176 



HANDBOOK OF MECHANICAL DESIGN 



SHORT-CENTER BELT DRIVES 



Calculations for the Arc of Contact and Length of Belts Having an Idler Pulley 



When an idler pulley is used to increase the When A is above the center line, angle A will be 

 arc of belt contact on the driving pulley, it minus, and, if 4 is below the center Une, angle A 

 becomes necessary to calculate that increase to will be plus. The scale A in the chart can be 



obtain the belt length. In the figure below, 

 center lines are drawn connecting pulley centers. 

 Solving for the belt wrap 6 on pulley d, 



.sin (0 + A) = 



d 

 2 sin {4> + A) 



2 sin (0 + A) 



{d + D2) 

 2 V^^ + 52 



{d + D2) 



d-\-D^ 



2 sin {4> + A) 



<i> 



sin~ 



<l> = sm~^ 



A = sin~' 



, d + D^ 



2 y/A^ + 52 



A 

 \/A'' + B^ 



- A 



2 V^' + B' 



sm" 



VA^ + W 



The angle of belt contact on the driving pulley 

 d will then be 



d = 180 deg. 

 = 180 deg. 



a + (<#. + A) + A 

 01 + <t> 



in which the angle of approach a is 

 D-d 



sin a 



2C 



or 



a = sm" 



D-d 



2C 



used for either plus or minus values but the sign 

 preceding the angle A must be kept in mind. 

 When values of A are less than 1, values of angle 

 A must be interpolated. For e.xample, when A 

 is between + 0.5 in., angle A is less than +2 deg. 

 and is read on the scales "A" and "angle A in 

 deg." by interpolating. 



For the example shown on the chart on the 

 next page, the arc of belt contact on pulley d 

 will be 



= 180 deg. - a+ {<f> + A) - A 



= 180 deg. - 14.5 deg. + 33 deg. - (-4.5 deg.) 



= 203 deg. 



Equation for the length L of belt is 

 L = E + e +F +G + H + J 



wnere £, ^ 57.3 



d (180 deg. - a + <t>) 

 ^ ~ 2 57.3 



2 57.3 



G = C cos a 



H = Dta.n (90 deg. - V- + X) 



J = dtun (90 deg. - 4> + y) 



In the foregoing equations, values for the 

 The angles 4> and A can be found on the chart, various symbols are calculated as follows: 



, . _i D + d 



B-- 



a = sin 



2 Va + (c - By 



D-d 



4> 



sin" 



tan'^ 



2C 



d + D, 



tan^^ -5 



y = sm 



2 VA- + B^ 



, / 2)2/2 cos lA - A \ 

 \(C - B)- D2/2 sin ^/ 



/ D2/2 cos <» - i4 \ 

 \B - £>2/2sin <^/ 



