
Y
PMin1 PMinC
PIC
13
2 Synthesis method
Sorting Criteria (½2/2) Five performance functions Input location quality F I = ∣NX IC− NX IS∣∣NY IC− NY IS∣∣NZ IC− NZ IS∣ /3
with
NX IC =
X IC− X MinC X MaxC− X MinC
and
NX IS =
X IS − X MinS X MaxS− X MinS
NXIC represent the non dimensional position in % of PIC along the specification box FI = 0 for an input perfectly fitting specifications FI = 1 for an input diagonally opposed to requirements
Output location quality b
Overall dimensions of a mechanism configuration
Relative I/O location quality F IO=1−cos
a
PMaxC
Specifications
POC
B
FIO = 0 when relative I/O location perfectly fit specifications FI = 1 when I/O is very far from expected
ZOC A
ZIS
PMinS PIS
ZIC PMaxS
C Z P MinC
POS
γ
O X ZOS
c
Y
PIC
14
3 Concluding example
Example
PMaxS
Design of the following mechanism : Orthogonal I/O shafts Efficiency > 90% Speed ratio around 47
Reversing sense No more than 4 stages
ZIS
PMinS
POS ZOS
PIS
Results :
10 240 000 combinations Less than 5 seconds on a PIII 650 Mhz No solutions with ∆U = 0 5723 solutions with ∆U = 1 (space reduced by a factor 2000) Most of time spent in Exploration and Elimination phases High efficiency of qsort method (average running time N.log(N)) Example : 106 combinations sorted in 10 seconds Time may be cut down by changing elimination rule order Example : Orientation rule from first to last place Computing time from 45 to 4 seconds 7 12 0° 90°
6 0°
6 0°
11 8 6 6 90° 270° 270° 0°
14 90°
4 0°
8 5 0° 0°
Demo
15
3 Concluding example
Conclusion A new method for designing transmission mechanisms standard components standard orientations with standard components and standard orientations Advantages Standard components : more precise and realistic particularly for transmission ratio and efficiency Solutions are better defined with more qualitative information Realistic diameters and part orientations Compact mechanisms are exhibited Divides by several thousands the initial combination space
To be Improved Elimination rules may be refined Sorting criteria may be refined Faster computations with large databases
An efficient tool for suggesting ideas to designer with exhaustivity 16
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