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RESEARCH PAPERS

# Teeth-Number Synthesis of a Multispeed Planetary Transmission Using an Estimation of Distribution Algorithm

[+] Author and Article Information
P. A. Simionescu

Department of Mechanical Engineering, The University of Tulsa, 600 S. College, Tulsa, OK 74104

D. Beale

Department of Mechanical Engineering, Auburn University, 202 Ross Hall, Auburn, AL 36849

G. V. Dozier

Department of Computer Science, Auburn University, 109 Dunstan Hall, Auburn, AL 36849

J. Mech. Des 128(1), 108-115 (Apr 01, 2005) (8 pages) doi:10.1115/1.2114867 History: Received August 24, 2004; Revised April 01, 2005

## Abstract

The gear-teeth number synthesis of an automatic planetary transmission used in automobiles is formulated as a constrained optimization problem that is solved with the aid of an Estimation of Distribution Algorithm. The design parameters are the teeth number of each gear, the number of multiple planets and gear module, while the objective function is defined as the departure between the imposed and the actual transmission ratios, constrained by teeth-undercut avoidance, limiting the maximum overall diameter of the transmission and ensuring proper spacing of multiple planets. For the actual case of a $3+1$ speed Ravigneaux planetary transmission, the design space of the problem is explored using a newly introduced hyperfunction visualization technique, and the effect of various constraints highlighted. Global optimum results are also presented.

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## Figures

Figure 1

Ravigneaux planetary gear (36): 1 small sun gear; 2-3 broad planet gear; 4 large sun gear; 5 narrow planet gear; 6 ring gear

Figure 2

Kinematic diagram of a 3+1 transmission ratios Ravigneaux planetary transmission. Note that the broad planet 2-3 consists now of two distinct gears

Figure 3

Schematic for calculating distances d22, d34, d35, and d35. Notice that one of the idler planets 5 has been removed for clarity.

Figure 4

Projection of the lower envelope of objective function f1=Err max with N2≠N3 on the (m1,m3,f1) space (a), and plot of the corresponding outer diameter of the transmission (b) for the case of equally spaced identical planets

Figure 5

Projection of the lower envelope of objective function f1=Err max with N2≠N3 on the (m1,m3,f1) space (a), and plot of the corresponding outer diameter of the transmission (b) for the case of equally spaced nonidentical compound planets 2-3

Figure 6

Projection of the lower envelope of objective function f1=Err max with m1=m3 and N2=N3 on the (m1,f1) plane (a), and plot of the corresponding outer diameter of the transmission (b) for the case of equally spaced identical planets

Figure 7

Projection of the lower envelope of objective function f1=Err max with m1=m3 and N2=N3 on the (m1,f1) plane (a), and plot of the corresponding outer diameter of the transmission (b) for the case of equally spaced nonidentical planets 2-3

Figure 8

Front view of the transmission with N1=34, N2=25, N3=33, N4=32, N5=23, N6=116, p=4 and m1=2.5 and m3=1.75mm (variant number 3 in Table 2)

Figure 9

Front view of the transmission with N1=35, N2=N3=28, N4=25, N5=28, N6=91, m1=m3=2mm (variant number 7 in Table 2)

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