Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 21

To have the belt equally tight on each pair of opposite steps, the total length of the
belt must be kept constant for all the output speeds. This can be ensured by satisfying
the following equality constraints:

C 1 −C 2 = 0 (E 2 )

C 1 −C 3 = 0 (E 3 )

C 1 −C 4 = 0 (E 4 )

whereCidenotes length of the belt needed to obtain output speedNi(i= 1 , 2 , 3 , 4 )
and is given by [1.116, 1.117]:

Ci≃

πdi

2

(

1 +

Ni

N

)

+

(

Ni

N

− 1

) 2

di^2

4 a

+ 2 a

whereNis the speed of the input shaft andais the center distance between the shafts.
The ratio of tensions in the belt can be expressed as [1.116, 1.117]

T 1 i

T 2 i

=eμθi

whereT 1 iandT 2 iare the tensions on the tight and slack sides of theith step,μthe
coefficient of friction, andθithe angle of lap of the belt over theith pulley step. The
angle of lap is given by

θi= π− 2 sin−^1

[

(

Ni

N

− 1

)

di

2 a

]

and hence the constraint on the ratio of tensions becomes

exp

{

μ

[

π−2 sin−^1

{(

Ni

N

− 1

)

di

2 a

}]}

≥ 2 , i= 1 , 2 , 3 , 4 (E 5 )

The limitation on the maximum tension can be expressed as

T 1 i= stw, i= 1 , 2 , 3 , 4 (E 6 )

wheresis the maximum allowable stress in the belt andtis the thickness of the belt.
The constraint on the power transmitted can be stated as (using lbffor force and ft for
linear dimensions)
(T 1 i−T 2 i)π d′i( 503 )
33 , 000

≥ 0. 75

which can be rewritten, usingT 1 i= stwfrom Eq. (E 6 ), as

stw

(

1 −exp

[

−μ

(

π−2 sin−^1

{(

Ni

N

− 1

)

di

2 a

})])

π di′

×

(

350

33 , 000

)

≥ 0. 75 , i= 1 , 2 , 3 , 4 (E 7 )