9.10 Additional Applications 579
9.10.3 Optimal Design of a Gear Train
Consider the gear train shown in Fig. 9.20, in which the gear pairs are numbered from
1 ton. The pitch diameters (or the number of teeth) of the gears are assumed to be
known and the face widths of the gear pairs are treated as design variables [9.19, 9.20].
The minimization of the total weight of the gear train is considered as the objective.
When the gear train transmits power at any particular speed, bending and surface wear
stresses will be developed in the gears. These stresses should not exceed the respective
permissible values for a safe design. The optimization problem can be stated as
FindX= {x 1 , x 2 ,... , xn}Twhich minimizes
∑n
i= 1
Ri(X) (9.42)
subject to
σbi(X)≤σb axm , σwi(X)≤σw axm , i= 1 , 2 ,... , n
wherexiis the face width of gear pairi,Rithe weight of gear pairi,σbi(σwi) the
bending (surface wear) stress induced in gear pairi, andσb axm (σw axm ) the maxi-
mum permissible bending (surface wear) stress. This problem can be considered as a
multistage decision problem and can be solved using dynamic programming.
9.10.4 Design of a Minimum-Cost Drainage System
Underground drainage systems for stormwater or foul waste can be designed efficiently
for minimum construction cost by dynamic programming [9.14]. Typically, a drainage
system forms a treelike network in plan as shown in Fig. 9.21. The network slopes
downward toward the outfall, using gravity to convey the wastewater to the outfall.
Manholes are provided for cleaning and maintenance purposes at all pipe junctions.
A representative three-element pipe segment is shown in Fig. 9.22. The design of an
Figure 9.20 Gear train.
