45 kW are transmitted by 2 m long wedge belts which operate at full capacity. The diameter of the motor pulley is 180 mm, and it rotates at 1440 rpm; the output pulley is 400 mm diameter.
Determine the effect of belt number on life if the belts are (a) SPA;   (b) SPB.

Given   :     P   =   45 kW   ;     n₁   =   24 Hz   ;     so, from ( 1)     v   =   πD₁ n₁   =   π (0.18)(24)   =   13.6 m/s

Geometry   :     D₁   =   180 mm   ;     D₂   =   400 mm   ;     L   =   2000 mm       so, from ( 2a) ( 2b) and ( 2c) :-
                              C   =   533 mm   ;     γ   =   0.208 rad   ;     θ₁   =   θ_min   =   2.726 rad

Since the belts operate at full capacity, slip is imminent and ( 5a) applies. Assume   f = 0.512, so
                              k_θ   =   1 - e ^{- fθ_min}   =   1 - e^{- 0.512 ∗ 2.726}   =   0.752
and, in ( 5a)     P/k_θ v   =   45 ∗ 10³ / 0.752 ∗13.6   =   4400 N

Properties of SPA from Table 1   ;     F   =   6235 N   ;     M   =   87.48 Nm   ;     ρ   =   0.1287 kg/m   ;     m   =   13.0
and, in ( 5a)     ρv²   =   0.1287 ∗ 13.62   =   23.8 N
                              ( P/k_θvz + M/D₁ + ρv² )/F   =   ( 4400/z +87.48/0.180 +23.8 )/ 6235   =   0.7057/z + 0.0818
                              ( P/k_θvz + M/D₂ + ρv² )/F   =   ( 4400/z +87.48/0.400+23.8 )/ 6235   =   0.7057/z + 0.0389

So, from ( 5a) with   z = 6 belts for example :-
                              L/vT   =   ( 0.7057/6 + 0.0818 )^13.0 + ( 0.7057/6 + 0.0389 )^13.0   =   8.23 E-10
                          ∴   T   =   L / 8.23 E-10 v   =   2.0 / 8.23 E-10 ∗ 13.6   =   1.79 E8 s   =   50 k.hours (kh)

Repeating these latter steps with different numbers of belts, and then repeating all calculations in a similar manner for SPB belts, illustrates the marked effect of number of belts upon the resulting belt life, thus :

	Number of belts	1	2	3	4	5	6	8	10	12
	SPA life ( kh )	6.1E-7	1.6E-3	0.11	1.6	11	50	400	1700	4800	[ 26kh =
	SPB life ( kh )	3.0E-5	0.035	1.1	9.5	40	120	500	1300	2600	10 years ]

Only one significant figure is really justifiable when quoting these lives, since the analysis is so approximate as mentioned.
The general trend of the results is surprising - at normal lives and as expected, the larger SPB belts live longer than SPA for a certain duty, or, alternatively, fewer SPB belts are required for a given life. However the reverse is indicated when loads are heavy and large numbers of belts are necessary. This is due to the changing relative significance of the power transmission, bending and centrifugal terms in the design equation. In fact, when loads are heavy and large numbers of small belts are indicated, then fewer large belts are usually more economical.

The trends exemplified here - in particular the marked effect of belt number on life - should be borne in mind when selecting a belt drive without the aid of software.